No title

Letters to the Editor

The Mathematical Intelligencer encourages comments about the material in this issue. Letters to the editor should be sent to the editor-in-chief, Chandler Davis.

Can Two Periodic Functions with Incommensurable Periods Have a Periodic Sum? This note adds to the arsenal of coun­ terexamples in elementary analysis. It is well known and easily proved that the sum of two periodic functions on the real line is periodic if the periods of the

two functions are commensurable, i.e.,

their quotient is a rational number. It seems reasonable to assume that the sum of two periodic functions with in­ commensurable periods must be aperi­ odic, as has been done in at least one

textbook on differential equations. 1 But

Proof Choose y such that a,{3, yare lin­ early independent over IQ. Then, if l, m, and n are integers for which la + mf3 + ny = 0, we must have l m = n 0. On this foundation I construct!, g, and h. =

=

Define G G(a,{3,y) = { lt,m,n = la + m{3 + nyl {l,m,nj C Z}, and let!, g, and h be functions that vanish on the com­ plement of G, i.e., for all t ft. G, but are =

.f(t)

= mf3 + ny g(t) la- ny h(t) = la + mf3 =

this assumption is incorrect. My examples are everywhere dis­ continuous. This suggests the question whether less pathological examples exist. I will give a partial answer to this question at the end. To be precise, saying thatj:

IR � IR

is a periodic function of period U

.f(t) for all t E ( -x,oo) , and that U is the smallest pos­ means that f(t + U)

=

itive value for which this is true. Note that by this definition a constant func­ tion is not periodic, nor is the charac­ teristic function of the rationals.

lt,m,n E G

(1.1)

termines the values of m and n. Similarly,

a non-zero value of g(t) or h(t) uniquely

determines the values of l and n, or l and

m, respectively. Thus, for a fixed pair of integers m and n, not both of which are zero, j(t) = mf3 + ny only at the points t la + m{3 + ny, for arbitrary l E 1:'. Since j( t) = j( t + a) = 0 for all t ft. G, this shows that .f has period a. Similar remarks apply to g and h. In summary: =

1. the period off is a, 2. the period of g is {3,

there exist

But h

and

=

The number y was chosen to ensure

3. the period of

JIIR � IR

t

that a non-zero value ofj( t) uniquely de­

THEOREM 1. Let a > 0 and {3 > 0 be in­ commensurable real numbers. Then functions

)

otherwise defined as follows:

=

h is y.

.f + g.The theorem is proved.

g IR � IR with periods a and {3, respec­ tively, such that h = f + g is periodic.

D

Comment. Since the set {qta + Q2f31{qt,Q2} C IQ} is countable, there ex­ ist uncountably many real numbers y such that the set {a,{3,y} is linearly in­

The class of triplets f, g, and h con­

dependent over IQ. We will need the ex­

istence of such a y below. For a simple example, take

Vs.

VJ;;,

a =

1, {3

=

V2

and y

For a class of examples, let a

{3

=

VJ;;.,

y

=

�.

= =

for distinct

primes Pi· Or, let o E IR be transcen­

Extending the class of examples

structed in the proof above can be en­ larged. Let


be a bijection,

where IQ0 is any infinite subset of the rationals. Now modify the definitions given above in the following way:

f(t)

= cfJ(m)f3 + cfJ(n)y g(t) cfJ(l)a- cfJ(n)y h(t) = cfJ(l)a + cfJ(m)f3 =

t

=

lt,m,n E G (1.2)

82, y = o3. (I

By reasoning analogous to that in

thank Basil Gordon for this example

the proof of Theorem 1, a particular

dental, then set a = o, {3

=

and for suggesting improvements to ----

non-zero value of .f recurs with period

a, with similar remarks holding for g

the text.)

�------�----------·

-------

1Borrelli, Robert L and Courtney S. Coleman, Differential Equations: A Modeling Perspective, 1 st ed., New York: John Wiley & Sons, Inc., 1 998, Prob. 1 "Periodic Function Facts," 189. This error does not appear in the second edition�

4

THE MATHEMATICAL INTELLIGENCER © 2005 Springer SC1ence+Bus1ness Media, Inc.

and h. Thus again the periods off, g, and h are a, {3, and y, respectively. By confining lDo to a bounded interval, we see thatf, g, and h themselves can be bounded. A question remains. Is it possible to have counterexamples in which one of the functions is continuous at a point or on a larger set? (I thank a referee for the question and for finding an error in a previous version of Theorem 2.) Here is a partial answer. THEOREM 2. Let H(t) F(t) + G(t), where F and G are bounded and peri­ odic with incommensurable periods U and V, andF is continuous everywhere. Then H is not periodic. =

Proof I show that the assumption that His periodic, say of period W > 0, leads to a contradiction. Note that W must be incommensurable with both U and V. The function F attains a maximum F*, and G and H have finite least up­ per bounds, say G* and H*, respec­ tively. I will show that H* = F* + G*. Because W must be incommensurable with U, the latter equality places high demands on F, sufficient to prove that F must be constant. Given s > 0, choose x and y such that F(x) = F* and G(y) > G* - s. Since F is uniformly continuous, there is a 15 > 0 such that F I (t) - F(t' } < s for I t - t' l < 15. I need two facts concerning a pair of incommensurable numbers (,TJ E lhbo: (1) 7L( mod TJ is dense in the in­ terval [0, TJL and hence (2) for any s1 > 0, there exist positive integers m and n such that jx + m( - y - nTJI <

s1.

(1.3)

These facts follow from the well­ known fact that, for irrational ( > 0, the set 7L( mod 1 is dense in [0,1]. According to inequality (1.3), we may choose m and n so that

lx + mU -

y- n vj < 15.

(1.4)

By the choice of 15, IF(x + mU)­ F(y + n V) l < s, and therefore H*::::: H(y + nV) = F(y + nV) + G(y + nV) > F* + G* - 2s.

(1.5)

As clearly H* :s: F* + G*, it follows that H* = F* + G*. We will exploit the fact that at val­ ues of the argument for which H is near its maximal value, so must F and G be near their maximal values. As evident in (1.5), such a value of the argument is afforded by z = y + nV. Thus, be­ cause H is presumed to have period W, and referring again to (1.5), H(z + jW) = F(z + jW) + G(z + jW) = H(z) > F* + G* - 2s (Vj E 7L). It follows that F(z

+

jW) >F* + G* - G(z + jW)- 2s >F*- 2s (Vj E 7L). (1.6)

Finally, the period ofF is U, U and W are incommensurable, and so the set ltJ = (z + jW) modU I J E 7L} is dense in [O,U]. And, by periodicity of F, in­ equality (1.6) implies that F(t) >F*2s for all t in the dense set l tJl· This is possible only if F = F*. This contradiction proves that H cannot be periodic. D Michael R. Raugh Department of Mathematics Harvey Mudd College Claremont, CA 9 1 7 1 1 -0788 USA e-mail: [email protected]

The Pythagorean Theorem Extended- and Deflated

In my paper "N-Dimensional Variations on Themes of Pythagoras, Euclid, and Archimedes" (Mathematical Intelli­ gencer 26 (2004), no. 3, 43-53), I pro­ posed a generalisation of the usual Pythagorean theorem in the form THEOREM OF PYTHAGORAS ND. The square of the (N-1)-dimensional vol­ ume of the hypoteneusal face of an N-dimensional orthosimplex is equal to the sum of the squares of the vol­ umes of its N orthogonal faces. This was a rediscovery; I mentioned I had been anticipated by H. S. M. Cox­ eter & P. S. Donchian, Math. Gazette 19 (1935), 206. And by many others! Rajendra Bhatia traced out an In­ dian path-which is, after all, satisfy-

------- ·----·-----

ing, given that ancient Indian mathe­ maticians seem to have known what we call the Pythagorean Theorem well before the Greeks. K. R. Parthasarathy published a proof based on volume in­ tegrals calculated by using Gauss's for­ mula for the volume of convex poly­ topes: "An n-dimensional Pythagoras Theorem," Math. Scientist 3 (1978), 137-140. This impelled S. Ramanan to give a simpler (unpublished) proof us­ ing antisymmetric tensors. After an­ other elaborate proof was indepen­ dently published by S. Y. Lin and Y. F. Lin in Lin. Multilin. Algebra 26 (1990), 9-13, R. Bhatia sent a letter giving Ra­ manan's proof (Lin. Multilin. Algebra 30 (1991), 155), and included it as prob­ lem 1.6.6 in his book Matrix Analysis. More recently, French colleagues also stumbled on the results: J.-P. Quadrat, J. B. Lasserre, and J.-B. Hiriart­ Urruty, "Pythagoras' Theorem for Ar­ eas," American Mathematical Monthly 108 (2001), 549-551. They pointed out a French connection, at least for the 3dimensional case, which has been known for quite some time (though its analogy with the standard Pythagorean Theorem was apparently not stressed). The result was very likely known to R. Descartes himself, according to P. Costabel (see his edition of Descartes's Exercices pour la Geometrie des Solides (De Solidorum Elementis), Presses Universitaires de France, Paris, 1987). In any case, the (3-dimen­ sional) theorem is found in J.-P. Gua de Malves's memoirs of 17831, and L. N. M. Camot stated the result (re­ ferring to it as already known) in his Geometrie de Position, Crapelet, Paris, 1803. It also found its way into text­ books, such as P. Nillus, Ler;ons de cal­ cui vectoriel (t. I), Eyrolles, Paris, 1931. Now the publication of the paper by J.-P. Quadrat et al. brought new refer­ ences. The Editor, B. P. Palka, quotes but two comments (see "Editors' End­ notes" in the Monthly 109 (2002), 313-314). G. De Marco, from Padova, mentions an equivalent result involving N-dimensional parallelotopes, to be found in F. R. Gantmacher, Theone des Matrices (t. I), Dunod, Paris, 1966.

-----------

1The abbot Gua de Malves is a most interesting character. A typical polymath of the Enlightenment, he was in fact the first editor of the Encyc/opedie, before handing over the task to Diderot and D 'Aiembert.

© 2005 Springer Science+ Business Media, Inc., Volume 27,

Number 2 , 2005

5

J. Munkres recalls that a more general result is given in his book Analysis of Manifolds, Westview Press, 1991, pp. 184-187: THEOREM.

Let u be a k-simplex in W'. Then the square of the area of u equals the sum of the squares of the areas of the k-simplices obtained by projecting u orthogonally to the various coordi­ nate k-planes of W'. An elementary proof of a similar result for parallelotopes was published by G. J. Porter in the Monthly 103 (1996), 252-256. There have been many other publi­ cations of the Theorem. A very cursory Google search led me to a note by Eric W. Weisstein on MathWorld [http:// mathworld.wolfram.comldeGuasTheo­ rem.html]; the 3-dimensional case, re­ ferred to as "de Gua's Theorem," is said there to be a special case of a general theorem presented by Tinseau to the Paris Academy in 1774 (slightly before de Gua's own publication), quoted in the textbooks by W. F. Osgood and W. C. Graustein, Solid Analytic Geom­ etry, Macmillan, New York, 1930, Th. 2, p. 517, and N. Altshiller-Court, Modern Pure Solid Geometry, Chelsea, New York, 1979, pp. 92 and 300. As for the general case, I found a reference to a pa­ per by R. F. Talbot, "Generalizations of Pythagoras Theorem in n Dimensions," Math. Scientist 12 (1987), 117-121, probably following Parthasarathy's 1978 publication in the same journal. A charming sequel is the recent (2002) posting by Willie W. Wong, a Princeton University student, of his proof of "A generalized N-dimensional Pythagorean Theorem" on his site [sep.princeton. edu/papers/gp.pdf]. It may still be that Coxeter and Donchian have the first occurrence in print of the result for N > 3. We may well ponder the significance of the re­ curring rediscovery of this result-and of its remaining so little known; its aes­ thetic and didactic merits certainly earn it a high place in textbooks or in the oral tradition. The least we can say is that our recording and referencing system clearly shows here its lacunae. The irony of the situation is that this discussion amounts to much ado about little. Indeed, as pointed out by

6

THE MATHEMATICAL INTELLIGENCER

J. Munkres in the aforementioned book, the theorem holds not only for simplices and parallelotopes, but (surprisingly at first) also for arbitrary sets lying in a k­ plane of Rn (k < n)! This generalisation is all the more interesting in that it only takes a meaning for higher dimension­ alities than the k = 1, n = 2 case of the standard Pythagorean Theorem. How­ ever, far from being a deep theorem, it is almost trivial, at least in the case k = n - 1 considered up to now. LetS be an arbitrary set contained in an (n - 1 )­ plane P of Rn, and call its volume A. Let vp be the unit vector orthogonal to P. Consider now the n projections Si (i = 1,2, . . . n) of S onto the (n - I)-dimen­ sional subspaces orthogonal to the unit vectors vi (i = 1,2, . . . n) of an orthog­ onal basis of Rn. Their respective vol­ umes Ai are obtained by projection and are given by Ai = (vp. vi)A. Since, by the usual n-dimensional Pythagorean Theo­ rem (or the so-called cosine law), one has I11Cvi, Vp)l2 = llvPII2 = 1, we obtain immediately the result announced, to wit I1 A� = A2• Hardly more than a Lemma!

The crux of the matter is that going from a !-dimensional segment to a k­ dimensional simplex is not the relevant generalisation here. In the present con­ text, a !-dimensional segment should be considered as an arbitrary con­ nected !-dimensional set. Here, as so often, a result proved in special cases through rather sophisti­ cated means finds an elementary proof showing its intrinsic nature once it is formulated in more general terms. This anticlimax only deepens the question of why the result has not been better understood by its many rediscover­ ers-including the present one. It is a pleasure to thank R. Bhatia, J. Holbrook, and J.-B. Hiriart-Urruty for a first introduction to the literature I had overlooked. Jean-Marc Levy-Leblond Physique Theorique Universite de Nice Sophie-Antipolis Pare Valrose 061 08 Nice Cedex France e-mail: [email protected]

HOLY GRAIL OF MATHEMATICS FOUND FERMAT'S PROOF TO HIS "LAST THEOREM" (A Restoration]

After some 370 years a 17th-Century proof to

the greatest enigma in mathematics is presented as the restoration of Fermat's letter to a dear friend divulging the origin and rationale of both the

mathematical AND geometrical proofs as examples of his descent infinite/indefinite

discussed in his note on the impossibility of the area of a rectangular triangle being an integer (newly translated) and his August 1659letter to Carcavi (the only translation).

Traces the proof from Euclid and Pythagoras.

A MUST FOR EVERY MATHEMATICIAN vii+ 22 pp. +illustrations $12.00+ $2.50

S&H + NJ 6% tax (U.S. $'s only)

Institutional checks or money orders only

Akerue Publications LLC Elizabeth, NJ 07202

•

PO Box 9547

c.J.ii.U.J.M

Knowledge and Community in Mathematics Jonathan Borwein and Terry Stanway

Mathematical Knowledge-As We Knew It

interrelationship

between

language,

meaning, and society that are com­ monly considered to fall under the um­

Each society has its regime of truth, its "general politics" of truth: that is, the types of discourse which it accepts and makes function as true; the mech­ anisms and instances which enable one to distinguish true andfalse state­ ments, the means by which each is sanctioned; the techniques and proce­ dures accorded value in the acquisi­ tion of truth; the status of those who are charged with saying what counts as truth.1 (Michel Foucault)

brella of postmodernism. Stating that "attempts to make sense of this elu­ sive concept threaten to outnumber at­ tempts to square the circle," he focuses his attention on two relatively well­ developed aspects of postmodern the­ ory: "poststructuralism" and "decon­

struction."3 He argues that the develop­ ment of these theories, in the works of Derrida and others, resonates with the debates surrounding foundation­ ism which preoccupied the philosophy of mathematics in the early stages of

Henri Lebesgue once remarked that

The Opinion column offers mathematicians the opportunity to write about any issue of interest to the international mathematical

the last century and may even have

"a mathematician, in so far as he is a

been partly informed by those debates.

mathematician,

Our present purpose is not to revisit the

need not preoccupy

himself with philosophy." He went on to

connections between the foundationist

add that this was "an opinion, moreover,

debates and the advent of postmodern

which has been expressed by many

thought, but rather to describe and dis­

philosophers."2 The idea that mathe­

cuss some of the ways in which episte­

community. Disagreement and

maticians can do mathematics without

mological relativism and other post­

controversy are welcome. The views

a precise philosophical understanding

modern perspectives are manifest in the

of what they are doing is, by observa­

changing ways in which mathematicians

tion, mercifully true. However, while a

do mathematics and express mathemat­

and opinions expressed here, however, are exclusively those of the author, and neither the publisher nor the editor-in-chief endorses or accepts responsibility for them. An Opinion

neglect of philosophical issues does not

ical knowledge. The analysis is not in­

impede mathematical discussion, dis­

tended to be a lament; but it does con­

cussion about mathematics quickly be­

tain an element of warning. It is central

comes embroiled in philosophy, and

to our purpose that the erosion of uni­

should be submitted to the editor-in­

perforce encompasses the question of

versally fixed perspectives of acceptable

the nature of mathematical knowledge.

practice in both mathematical activity

chief, Chandler Davis.

Within this discussion, some attention

and its publication be acknowledged as

has been paid to the resonance between

presenting significant challenges to the

the failure of twentieth-century efforts

mathematical community.

to enunciate a comprehensive, absolute foundation for mathematics and the postmodern deconstruction of meaning

Absolutism and Typographic Mathematics

and its corresponding banishment of en­ compassing philosophical perspectives from the centre fixe. Of note in this commentary is the contribution of Vladimir Tasic. In his book, Mathematics and the Roots of Postmodern Thought, he comments on the broad range of ideas about the

I believe that mathematical reality lies outside us, that ourfunction is to discover or observe it, and that the theorems which we prove, and which we describe grandiloq u en tl y as our "creations," are simply the notes of our observations. 4 (G. H. Hardy)

1Michel Foucault, "Truth and Power," Power/Knowledge: Selected Interviews and Other Writings 1972-1977, edited by Colin Gordon. 2Freeman Dyson, "Mathematics 1n the Physical Sciences," Scientific American 21 1 , no. 9 (1964):130. 3VIadimir Tasic.;, Mathematics and the Roots of Postmodern Thought (Oxford: Oxford University Press, 2001 ), 5.

© 2005 Springer Science+ Business Med1a,

Inc., Volume 27, Number 2 , 2005

7

We follow the example of Paul Ernest and others and cast under the banner of

absolutism descriptions of math­

eprints which are read reprints which are cited is

junct between digitally "published" and typographically published

ematical knowledge that exclude any element of uncer­

quite striking. Rather, it is a description of a time-honoured

tainty or subjectivity. 5 The quote from Hardy is frequently

and robust definition of

cited as capturing the essence of Mathematical Platonism,

ing environment. In the latter part of the twentieth century,

merit in a typographical publish­

a philosophical perspective that accepts any reasonable

a critique of absolutist notions of mathematical knowledge

methodology and places a minimum amount of responsi­

emerged in the form of the experimental mathematics

bility on the shoulders of the mathematician.

methodology and the social constructivist perspective.

An undigested

Platonism is commonly viewed to be the default perspec­

In

the next section, we consider how evolving notions of

tive of the research mathematician, and, in locating math­

mathematical knowledge and new media are combining to

ematical reality outside human thought, ultimately holds

change not only the way mathematicians do and publish math­

the mathematician responsible only for discovery, obser­

ematics, but also the nature of the mathematical community.

vations, and explanations, not creations. Absolutism also encompasses the logico-formalist schools as well as intuitionism and constructivism-in short, any perspective which strictly defines what constitutes mathe­ matical knowledge or how mathematical knowledge is cre­ ated or uncovered. Few would oppose the assertion that an absolutist perspective, predominately in the de facto Pla­ tonist sense, has been the dominant epistemology amongst working mathematicians since antiquity. Perhaps not as ev­ ident are the strong connections between epistemological perspective, community structure, and the technologies which support both mathematical activity and mathemati­ cal discourse. The media culture of typographic mathe­ matics is defined by centres of publication and a system of community elites which determines what, and by extension

wlw, is published. The abiding ethic calls upon mathemati­ cians to respect academic credentialism and the systems of publication which further refine community hierarchies. Community protocols exalt the published, peer-reviewed ar­ ticle as the highest form of mathematical discourse. The centralized nature of publication and distribution both sustains and is sustained by the community's hierar­ chies of knowledge management. Publishing houses, the peer review process, editorial boards, and the subscription­ based distribution system require a measure of central con­ trol. The centralized protocols of typographic discourse resonate strongly with absolutist notions of mathematical knowledge. The emphasis on an encompassing mathemat­ ical truth supports and is supported by a hierarchical com­ munity structure possessed of well-defined methods of knowledge validation and publication. These norms sup­ port a system of community elites to which ascension is granted through a successful history with community pub­ lication media, most importantly the refereed article. The interrelationships between community practice, structure, and epistemology are deep-rooted. Rigid episte­ mologies require centralized protocols of knowledge vali­ dation, and these protocols are only sustainable in media

Towards Mathematical Fallibilism

This new approach to mathematics-the utilization of ad­ vanced computing technology in mathematical research­ is often called experimental mathematics. The computer provides the mathematician with a laboratory in which he or she can perform experiments: analyzing examples, testing out new ideas, or searching for patterns. 6 (David Bailey and Jonathan Borwein) The experimental methodology embraces digital com­ putation as a means of discovery and verification. De­ scribed in detail in two recently published volumes, Math­ ematics by Experiment: Plausible Reasoning in the 21st Century and Experimentation in Mathematics: Compu­ tational Paths to Discovery, the methodology as outlined by the authors Uoined by Roland Girgensohn in the later work) accepts, as part of the experimental process, stan­ dards of certainty in mathematical knowledge which are more akin to the empirical sciences than they are to math­ ematics.

As an experimental tool, the computer can pro­

vide strong, but typically not conclusive, evidence regard­ ing the validity of an assertion. While with appropriate validity checking, confidence levels can in many cases be made arbitrarily high, it is notable that the concept of a

"confidence level" has traditionally been a property of sta­ tistically oriented fields. It is important to note that the au­ thors are not calling for a new standard of certainty in mathematical knowledge but rather the appropriate use of a methodology which may produce, as a product of its methods, definably uncertain transitional knowledge. What the authors do advocate is closer attention to and acceptance of degrees of certainty in mathematical knowl­ edge. This recommendation is made on the basis of argued assertions such as: 1. Almost certain mathematical knowledge is valid if treated

appropriately;

As an aside, we emphasize that this

2. In some cases "almost certain" is as good as it gets; 3. In some cases an almost certain computationally derived

is not meant as an indictment of publishers as bestowers

assertion is at least as strong as a complex formal as­

of possibly unmerited authority-though the present dis-

sertion.

environments which embrace centralized modes of publi­ cation and distribution.

4G. H. Hardy, A Mathematician's Apology (London: Cambridge University Press, 1 967), 21. 5Paul Ernest, Social Constructivism As a Philosophy of Mathematk:;s (Albany: State University of New York Press, 1998), 13. 6J. M. Borwein and D. H. Bailey, Mathematics by Expenment: Plausible Reason1ng 1n the 21st Century, A. K. Peters Ltd, 2003. ISBN: 1-56881 -21 1 -6, 2-3.

8

THE MATHEMATICAL INTELLIGENCER

The first assertion is addressed by the methodology it­ self, and in

Mathematics by Experiment, the authors dis­

cuss in detail and by way of example the appropriate treat­

Zucker.1° Computational confirmation to very high preci­ sion is, however, easy. Further experimental analysis involved writing

w3 as a

ment of "almost certain" knowledge. The second assertion

product of only r-values. This form of the answer is then

is a recognition of the limitations imposed by Gi:idel's In­

susceptible to integer relation techniques. To high preci­

completeness Theorem, not to mention human frailty. The

sion, an

third is more challenging, for it addresses the idea that cer­

0= -1.* log[w3] + -1.* log[gamma[l/24]] + 4.*log[gamma[3/24ll

tainty is in part a function of the community's knowledge validation protocols. By way of example, the authors write,

. . . perhaps only 200 people alive can, given enough time, digest all of Andrew Wiles' extraordinarily sophisticated proof ofFermat's Last Theorem. If there is even a one per­ cent chance that each has overlooked the same subtle er­ ror (and they may be psychologically predisposed so to do, given the numerous earlier results that Wiles' result relies on), then we must conclude that computational re­ sults are in many cases actually more secure than the proof of Fermat's Last Theorem. 7

Our first and pithiest example answers a question set by Donald Knuth,8 who asked for a closed form evaluation of the expression below.

00

I

k�l

{

-

"

}

1 = -o.o840695o872765599646t ,v;c;-;2 �7Tk

It is currently easy to compute

.

.

0.0840695087276559964

=

c\13- 1)2/96.

Similar searches suggest there is no similar four-dimen­

W4.

Fortunately, a one-variable inte­

W4 := fo exp( -4t)Ig(t)dt, I0 is the Bessel integral of the first kind. The high

gral representation is at hand in where cost

of

four-dimensional

exp( -t) for

numeric

integration is

thus

100 places in under 6 seconds and to 500 in 40 sec­ 0

The second example originates with a multiple integral which arises in Gaussian and spherical models of ferromag­ netism and in the theory of random walks. This leads to an impressive closed form evaluation due to G.

N. Watson:

I0(t), using

Io(t) = exp( -t)

t2n n�o 22n (n!)2

I

t up to roughly 1.2 d, where d is the number of signif­ ·

icant digits needed, and exp( -t)

+

onds. Arguably we are done.

Io( t) =

1

--

�

L

(2k- 1)2 (8t) nn!

II��1

V2m n�o

for larger

t, where the limit N of the second summation n such that the summand is less than w-d. (This is an asymptotic expansion, so taking more terms than N may increase, not decrease the

is chosen to be the first index

error.) Bailey and Borwein found that a product of powers of

W4 is not expressible as

f(k/120) (for 0 < k < 120) with co­ efficients of less than 12 digits. This result does not, of

course, rule out the possibility of a larger relation, but it

Example 2:

I-TT I-TT I-TT 3 TT TT TT

representation of

� �.

We thus have a prediction which Maple 9.5 on a laptop con­

�

Proving this discovery is achieved by comparing the out­ come with Watson's result and establishing the implicit[­

.

20 or 200 digits of this Sym­

bolic Calculator9 rapidly returns

%=

+ 18. *log[gamma[15 I 24]] + -2.*log[gamma[l7/24]]-7.*log[gamma[l9/24]]

careful computation of exp( - t)

sum. Using the "smart lookup" facility in the Inverse

firms to

+ -6.*log[gamma[ll/24ll + -9.*log[gamma[l3/24]]

avoided. A numerical search for identities then involves the

1 : Evaluate

kk --,-::k k. (::"

+ -8. *log[gamma[5/24ll + l.* log[gamma[7/24]] + l4.*log[gamma[9/24]]

sional closed form for

Three mathematical examples

Example

Integer Relation algorithm returns:

does cast experimental doubt that such a relation exists­

1 -

cos(x) - cos(y) - cos(z)

=

cV3 96

1

r2

���

( ) ( )

_!__ r2 .!.!. " 24 24

more than enough to stop one from looking.

0

The third example emphasizes the growing role of visual discovery. Example

3: Recent continued fraction work by Borwein

The most self-contained derivation of this very subtle

and Crandall illustrates the methodology's embracing of

Green's function result is recent and is due to Joyce and

computer-aided visualization as a means of discovery. They

7Borwein and Bailey, p. 10. 8Posed as MAA Problem 1 0832, November 2002. Solution details are given on pages 1 5-1 7 of Borwein, Bailey, and Girgensohn. 9At www.cecm.sfu.ca/projects/ISC/ISCmain.html

10See pages 1 1 7-1 21 of J. M. Borwein, D. H. Bailey, and R. Girgensohn, Experimentation in Mathematics: Computational Paths to Discovery, A.K. Peters Ltd, 2003. ISBN: 1 -56881 - 1 36·5.

© 2005 Springer Science+ Business Mecia,

Inc., Volume 27, Number 2, 2005

9

For our purpose, it is useful to note that the philosoph­ ical aspects of the experimental methodology combined

...

with the social constructivist perspective provide a prag­ matic alternative to Platonism-an alternative which fur­ thermore avoids the Platonist pitfalls. The apparent para­ dox in suggesting that the dominant community view of

.01

mathematics-Platonism-is at odds with a social con­ structivist accounting is at least partially countered by the observation that we and our critics have inhabited quite distinct communities. The impact of one on the other was

Fig. 1 . The starting point depends on the choice of unit vectors, a and

b.

investigated the and

tn where

wn

=

dynamical system defined by: to : = t 1

�

�

tn�l + Wn�l

(1- �)

:=

1

tn�2,

a2,b 2 are distinct unit vectors, for n even, odd,

respectively-that occur in the original continued fraction. Treated as a black box, all that can be verified numerically is that

1.

tn � 0 slowly. Pictorially one learns more, as illus­

trated by Figure Figure

2 illustrates the fine structure that appears when

the system is scaled by coloured distinctly.

Vn

and odd and even iterates are

With a lot of work, everything in these pictures is now explained. Indeed from these four cases one is compelled to conjecture that the attractor is finite of cardinality N ex­ actly when the input,

a or b, is an Nth root of unity; other­

wise it is a circle. Which conjecture one then repeatedly may test.

D

well described by Dewey a century ago:

Old ideas give way slowly; for they are more than ab­ stract logical forms and categories. They are habits, pre­ dispositions, deeply engrained attitudes of aversion and preference. . .. Old questions are solved by disappearing, evaporating, while new questions corresponding to the changed attitude of endeavor and preference take their place. Doubtless the greatest dissolvent in contemporary thought of old questions, the greatest precipitant of new methods, new intentions, new problems, is the one effected by the scientific revolution that found its climax in the "Origin of Species. "13 (John Dewey)

New mathematics, new media, and new community protocols

With a proclivity towards centralized modes of knowledge validation, absolutist epistemologies are supported by well­ defined community structures and publication protocols. In contrast, both the experimental methodology and social constructivist perspective resonate with a more fluid com­ munity structure in which communities, along with their implicit and explicit hierarchies, form and dissolve in re­

The idea that what is accepted as mathematical knowl­

sponse to the establishment of common purposes. The ex­

to some degree, dependent upon a community's

perimental methodology, with its embracing of computa­

methods of knowledge acceptance is an idea that is cen­

tional methods, de-emphasizes individual accomplishment

edge is,

tral to the social

constructivist school of mathematical phi­

losophy.

by encouraging collaboration not only between mathe­ maticians but between mathematicians and researchers from various branches of computer science.

The social constructivist thesis is that mathematics is a social construction, a cultural product, fallible like any other branch of knowledge. 11 (Paul Ernest)

munities, the social constructivist perspective is inherently

Associated most notably with the writing of Paul Ernest,

from easily identified elites and in the direction of those

Conceiving of mathematical knowledge as a function of the social structure and interactions of mathematical com­ accepting of a realignment of community authority away

an English mathematician and Professor in the Philosophy

who can most effectively harness the potential for collab­

of Mathematics Education, social constructivism seeks to

oration and publication afforded by new media. The ca­

define mathematical knowledge and epistemology through

pacity for mass publication no longer resides exclusively

the social structure and interactions of the mathematical

in

Social Construc­ tivism As a Philosophy of Mathematics, Ernest carefully

equipped with a LATEX compiler and the appropriate in­ terpreters is all that is needed. The changes that are oc­

traces the intellectual pedigree for his thesis, a pedigree

curring in the ways we do mathematics, the ways we

that encompasses the writings of Wittgenstein, Lakatos,

publish mathematical research, and the nature of the math­

Davis, and Hersh among others. 12

ematical community leave little opportunity for resistance

community and society as a whole. In

11 Ernest, p. 39ff. 12Ernest, p. 39ff. 13Quoted from The Influence of Darwin on Philosophy, 1 9 1 0.

10

THE MATHEMATICAL INTELLIGENCER

the

hands

of publishing

houses;

any workstation

Fig. 2. The attractors for various

ial lbl =

=

1.

or nostalgia. From a purely pragmatic perspective, the com­ munity has little choice but to accept a broader definition

In the epistemological universe, mathematics is con­ ceived as a large mass about which orbit many other bod­

of valid mathematical knowledge and valid mathematical

ies of knowledge and whose gravity exerts influence

publication. In fact, in the transition between publishing protocols based upon mechanical typesetting to protocols

throughout. The medieval recognition of the centrality of mathematics was reflected in the quadrivium, which as­

supported by digital media, we are already witnessing the

cribed to the sciences of number-arithmetic, geometry,

beginnings of a realignment of elites and hierarchies and a corresponding re-evaluation of the mathematical skill-set.

astronomy, and music-four out of the seven designated

curring in mathematics, we turn our attention to some per­

liberal arts. Today, mathematics is viewed by many as an impenetrable, but essential, subject that is at the founda­ tion of much of the knowledge that informs our under­

haps immutable aspects of mathematical knowledge.

standing of the scientific universe and human affairs. We

Before considering more carefully the changes that are oc­

are somehow reassured by the idea of a Federal Reserve

Some Societal Aspects of Mathematical Knowledge

The question of the ultimate foundations and the ultimate meaning of mathematics remains open: we do not know in what direction it will find its final solution or even whether a final objective answer can be expected at all. "Mathematizing" may well be a creative activity of man, like language or music, of primary originality, whose historical decisions defy complete objective rationalisa­ tion.14 (Hermann Weyl)

Chairman who purportedly solves differential equations in his spare time. The high value that society places on an understanding of basic mathematics is reflected in UNESCO's specifica­ tion of numeracy, along with literacy and essential life skills, as a fundamental educational objective. This place of privilege bestows upon the mathematical research com­ munity some unique responsibilities. Among them, the ar­ ticulation of mathematical ideas to research, business, and public policy communities whose prime objective is not the

furthering of mathematical knowledge. As well, as con­ cerns are raised in many jurisdictions about poor perfor­

Membership in a community implies mutual identification with other members which is manifest in an assumption of

mance in mathematics at the grade-school level, research communities are asked to participate in the general dis­

some level of shared language, knowledge, attitudes, and

cussion about mathematical education.

practices. Deeply woven into the sensibilities of mathemati­ cal research communities, and to varying degrees the sensi­

The mathematical canon

bilities of society as a whole, are some assumptions about the

role of mathematical knowledge in a society and what con­ stitutes essential mathematical knowledge. These assump­ tions are part of the mythology of mathematical communities and the larger society, and it is reasonable to assume that they will not be readily surrendered in the face of evolving ideas about the epistemology of mathematics or changes in the

I will be glad if I have succeeded in impressing the idea that it is not only pleasant to read at times the works of the old mathematical authors, but this may occasionally be of use for the actual advancement of science. 16 (Con­

methods of practicing and publishing mathematics.

The mathematical community is the custodian of an ex­ tensive collection of core knowledge to a larger degree than

Mathematics as fundamental knowledge

stantin Caratheodory)

any other basic discipline with the arguable exception of the combined fields of rhetoric and literature. Preserved

Mathematics is the tool specially suited for dealing with abstract concepts of any kind and there is no limit to its power in this field.15 (Paul Dirac)

largely by the high degree of harmonization of grade-school and undergraduate university curricula, this mathematical canon is at once a touchstone of shared experience of com-

14Cited in: Obituary: David Hilbert 1862-1943, RSBIOS, 4, 1 944, pp. 547-553. 15Dirac writing in the preface to The Prn i ciples of Quantum Mechanics (Oxford, 1 930). 16Speaking to an MAA meeting in 1 936.

© 2005 Springer Science+Business Media, Inc., Volume 27, Number 2, 2005

11

munity members and an imposing barrier to anyone who might seek to participate in the discourse of the commu­ nity without having some understanding of the various re­

To see a World in a Grain of Sand; and a Heaven in a Wild Flower; Hold Infinity in the palm of your hand; And Eternity in an hour. (William Blake)

lationships between the topics of core knowledge. While the exact definition of the canon is far from precise, to vary­ ing degrees of mastery it certainly includes Euclidean

Freedom and Discipline

geometry, differential equations, elementary algebra, num­

In this section, we make some observations about the ten­

ber theory, combinatorics, and probability. It is worth not­

sion between conformity and diversity which is present in

ing parenthetically that while mathematical notation can

the protocols of both typographically and digitally oriented

act as a barrier to mathematical discourse, its universality

communities.

helps promote the universality of the canon. At the level of individual works and specific problems, mathematicians display a high degree of respect for histor­ ical antecedent. Mathematics has advanced largely through the careful aggregation of a mathematical literature whose

The only avenue towards wisdom is by freedom in the presence of knowledge. But the only avenue towards knowledge is by discipline in the acquirement of ordered fact.19 (Alfred North Whitehead)

reliability has been established by a slow but thorough sciences, mathematical works and problems need not be re­

The Rhythmic Claims ofFreedom and Discipline, Whitehead's comments

process of formal and informal scrutiny. Unlike the other

Included in the introduction to his essay

cent to be pertinent. Tom Hales's recent computer-assisted

about the importance of the give and take between free­

solution of Kepler's problem makes this point and many oth­

dom and discipline in education can be extended to more

ers. Kepler's col\iecture-that the densest way to stack

general domains. In the discourse of mathematical re­

spheres is in a pyramid-is perhaps the oldest problem in

search, tendencies towards freedom and discipline, decen­

discrete geometry. It is also the most interesting recent ex­

tralization and centralization, the organic and the ordered,

ample of computer-assisted proof. The publication of

coexist in both typographic and digital environments. While

Hales's result in the

Annals of Mathematics, with an "only

ggo,.b checked" disclaimer, has triggered varied reactions.17

it may be true that typographic norms are characterized by centralized nodes of publication and authority and the com­ munity order that they impose, an examination of the math­ ematical landscape in the mid-twentieth century reveals

The mathematical aesthetic

strong tendencies towards decentralization occurring in­

mathematician's patterns, like the painter's or the poet's, must be beautiful; the ideas, like the colors or the words, must fit together in a harmonious way. Beauty is the first test: there is no permanent place in this world for ugly mathematics. 18 (G. H. Hardy) The

dependently of the influence of digital media. Mutually re­ inforcing trends, including an increase in the number of PhD's, an increase in the number of journals and published articles, and the application of advanced mathematical methods to fields outside the domain of the traditional mathematical sciences combined to challenge the tendency

Another distinguishing preoccupation of the mathemat­

to maintain centralized community structures. The result

ical community is the notion of a mathematical aesthetic.

was, and continues to be, a replication of a centralized com­

It is commonly held that good mathematics reflects this

munity structure in increasingly specialized domains of in­

aesthetic and that a developed sense of the mathematical

terest. In mathematics more than in any other field of re­

aesthetic is an attribute of a good mathematician. The fol­

search, the knowledge explosion has led to increased

"infinity in the palm of your hand"

specialization, with new fields giving birth to new journals

lowing exemplifies the

encapsulation of complexity which is one aspect of the aes­ thetic sense in mathematics.

1

1

1

+ 22 + 33 +

1 44

+

.

.

.

=

11 0

and the organizational structures which support them. While the structures and protocols which describe the digital mathematical community are still taking shape, it

1

would be inaccurate to suggest that the tendency of digital

:ff dx

media to promote freedom and decentralized norms of

Discovered in

1697 by Johannes Bernoulli, this formula has been dubbed the Sophomore's Dream in recognition of the

knowledge-sharing is unmatched by tendencies to impose

surprising similarities it reveals between a series and its in­

typographic mathematics manifest decentralization

tegral equivalent. Its proof is not too simple and not too

knowledge fragmentation, we are presently observing ten­

hard, and the formula offers the mix of surprise and sim­

dencies emerging from digital mathematics communities to

plicity that seems central to the mathematical aesthetic. By

find order and control in the

contrast several of the recent very long proofs are neither

results from the codification of mathematical knowledge at

simple nor beautiful.

the level of micro-ontologies. The World Wide Web Con-

17See "In Math, Computers Don't Lie. Or Do They?", The New York Times, April 18G. H. Hardy, A Mathematician's Apology (London: Cambridge University Press, 19Aifred North Whitehead, The Aims of Education (New York: The Free Press,

12

THE MATHEMATICAL INTELLIGENCER

control and order. If the natively centralized norms of

6, 2004.

1 967), 21.

1957), 30.

as

knowledge atomization that

MathML initiative and the European Union's OpenMath project are complementary efforts to

thorship, such as are common in the Open Source pro­

construct a comprehensive, fine-grained codification of

cal research. Michael Kohlhase and Romeo Anghelache

sortium (W3C)

gramming community, may find a place in mathemati­

mathematical knowledge that binds semantics to notation

have proposed a version-based content management

and the context in which the notation is used. 2 The tongue­

system for mathematical communities which would per­

0

in-cheek indictment of typographic subject specialization as producing experts who learn more and more about less

mit multiple users to make joint contributions to a com­

mon research effort. 23 The system facilitates collabora­

and less until achieving complete knowledge of nothing-at­

tion by attaching version control to electronic document

all becomes, under the digital norms, the increasingly de­

management. Such systems, should they be adopted,

tailed description of increasingly restricted concepts until

challenge not only the notion of authorship but also the

one arrives at a complete description of nothing-at-all. On­

idea of what constitutes a valid form of publication.

"non­

2. The ascendancy of gray literature : Under typo­

tologies. " If typographic modes of knowledge validation and

graphic norms, mathematical research has traditionally

tologies become micro-ontologies and risk becoming

publication are collapsing under the weight of subj ect spe­

been conducted with reference to j ournals and through

cialization, the digital ideal of a comprehensive meta-math­

informal consultation with colleagues. Digital media,

ematical descriptive and semantic framework which em­

with its non-discriminating capacity for facilitating in­

braces all mathematics may also prove to be overreaching.

stantaneous p ublication, has placed a wide range of

Some Implications

Ranging from Computer Algebra System routines to

sources at the disposal of the research mathematician. Home

Communication of mathematical research and scholar­ ship is undergoing profound change as new technology creates new ways to disseminate and access the litera­ ture. More than technology is changing, however, the cul­ ture and practices of those who create, disseminate, and archive the mathematical literature are changing as well. For the sake of present and future mathematicians, we should shape those changes to make them suit the needs of the discipline. (International Math Union Committee

21

on Electronic Information and Communication)

Pages

and

conference

programmes,

these

sources all provide information that may support math­ ematical research. In particular, it is possible that a p ublished paper may not be the most appropriate form of p ublication to emerge from a multi-user content management

such as proposed

by Kohlhase

and

Anghelache. It may be that the contributors deem it more appropriate to let the result of their efforts stand with its organic development exposed through a his­ tory of its versions. 3. Changing modes of knowledge authentication: The refereeing p rocess, already under overload-induced

. . . to suggest that the normal processes of scholarship work well on the whole and in the long run is in no way contradictory to the view that the processes of selection and sifting which are essential to the scholarly process are filled with error and sometimes prejudice. 22 (Kenneth Arrow)

stress, depends upon a highly controlled publication process. In the distributed p ublication environment af­ forded by digital media, new methods of knowledge au­ thentication will necessarily emerge. By necessity, the idea of authentication based on the ethics of referees will be replaced by authentication based on various types of valuation p arameters. Services that track ci­

Our present idea of a mathematical research community is built on the foundation of the protocols and hierarchies which define the practices of typographic mathematics. At this p oint, how the combined effects of digital media will affect the nature of the community remains an open ques­ tion; however, some trends are emerging:

1. Changing modes of collaboration: With the facilita­ tion of collaboration afforded by digital networks, indi­

tations are currently being used for this p urpose by the Web document servers

CiteSeer and citebase, among

others.24 Certainly the ability to compute informedly with formulae in a preprint can dramatically reduce the reader's or referee's concern about whether the result is reliable. More than we typically admit or teach our students, mathematicians work without proof if they feel

secure

in

the

correctness

of

their

thought

processes.

vidual authorship is increasingly ceding place to joint

4. Shifts in epistemology: The increasing acceptance of

authorship. It is possible that forms of community au-

the experimental methodology and social constructivist

2°For background on these projects, see: www.w3.org/Math/ and www.openmath.org, respectively. 21The IMU's Committee on Electronic Information and Communication (CEJC) reports to the IMU on matters concerning the digital publication of mathematics. See www.ceic.math.ca/Publications/Recommendations/3_best_practices.shtml 22 E. Roy Weintraub and Ted Gayer, "Equilibrium Proofmaking," Journal of the History of Economic Thought,

23 (Dec. 2001), 421--442. This provides a remarkably de­

tailed analysis of the genesis and publication of the Arrow-Debreu theorem. 23Michael Kohlhase and Romeo Anghelache, "Towards Collaborative Content Management and Version Control for Structured Mathematical Knowledge," Lecture Notes in Computer Science no.

2594: Mathematical Knowledge Management: Proceedings of The Second International Conference, Andrea Asperti, Bruno Buchberger, and

James C. Davenport editors, (Berlin: Springer-Verlag,

2003) 45.

24citeseer.ist.psu.edu and citebase.eprints.org, respectively.

© 2005 Springer Science+ Business Media, Inc., Volume 27, Number 2, 2005

13

', )

·�

' .,

:

,,

.·

,I

' ,

Fig. 3. What you draw is what you see. Roots of polynomials with coefficients 1 or -1 up to degree 1 8. The coloration is determined by a normalized sensitivity of the coefficients of the polynomials to slight variations around the values of the zeros, with red indicating low sen­ sitivity and violet indicating high sensitivity. The bands visible in the last picture are unexplained, but believed to be real-not an artifact.

perspective is leading to a broader definition of valid

ships. Collaborations, when they arise, are often career­

knowledge and valid forms of knowledge representa­

long, if not life-long, in their duration. The highly pro­

tion. The rapidly expanding capacity of computers to fa­

ductive friendship between

cilitate visualization and perform symbolic computa­

wood provides a perhaps extreme example.

tions is placing increased emphasis on visual arguments

long-term collaborations are not excluded, the form of

G. H.

Hardy and J. E. Little­ While

and interactive interfaces, thereby making practicable

collaboration supported by digital media tends to admit

the call by Philip Davis and others a quarter-century ago

a much more fluid community dynamic. Collaborations

to admit visual proofs more fully into our canon.

and coalitions will form as needed and dissolve just as

The SIAM 100-digit Chal­ lenge: A Study In High-accuracy Numerical Comput­ ing25 never met while solving Nick Trefethen's 2002 ten

quickly. The four authors of

The price of metaphor is eternal vigilance

(Arturo

Rosenblueth & Norbert Wiener) For example, experimentation with various ways of rep­ resenting stability of computation led to the four images in Figure

3. They rely on perturbing some quantity and

recomputing the image, then coloring to reflect the change. Some features are ubiquitous while some, like

the bands, only show up in certain settings. Nonethe­ less, they are thought not to be an artifact of roundoff or other error but to be a real yet unexplained phe­ nomenon.

5. Re-evaluation of valued skills and knowledge: Com­ plementing a reassessment of assumptions about math­

challenge problems which form the basis for their lovely book. At the extreme end of the scale, distributed com­ puting can facilitate virtually anonymous collaboration. In

2000, Colin Percival used the Bailey-Borwein-Plouffe 1,734 machines from 56 coun­

algorithm and connected

tries to determine the quadrillionth bits of

7T.

Accessing

an equivalent of more than

250 cpu years, this calcula­ tion (along with Toy Story Two and other recent movies) ranks as one of the largest computations ever. The computation was based on the computer-discov­ ered identity

ematical knowledge, there will be a corresponding reassessment of core mathematical knowledge and methods. Mathematical creativity may evolve to depend less upon the type of virtuosity which characterized twentieth-century mathematicians and more upon an ability to use a variety of approaches and draw together and synthesize materials from a range of sources. This

which allows binary digits to be computed indepen­ dently.26

A Temporary Epilogue

is as much a transfer of attitudes as a transfer of skill sets; the experimental method presupposes an experi­

The plural of "anecdote" is not "evidence." 27 (Alan L. Lesh­

mental mind-set.

ner)

6. Increased community dynamism: Relative to com­ puter- and network-mediated research, the static social

These trends are presently combining to shape a new

entities which intermesh with the typographic research

community ethic. Under the dictates of typographic norms,

environment extend the timeline for research and pub­

ethical behaviour in mathematical research involves ad­

lication and support stability in inter-personal relation-

hering to well-established protocols of research and publi-

25Folkmar Bornemann, Dirk Laurie, Stan Wagon, Jbrg Waldvogel, SIAM 2004. 26See Borwein and Bailey, Chapter 3.

27The publisher of Science speaking at the Canadian Federal Science and Technology Forum, Oct 2, 2002.

14

THE MATHEMATICAL INTELLIGENCER

cation. While the balance of personal freedom against com­

taneous weakening of community authority structures as

munity order which defines the ethic of digitally oriented

typographic elites are rendered increasingly irrelevant by

it seem as though

mathematical research communities may never be as firm

digital publishing protocols may make

or as enforceable by community protocols, some principles

the social imperatives that bind the mathematical commu­

best current prac­

nity have been weakened. Any sense of loss is the mathe­

provides a snapshot of the de­

matician's version of postmodern malaise; we hope and

are emerging. The CEIC's statement of

tices for mathematicians

"those who write, disseminate, and store mathematical litera­ ture should act in ways that serve the interests of math­ ematics, first and foremost, " the recommendations advo­

veloping consensus on this question. Stating that

predict that, as the community incorporates these changes, the malaise will be short-lived. That incorporation is tak­ ing place, there can be no doubt. In higher education, we now assume that our students can access and share infor­

cate that mathematicians take full advantage of digital

mation via the Web, and we require that they learn how to

media by publishing structured documents which are ap­

use reliably vast mathematical software packages whose

propriately linked and marked-up with meta-data.28 Re­

internal algorithms are not necessarily accessible to them

searchers are also advised to maintain personal homepages

even in principle.

with links to their articles and to submit their work to pre­ print and archive servers.

One reason that, in the mathematical case, the "unbear­ able lightness" may prove to be bearable after all is that

Acknowledging the complexity of the issue, the final CEIC recommendation concerns the question of copyright:

while

fundamental

assumptions

about

mathematical

knowledge may be reinterpreted, they will survive. In par­

it makes no attempt to recommend a set course of action,

ticular, the idea of mathematical knowledge as being cen­

but rather simply advises mathematicians to be aware of

tral to the advancement of science and human affairs, the

copyright law and custom and consider carefully the op­

idea of a mathematical canon and its components, and the

The

idea of a mathematical aesthetic will each find expression

tions. Extending back to Britain's first copyright law,

Statute of Anne,

enacted in 1710, the idea of copyright is

in the context of the emerging epistemology and protocols

historically bound to typographic publication and the pro­

of research and publication. In closing, we note that to the

tocols of typographic society. Digital copyright law is an

extent that there may be an opportunity to shape the epis­

emerging field; it is presently unclear how copyright, and

temology, protocols, and fundamental assumptions that

the economic models of knowledge distribution that de­

guide the mathematical research communities of the future,

pend upon it, will adapt to the emerging digital publishing

that opportunity is most effectively seized upon during

environment. The relatively liberal epistemology offered

these initial stages of digital mathematical research and

by the

experimental method and the social constructivist per-spective and the potential for distributed research and

publishing.

publication afforded by digital media will reshape the pro­

Whether we scientists are inspired, bored, or infuriated by philosophy, all our theorizing and experimentation de­ pends on particular philosophical background assump­ tions. This hidden influence is an acute embarrassment to many researchers, and it is therefore not often ac­ knowledged. Such fundamental notions as reality, space, time, and causality-notions found at the core of the sci­ entific enterprise-all rely on particular metaphysical as­ sumptions about the world. 30 (Christof Koch)

tocols and hierarchies of mathematical research commu­ nities. Along with long-held beliefs about what constitutes mathematical knowledge and how it is validated and pub­ lished, at stake are our personal assumptions about the nature of mathematical communities and mathematical knowledge. 29 While the norms of typographic mathematics are not without faults and weaknesses, we are familiar with them to the point that they instill in us a form of faith; a faith that if we play along, on balance we will be granted fair ac­

The assumptions that we have sought to address in this

As the centralized protocols of typo­

article are those that define how mathematical reality is in­

graphic mathematics give way to the weakly defined pro­

vestigated, created, and shared by mathematicians work­

cess to opportunity.

tocols of digital mathematics, it may seem that we are

ing within the social context of the mathematical commu­

ceding a system that provided a way to agree upon math­

nity and its many sub-communities. We have maintained

ematical truth for an environment undermined by rela­

that those assumptions are strongly guided by technology

tivism that will mix verifiably true statements with state­

and epistemology, and furthermore that technological and

ments that guarantee only the probability of truth and an

epistemological change are revealing the assumptions to be

environment which furthermore is bereft of reliable sys­

more fragile than, until recently, we might have reasonably

tems for assessing the validity of publications. The simul-

assumed.

28CEIC Recommendations. See: http://www.ceic. math.ca 29As one of our referees has noted, "The law is clearly 25 years behind info·technology. " He continues, "What is at stake here is not only intellectual property but the whole system of priorities, fees, royalties, accolades, recognition of accomplishments, jobs." 301n "Thinking About the Conscious Mind," a review of John R. Searle's Mind. A Brief Introduction, Oxford University Press, 2004.

© 2005 Springer Science+ Bus,ness Media,

Inc., Volume 27, Number 2, 2005

15

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Sharing Your Work Is Easier

16

THE MATHEMATICAL INTELLIGENCER

•

U1i

Typesetting

•

Computer Algebra

c.m;.u.u;

Evolution and Design Inside and Outside Mathematics Eric Grunwald

L

et me start with three dichotomies.

political views, they are a useful lan­

Although the first one (at least) is

guage for describing real-life atti­

very old, I will refer to them for conve­

tudes and positions.

nience by the names of twentieth­

Both Gowers and Atiyah recog­

century mathematicians who have dis­

nize this explicitly. Gowers points

cussed them.

out

that

most

mathematicians

would say that there is truth in both

Hardy's Dichotomy

points of view. The distinction is, as

. . . there is no sort of agreement about the nature of mathematical reality among either mathematicians or philosophers. Some hold that it is 'mental' and that in some sense we construct it ourselves, others that it is outside and independent of us [ 1}.

Gowers says, between the priori­ ties of mathematicians. Some math­ ematicians prefer to develop a gen­ eral understanding of mathematics and mathematical theories; others, to solve specific problems; but it's hard to imagine an intelligent math­ ematician thinking that theories are

Gowers's Dichotomy

bunk or that problems don't matter.

Loosely speaking . . . the distinction between mathematicians who regard their central aim as being to solve problems, and those who are more concerned with building and under­ standing theories [2}.

The extreme points of Atiyah's dichotomy also tend not to exist in isolation. One of the great glories of mathematics is the interconnected­ ness of algebra and geometry, to the extent that it's sometimes hard to tell whether a piece of mathematics

Atiyah's Dichotomy

Geometry and algebra are the two formal pillars of mathematics. . . . Geometry is, of course, about space. . . . Algebra, on the other hand . . . is concerned essentially with time. Whatever kind of algebra you are do­ ing, a sequence of operations is per­ formed one after the other and 'one af­ ter the other' means you have got to have time. In a static universe you cannot imagine algebra, but geometry is essentiaUy static [3}. My questions in this paper are these: are the three dichotomies related, and are they special cases of a wider di­ chotomy that operates outside as well as inside mathematics? My answer to both questions is yes.

is actually geometry or algebra. Nevertheless, some mathematicians certainly prefer geometry and think geometrically, whereas others are happier with algebra. It does seem possible, on the other hand, for an intelligent person to take an extreme position on Hardy's dichotomy. Indeed, Hardy himself goes on to say, " . . . I will state my own position dogmatically. I believe that mathematical reality lies outside us, that our fimction is to discover or observe it, and that the theorems

which

we

prove,

and

which we describe grandiloquently as our 'creations', are simply the notes of our observations." For an al­ ternative view, that mathematics is a social and cultural creation, see [4].

The Three Dichotomies Notice the following about these di­ chotomies:

2. The dichotomies operate on different levels. Hardy's dichotomy is at the top level. It is concerned with mathemat­ ics as a whole and the question, what

1. Cowers's and Atiyah's dichotomies describe extreme points on a spec­ trum. These extreme points can

acts at the level of mathematicians

probably not be attained, but like

tions. And Atiyah's dichotomy deals

"left-wing" and "right-wing," over­

with the structure of mathematics.

simplifications of highly complex

When we do mathematics we move

is mathematics? Gowers's dichotomy and their preferences and motiva­

© 2005 Spnnger Sc1ence+ Business Media, Inc., Volume 27, Number 2, 2005

17

around between these levels. Mathe­ maticians may only rarely think ex­ plicitly about the Hardy question, but at a deep level their attitude to it must affect everything they do. Atiyah, for example, likes to "move around in the mathematical waters" [5]: but in or­ der to do that, he must think of the waters as being there for him to move around in. When mathematicians do mathematics their

choices

about

what to do and the ways they go about doing it are shaped by their po­ sition on Gowers's spectrum, and

particularly relevant to the main aims of mathematics {2}.

dichotomy.

3. The three dichotomists are not neu­ tral

dichotomy-observers.

They

Evolution and Design For thousands of years, the stories through which we made sense of the

your sales and minimise your costs. The

world were founded on the idea of a solid design. Heroes did great deeds; vil­ lains tried to stop them. The gods, or the one God, duly rewarded or punished

speaks from the heart:

. . . algebra is to the geometer what you might call the 'Faustian Offer'. As you know, Faust in Goethe's story was offered whatever he wanted . . . by the devil in return for selling his soul. Algebra is the offer made by the devil to the math­ ematician. The devil says: 'I will give you this powerful machine, it will answer any question you like. All you need to do is give me your soul: give up geometry and you will have this marvellous machine'. . . . the danger to our soul is there, be­ cause when you pass over into al­ gebraic calculation, essentially you stop thinking; you stop thinking geometrically, you stop thinking about the meaning {3}.

butcher needs to cut as much meat as he can of a type that people are prepared to pay a high price for, using the most efficient equipment he can afford, and to beat down as much as possible the price

wrong, nothing fundamentally changed, and we all knew where we stood.

farmers, the slaughterers, and the dis­

Zeus

spoke, and nodded with his dark­ ish brows, and immortal locks feU for­ wardfrom the lord's deathless head, and he made great Olympus tremble {6}. When Zeus nodded you were well ad­

vised to jump to it. As late as the sev­

of the carcasses he buys. In this way, the

tributors will have to become more effi­ cient, and the butcher's customers will enjoy good nourishing food, and, so for­

tified, go out and enrich themselves and the rest of us. And so on. There is no need at all for Zeus to tell us what to do-all he needs to do is to set up the rules of competition, make sure we all want to better ourselves, and let rip. We

enteenth century, Newton seemed to

are guided, not by Zeus, but by the in­

have shown that the world operated me­

visible hand detected by Adam Smith.

chanically, like a clock. There may have been superficial variations as planets moved and apples fell, but the funda­ mental design was unchanging, perma­ nent. Newton himself was never a fully

And this hand will make sure that the

world won't stay the same: it will get richer. Smith's story is based on the al­ gebra of Atiyah's dichotomy: a process

committed Newtonian, by the way. Like

that moves over time. In this algebraic story there is no end-point, no objective,

some other great scientists, he felt un­

only rules of procedure.

comfortable with the consequences of his own discoveries. He wanted to look

In the nineteenth century, Charles Darwin did the same trick for biology. His story told how life on earth, with

through nature to see God. He believed that God, like Zeus, acted in the world and retained a perpetual involvement

its astonishing variety and beauty, was

can't predict. Newton's world was de­ signed, redesigned, and re-redesigned. Then, in the eighteenth century, Adam Smith introduced the idea of the

the result of random variations to­ gether with a rule of procedure called natural selection. This ensures that or­ ganisms that manage to survive and re­ produce will produce more organisms much like themselves, while the ones that can't feed themselves or find mates will simply die out. Despite ap­ pearances, we are, according to Dar­ win, not the result of a design at all, but

makes a plea on behalf of the prob­

invisible hand. According to Smith,

the product of a battle for survival and

lem-solvers:

people's desire for self-betterment,

resources, rather like Adam Smith's

And Gowers calls his paper The

Two Cultures of Mathematics, and

. . . the subjects that appeal to the­ ory-builders are, at the moment, much more fashionable than the ones that appeal to problem­ solvers. Moreover, mathematicians in the theory-building areas often regard what they are doing as the central core (Atiyah uses this ex­ act phrase) of mathematics, with subjects such as combinatorics thought of as peripheral and not 18

der to get rich you need to maximise

them, and the world stayed much the same. Right was right, wrong was

make their own positions pretty clear. Hardy is dogmatic. Atiyah

less design. There was no geometry, fixed in space, in Smith's story. Rather, it was about rules of procedure. In or­

when we analyse the mathematics they produce we are deep in Atiyah's

tremble as he placed us all in his death­

THE MATHEMATICAL INTELLIGENCER

with and control over His creation [ 19]. Newton's God didn't merely set up the machine and press the "start" button: He was always fiddling with the world and adjusting it in ways that mortals

guided by their reason, together with

competitive game involving bakers,

the forces of competition that ensure

farmers, and bread-eaters. We are not,

ever greater efficiency, ensure the well­

in Darwin's story, going anywhere, let

being of all:

alone trying to achieve perfection in or­

It is not from the benevolence of the butcher, the brewer, or the baker, that we expect our dinner, but from their regard to their own interest [7}. The story Adam Smith was telling, unlike Homer's, had no Zeus making us

der to deflect the nods from a god's darkish brows. We are merely trying to compete: to be more attractive than the next man or woman, and to survive, if necessary at the expense of others. And in the twentieth century, Karl Popper told an evolutionary story about science itself. According to his

ideas, scientists do not use a technique called "induction" to analyse data and approach ever closer to the true design of nature. Rather, they create theories about the world that compete with each other according to certain crite­ ria, the winner emerging to face chal­ lenges from new rivals. The evolutionary stories, like alge­ bra, tell of processes through time, op­ erating according to rules of proce­ dure. The design stories are based on the assumption of a timeless geomet­ ric structure, often a goal, an ideal of perfection towards which we are en­ joined to strive. Just as algebra is a more recent dis­ covery (to use the language of Hardy's dogma) or invention than geometry, so the algebraic, evolutionary stories, such as Adam Smith's invisible hand and Charles Darwin's natural selection, are much more recent than the ancient design stories. Betty S. Flowers [8] tells the story of how American society has developed under the influence of a se­ quence of myths (where the word "myth" is not intended to imply either truth or falsity), from the hero myths of Greek warriors and gun-slinging white-hatted cowboys through the "re­ ligious myth," the "enlightenment myth" or "democratic myth," to what she believes to be the currently pre­ dominant "economic myth." In the lan­ guage of Atiyah's dichotomy, the old hero myths are geometric in nature, based on a fixed design of right and wrong, good and evil. The religious myth is still geometric. The enlighten­ ment/democratic myth also involves an element of geometry through the idea of a search for pre-Popper scientific "truth" and the "best" solutions, al­ though it also contains a considerable element of algebra: democracy is a process of votes, elections, changes of policy and rulers in the light of experi­ ence, more elections and so on, with no fixed geometric end in view. The economic myth, which is so influential nowadays, is pure algebra, pure evolu­ tionary process. There is no end-point: the imperatives are to get more effi­ cient, get bigger, get richer! The evolutionary stories have a pe­ culiar power to discomfort us. Many of us are still trying to come to terms with their real meaning. We feel uncomfort-

able with stories that don't give us the comfort of clear moral rules, elegant de­ signs, and God-given systems of rewards and punishments. The initial objections to Darwin's evolutionary theory were not principally against the notion that species vary or even that we are de­ scended from apes, ideas that had been current in the scientific community for some time. Darwin's predecessors had assumed that evolution was a goal-di­ rected process, that each stage of evo­ lutionary development was a more per­ fect realisation of a plan that had always been in existence: a plan, perhaps, pres­ ent in the mind of God. Darwin disposed of this comforting idea [9] . It was the lack of a fixed goal that shocked Dar­ win's contemporaries, and continues to shock today. The fundamental objection is that Darwin moved us from geometry to algebra. Protesters who disrupt interna­ tional meetings to agitate against something called "globalisation" are also shocked by a shift from geometry to algebra. They are protesting against the economic myth. Statesmen and ex­ ecutives of multi-national companies may explain and illustrate the benefits of joining the global economy, but they find their motives questioned. From a geometric point of view, it is necessary to describe one's overall design, which soon acquires the status of a moral code. To a geometer, the algebraist's simple answer "The process works!" cuts no ice.

in the southern states of the USA, where it can be spelled you aU, you-aU, ya 'U, yawl, or yo-aU [ 10, p.449]. Nevertheless, in the linguistic equivalent of Hardy's di­ chotomy, Crystal stands firmly on the side of evolution. There is no fixed de­ sign for the English language, he says, to which we should all aspire. The lan­ guage is constantly changing through complex social processes of which we have only a limited understanding, and it has no end to aim at. Crystal tells how English has gone through phases when it has split into different species, and other periods when dialects have been brought closer together, sometimes prescrip­ tively. In mediaeval times, for example, regional variation was rife in England, but the introduction of printing, to­ gether with the establishment of writ­ ten laws, made standardisation impor­ tant: for the rule of law to work, it was essential that people could understand what was written and interpret it as far as possible in exactly the same way. The most highly prescriptive period reached its zenith in the eighteenth and nineteenth centuries, when English dictionaries were first published, and people could write things like

Language

According to Crystal, English is un­ dergoing a period of rapid evolution, with many new species developing round the world out of British and Amer­ ican English, the first two to emerge. The proponents of design in English tend to conflate their own strict views with a moral code. The following letter to a newspaper, with its morally loaded words, is a typical example:

The linguistic scientist David Crystal has written a book called The Stories of English [ 10]. The plural in the title de­ liberately suggests his viewpoint: that there is no such thing as "standard" Eng­ lish, spoken in England, with somewhat defective versions in the USA, India, Nigeria, and so on. Rather, English is an evolutionary system with different vari­ ants spoken in different places and at different times. The rules of procedure that underlie this evolution are not clearly understood: as yet, we haven't found the linguistic equivalent of Smith's invisible hand or Darwin's natural se­ lection. We don't know why, for exam­ ple, the second person plural has be­ come youse in Liverpool, ye and yiz in parts of Ireland and Scotland, and y'aU

harsh as the sentence may seem, those at a considerable distance from the capital, do not only mispronounce many words taken separately, but they scarcely pronounce, with purity, a single word, syllable, or letter [ 1 1 j.

. . . Oh, and the phrase "dumbing down. " A wholesale and quite horrible corruption of the verb, filtered infrom the US, I accept, but no more forgiv­ able for that. Dumb is a perfectly ho­ nourable word, meaning an inability to speak, and its modern usage is very lazy, unacceptable and a sign of ver­ bal and literate degeneracy [ 12].

© 2005 Spnnger Sc1ence+Business Med1a,

Inc., Volume 27. Number 2, 2005

19

The novelist Kingsley Amis has identified the two extreme points on Gowers's spectrum as they apply to the use of the English language. He calls them berks and wankers ( 13] . Berks don't care about the language's design, they are content to let it evolve hap­ hazardly. They are "careless, coarse, crass, gross. . . . They speak in a slip­ shod way." Left to berks, a language would "die of impurity. " They are ex­ treme problem-solvers in Gowers's language, using whatever expressions come to mind: they certainly don't care about grammar. Wankers, on the other hand, are obsessed with the mainte­ nance of what they mistakenly believe to be timeless rules of English. They are "prissy, fussy, priggish, prim. . . . They speak in an over-precise way. . . . " They write letters to newspapers, such as the one above, complaining about new usages. Left to wankers, a lan­ guage would "die of purity." Wankers are on the theory-building extreme of Gowers's spectrum: mathematical wankers (if they existed) would take so much care of the structural unity of mathematics that it would become too inward-looking to have any relevance outside itself. What about Atiyah's dichotomy? Can we fmd linguistic equivalents of fixed geometric space and an algebraic process over time? There is some evi­ dence that our brains do operate in time-like and space-like ways when we use regular and irregular verbs. (Regu­ lar verbs in English are those, such as "continue" or "risk," that add -d or -ed to form the past tense and past partici­ ple, as opposed to irregular verbs such as go-went-gone and ring-rang-rung.) The experimental psychologist Steven Pinker reckons that the brain produces regular and irregular verbs in two quite different ways [ 14]. Accord­ ing to him, regular verbs are produced through a series of rules, which the brain knows. One such rule is: "add -ed to make the past tense and the past participle." There's no need to look up the past forms in your memory: merely follow the rules. Irregular verbs, on the other hand, are retrieved from memory. But, ac­ cording to Pinker, this memory isn't a mere string of words, it includes a net­ work of connections between them. 20

THE MATHEMATICAL INTELLIGENCER

That's why irregular verbs show so many patterns: blow-blew, grow-grew, throw-threw, and so on. If irregular verbs were stored as an unconnected list, we would expect them to be quite random. It also explains why many verbs have become irregular. Ring-rang used to be ring-ringed, but it became at­ tracted to ing-ang-ung by analogy with words like "sing." Quit-quit became pop­ ular only in the nineteenth century: Jane Austen used "quitted." Light-lit, creep­ crept, kneel-knelt, dive-dove, catch­ caught are other examples. And sneak­ snuck is in the process of becoming standard in America. We can also see this process at work in experiments with meaningless verbs. For instance, 800/o of people suggest that the past tense of "spling'' is "splang" or "splung." I like to think of the irregular verb forms as crystals. These crystals have formed, somehow or other, out of the surrounding flow of general rules. Once a crystal has become established, it can grow by attracting other similar forms. Bear-bore attracted wear-wore, for example. A crystal will form more easily, and an irregular verb enter the mental dictionary more easily, if its pattern is repeated more frequently. That's why the irregular verbs tend to be frequently used: of the most com­ monly used verbs in English, the first regular ones are "use" (fourteenth place) and "seem" (sixteenth place) [ 15]. Most of the crystals that spring into existence by the sporadic use of a new irregular form of a verb will die away. In order to grow and survive, these crystals need a lot of use. The crystals are examples of design. Of course the crystals originally sprang up as the result of some sort of evolu­ tionary process, but once they have been created, their designs are so powerful that they become fixed and draw other words towards them, making them more like themselves. Like design stories, such as religions or Zeus with his im­ mortal locks, these designs are hard to resist. People who say that the past tense of "spling" is "splung" are partici­ pants in an old and gripping design story. So our brains use two ways of work­ ing-at least, as far as producing past tenses of verbs is concerned. Either we take an evolutionary path, consult our rule book, follow its instructions, and

add -ed or -d to the end of the verb; or we look up our special dictionary of de­ signs, with its network of connections, for the irregular verbs. These two modes of operation can both be highly creative. As we have seen, the networked dictio­ nary creates new irregular verbs all the time. And, although the word "rule" may sound dull and constricting, the general rule for past tenses can, and often does, lead to the creation of completely new expressions. Pinker has noted "he out­ Clintoned Clinton," and made-up chil­ dren's words such as "spidered" and "lightninged." Living in Malaysia, I was delighted when people "onned" and "offed" the lights. There seems, therefore, to be a "di­ chotomorphism" between mathemat­ ics and language under which the six dichotomists are mapped onto each other (Fig. 1). Discovery and Synthesis

Leaders in business organisations can, and do, use two kinds of approach. They can set up a series of rules to determine, for example, the processes by which a business should run. They might specify the circumstances under which parts of the business should be divested or, al­ ternatively, given extra resources. These rules might be in terms of return on cap­ ital, or profit, for example. And then they can simply allow the different parts of the business to have their heads but sub­ ject them to the rigour of the rules. This is an evolutionary approach: there is no objective, merely a process. The overall business is broken into separate parts, each of which is free to go its own way within the rules. This approach is simi­ lar to the way we make regular verbs: we break the word up into its con­ stituent parts, and apply a series of rules to each part. An extreme kind of holding com­ pany would use an evolutionary ap­ proach, an algebraic process though time. The Chief Executive Officer (CEO) would, in principle, say to each of the business managers something like, "Make a business plan that brings in a return on capital of at least 12%. Then go away and implement the plan. Come and see me again next year with your 12% return and with the next plan. If during the course of the year you can see that you won't make your 12%, let

Language

Mathematics

Level

when great chunks of the British econ­ omy were moved from the geometric central control of government owner­

Whole

Hardy

Gowers

People

B

Crystal

B

Arnis

B

Pinker

ship and made to ply their trades ac­ cording to the algebraic rules of the mar­ ket, with no destination in view. It was a

time

when

government

gradually

stopped trying to pick the economic sec­ tors in which Britain should be a big player-steel or car manufacture, for ex­

Structure

Atiyah

ample-and started trying instead to set a framework, a process within which the players would determine their own fu­ ture. The Department of Trade and In­

Fig. 1. The "dichotomorphism" between mathematics and language.

dustry was transformed (to a large ex­ tent) from the British economy's

me know. But, unless there's a very

the business, and at different times. The

into its

good reason for it, watch out!"

regulator.

owner

It used to do geome­

skill is to know when to break up and al­

try: to try, at least, to determine the con­

Such a CEO would have little idea of

low to evolve and when to bring together

figuration of Britain's industrial econ­

the overall shape of the business in

into a known crystalline pattern which

omy. But now it does algebra: it sets the

three or four years' time. He or she

will grow in a relatively controlled way.

rules and allows the economy to develop

might not even care: as long as we're

Many companies go through phases of

over time. In Amis's linguistic terms, one

making a good return, why worry about

evolution,

might say that where wankers once tried

of breaking apart existing

what we're selling? This is a highly Dar­

structures, when they give individual

unsuccessfully

winian, evolutionary approach. Nature

business managers freedom to diversify

economy, now berks don't care what it

to

shape the

British

doesn't care where each species is go­

into new, apparently profitable, busi­

looks like as long as it works.

ing. And, like the rule of natural selec­

nesses. They generally do this in good

Just as Atiyah sees algebra as the

tion and the rules which shape regular

times, when the risk of a few loss-mak­

devil's Faustian offer, many people felt

verbs, the rules of the market bring us

ing operations can be borne in view of

that some sort of Faustian bargain was

extraordinary new species.

the possibility of some bonanzas. And

being struck in 1980s Britain: that the

they also go through phases of design, of

country was giving something up, per­

On the other hand, business leaders approach. They can

bringing together. In harder times, when

haps social cohesion or even decency

view the overall business as a whole and

companies decide they can no longer af­

and kindness, by its pursuit of wealth in the turbulent marketplace. It felt at

can use a

design

impose a structure on it, fixed in space

ford to wait for experimental businesses

like a geometrical object. This kind of

to become profitable, businesses are

the time like a turning point, and with

approach leads to concepts like the

sold off or shut down, and the company

the benefit of hindsight we can see that

"core business." CEOs say, in effect,

retreats to its "core business."

indeed it was. Britain's long relative

"This is what we do, trying out other

Gary Hamel, well-known in the world

economic decline stopped. In 2002, for

lines of business diverts too much man­

of business as a strategy expert, has de­

the first time, the UK rate of unem­

agement effort, we should stick to what

scribed strategy as a process of discov­ enJ and synthesis [16]. Discovery is the

ployment was the lowest of all the G 7

we do best." The parts of such a busi­ ness come together. It is reminiscent of

algebraic process of moving through

the 1970s, the UK economy overtook

countries. And for the first time since

the way our brains deal with irregular

time, of following clear rules of engage­

that of France. This may explain why

verbs. First, you spot a crystalline pat­

ment that allow businesses freedom to

Britain has, ever since, voted for more

tern: widget factories in West Europe

evolve and determine, on the basis of

or less Thatcherite governments led by

are good business. Then you focus on

their performance, whether they are to

Margaret Thatcher herself, John Major,

that pattern: our core business is widget

be shut down or expanded.

Synthesis is

and Tony Blair. If the British people

manufacture in West Europe. Then you

the geometric redesign in space, the

have sold their soul, at least the devil

grow the crystal slowly: we will invest a

bringing together of current knowledge

has kept his or her side of the bargain.

little in splodget factories in West Eu­

and insights into what appears to be the

It's worth pointing out that, although

rope and, maybe, widget factories in

best business configuration at a partic­

the British economy moved towards an

East Europe.

ular point in time.

evolutionary story,

through privatisa­

tions and increased labour-market flexi­

The two approaches--evolution and design-are quite different. But, just as

Right and Left

bility, in another respect the country be­

the brain needs both a set of rules for

Another complex system, the British

came more centralised, more geometric.

regular verbs and a dictionary with con­

economy, made a significant move from

By 2002, government in Britain raised

nections for irregular verbs, corpora­

design towards evolution in the 1980s,

only

tions need to use both evolutionary and

under Margaret Thatcher. This was the

ment, compared with 1ZO/o in the US and

design approaches in different parts of

beginning of the era of privatisation,

100/o in France. Mrs Thatcher didn't trust

4% of taxes through local govern­

© 2005 Spnnger Sc1ence+Bus1ness Media, Inc., Volume 27, Number 2, 2005

21

local councils to push the evolutionary

tionary world. The politics of evolution

revolution

versus design is being replaced by the

deed, geometry is sometimes used as a

search for design in an evolutionary

metaphor in this very sense:

through-probably

rightly.

Neither did her successors [ 1 7]. a Liberal Democrat

world. And we can see in the aftermath

Member of Parliament in the UK, has

Vince

Cable,

of the twentieth century's great design

written about the distinction between

theories-communism, socialism, and

left and right in politics, how it has

fascism-the beginnings of some design

careful, degenerate into stagnation. In­

The impression is of two men at the height of their abilities but exhausted and immobilised by the fixed geome­ try of their power [21].

blurre d and lost meaning and signifi­

theories for the twenty-first century: re­

cance in recent years [ 18]. Following the

ligious fundamentalism and a strong

end of the Cold War, the issues that ex­

form of environmentalism known to its

cite and divide people don't fall within

detractors as "eco-fascism."

nies that concentrate on synthesis to the

wing. Such questions as regional inte­

Back to Mathematics

tant to

gration and loss of sovereignty (in the

The dichotomy of evolution and design

picked

EU, for example), minority rights, and

provides different points of view on a

Crystal [10] observed that the vast ma­

Large, overly bureaucratic compa­ exclusion of discovery become resis­

the framework of left-wing and right­

change,

and are eventually

off by nimbler competitors.

immigration are not left-right matters,

number of complex systems including

jority of comments he received from lis­

and the existing political parties, having

mathematics, science, language, busi­

teners to his radio series about English

grown up in the old left-right era, are of­

ness

and

were complaints about linguistic evolu­

ten thoroughly divided when it comes to

politics. The design point of view, like

tion: nobody seemed to be worried that the language might be too static.

organisations,

economics,

these issues. Many of the "new fissures"

geometry, centres on an image fixed in

in the world, as Cable terms them, are

space, often one towards which we

People move between the evolution­

along questions of identity. What nation

strive. The evolutionary point of view,

ary and design points of view, just as

do I feel I belong to? In what minority

on the other hand, has no end-point,

mathematicians switch between algebra

group, linguistic or racial or sexual or

and deals with a process through time.

and geometry: we all use both regular

professional, do I feel most at home?

In the case of mathematics, the two

and irregular verbs and have moments of

points of view are reflected in the three

linguistic experimentation and pangs of

We can now think of the old left-right distinction as a question of evolution

dichotomies with which this paper was

irritation when other people push the

versus design. The Left had a solid, so­

introduced: they describe the distinc­

boundaries of language. A mathemati­

cialist design in mind. They knew what

tion between evolution and design as

cian need not be on the same side of each

they wanted the world to look like, and

it impinges upon the overall philoso­

of the three dichotomies. Complex busi­

were less clear on how to get there. The

phy of mathematics, the motives and

ness organisations can operate in evolu­

Right, on the other hand, wanted a cer­

interests of mathematicians, and the

tionary, discovery mode in one part of

tain process through time, a world op­

structure of mathematics itself.

the business while retreating to the core

erating largely through the evolutionary

The evolutionary point of view is

business, in synthesis mode, in another.

processes of free markets, but they were

more recent than the design idea. Al­

The British economy moved towards

unclear about where the world should

gebra was invented/discovered long af­

evolution when state industries were

be heading. The Left did geometry. The

ter geometry, and the apparently sim­

privatised while simultaneously moving

Right did algebra. One might say that

ple evolutionary ideas of Smith and

towards design when power moved

while the Left sometimes thought the

Darwin emerged thousands of years

from local to central government. We

ends justified the means, the Right be­

later than the religious and mythical

shouldn't think of evolution and design

lieves the means justify the ends. By the

theories and stories. Science managed

as mutually exclusive modes of opera­

end of the twentieth century, in the post­

things an order of magnitude more

tion, rather as two distinct points of view

Cold War world, this conflict essentially

quickly: the gap from Bacon to Popper

that are often relevant in understanding

ended in victory for algebra. (A big sur­

was merely hundreds of years.

and describing what's going on.

prise for many people who, like me, en­

Evolutionary stories have the power

tered university in 1968, the Year of Rev­

to discomfort us. Darwin's ideas are

ical

olutions.)

still the subject of heated debate, and

points of view do not have equal sta­

In some fields, the theory of biolog­ species for example,

the two

Following the victory of evolution,

"globalisation" invokes fierce protests.

tus: as evidence is accumulated one

people are yearning for the comfort of a

Design theories often become associ­

point of view becomes preferable to

solid design. An evolutionary world is a

ated with moral codes, so that Dar­

the other. But in many cases, such as

cold, soulless place. The unease people

win's theory is seen as "anti-religious,"

mathematics,

feel with evolutionary theories like Dar­

capitalism can be thought to be im­

points of view enriches our under­

win's natural selection, and the repulsion

moral, algebra can be said to be a Faus­

standing of the whole.

with which many view the evolution of

tian offer, and Gowers writes a paper

Atiyah's dichotomy describes the

free markets under Smith's invisible

imploring theory-builders not to look

most exquisite conjunction between

hand, leads them to search for a new de­

down on problem-solvers.

evolution and design that has ever been

the

existence

of two

sign, an identity through nationhood or

Geometry/design, on the other hand,

achieved in any complex system: the

language or colour or sexual orientation,

while generally more comforting than

interplay between algebra and geome­

in an algebraic, ever-changing, evolu-

algebra/evolution, can, if we're not

try. Although geometry existed for a

22

THE MATHEMATICAL INTELLIGENCER

long time before the evolutionary in­ terloper appeared, and although there are no doubt many local skirmishes, peace has essentially broken out be­ tween algebra and geometry to the benefit of everyone: many of the liveli­ est parts of mathematics have names like algebraic topology, differential geometry, and analytic number theory. Gower is talking about evolution and design as they affect the politics of math­ ematics. Who sneers at whom? Who gets the plum jobs? Who gets the money? The situation he discusses seems eerily fa­ miliar to observers of complex business organisations. The people running the core businesses are trying to perfect a design. One senior executive even told me once that he thought of himself as the steward of a stately home: his role was to hand his successor the business in inunaculate condition. The evolution­ ists in big companies are the people dri­ ving the diversifications, people in an en­ ergy company trying to develop a timber business, for example, or those in a med­ ical insurance company developing a care homes business. Like Gowers's problem-solvers, they don't want to per­ fect the current design, they want to strike out in new directions, not caring much about the connections with the present design: they merely want to solve the problem of making more money for the business. To adapt Amis's words, left to the stately home stewards, the business would die of purity; left to the diversifiers it would die of impurity. The power in large companies usu­ ally lies with the big battalions of the core business, because, inevitably, that's where most of the money and the people are. Most of the new ventures fail, so the safest thing to do when times are hard is to abandon them al­ together. (I might add that it's gener­ ally easier to have an intelligent con­ versation about these matters in a business context than in mathematics, because in business there are more or less agreed financial criteria against which to judge proposals.) I find it hard to see how the situation could be much different. The evolutionists, mutating on the fringes, are in an uncomfortable and dangerous region. Most of them will probably fail. The designists in the core business of mathematics may make life difficult for them, and per-

haps one should view this as part of the competitive, evolutionary environment that ensures that the unsuccessful mu­ tations die quickly. In [22], Atiyah recognises the need for a balance between ensuring the continuity and unity of mathematics (i.e., taking care of its design) and al­ lowing the evolutionary possibility of exciting new discoveries that might at first appear to be disjoint from the mathematical core. In practice, when, for example, specific funding decisions have to be made that might affect this balance, the outcome may well depend on whether the decision-makers are natural evolutionists or designists. On the two sides of Hardy's di­ chotomy are people who believe in a timeless design of mathematics that it is our duty to discover, and those who think of mathematics as a human creation evolving through complex processes over time. Of the three di­ chotomies, this is the least capable of discussion on the basis of evidence or facts. It is truly a matter of point of view: what inspires you to do mathe­ matics? To what extent is it to under­ stand and explain the design of the world outside us, including mathemat­ ics itself; and to what extent is it to cre­ ate something beautiful and remark­ able that evolves and grows over time? Hardy's dichotomy generates a good deal of heat because, when people ar­ gue about it, they are arguing not about mathematics or even prestige, power, or money; they are arguing about them­ selves. The issues are their own deeply held beliefs. The stakes are really much higher than in debates on the other two levels. Lose the argument and what's left of you? Thus Hardy, sensibly enough, is merely dogmatic, while Mar­ tin Gardner (referring to the mathe­ matics of elementary particles) ad­ monishes, "To imagine that these awesomely complicated and beautiful patterns are not 'out there,' indepen­ dent of you and me, but somehow cob­ bled by our minds in the way we write poetry and compose music, is surely the ultimate in hubris'' [20]. Some be­ lievers in the design theory of mathe­ matics may want to batter the evolu­ tionists into submission. Judging from the infiltration of algebra into geome­ try's domain, and the successes of the

evolutionists in economics, politics, bi­ ology, epistemology, and linguistics, perhaps they have a point. REFERENCES

[ 1 ] G. H. Hardy, A Mathematician's Apology, Cambridge University Press, 1 969, 1 23. [2] W T. Gowers, The Two Cultures of Math­ ematics , Mathematics: frontiers and per­

spectives, 65-78, Amer. Math. Soc. Provi­ dence, R . I . , 2000. [3] M. F. Atiyah, Mathematics in the Twentieth Century, Am. Math. Monthly (200 1 ) , 1 08,

no. 7, 654-666. [4] R. Hersh, What is Mathematics, Really?, Jonathan Cape, 1 997. [5] M . F. Atiyah, An interview with Michael A tiyah , Math. l ntelligencer 6 (1 984), no.

9-1 9. [6] Homer, The Iliad. [7] A. Smith, Wealth of Nations. [8] B. S. Flowers, The Economic Myth , Center for International Business Education and Research, University of Texas at Austin. [9] T. Kuhn, The Structure o f Scientific Revo­ lutions, 1 962, 1 7 1 .

[1 0] D. Crystal, The Stories of English, Allen Lane, 2004. [1 1 ] J. Walker, Pronouncing Dictionary of Eng­ lish, 1 774, quoted in [1 0] , 408.

[1 2] D. Taylor, The Times, letter, 21 Septem­ ber 2004, 32. [1 3] K. Amis , The King's English, Harper Collins, 1 997. [1 4] S. Pinker, Words and Rules, Weidenfeld & Nicholson, 1 999. [1 5] N. Francis and H . Kucera, Frequency Analysis of English Usage; Lexicon and Grammar, 1 984.

[ 1 6] G. Hamel, Masterclass at CBI Conference, November 1 999. [1 7] The Economist, 4 May 2002, 29. [1 8] V. Cable, The World's New Fissures: Iden­ tities in Crisis , Demos, 1 994.

[1 9] B. J . T. Dobbs, The Janus Faces of Ge­ nius, Cambridge U niversity Press, 1 99 1 .

[20] M. Gardner, Math. lntelligencer 23 (2001 ) , no. 1 , 7 . [2 1 ] S. Jenkins, The Times, 2 9 September 2004. [22] M. Raussen and C. Skau, Interview with Michael Atiyah and Isadore Singer, EMS

September 2004. Perhelion Ltd. 1 87 Sheen Lane London SW1 4 8LE UK e-mail: [email protected]

© 2005 Springer Sc1ence+ Business Media. Inc., Volume 27, Number 2, 2005

23

M a thematic a l l y Bent

The proof is in the pudding.

Col i n Ada m s , Editor

rienced the floor show. So it could not

The Mathematical Ethicist

influence their survey responses, a complete waste of what must be a ma­ jor component of your budget. Com­ mon sense, fella? -Dr. Brad

() ()() Dear Dr. Brad, I was recently invited to give a talk at

Colin Adams

the prestigious Oberwolfach Tricen­

Opening a copy of The Intelligencer you

tennial Conference on Number Theory. Mathematical

may ask yourself

uneasily, "lt?tat is this anyway-a mathematical journal, or what?" Or you may ask, "lt?tere am I?" Or even "lt?to am I?" This sense of disorienta­ tion is at its most acute when you

This was quite an honor for me, par­

Dear Dr. Brad, I have gotten in the habit of throwing a lavish banquet for my students the evening before I hand out the student course evaluations each semester. My question is whether or not it is appro­ priate to have the students fill out the forms at the banquet. There is a break in the festivities after the dinner but

open to Colin Adams's column.

preceding the floor show, which could

Relax. Breathe regularly. It's mathematical, it's a humor column, and it may even be harmless.

serve for this purpose.

ticularly since I am not a number the­ orist. However, when it was time for me to give my talk, the conference or­ ganizer introduced me by saying, "And here is the man who proved Fermat's Last Theorem, Andrew Wiles. " Unfor­ tunately, I am not Andrew Wiles, and it was at this point that I realized that a mistake had been made. It was never their intent to invite me at all. I looked out at the sea of expectant faces, and did the only thing I could do. I pre­

Best,

tended to be Andrew Wiles for the next

Waldo Wendt

hour, receiving a standing ovation at

University of Westport

the end. Did I do the right thing? Andre Wilson

Dear Waldo I am assuming that the course evalua­ tions play a substantial role in the tenure process at your institution, and

Prinsetown University Dear Andre,

further, that you are junior faculty. It is

Under the

a rare senior faculty member who

the right thing. You certainly wouldn't

circumstances,

you

did

throws a banquet with floor show for

want to disappoint the audience, many

his or her students.

of whom had come a long way to hear it be­

Dr. Wiles. Luckily, mathematics is eso­

hooves you to behave in a manner that

teric enough that you can make terms

cannot possibly be interpreted to sug­

up on the fly, and the audience mem­

gest, in the slightest way, that you are

bers will be too embarrassed to ac­

Given these

assumptions,

attempting to influence the outcome of

knowledge that they have no idea what

the student course surveys. Even the

the heck you are talking about. So they

hint of such impropriety could be­

applaud at the end, even though they

smirch your career permanently. In

haven't understood. Would you con­

Column editor's address: Colin Adams,

other words, do not give the surveys

sider giving a talk at my institution? We

Department of Mathematics, Bronfman

out between dinner and the floor show.

can't afford Wiles.

Science Center, Williams College,

Wait until the next day. Besides, if they

Williamstown, MA 01 267 USA

did fill them out right after the banquet,

e-mail: [email protected]

the students would not yet have expe-

24

THE MATHEMATICAL INTELLIGENCER

© 2005

Springer Science+ Business Media, Inc.

-Dr. Brad ()()()

Dear Dr. Brad, A paper of mine was recently published in the

Journal of Algebra.

pensive presents to your new Bulgar­

theorem, two theorems? Brush. And

ian friend.

what about a corollary? Spit. Some­

translation of a paper by an obscure Bulgarian author that appeared in 1972

times I find myself lost in thought, froth

()()()

It was a

dribbling from my open mouth, the sound of banging on the bathroom door

Dear Dr. Brad,

the Journal of the

At a recent conference, I saw a very

Bulgarian Academy of Science. How­

nice talk on laminated deck transfor­

ever, I did make a few changes in the

mations. Afterward, I suggested to the

spacing and in the numbering of the

speaker that he might want to extend

theorems. Unfortunately, the Bulgarian

his results to polylaminated deck trans­

author discovered my translation (turns

formations. I was chagrined, six weeks

in

Bioavtomatika,

out he speaks English, who knew?) and

later, when the editor of a prestigious

he seems to be upset. Should I have ref­

journal asked me to referee a paper by

erenced his paper in the bibliography?

this same speaker, in which he ex­

Hortense Galbloddy College of St. Geronimo

plained laminated deck transforma­ tions and the extension to polylami­ nated deck transformations, with no mention of me whatsoever. I see that I

Dear Hortense,

have three alternatives.

Do ideas transcend the language in

1. I could contact the author and let

from desperate members of my family echoing in the background. But in your case, the time it takes to floss should suffice. It boils down to a single word. Polylaminated. In fact, your contribution wasn't even the whole word. It was actually just the prefix. Does a four-letter pre­ fix justify inclusion as a co-author in a paper. It turns out that there is a prece­ dent. In its landmark ruling of 1967, the Ethics Committee of the Canadian Mathematical Society determined that a prefix of three or fewer letters does not suffice to presume co-authorship.

which they are stated? And is it the ideas

him know I am the referee, making

with which we credit a creator? These

it clear there is no way I will rec­

are questions that I ask myself some­

ommend the paper for publication

times, when I am in the shower, and I

without my own name on it as co­

don't feel like going to work After sev­

author.

Hence the prefixes sub-, dis-, ir-, bi-, in-, co-, and non- do not cut the mustard. However, in an intricate argument I

will not attempt to recreate here, they determined that a prefix of four or more

the shower, a complete prune, and think

recommend that the editor reject

as long as at least one letter is from the last ten letters of the alphabet, does suffice. Hence, any of para-, trans-,

about what to have for lunch. But

the paper, and in the meantime,

null-, pseudo-, semi-, ortho-, or quasi­

enough about me and my day.

write up my own version and sub­

will do nicely. But endo- doesn't quite

mit it elsewhere.

make the cut. Of course, if the prefix

eral hours of consideration, I step out of

The short answer is yes, ideas do

letters,

2. I could submit my referee's report,

transcend the language in which they

3. I could send out a blanket e-mail to

are created. Hence, if the ideas in your

everyone in the field explaining how

is no lower limit on the number of let­

paper are exactly the same as the Bul­

this cretin tried to steal my idea.

ters necessary to warrant co-authorship.

garian's ideas, you must credit him with their discovery. But what is an idea? How does one decide if two ideas are identical? These are questions I reserve for my time in the tub. My con­ clusion is that there is no idea yardstick which can be utilized to determine the size of an idea, and to compare it with the size of other ideas. So who is to say if your translation actually captures the same ideas that the Bulgarian was at­ tempting to express? How can you

But whatever happens, I want to make sure that my actions are completely ethical and above reproach. I look for­ ward to hearing from you as soon as possible.

a-,

there

For God's sake, it's a Greek letter. In your case, poly- does the trick. This means that any of the three alter­ natives you outlined above would be fully justified, and you can rest assured, if nowhere else, you will find support on the Ethics Committee of the Cana­

With great respect,

dian Math Society.

Dr. Donald Dumpstead Ullalah U.

-Dr. Brad

()()()

Dear Don,

know what was in his head? You can't.

The question of what constitutes a suf­

So sleep easy. You need not feel guilty

ficient contribution to a paper to jus­

for neglecting to include his paper in

tify inclusion as a co-author is one of

your references.

the most difficult and slippery in all of

-Dr. Brad

contains a Greek letter, such as

mathematics. It is a question that oc­ cupies my thought processes when I

This concludes another column. But re­ member, when you find yourself tan­ gled in the morass of mathematical morality, you are only an e-mail away. I hope you don't have to write often. But a letter once in a while wouldn't hurt.

P. S. Of course, the Ethics Committee

brush my teeth every morning. I reserve

at your institution may see it differ­

that time to consider it. Can a single

Conscientiously yours,

ently. You might want to send some ex-

lemma be enough? Brush, brush. One

Dr. Brad Dearborn, Ph.D.

© 2005 Springer Science+Business Media, Inc , Volume 27, Number 2 , 2005

25

GOVE EFFI NGER, KENNETH HICKS, AND GARY L. MULLEN

I ntegers and Po ynom i a s · Com pari ng the Close Cous i ns Z and Fq [x]

umber theory is an enigmatic discipline; its fundamental simplicity is tempered by a very rich complexity. Few areas of human inquiry give rise to questions which are so easy to ask but so dif ficult to answer. The simplicity is obvious in the straightforward, even intuitive, definition of the

positive integers. The complexity arises when, among other things, one attempts to isolate the fundamental building blocks (the prime numbers) and their distribution. How­ ever, the integers are not the only ring with this enigmatic property. In this paper, we contrast and compare the ring of integers and the ring of polynomials in a single variable over a finite field. We explore a number of questions which have analogous versions in both settings. While these ques­ tions are quite easy to state, they are at the same time not easily answered in either setting. Why Fq[x] Is Special

As is standard, we denote the integers as Z and the posi­ tive integers as z + . Viewed algebraically, Z is a commuta­ tive ring, and, in fact, Z is a very nice ring in the sense that it is a unique factorization domain, which means that any integer can be written uniquely (up to the ordering of the factors) as the product of a unit and powers of prime num­ bers (the units, i.e., elements possessing multiplicative in­ verses in the ring, of Z are simply 1 and - 1) This unique factorization in Z is precisely the content of the Funda­ mental Theorem of Arithmetic. The basic idea in any unique factorization domain is that there are certain (non­ unit) elements, called irreducible elements or, in the case .

26

THE MATHEMATICAL INTELLIGENCER © 2005 Springer Science+ Business Media, Inc

of Z, primes, which form the "multiplicative building blocks" for the ring. Notice, by the way, that z+ is exactly those elements of Z which have 1 as the unit in their mul­ tiplicative representation. Because they do form the foundation for the multi­ plicative structure of Z, prime numbers play a central role in many of the questions which arise in classical number theory. For example, we know the role of primes in multi­ plication, but what about in addition? Is it true, for exam­ ple, that there are enough primes so that every even posi­ tive integer can be written as sum of two of them? This is precisely the famous Goldbach Conjecture, first formulated in 1 742. It is an easy question to ask, but it remains un­ solved to this day. For another example, two primes are called twins if they are two apart (e.g., 3 and 5, 5 and 7, 1 1 and 13, etc). Easy question: Are there infinitely many such pairs? The answer to this "Twin Primes Question" is also unknown. Generalizing now, we note that while Z is one example of a unique factorization domain, we have many other ex­ amples which may be less fundamental than Z but nonethe­ less are interesting and important in their own right. So let us investigate one class of such examples: polynomials in one variable x with coefficients in a field K. We denote this

set by K{x] and observe that it is in fact, like Z, a unique factorization domain (see, for example, [14], pages 289 and 3 19). What are the units of this ring? Well, they are pre­ cisely the (non-zero) elements of the field, i.e., the non-zero constant polynomials. This tells us then that the correct analogue to z+ in this domain is precisely the set of monic polynomials, i.e., polynomials whose leading coefficient is 1. Now, we see that the analogues to prime numbers in Z are simply monic irreducible polynomials in K{x] . All right, can we now do "number theory" in this new domain? For example, what would be a reasonable ana­ logue to the Goldbach Conjecture in this setting? Maybe one should ask, "Can every even monic polynomial be writ­ ten as a sum of two monic irreducible polynomials?" Though the meaning of this question is not completely clear (for example, what is an "even" polynomial?), it does seem to make some sense. If, however, we try the Twin Primes Conjecture in this setting, it's not even clear what "twin irre­ ducible polynomials" might be. Hence we see that questions which are asked in classical number theory about the inte­ gers may or may not even make sense in our new setting. If we want to raise or, even better, answer questions like the two just presented about polynomials, we need to know something about the irreducible ones. The basic idea would be: If irreducibles are more dense among the polynomials than primes are among the integers, then the answers should be yes; if irreducibles are less dense than primes, the answers may be no; and if the densities are similar, then the questions may be comparably hard to answer. But the nature and density of irreducible polynomials depends completely on the coefficient field K, and so we need to consider some different, common fields of coefficients. Those which come to mind immediately are the complex numbers C, the real numbers R, and the rational numbers Q. Let's think a little about irreducibles in the domains of polynomials over these three fields. The Fundamental Theorem of Algebra quickly settles the issue of irreducible polynomials in C [x] . Since every poly­ nomial of degree n over C has n roots in C, we see that every polynomial over C factors completely into linear polynomials, that is, the only irreducibles over C are the linear polynomials. Turning now to the reals R, we note that since complex conjugation (i.e., the function which takes a complex number a + bi to a - bi) is a field homo­ morphism, its natural extension to C [x] will take a poly­ nomialf(x) in R[x] to itself. But now factoring j(x) over C and applying conjugation, we see that any non-real roots f(x) might possess must occur in conjugate pairs. If such a pair is a + bi and a - bi, then (x - (a + bi))(x - (a ­ bi)) = x2 - 2ax + a2 + b 2 is irreducible over R. We see then that over R,j(x) factors into a product of linear and/or quadratic polynomials; that is, all irreducibles over R are either linear or quadratic. Thus we see that, in some sense, irreducibles in C[x] and R[x] are relatively scarce. In particular, it is obvious that no analogue to the Goldbach Conjecture could possi­ bly be true in these domains. More generally, the obviously wide gap between the multiplicative structures of Z on the

one hand and C [x] and R[x] on the other make any com­ parable "number theory" of those domains unlikely. The situation with Q[x] is not so clear. There exist few criteria for determining the irreducibility of polynomials over Q, the Eisenstein Criterion being one of those few. It states that ifj(x) has integer coefficients and if there ex­ ists a prime number p which does not divide the leading coefficient of f(x), which does divide all the other coeffi­ cients, but whose square does not divide the constant co­ efficient, then}t:x) is irreducible. For example, x" + p is ir­ reducible for all exponents k and any prime p. In particular then, the Eisenstein Criterion shows us that, unlike the cases of C [x] and R[x] , there are in Q[x] irreducibles of ar­ bitrarily high degree. But are they more or less dense in the set of all polynomials in Q[x] than the primes in Z? This is difficult to answer precisely, but a strong hint is the fol­ lowing result of D. R. Hayes [20], whose proof is quite short and uses primarily the Eisenstein Criterion: Theorem 1 (A Goldbach Theorem for Q[x]) Every poly­ nomial over Q can be written as a sum of two irreducible polynomials.

The ease with which this result comes forth, in contrast to the great difficulty of the classical Goldbach Conjecture, certainly leads us to suspect that irreducibles are quite dense in Q[x] and that any "number theory" of Q[x] would (as in the cases of C[x] and R[x] but, in a sense, for the op­ posite reason) be quite different from that of Z. Though perhaps not as "famous" as the fields C, R, and Q, another important collection of fields is thefinitefields. If q is any power of a prime number p, then there exists a (unique up to field isomorphism) finite field, denoted Fq, which has q elements. Once again, for any such q, the ring Fq[X] is a unique factorization domain. What about its ir­ reducibles? We shall discuss these at some length in this paper, but let us start by saying that although recognizing irreducible polynomials over Fq can be tricky, counting them is not hard. Restricting our attention to monic poly­ nomials (as previously justified), it is obvious that the num­ ber of such of degree r over Fq is exactly qT since there are r coefficient slots and q choices for each. Of these qr poly­ nomials, how many are irreducible? It turns out not to be difficult to see that there are precisely

Nq(r)

=

_!_ ) JL(d)qTid r d[';.

( 1 . 1)

monic irreducibles of degree r over Fq, where f.L is the so­ called Mobius function, whose values are 0, - 1 and + 1 . See, for example, Theorem 3.25 o f [23] . The order (i.e., largest term) of the sum is qr, and hence we see that the order of the number of monic irreducibles of degree r over Fq is simply qr/r. In other words, about one out of every r monic polynomials of degree r over Fq is irreducible. How does this density compare with the density of primes in z + ? According to the famous Prime Number The­ orem of classical number theory, the number of primes less than a positive integer n is of order nllog(n), where log in-

© 2005 Springer Sc1ence+Business

Media, Inc . Volume 27, Number 2, 2005

27

dicates the natural logarithmic function. In fact, a more ac­ curate estimate for the number of primes below a real num­ ber y is the so-called logarithmic integral

(Y . Ll(y) =

Jz

dt log(t) '

(1 . 2)

which makes it clear that "near" an integer n, the density of primes is approximately 1/log(n). For an excellent dis­ cussion of these ideas, see Section I of Chapter 4 of [29]. Now, back in Fq [x] , we would like to have a way to mea­ sure the "size" of a monic polynomial of degree r. A nat­ ural such measure would be the following: Definition 2 The absolute value of a polynomial A of de­ gree r in Fq[x], denoted !AI, is qr.

Note that whereas only one positive integer n has (in the standard sense) absolute value n, numerous (in fact qry monic polynomials of degree r over Fq have (in this new sense) absolute value qr. Using this notion and observing that r logq(q1, we see the following fascinating and im­ portant fact: In both Z and Fq[X], the density of irreducible elements "near" an element is approximately 1 over the log of the absolute value of that element. In the case of Z the log is natural; in the case of Fq[x] the log is base q. This close connection between these two quite different looking unique factorization domains Z and Fq[X] means that their number theories may very well be quite similar. In the remainder of this paper we shall investigate these similarities, together with the inevitable differences which arise. Can we state and prove a precise "Goldbach Con­ jecture for Fq[x]"? What about a precise "Twin Irreducibles Conjecture for Fq[x]"? In what follows we consider these and the analogues of various other questions, answered and unanswered, from classical number theory. =

Factorization Both Z and Fq[X] being unique factorization domains, every element in each can be factored in a unique way, apart from order of the factors, into "prime" elements. In the ring Z this factorization is into a product of prime numbers while in Fq[X] it is into a product of irreducible polynomials. Given any unique factorization domain, some basic ques­ tions arise quite naturally: How can we determine if a given element is irreducible, and are there efficient algo­ rithms to make this determination? More generally, we

can ask the harder questions: Given an element of our do­ main, how can we determine its unique factors, and are there efficient algorithms to make this determination? In this section we address these questions briefly in the cases Z and Fq[x], saying more about the polynomial case since it is less well known. A partial answer to all of our questions in the case of Fq[X] is as follows: There are at least two efficient linear algebraic methods which can be used to factor polynomi­ als over Fq (and hence, of course, determine irreducibility),

28

THE MATHEMATICAL INTELLIGENCER

at least in the case when the field is "small," i.e., when the cardinality q of the field is smaller than the degree r of the polynomial. We deal here primarily with polynomial fac­ torization over small fields since in most applications the field size is small (e.g., in algebraic coding theory for the error-free transmission of information and in cryptography for the secure transmission of information, q is usually 2). The first of these two methods is known as Berlekamp 's algorithm, and is described in detail in Section 4.1 of [23] . The second method, developed by Niederreiter in [24] , uses non-linear ordinary differential equations in characteristic p. In both cases, in order to factor a polynomial A(x) of de­ gree r, one sets up an r X r matrix MA over Fq. One then uses linear algebra to reduce the matrix MA - I, where I is the identity matrix. It turns out that the rank of MA - I is r - k, where k is the number of irreducible polynomials in the factorization of A(x). Note then that the polynomial A(x) is irreducible if and only if the rank of the matrix MA - I is r - 1. Moreover, with further work we can ob­ tain the actual irreducible factors using the vectors in a ba­ sis of the nullspace of MA - I. Again, these algorithms are efficient; i.e., their running times are bounded by a poly­ nomial function of r. To contrast now with the case of Z, we remark also that the Berlekamp and Niederreiter algorithms are not only ef­ ficient but are also deterministic (as opposed to proba­ bilistic) in the sense that they are guaranteed to give a def­ inite and correct output upon any appropriate input. Thus, as far as Fq[x] with small q goes, things are as good as they can be in terms of irreducibility testing and factorization. This however, is not the case for integers. First, for pri­ mality testing, up until very recently the existence of a comparable algorithm for integers had eluded mathe­ maticians, though there are various deterministic but not efficient algorithms as well as efficient but probabilistic al­ gorithms for proving primality (see for example [ 1]). How­ ever, a deterministic polynomial time algorithm for prov­ ing primality was announced in August 2002 and will be published in [2], providing a significant break-through in the area of computational number theory. Nonetheless, it

would appear that irreducibility testing in Fq[x] is a somewhat easierproblem than primality testing in Z. The phenomenon of problems seeming to be a bit easier in the polynomial setting than in the integer setting is a theme we shall see repeatedly in this paper. Turning to factorization, the problem for Z is a notori­ ously difficult one; there is no known efficient algorithm to factor integers. This difficulty lies at the heart of modem cryptography, for current cryptographic systems such as RSA depend on the fact that multiplication is easy but fac­ torization is hard. The fastest methods known are based upon number field sieves or on elliptic curves; see [7], [22], and [33] , for example. If one wishes to speculate about the use of quantum computers (not a current reality), then Shor has provided a polynomial time algorithm in his 1994 pa­ per [32]. However, there are not even any efficient proba­ bilistic algorithms for integer factoring as yet! On the poly-

nomial side, we have seen that for small q, all is well, but if we remove restrictions on q, then there are again no known efficient deterministic polynomial factorization al­ gorithms, even using the generalized Riemann Hypothesis (see for example [ 15]). However, there are efficient prob­ abilistic polynomial time algorithms for polynomial factor­ ing, some with near quadratic running times (see [ 16]). In other words, factorization appears to be, in general, a hard problem in both Z and Fq[x], but, again, probably "less hard" for polynomials than for integers, and definitely not hard if q is small, i.e., less than r. The Distribution of Primes and Irreducible& When studying the distribution of primes in Z or of irre­ ducible polynomials in Fq[x], one can be concerned with "large-scale" or "small-scale" distribution. As noted in the opening section, the large-scale distribution is essentially understood in both domains. The Prime Number Theorem tells us that the number of primes below an integer n is asymptotically 1T (n)

�

n

--

log n

,

and the number of monic irreducible polynomials at or be­ low a given degree r is asymptotically

Nq(r)

� qr!r.

We observed there the remarkable similarity of these esti­ mates. We should also emphasize, in support of one of our themes, that whereas the proof of the Prime Number The­ orem is extremely deep and complex, the estimate for Nq(r) is quite easy to establish using basic facts from the field theory. The small-scale distribution of primes and irreducibles seems to be a much more difficult problem in both domains. In this and the next section we shall look at two problems which illustrate this difficulty: the classical and polynomial versions of the Twin Primes Conjecture and the Goldbach Conjecture. In each of these cases, it is the small-scale dis­ tribution which is at issue. Here the matter of the Riemann Hypothesis, which ties the small-scale distribution of primes to the zeros of the Riemann Zeta-function, and the analogue of that hypothesis in the polynomial setting must come into play. So let us consider now twin primes and irreducibles. The Twin Prime Conjecture (see, e.g., [ 19]) states that there are an infinite number of consecutive primes with a difference of two, that is, Pm+ 1 Pm 2 for infinitely many m. More specifically, it is conjectured that the number 1r2 (n) of such twin prime pairs less than n is asymptotically -

==

(3.3) The product is known as the "twin prime constant," de­ noted C2, whose value is about 0.66016. In other words then, the density of twin primes near an integer n is ap­ proximately 1 .32/(log n)2. The Twin Primes Conjecture has

never been proved, but there is excellent numerical evi­ dence in support of its truth (see, e.g., [25]). It is thought-provoking to note that if the probability of finding a prime near n is of order 1/log(n), then the prob­ ability of finding a twin prime pair near n should be of or­ der 1/(log n)2• A wonderful discussion of heuristic reason­ ing in the theory of numbers and the apparent connection (perhaps midleading) between these probabilities is given by P6lya [28] . It may seem remarkable that while the reciprocal sum of the primes diverges, the reciprocal sum of the twin primes converges, whether or not there are an infinite num­ ber of twin primes. This was proved by Viggo Brun in 1919 the sum, known as Brun's constant and denoted Bz has value approximately 1.90216 and has been tabulated to con­ siderable precision [25] . This result expresses the scarcity of twin primes compared with the primes, much like the divergence of the harmonic series as compared to the con­ vergence of the reciprocal sum of squares. We tum now to an analogous question in Fq[x]. Can one define "twin" irreducible polynomials, and if so, does the dis­ tribution of twin irreducibles mimic the distribution of twin primes? The answers to these questions are "yes" and have been presented in some detail in our recent paper [ 13]. Again, we will consider only monic polynomials, for these are the correct polynomial analogues to positive integers. In anal­ ogy to the integer case, we define two irreducible polyno­ mials to be "twins" if they differ by as little as possible.

[4];

Definition 3 Two polynomials P1 and Pz, both of degree r over Fq, are said to be twin irreducible polynomials, or simply twin irreducibles, provided that IPz - P11 == 4 if q = 2 or IPz - P11 = 1 otherwise.

In fact, it is easy to show [ 13] that for any fixed degree

r, there are infinitely many twin irreducibles as the field or­ der q goes to infinity. However, the true polynomial ana­ logue of the Twin Primes Conjecture is when the order q is fixed and the degree r goes to infinity. Using a line of reasoning similar to that used by Hardy and Wright [ 19], it is possible to derive an analogue of the Twin Primes Con­ jecture for irreducible polynomials. From [ 13] , this con­ jecture is:

Nz,q(r)

(

� 0

q - 1 -2

)

qr

r2

I)

(

1

-

ciPI

1 -

)

1)2 '

where Nz,q(r) is the number of twin irreducible pairs of de­ gree r over F q[x], and the product extends over all irre­ ducibles P provided q > 2 but does not include linear irre­ ducibles if q = 2. Also, o = if q = 2 and o 1 otherwise.

4

=

The infinite product on the right converges to a number which depends on q. The reader should note the intriguing similarity between N2,q(r) and 1r2 (n) (Equation 3.3), giving another nice example of the closeness of Fq[x] and Z. Though well supported by numerical evidence (see [ 13] for the polynomial case), both of these results remain conjec­ tures for now.

© 2005 Springer Science+ Bus1ness Med1a, Inc., Volume 27,

Number 2, 2005

29

We remark that it is necessary to separate out the case for the field Fz because only for this field is the smallest difference between irreducibles not a simple constant term of the polynomial. For example, when q = 2 and r 3, the irreducibles :r3 + x + 1 and :r3 + x2 + 1 differ by x2 + x, the smallest difference between irreducibles for all r > 2. In contrast, for q = 3 and r 3, :r3 + 2x + 1 and :r3 + 2x + 2 are twins differing by the constant 1 . Finally, for any given q, one can estimate an appropri­ ate analogue B2 ,q of Brun's constant. For example, in [ 13] it is shown that B2, 2 1.059 1 . . . . Thus it is true that whereas both the sum over reciprocal primes and the sum over reciprocal absolute values of irreducibles over Fq[x] (which the reader should check is essentially the harmonic series) diverge, both the sum over reciprocal twin primes and the sum over reciprocal absolute values of twin irre­ ducibles converge. This is another indication of the close connection between the distribution of primes in Z and the distribution of irreducibles in Fq[X]. One wonders if the Twin Irreducibles Conjecture may be easier to settle than the notorious unsolved Twin Primes Conjecture. =

=

=

The Goldbach Conjecture and the Riemann Hypothesis In 1742 the German mathematician Christian Goldbach conjectured in a letter to Leonard Euler that every positive integer (greater than 5) is a sum of three prime numbers. Euler observed in reply that this is equivalent to every even positive integer (above 2) being a sum of two primes (the reader should check this equivalence), and it is this latter formulation which is now commonly referred to as the Goldbach Conjecture. There are many excellent discus­ sions of the complicated mathematical history of this con­ jecture (see, for example, [29]); we will focus here on a spe­ cific strand of that history which is then tied to the corresponding problem for polynomials. In 1912, 170 years after Goldbach's letter, the distin­ guished German number theorist Edmund Landau declared in a lecture in London that the Goldbach problem was

"beim gegenwtirtigen Stande der Wissenschaft unan­ greijbar"-"in the current state of knowledge intractable." Two young British mathematicians, G. H. Hardy and J. E. Littlewood, took up the challenge and, over the next 15 years, produced a seminal series of papers entitled "Some Problems of 'Partitio Numerorem.' " In the third of these, subtitled "On the Expression of a Number as a Sum of Primes," they attacked the Goldbach problem with a brand new method and, though still failing to solve it, were able to " . . . show, however, that the problem is not 'unan­

greijbar', and bring it into contact with the recognized methods of the Analytic Theory of Numbers" ([18], page 2). This method, known today as the "Circle Method" or (ap­ propriately) the "Hardy-Littlewood Method" (first used by Hardy and Ramanujan, see page 698 of [31]), uses complex analysis (more specifically, complex line integrals around the unit circle in the complex plane) to obtain asymptotic formulas for the number of representations of integers in various forms, including as sums of primes. It turns out, as

30

THE MATHEMATICAL INTELLIGENCER

we shall see, that the method can be fruitfully applied to other domains by using appropriate analogues to the com­ plex plane, its compact unit circle, and line integrals. Saving some details for a bit later, here is what Hardy and Littlewood discovered. Let us denote by Mk(n) the number of representations of a positive integer n as a sum of k odd primes. Assuming what they called "Hypothesis R," a generalization of the Riemann Hypothesis to Dirich­ let L-functions, they obtain an asymptotic formula for Mk(n), meaning a formula which contains a main term, which measures Mk(n) with greater and greater relative ac­ curacy as n � oo, and an error term, whose magnitude is given using "Big-0" notation. That formula can be summa­ rized as follows:

Mk(n)

=

Ck(n)

nk- 1 + O (nk/2 + nk - 1 - <\; , (log n)k

where Ck(n) is a positive bounded function whose exact value depends on k and on n's prime factorization, where n and k have the same parity (otherwise Mk(n) is evidently 0), and where 0 < E :=::; 1/4. The key to success for any asymptotic analysis is that the main term must grow at a rate strictly greater than that of the error term as the key parameter (in this case n) goes to infinity. First let's set k = 2, that is, let's consider the case with which the Goldbach Conjecture deals. Here, since 2/2 1 > 1 - E 2 - 1 - E, we see that the formula says that =

=

M2 (n)

=

C2 (n)

n

(log n)Z

+

O (n),

from which, unfortunately, we learn nothing, for the growth rates of the main and error terms are the same. Hence Hardy and Littlewood's analysis, even assuming the un­ proven Hypothesis R, fails to solve the Goldbach Conjec­ ture. However, if k 2: 3, then k > 2 + 2E, 2k - 2 - 2E > k k - 1 - E > k/2, and so we obtain, for k 2: 3,

Mk(n)

=

Ck(n)

nk - 1 + O (nk- 1 - E). (log n)k

Now the asymptotics have "kicked in," and we can use Hardy and Littlewood's analysis to prove the existence of representations of positive integers as the sum of k primes provided k 2: 3, and provided, of course, that Hypothesis R is true. The case k = 3 is of particular interest. We shall call this the Odd Goldbach or 3-Primes Conjecture: Every odd num­ ber greater than 5 is a sum of three primes. Let us write down here exactly what Hardy and Littlewood proved for this important case: Theorem 4 (Asymptotic 3-Primes) If Hypothesis R

is true, then every sufficiently large odd number can be repre­ sented as a sum of three odd primes, and the number of such representations is given asymptotically by:

M (n) 3

; Jl ( 2

�

(lo

n)3

1

+

(p

� 1)3 ) ·

!] (

als, we very briefly summarize the current state of knowl­ edge about the Goldbach Conjecture. In 1937 Vinogradov, 1

-

�

p2 - p + 3

)

where p runs over prime numbers as specified.

using a modification of the Hardy-Littlewood method to­ gether with numerous ingenious estimates of trigonomet­ ric sums, proved without

hypothesis that every sufficiently

large odd number is a sum of three primes [35]. His analy­ sis has been refined to the point that "sufficiently large" to­

The latter two products here are the explicit form of

day means greater than about 1043000 [6], which is of course

C (n ). The first product does not depend on n and has value 3 about 1 . 15. The second does depend on n's prime factor­

still far beyond the possibility of checking the cases below

ization and has a value at or just below .67 if 3 1 n, and just

it using computation. In 1997 it was shown that if the full GRH holds, then every odd number above 5 is a sum of

below 1 otherwise, provided n is odd. Notice also that if n

three primes [8]; that is, assuming the GRH allows us to

is even, this factor is 0 and so M (n) = 0, as it must be. 3 One can use a simple heuristic to see in general that the basic order of the main term for Mk (n) should be nk - 1/(log n)k . Here is the argument for the case k = 3. To write an

eliminate the "sufficiently large" part of the Odd Goldbach

nal Goldbach Conjecture, the best current result, obtained

odd number n as a sum of three odd primes, by the Prime

in 1973 by Chen [5] using sieve methods, is that every suf­

Conjecture-this will be significant when we discuss the polynomial analogue below. Finally, concerning the origi­

Number Theorem we have about nllog n choices for primes

ficiently large even number is a sum of a prime and an "al­

below n in each of the 3 slots. Each of these n3/(log n)3

most-prime" (a number which is either prime or the prod­

combinations adds up to an odd number below 3n, so if

uct of two primes).

these sums are uniformly distributed, about 1 out of every

Let us tum now to the polynomial case. If we hope to

3n/2, or 2n2/3(log n)3 of them, will add up to n itself. The

write a monic polynomial of degree

2/3 part of this obviously needs closer analysis, but the

three irreducible monic polynomials, then, in general, one

n2/(log n)3 part is now clear.

r as a sum of two or

of the irreducibles must also be of degree r and the other(s)

We now tum to the connection between Hardy and Lit­

must be of lesser degree. It is not clear that the concepts

tlewood's analysis and the Riemann Hypothesis, which is

of "even" and "odd" make any sense in the polynomial con­

the source of the

text, but, surprisingly, they not only make sense but also

in the error terms above. If

E

x is

a "nu­

merical character" mapping

come into play, much as in the integer case. We make the

page 418), then the

following definition:

x,

denoted

isfying

Z to C (see, for example, [3], Dirichlet L-function associated with

L(s,x), is ffi(s) > 1, by

defined, for a complex number s sat­

L(s,x)

= I x(n�) . n�l

Like their close relative the Riemann Zeta-function, the L­ functions can be extended analytically to the whole com­ plex plane with a unique pole at ( 1 , 0) and with "non-triv­ ial" zeroes in the strip 0 ::S ffi(s) ::S 1 . The Generalized Riemann Hypothesis (GRH) is that these functions all have the property that all their non-trivial zeros z actually sat­ isfy ffi(z) = 112. Hardy and Littlewood's "Hypothesis R" (or a Weak Generalized Riemann Hypothesis) is that there ex­ ists a ®, 1/2 ::S ® < 3/4, such that every such zero z satis­ fies ffi(z) ::S ®. Finally, E above is simply 3/4 - ®. The connection between the locations of the zeros of the Dirichlet L-functions and the numbers Mk(n) is far from transparent and can only be hinted at here. The interested

Definition 5 A polynomial A over Fa is called even if it is divisible by an irreducible whose absolute value is 2. Otherwise A is called odd. It is evident from the definition of absolute value that even polynomials exist

only over the field F2 and are pre­ cisely those polynomials which are divisible by x or x + 1 , the two even irreducibles. Hence a polynomial over F

is 2 even if it lacks a constant term or if it has an even number

of non-zero terms (check). Unlike

Z, it is not true, for ex­ like Z, the

ample, that "even plus odd is odd. " However

reader should check that no even polynomial can be the sum of three odd irreducibles, and so oddness is a neces­ sary condition for our desired "3-irreducibles representa­ tion. " In a similar vein, polynomials which are too "small, " e.g., linear ones, won't have such a representation, and in fact it turns out that if q is even, then A

=

x2 + a (a E Fq)

reader can profitably consult the original work [ 18] or sec­

also won't have such a representation (check), being also

ondary sources such as [34]. We say here only that in the

a little too "small." But those tum out to be the only ex­

course of the difficult but beautiful analysis done by Hardy

ceptions:

and Littlewood, the L-functions and in particular their log­ arithmic derivatives

L'(s,x)IL(s,x)

arise in a natural way.

The locations of the zeros of L(s,x) bear directly on the es­ timates being made, and what is of crucial importance is that the real parts of those zeros stay away from 1, in fact stay strictly to the left of 3/4. Unfortunately, this was un­ proved in 1922, and it remains unproved today. Before moving to the analogous problem for polynomi-

6 (A Complete "3-Irreducibles" Theorem) Every odd monic polynomial of degree r 2: 2 over every finite field Fq (except for the case of x2 + a when q is even) is (t sum of three monic irreducible polynomials. Theorem

The proof of this result, which is somewhat lengthy, is contained mostly in [ 12] and completed in [9], [ 10], and [ 1 1 ] .

© 2005 Springer Science +Bus1ness Media, Inc., Volume 27, Number 2 , 2005

31

Actually, the first asymptotic result in this direction was David Hayes's 1966 paper [21], but a drawback there was that it dealt with representations of A (of degree r) of the form aP1 + f3P2 + yP3, where P1 , P2, and P3 are all of de­ gree r and a + f3 + y = 1. Nonetheless, it pointed the way to the analysis done in [12], which we now discuss briefly. To apply the Hardy-Littlewood method in the polyno­ mial setting, one first needs an appropriate analogue to the unit circle in C on which to carry out the analysis. In [12], that analogue is the compact adele class group Aklk, where k is Fq(x), the field of rational functions over the finite field Fq, and Ak is the adele ring. This latter object Ak is a re­ stricted direct product of the completions of k at all its places, including the infinite place. The reader should con­ sult Chapter 4 of [12] for the details. Now the complex path integral around the unit circle used by Hardy and Little­ wood is replaced by the Haar integral on A�, and the analysis can go forward in a similar fashion. The classical Dirichlet £-functions come over into this setting in a way which is natural but somewhat complicated (see Chapter 5 of [ 12]). But now a major advantage occurs because of the pioneering work of Andre Weil, who in 1948 proved an analogue of the Generalized Riemann Hypothesis in this function field setting [36]. We write now the implication of his remarkable result for our analysis:

We hope the reader, in comparing this result to Hardy and Littlewood's Theorem 4, cannot help but be impressed with their remarkable similarities. Z and Fq[X] are indeed close cousins! Some Concluding Thoughts on Z and Fq[x]

Despite the many similarities between Z and Fq[x], there are differences (1) in structure and (2) in the depth of analysis needed to uncover that structure. Two obvious examples of the former are that Z has characteristic 0 while F q[x] has characteristic p (where q = pk for some k), and that whereas each positive integer has its unique ab­ solute value, qr monic polynomials over Fq have absolute value qr. Two examples of the latter which we have seen are the Prime Number Theorem (very deep) versus for­ mula (1.1) for the number Nq(r) of monic irreducibles in Fq[x] of degree r (not deep), and the Riemann Hypothesis (unsolved) versus Weil's Riemann Hypothesis for function fields (deep but solved). We present now another exam­ ple where things are a little bit easier to understand in Fq[x] than in Z. For d 2: 1, let P(x) = PaXd + + Po E Z[x] be a poly­ nomial of degree d with Pa = 1. If Q = Z [x]I(P(x)) denotes the quotient or factor ring of Z [x] modulo the ideal gener­ ated by the polynomial P(x), then each a E Q has a repre­ sentation of the form a = ao + a 1x + + aa - 1xd- l with ai E Z, 0 :::; i :::; d - 1. The pair (P(x), M) with M = (0, 1, . . . , !Pol - 1}, is called a canonical number system if each a E Q admits a unique representation in the form a = h + ahx for some h 2: 0 with each ai E M, a0 + a 1x + 0 :::; i :::; h and ah =t- 0 for h =F 0. The problem of characterizing canonical number sys­ tems over Z is a very difficult one. As an indication of this difficulty, for quadratic polynomials over Z the problem has been solved, but only partial results are known for cubic polynomials; see [30]. On the other hand, the following re­ sult from [30] provides a complete characterization of anal­ ogous digit system polynomials in the setting of Fq[x]. For the finite field analogue, let x, y be transcendental + bo E over Fq and let P(x,y) = bn y n + bn - lyn - l + Fq[x,y] with bi E Fq[x], bn =F 0, deg bn = 0, and deg bo > 0. Let N = (P E Fq[X] : deg p < deg b0} and let R denote the quotient ring Fq[x,y]I(P(x,y)). Then each r E R can be rep­ resented as r = r0 + r1y + + rn l Y n - l with ri E Fq[x]. Further, we say that r E R has a finite y-adic representa­ tion if r admits a representation of the form r = r0 + r1 y + h + rh y for some h 2: 0 with ri E N for 0 :::; i :::; h and rh =F 0 for h =F 0. Finally the pair (P(x,y), N) is called a digit system in R if each r E R has a unique finite y-adic repre­ sentation. A complete characterization of finite field digit systems is given as follows in [30]: A polynomial P(x,y) as above is a digit system polynomial if and only if n n:tax{deg bil < deg bo. ·

·

·

·

Theorem 7

(GRH for Function Fields) Given suitable re­ strictions on the character x, the function field £-func­ tion L(s, x) is a complex polynomial Px in q-s, and when factored into

each

n

satisfies

l ril

=

1 q 12•

Though the details are many, this result allows the analy­ sis to proceed very smoothly and eventually yields not only the asymptotic result below but also very sharp numerical estimates for the error terms (see Chapters 6 and 7 and the Appendix of [ 12]). These estimates then lead to the "com­ plete" Theorem 6. The asymptotic result is (Asymptotic "3-Irreducibles") Ifeither thefield order q or the degree r is sufficiently large, then every odd polynomial A of degree r over Fq is a sum of three odd ir­ reducible polynomials, and the number M3(A) of such rep­ resentations is given asymptoticaUy by Theorem 8

q�

M3 CA) - �

J}2(

1

+

1

c�l - 1)3

TI (

1-

1

�12 - 3�1 + 3

where P runs over irreducible polynomials over Fq specified.

)

,

as

Note once again that if A were an even polynomial, then the second product, and hence M3(A) , would be 0, as ex­ pected.

32

THE MATHEMATICAL INTELLIGENCER

·

·

·

·

·

·

·

·

·

·

·

·

·

·

t�l

Despite examples like this one where the gap between Z and Fq[x] seems somewhat wide, we hope that by this time the reader is struck more by the similarities between

these two domains than by their differences. Though we have gone into some detail on certain points, we have not attempted to be comprehensive in our comparisons, and so the reader can certainly pursue further comparisons on his or her own-in fact such studies might make excellent un­ dergraduate research topics. For example, what about a "Waring Problem" for polynomials-can a given monic polynomial over

Fq be written as a sum of a fixed number

[1 4] J .A. Gallian, Contemporary Abstract Algebra , Houghton-Mifflin, 2002. [ 1 5] S. Gao, On the Deterministic Complexity of Polynomial Factoring, J. Symbolic Computation, 31 (2001 ), 1 9-36. [1 6] J. von zur Gathen and J. Gerhard, Modern Computer Algebra , Cambridge Univ. Press, 1 999. [1 7] H . Halberstan and H.E. Richert, Sieve Methods , Academic Press, New York, 1 974.

of k-th powers of polynomials? What about a polynomial

[1 8] GH Hardy and J.E. Littlewood, Some Problems of 'Partitio Nu­

analogue to Dirichlet's Theorem on primes in arithmetic

rnerorurn'; Ill: On the Expression of a Number as a Sum of Primes,

progression? And so on. Here we have simply tried to shed a little light on the wonderful and mysterious relationship between these two domains.

Acta Mathernatica 44 (1 922), 1 -70. [1 9] G . H . Hardy and E.M. Wright, An Introduction to the Theory of Num­ bers, 4th Edition, Clarendon Press, Oxford, 1 959.

[20] D.R. Hayes, A Goldbach Theorem for Polynomials with Integer Co­ efficients, Arner. Math. Monthly, 72 (1 965), 45-46. REFERENCES

[2 1 ] D.R. Hayes, The Expression of a Polynomial As a Sum of Three

[I ] L. M. Adleman and M . -D.A. Huang, Primality Testing and Abelian Varieties over Finite Fields, Lect. Notes in Math., Vol. 1 51 2,

the Number Field Sieve, Lect. Notes in Math., 1 554, Springer-Ver­

Springer-Verlag , 1 992. [2] M . Agrawal , N . Kayal, and N. Saxena, PRIMES is in P, Annals of [3] Z. I. Borevich and I. R. Shafarevich, Number Theory, New York,

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[4] V. Brun, La serie 1 /5

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. . . ou les denominateurs

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trnicely.net.

42 1 -430. [6] J. R. Chen and Y. Wang , On the Odd Goldbach Problem , Acta Math. Sci. Sinica 32 (1 989), 702-7 1 8. [7] R. Crandall and C. Pomerance, Prime Numbers: A Computational Perspective, Springer-Verlag, New York, 2001 .

[8] J.-M. Deshouillers, G. Effinger, H. te Riele, and D. Zinoviev, A Com­ plete Vinogradov 3-Primes Theorem under the Riemann Hypothe­ sis, Electronic Research Announcements of the AMS, 3 (1 997),

99-1 04. [9] G. Effinger, A Goldbach Theorem for Polynomials of Low Degree over Odd Finite Fields , Acta Arithmetica, 42 (1 983), 329-365.

x

1 01 6 is available from web site http://www.

[26] A M . Odlyzko, M. Rubinstein, and M. Wolf, Jumping Champions, Exp. Math. 8 (1 999) 1 07-1 1 8. [27] I. Peterson, Cranking Out Primes, Science News, Oct. 1 , 1 994, 2 1 7 . [28] G . P61ya, Heuristic Reasoning in the Theory o fNumbers, Am. Math. Monthly, 66 (1 959), 375-384. [29] P. Ribenboirn, The New Book of Prime Numbers , Springer-Verlag, 1 995. [30) K. Scheicher and J.M. Thuswaldner, Digit Systems in Polynomial Rings over Finite Fields , Finite Fields Appl. 9 (2003), 322-333.

[31 ) A Selberg, Reflections around Ramanujan Centenary, Paper 41 in Collected Papers/At/e Selberg, Vol. I , Springer-Verlag, Berlin, 1 989.

[1 OJ G. Effinger, A Goldbach 3-Primes Theorem for Polynomials of Low

[32] P.W. Shor, Polynomial-time Algorithms for Primie Factorization and

Degree over Finite Fields of Characteristic 2, Journal of Number

Discrete Logarithms on a Quantum Computer, SIAM J. Cornp. 26

Theory, 29 (1 988), 345-363.

(1 997), 1 484-1 509.

[1 1 ] G. Effinger, The Polynomiai 3-Primes Conjecture, in Computer As­ sisted Analysis and Modeling on the IBM 3090, Baldwin Press,

Athens, Georgia 1 992. [1 2] G. Effinger and D.R. Hayes, Additive Number Theory of Polyno­ mials over Finite Fields, Oxford University Press, 1 991 .

[1 3] G. Effinger, K. Hicks, and G . L. Mullen, Twin Irreducible Polynomi­ als over Finite Fields , in: Finite Fields with Applications in Coding

Theory, Cryptography, and Related Areas, Springer, Heidelberg, 2002, 94-1 1 1 '

[33] J. Teitelbaum, Review of Prime Numbers: A Computational Per­ spective, Bull. Amer. Math. Soc . , 39 (2002), 449-454.

[34] R.C. Vaughan, The Hardy-Littlewood Method (2nd ed .), Cambridge University Press, 1 997. [35] I . M . Vinogradov, Representation of an odd number as a sum of three primes , Cornptes Rendus (Doklady) de I'Academia des Sci­

ences de I' USSR 15 (1 937), 291 -294. [36] A Weil , Sur les courbes algebriques et les varietes qui s 'en de­ duisent, Hermann, 1 948.

© 2005 Springer SC1ence +Bus1ness Media, Inc., Volume 27, Number 2, 2005

33

A U T H O R S

QOVE EFFINGER

KENNETH HICK

Depart

ol

OM lkWersi1y

rOilOf'llV

, OH 45701

USA

USA

IIXEIS out

spare t

l)iafiO �rod doing com·

Parallel Histories

34

THE MATHEMATICAL INTELLIGENCER

M?•ffiJ•i§::Gihfii@i§#fii.i,i§lid

M i chael Kleber and Ravi Vaki l , Editors

Cartographiana M Michael Kleber

aps maps maps, we love maps. Herein a collection of carto­ graphical miscellania, each of which appeals to us mathematically for its own reason. Not Looking Over a Four-Leaf Clover

This column is a place for those bits of contagious mathematics that travel from person to person in the community, because they are so elegant, suprising, or appealing that one has an urge to pass them on. Contributions are most welcome.

Please send all submissions to the Mathematical Entertainments Editor, Ravi Vakil, Stanford University,

Department of Mathematics, Bldg. 380, Stanford, CA 94305-2 1 25, USA e-mail: [email protected] .edu

Item one is courtesy of the "Knotted Objects" section of University of Toronto professor Dror Bar-Natan's Image Gallery [ 1 ] . American drivers can generally tum right easily: at a standard intersection of two-way roads, a right tum does not require crossing any lanes of traffic. Turning left is more difficult; the "cloverleaf' highway interchange, for example, makes up for this by replac­ ing 90° left turns with 270° right turns. Dror points out that "in England, on the other hand, it is easier to make left turns than it is to make right turns. The lovely interchange of I-95 and I-695, northeast of Baltimore (Fig. 1), com­ bines the advantages of the two sys­ tems-it has an outer layer of Ameri­ can (right) turns, followed by a braiding of the lanes, followed by an in­ ner layer of English (left) turns!" Tom Hicks, the Director of Traffic and Safety for the Maryland State High­ way Administration, says this delight­ ful interchange probably would not be built today, primarily because it takes up too much space. The reliance on the unusual left exit ramp is also a bit of a strike against it, according to the Hu­ man Factors people (and couldn't we do with a few more of those in mathe­ matics?). But the interchange does work well. Geometrically, Hicks tells me that the curves of the roadway and the ramps are gentle enough that cars have to slow down only slightly from full high­ way speeds. It's also a win topologi­ cally. In a cloverleaf, two lanes of traf­ fic must cross one another, as cars coming off one ramp need to switch places with ones entering another ramp. This interchange avoids that

problem: drivers entering from any di­ rection can tum either left or right without crossing any other lanes of traffic . (Unlike the cloverleaf, though, one cannot use multiple exit ramps to make a free U-tum.) But I like it purely aesthetically. So does Dror, who writes, "Notice that the lanes of I-95 and of 1-695 are braided in a non-trivial way, proving that the de­ signer of the interchange cared about el­ egance. " We're not the only ones: this same interchange put in an appearance as the "Michael G. Koerner Highway Feature of the Week" in January 1999 [4], and from Koerner I also learned that there is at least one more instance of this design, where 1-2011-59 crosses 1-65 just west of Birmingham, Alabama [5]. (In Birmingham, though, the lanes ofthe individual highways are unbraided.) Hicks tells me that you can drive through the Baltimore interchange without ever noticing there's some­ thing unusual going on: to the driver it just looks like a left exit and a few ex­ tra bridges, and only the sharp-eyed will notice that the first bridge's cross traffic is going the wrong way. Hooray for maps. Cartograms

Item two is courtesy of the 2004 U.S. election. In the past several years, the media have taken to referring to "blue states" and "red states," those which respec­ tively vote Democrat or Republican; televised election-night returns fea­ tured maps which converged (albeit slowly!) to one like Figure 2a. Robert Vanderbei, a professor of Operations Research and Financial Engineering at Princeton, has championed in re­ sponse a Purple America map, linearly interpolating the colors on a county-by­ county level to show that we're not so divided after all. (His map differs from the one reproduced in Figure 2b in that this one is higher contrast: a county here is colored pure red or blue if its vote split is 700h-300,6 or more.)

© 2005 Springer Science+ Business Media. Inc . , Volume 27,

Number 2, 2005

35

Fig. 1 . Cartographic and photographic views of an inter­ change northeast of Baltimore.

36

THE MATHEMATICAL INTELLIGENCER

Fig. 2a. The standard red/blue map of the election.

Fig. 2b. Purple America, with county colors interpolated.

© 2005 Springer Science+ Business Media, lnc.,

Volume 27, Number 2, 2005

37

j

t

Fig. 3. Name this city. (For a hint, see "Where in the World," below.)

This coloration is more informative than the winner-take-all version, but it still looks quite unbalanced, for an elec­ tion where only a few percentage points separated the popular vote for the two candidates. The problem, of course, is that the map is an equal-area projection, while we would need one with counties sized in proportion to population. Such a map is called a cartogram. Given a map and a population density function on it, a cartogram is a trans­ formation of the map whose Jacobian at each point is proportional to the den­ sity function-so after transformation, the population is uniformly distributed. Of course, this is not enough to deter­ mine the map, and the art of car­ togramy to date has been in trying to

develop a way to find such a transfor­ mation which also looks good. Figure 2c is the work of Michael Gastner, Cosma Shalizi, and Mark Newman, of the Center for the Study of Complex Systems and Department of Physics at the University of Michi­ gan. The technique is a new algorithm, by Gastner and Newman [3], which al­ lows the population density to equili­ brate by flowing according to basic linear diffusion, with the map bound­ aries being dragged along for the ride. The physics does a remarkably good job; I think their map is beautiful. Their Web page [2] contains more pictures. Dror (see item one) tells me he has heard, from both Bill Thurston and

Yael Karshon, that "given a smooth density function on S2 there is a canonical (up to rotations) diffeomor­ phism of 82. that takes it to the uniform area density." So there is, for example, a canonical way to smoothly redraw the globe with each country's area proportional to its population-well, once you figure out how to populate the oceans. I wonder how it compares to Gastner and Newman's diffusion technique. Where in the World?

And finally courtesy of University of Toronto professor Balint Virag: what city is shown in the map in Figure 3? The answer appears below, slightly ob­ scured.

© 2005 Springer Science +Business Media, Inc., Volume 27, Number 2, 2005

39

Credits, thanks, answer

Thanks to Greg Slater of the Maryland State Highway Administration for the aerial photograph in Figure lb and the data behind Figure la, and to Lisa Sweeney, Head of GIS Services at MIT, for help in handling it. See the Web sites listed in the references for more interchange maps and aerial photos. The maps in Figure 2 are by Michael Gastner, Cosma Shalizi, and Mark Newman, and are reproduced here with their permission. Moreover, the work is licensed under a Creative Com­ mons License, and the images, includ­ ing the ones which appear here, may

t

\

I t I I I ">

be freely distributed and used to make derivative works for any purpose, as long as the original authors are given proper credit and any redistribution passes on these terms. The city pictured in Figure 3 is of Lpmohdnrth (eoyj drbrm ntofurd), if you type the city's name and notable mathematical attribute with your fin­ gers shifted one key to the right. The map is copyright Ardis Media Group, and used with permission. REFERENCES

Both editions have sewn bindings

40

THE MATHEMATICAL INTELLIGENCER

http://www-personal. umich .edu/� mejn/ election/ [3] Michael T. Gastner and M. E. J. Newman, Diffusion-based method for producing den­ sity equalizing maps, Proc. Nat/. Acad. Sci. USA 1 01 (2004), 7499-7504.

[4] Michael Koerner, "The Michael G. Koerner Highway Feature of the Week, 9 January 1 999." http://www.gribblenation.corn/hfotw/ [5] Michael Koerner, "The Michael G. Koerner

[1 ] Dror Bar-Natan. "Dror Bar-Natan's Image Gallery."

http://www.math.toronto.edu/�

drorbn/Gallery/

Highway Feature of the Week, 21 Novem­ ber 1 998." http://www.gribblenation.com/ hfotw/exit_33.html

A CALCULUS BOOK WORTH READING •

•

537 pp $46 softcover (98 1 -02-4904-7) $82 hardcover (981 -02-4903-9)

Newman, "Maps and cartograms of the 2004 US presidential election results."

exit_40.html

•

Calculus: The Elements MICHAEL COMENETZ

[2] Michael Gastner, Cosma Shalizi, and Mark

Clear narrative style Thorough explanations and accurate proofs Physical i nterpretations and applications

"Unlike any other calculus book I have seen . . . Meticulously written for the i ntelligent person who wants to understand the subject. . . Not only more intuitive in its approach to calculus, but also more logically rigorous in its discussion of the theoretical side than is usual. . . This style of explanation is well chosen to guide the serious beginner . . . A course based on it would in my opinion definitely have a much greater chance of producing students who understand the structure, uses, and arguments of calculus, than is usually the case . . . Many recent and popular works on the topic will appear intellectually sterile after exposure to this one." -Roy Smith, Professor of Mathematics, University of Georgia (complete review at publisher's website) "One has the feeli ng that it is a work by a mathematician still in close touch with physics . . . The author succeeds well in giving an excellent intuitive introduction while ultimately maintaining a healthy respect for rigor." -Zentralblatt MA TH (online)

A selection of the Scientific American Book Club

World Scientific Publishing Company http://www. worldscientific. com 1 -800-227-7562

Leray in Edelbach Anna Maria Sigmund, Peter Michor, and Karl Sigmund

Does your hometown have any mathematical tourist attractions such as statues, plaques, graves, the cafe where the famous conjecture was made, the desk where the famous initials are scratched, birthplaces, houses, or memorials? Have you encountered a mathematical sight on your travels?

If so, we invite you to submit to this

column a picture, a description of its

mathematical significance, and either a map or directions so that others may follow in your tracks.

Please send all submissions to Mathematical Tourist Editor, Dirk Huylebrouck, Aartshertogstraat 42,

8400 Oostende, Belgium e-mail: [email protected]

T

his is a most unlikely place for the mathematical tourist to visit In fact, it is off-limits for tourists of any kind. Photographing, filming, even drawing, is prohibited by law, as signposts tell you sternly, and trespassers will be punished. If they survive at all, that is. Indeed, the signposts also warn you of LEBENSGEFAHR, meaning mortal danger. You are in a military zone, and had better watch out. Don't step on any mines, and avoid getting shot, says an urgent inner voice. But this is ridiculous. We are in Aus­ tria, after all, with almost sixty years of peace and prosperity behind us. No­ body wants any trouble. Let's not get caught, that's all. Welcome to Edelbach, or what is left of it. The place is not easily found on a map: it ceased to exist many years ago, during the darkest days of Aus­ trian history. Nobody lives here any longer. The main road between Vienna and Prague is a couple of miles to the north, but it can be neither seen nor heard. An eerie silence hangs over the place. All that remains of the former village are a few stone-heaps between thickets of fir trees, and a small, aban­ doned graveyard. To the north of it, a modem fence surrounds a vast ammu­ nition depot. It is very well guarded, and you can be sure, by now, that binoculars are fixed on you.

This place was once a camp for pris­ oners of war, mostly French officers. An "Offizierslager"-or Oflag for short: the bureaucrats of the Third Reich were fond of abbreviations. Oflag XVIIA was the birthplace of a substan­ tial part of algebraic topology. Spectral sequences and the theory of sheaves were fathered here by an artillery lieu­ tenant named Jean Leray, during an in­ ternment lasting from July 1940 to May 1945 ([Sch 1990] [Eke 1999] [Gaz 2000]). In the annals of science one finds several examples of first-rate mathe­ matical research conducted by prison­ ers of war. The Austrian Eduard Helly, for instance, wrote a seminal paper

©

on functional analysis in the Siberian camp of Nikolsk-Ussurisk, during World War I; and a century before, the Napo­ leonic officer Jean-Victor Poncelet de­ veloped projective geometry while in Russian captivity for five years. This may sound as if the monastic reclusion and monotonic regularity of confmed life provided ideal conditions for con­ centrating the mind. And indeed, An­ dre Weil wrote that "nothing is more favourable than prison for the abstract sciences" [Weil 199 1 ] . He wrote this while he was in prison, and managed, during his months of captivity, to find some of his major theorems. But he had a prison cell for himself, could re­ ceive visits from his family, and knew assuredly, to use his words, "captivity from its most benign side only." The physical and psychic deprivations of years in a POW camp, with its over­ crowding, sickness, hunger, and biting cold, on top of the boredom and un­ certainty, were something else: in these conditions, intense intellectual pursuit must have been a desperate means for keeping hold of sanity. The prisoners of Edelbach founded a "University in Captivity." Of the 5,000 inmates of the camp, of which a few hundred were Polish and the rest French, almost 500 got degrees, and their diplomas were all officially con­ firmed in France after the war. The fact that Jean Leray had been the director, or recteur, of this impromptu univer­ sity must have helped with the French authorities. His academic credentials were impressive: he had received his doctorate at the elite Ecole Normale Superieure in Paris, and had been pro­ fessor at the Universite de Nancy before being drafted into the war. His joint work with the Polish mathematician Juliusz Schauder Oater a victim of the Holocaust) developed a topological in­ variant to prove the existence of solu­ tions of partial-differential equations. This earned him in 1940 the Grand Prix in mathematics from the Acadbnie des Sciences de Paris.

2005 Springer Science+ Business Media, Inc., Volume 27, Number 2, 2005

41

M I LITX R I S C H E S S PE R R G E B I ET 1 L e b e n s gefah r ! Betreten und B efahren , Fotografi eren, Fi lmen u nd Zeichnen gesetzlich verboten und strafbar!

Fig. 1. Tourists are not exactly welcome in Edelbach nowadays, but what can you expect from an ammunition depot?

But Leray was not the only distin­ guished scientist in the Oflag. There was the embryologist Etienne Wolff, by all testimonies a driving force behind the university, but obliged, for racial reasons, to keep discreetly in the back­ ground. Etienne Wolff later became professor at the College de France, and member of the Academie des Sciences de Paris as well as of the Academie Franr;aise. Another luminary was Fran<;ois Ellenberger, a future presi­ dent of the Societe Geologique de France. The geologists at Oflag XVII had to content themselves with the stones they could find in the prison yard. Their laboratory was an old kitchen which they could use for a few hours daily. Eventually, friends and relatives from France were permitted to send books. Over the years, Leray received a small library from his former teacher Henri Villat [Sch 1990], [Ell 1948]. From eight in the morning to eight in the evening, Barrack 19 housed lec­ tures on law and biology, on psychol­ ogy and Arab language, on music and moral theology, on horse-raising (by a

42

THE MATHEMATICAL INTELLIGENCER

Polish fellow-officer, bien sur!), on public finances, and on astronomy. The course on probability was given by Lieutenant Jean Ville, who had pub­ lished, just before the war, an inge­ nious elementary proof of von Neu­ mann's minimax theorem [Poll 1989]. Recteur Leray lectured mostly on cal­ culus and topology. He had succeeded in hiding from the Germans the fact that he was a leading expert in fluid dynam­ ics and mechanics (a mecanicien, as he liked to say). He turned, instead, to algebraic topology, a field which he deemed unlikely to spawn war-like ap­ plications. This led, first, to some notes in the Comptes Rendus de l'Academie des Sciences de Paris, and eventually to a three-part work "Algebraic topology taught in captivity," which was submit­ ted in 1944 to the Journal des Mathe­ matiques Pures et Appliquees, through the good offices of Heinz Hopf from neutral Switzerland, who endorsed it enthusiastically. It was published, after Leray's release, in 1945 [CRAS 1942] [JMPA 1945]. The university's curriculum shows that on Sunday nights, the prisoners

could listen to a lecture giving "practi­ cal advice for constructing an inex­ pensive house," before having to return to their cheerless cold quarters. The barracks consisted of two rooms hous­ ing 100 inmates each, one small kitchen, and one toilet with eight wash-basins. There was a special building for the showers: each officer could use it twice a month. Half of one barrack was used as a chapel. More than seventy of the prisoners were priests, and each could say mass daily if he wished. The captives founded a first-rate choir and a theatre group, and soon set up their own sports stadium, named stade Pe­ tain. The prisoners even managed to produce, behind the back of their guards, a documentary film of about thirty minutes' length, entitled Sous le Manteau ("Beneath the Cloak," be­ cause the camera had always to remain hidden). Three versions of it have survived to this day ([Poll 1989] [Kus 2003]). As in many other POW camps, the captives printed their own newspaper, a weekly called Le Canard . . . en KG. KG is Wehrmacht shortspeak meaning

Kriegsgefangener, or prisoner of war, and the French would pronounce it as Le canard encage (The caged duck), a pun referring to the celebrated Le Ca­ nard Enchaine (The duck in chains), which was, and still is, a hugely popu­ lar satirical journal in France. The pris­ oners' version was not permitted to comment on politics, satirically or oth­ erwise: it was filled with harmless cari­ catures, theatre bills, sports news, cross­ word puzzles, and announcements of special lectures. Nothing about the war, or about the conflicts dividing the French community into what, with hindsight, was simply the issue of col­ laboration vs. resistance, but seemed much more confusing at the time. The Vichy regime tried to foster a network of "hommes de confiance," but an un­ derground resistance group, who called themselves the mafia, eventually be­ came the dominating force in the camp. For many of the prisoners, the dilemma was whether to become a civilian worker in Germany, with a freedom . . . of sorts, or to stick it out behind the barbed wire, in the hope that the legal status of a captive officer would pro­ tect them from the worst. For Leray, who in 1933 had witnessed in Berlin the accession of Hitler to power, col­ laboration was never an issue.

When Leray later spoke about Edel­ bach, he located it "near Austerlitz, in Austria" [Sch 1990]. Actually, Austerlitz is across the border, in Czechia, and not really nearby (some 83 kilometres away). Edelbach is closer to Vienna than to Austerlitz, but for the defeated French officers, the thought of being near the site of the great Napoleonic victory-"a portee de canon d'Austerlitz," as some liked to say-must have been a solace. At first, they all had hoped to be back in France by the end of 1940. The war seemed over. When this proved an il­ lusion, many fell prey to depression and to homesickness. Leray and his academic colleagues used to meet every evening in the highest, southern­ most comer of the camp, and watch, weather permitting, the sunset over "Ia petite France." Needless to say, the French did not merely bemoan their fate. Some tried to change it. The prison guards became experts at discovering tunnel en­ trances beneath the barracks. They were so good at it that they overlooked a tunnel entrance which was out in the open, right under their noses. It was through this 90-meters-long tunnel that on the nights of September 17 and 18, 1943, no fewer than 132 prisoners de­ camped. It was the greatest escape

from a POW camp in World War II, and its story is almost unknown [Kus 2004). The prisoners had established an open-air theatre, called Theatre de la Verdure. They were allowed to deco­ rate it with twigs and greenery, hiding it partially from the guard towers. Be­ cause delegates of the International Red Cross had found that the camp lacked protection against Allied air raids, the POWs were told to dig a few trenches, and were even provided with shovels and wheel-barrows. Under a plank bridging one of the dug-outs, they started burrowing in earnest. The tun­ nel grew quickly, by almost a metre per day, although water kept flooding in. After some time, ventilation became a problem: through a hose made from tin cans, fresh air had to be pumped into the gallery, which was less than two feet wide and three feet high. In paral­ lel, a tailor shop produced civilian clothes, and the printing press pre­ pared maps and forged documents. Canned food was hoarded in hidden depots. The first group left on a Saturday night. Their escape went unnoticed during Sunday, because some of the guards were on holiday. The second group left on the following night. Most of the runaways hoped to pass for

Fig. 2. Lieutenant Jean Leray, POW, became the rector of the "University in Captivity." The picture on the right shows him with his Edelbach colleagues. Some would later join him at the Sorbonne or the College de France [Gaz 2000].

© 2005 Springer Science+ Business Media,

Inc., Volume 27, Number 2. 2005

43

Fig. 3. The curriculum of the Universite en Captivite [Poll 1 989]. As Leray later said, "students had no other distraction than their studies. They had little to eat, and little to keep warm; but they were courageous." [Sch 1 990]

French civilians, of whom there were

time touching Edelbach. Nevertheless,

were soon to use were erected origi­

many working in Germany at that time.

the booming of great guns and the an­

nally to house the first German soldiers

The first escapees were arrested and

gry buzz of Stukas could be heard at aJl

claiming the exercise grounds. Very

returned to the camp by the police

times by the inmates of the camp. In­

soon, the

Truppenilbungsplatz proved

even before the break-out was discov­

deed, Oflag XVII was located within

an ideaJ stepping-stone for the armies

ered by the military guards. EventuaJly,

an evacuated zone, strictly off-limits

which were assembling to invade and

only two fugitives managed to reach

for civilians, the

Truppenilbungsplatz

France.

dismember nearby Czechoslovakia, in spring 1939, and for preparing the as­

Soon after, a panel of agitated Ger­ man officers, including severaJ gener­ aJs, visited the Oflag, where they were

"Breaking out is

sault on Poland during the following summer months [Poll 1989]. The fact that both the father and the

no longer a sport . "

mother of Adolf Hitler had been born in

play down the escape-it did not show

Dollersheim. This was the largest mil­

ruthlessly evacuated, gave rise to spec­

the Wehnnacht in a favourable light.

itary training ground in centraJ Europe,

ulations.

The prisoners were sternly told that

twenty kilometres in diameter, larger

sociates of the FUhrer, Hans Frank,

filmed surreptitiously by the French prisoners. The commission decided to

the region, which was so suddenly and One

of

the

closest

as­

they should not try it again. Handbills

than the dukedom of Liechtenstein.

would later write, in his death cell in

were distributed warning that "break­

A few months after the

Anschluss,

Nuremberg, that Hitler intended there­

ing out is no longer a sport" and that

Hitler's annexation of Austria in 1938,

by to erase all traces of his origins

death zones were waiting for the run­

the German army had taken over the

[Frank 1953]. He reported that these

aways. Half a year later,

76 British fly­

ground. Forty-five villages with more

traces could reveaJ a dark secret, the

ers escaped from Oflag III Luft in

than seven thousand inhabitants were

shame and scandaJ of the Third Reich:

Sagan. This time, the Wehrmacht could

hastily evacuated, and huge mechanised

namely, that Hitler had a Jewish grand­

no longer keep it a secret from Hitler

forces rattled across the fields, taking

father. This rumour, which had been

and Rimmler. Only three of the fugi­

little notice of the fact that the harvest

widespread in Nazi Germany and still

tives reached England; 50 were shot.

was not yet in. The Wehrmacht had to

finds adherents today, has been de­

live up to its new and as yet untested

bunked by scores of historians since.

spent in Oflag XVII, battles raged from

doctrine of the Blitzkrieg. The barracks

Hitler's father had been born out of

one end of Europe to the other, at no

which Leray and his fellow-prisoners

wedlock, as Alois Schicklgruber, and

During the five years that Leray

44

THE MATHEMATICAL INTELLIGENCER

was later to change his name, but the

regime, which was engaged in a hope­

thorities had other things on their minds.

FUhrer was far too powerful to have felt

less struggle against illegal Nazis, and

Eventually, the district of Lower Austria

threatened by slurs concerning his an­

fmally, right after the revels of the an­

was occupied by the Red Army, which

cestry. In fact, when the villages around

nexation,

could find good use for the vast training

Dollersheim were evacuated, all church

without further ado. No account was

opportunities filled with bunkers and

archives were properly stored. They are

taken of the fact that 220 out of the 220

artillery ranges. By 1955, the Allied oc­

expelled

from

their

land

cupation troops left Austria, but the

preserved to this day. For years, the Wehnnacht had been looking for a king­ its frantic growth, and to manoeuvre with its new weapons, whose range

evacuated region was not returned to

"Today only

its former dwellers. They had been

the c h u rch of

far too weak to succeed in their de­

sized training ground to accommodate

would not fit into existing exercise ar­

scattered all over the district and were mands for a return. The small new Aus­

Dol lershei m

eas. The Waldviertel (or woods district),

'

with its poor soil and its sparse, lowly

surv1ves . . .

population was perfectly suited: a hilly

trian army managed to keep the over­ sized training grounds for itself. Those

"

abandoned houses which were still standing, after the years of Nazi and

plateau, some 600 metres above sea level, with long, bitterly cold winters,

citizens of Grosspoppen had voted for

Soviet occupation, including Edelbach,

and no reputation for hospitality.

the

Anschluss. In fact their hamlet,

were now flattened in a remarkably

which obstructed a planned artillery

short time. The Austrian army had in­

It is clear that Hitler had no emo­

tional ties to the Waldviertel. The prop­

range, was the first to become

men­

herited an amazing amount of ammuni­

aganda from the Goebbels ministry had

schenrein (the callous Nazi expression

tion, and made a point of spending it lav­

hailed it as the Ahnengau, the cradle of

for "evacuated") and be knocked down.

ishly by shelling the empty settlements.

the ancestors, and the humble dwellers

A fortunate few were compensated

Today only the church of Dollersheim

of the tiny hamlet of Grosspoppen, led

with hastily built ersatz farms, not too

survives: its spire serves as a conve­

by their inn-keeper, had conferred hon­

far away. Others were given provi­

nient mark for ranging artillery sights.

orary citizenship on Hitler in 1932,

sional quarters and the promise of a

But during Leray's years of intern­

when he was a rising young politician

settlement after the war. In 1942, all

ment he was daily faced with the

and demagogue. In return, they first got

evacuees were offered a special re­

vacant houses of a seemingly intact,

scolded by the authorities of Lower

duction on a richly produced coffee­

menschenfrei Edelbach behind the

Austria (who pointed out that the ac­

table book,

Die alte Heimat, complete

barbed-wire fence. The chimneys did not smoke and the doors never opened.

tion was legally void because Hitler

with pictures of their empty villages, and

was no longer an Austrian citizen),

Hitler's family tree as a keepsake [Heim

The window-panes had been replaced

then frowned upon by the Viennese

1942).

by planks. A poem on the front page of

In

the ensuing years, Nazi au-

Fig. 4. Notes from captivity. KG Jean Leray reports, in this Comptes Rendus note from 1 942, that in his present condition, he is unable to guarantee the originality of his results [Gaz 2000] .

© 2005 Springer Sci811Ce + Business Media, Inc., Volume 27, Number 2 , 2005

45

Fig. 5. No open university, but a closed universe of 440

x

530 meters. The camp, and campus, of

Oflag XVII housed some 5,000 prisoners. Today, the barracks are gone: in their place one finds con­ crete, earth-covered ammunition dumps. The village of Edelbach is a rubble of stones covered by a dense forest.

the Canard en KG, with the title "Le viUage ignore," describes the mute bell-tower of the deserted hamlet, and the silence broken only by the wind [Poll 1989]. And while the Nazi picture book acknowledges that when Edel­ bach had to be cleared out, some left it with a bleeding heart, the captive French poet imagines how his heart, far from bleeding, "jumps with joy on

the day, known only to destiny," when he is released and the forsaken village vanishes behind the firs. The day known only to destiny was April 1 7, 1945. The camp had to be evacuated because the Red Army was perilously close. The Wehrmacht was by now out of gas and lorries. The Blitzkrieg days were over. The prison­ ers had to march, carrying their be-

longings on their backs. Some of the guards used bicycles, and their officers sat on underfed horses. The trek aimed for Linz, some 128 kilometers away to the west. The group covered, on aver­ age, less than ten kilometres a day, and dwindled rapidly in size. The marching column was long, the forest dense. Un­ derfed Fran<;ois Ellenberger schlepped a rucksack half his own weight: he had

Fig. 6. A room with a view. The barracks were originally built for the Wehrmacht soldiers claiming the grounds. Fences and watchtowers were added later [Poll 1989].

46

THE MATHEMATICAL INTELUGENCER

Fig. 7. Underground film. These clandestine stills show the escape tunnel, also known as metro pour Ia liberte. The clandestine movie Sous le manteau, shot in real time, is a French alternative to Holly­ wood's The Great Escape [Corr 1 954].

insisted on taking along his volumi­ nous mineralogical notes, a hand-made telescope, and his rock samples, some of which had come from the tunnel. He still found the strength to sketch the lines of the hills in his notebook, and the interiors of rural chapels. The pris­ oners had to look after their own food; some managed to get it from old wives and barefoot children, in exchange for soap, which they had produced in their camp. By May 10, the column had been reduced by half. This was the day the Wehrmacht surrendered.

After his liberation, Jean Leray be­ came professor, first at the University of Paris (which had appointed him in 1942), and then, in 1947, at the presti­ gious College de France. In 1953 he was elected to the Academie des Sciences de Paris (which had made him a cor­ responding member in 1944). He was showered with prizes: among them, the prix Ormoy in 1950, the Feltrinelli prize in 1971, the Lomonosov gold medal in 1988 Gointly with Sobolev), and in 1979, the Wolf prize, jointly with Andre Weil (who, incidentally, had also been

a candidate for that same chair at the College de France). In an obituary writ­ ten for Nature, Ivar Ekeland called Leray "the first modem analyst," and compared him with Weil, "the first modem algebraist" [Eke 1999]. The parallels, which also were stressed by Jean-Michel Kantor [Gaz 2000], are indeed intriguing: the two men share their year of birth, 1906, and their year of death, 1998. They both were among the very select few to at­ tend the Ecole Normale Superieure, and both did some of their best work

II Fig. 8. Cold feet and frosty advice. Unaware of being filmed, a Wehrmacht delegation decided to keep the news of the escape under wraps. But posters warned the French that henceforth, s'evader n'est plus un sport.

© 2005 Spnnger Science+Bus1ness Media, Inc., Volume 27, Number 2, 2005

47

Fig. 9. The church of Edelbach, in an already deserted village. The poem "Le village ignore" laments that in the humble church, no bell ever rings. In 1 957 the church was flattened by Austrian artillery.

in prison. But the differences are even

hair-raising risks to avoid waging war

man assault and throughout the pro­

more

against Hitler. Leray served as a patri­

tracted years of confinement. Whereas

striking.

Weil

followed

his

his

dharma (that is to say, he was a con­

otic officer and remained stolidly at

scientious objector) and therefore took

post to the end, during the swift Ger-

Weil studied abstract algebraic struc­ tures and shunned anything even re-

Fig. 1 0. Forty years after. This stone commemorates a visit in 1985 by some former inmates of the Oflag. The French prisoners had their own graveyard in Edelbach, complete with funeral statue.

48

THE MATHEMA1lCAL INTELLIGENCER

motely smacking of applications or

from ongoing research, in particular

daira, both based their work on Leray's

physical intuition, Leray was deeply

from contemporary, related work by

sheaves and spectral sequences.

steeped in physics and geometry. This

Eilenberg and Steenrod, and had to

makes all the more remarkable the fact

start from scratch.

As Armand

In the hands of Cartan and Oka, sheaves became an essential tool for

wrote,

the theory of several complex vari­

in the prison camp, and laid the basis

Leray's original concepts, based on a

ables. Weil used sheaf cohomology and

for what soon became a main item on

language of his own making, have been

spectral sequences on real manifolds

Bourbaki's menu, although he had left

strongly modified or have not survived

to give a lucid proof of de Rham's the­

that he switched to algebraic topology

Borel

later

2000). Leray's aim was to create

orem, generalising the Mayer-Vietoris

Changing direction seems to have

something similar to differential forms,

sequence from an open cover of two

posed no problem for Leray. "The es­

keeping their multiplicative algebraic

sets to one of infinitely many sets.

sential characteristic of my publica­

structure, but in a purely topological

Godement wrote the definitive treat­

tions is their diversity, " he later said,

framework His cohomology was simi­ lar to that created by Cech, and his re­

ogy for algebraic topology. Serre and

ics that obliged me to give new devel­

sults did not, as Borel wrote, "seem to

Grothendieck adapted the notion of

opments to mathematical analysis and

go drastically beyond those of main­

sheaves for algebraic geometry. Even

1990). Indeed,

stream algebraic topology." But the in­

the (still unfinished) theory of motives

the Bourbaki group in

1935.

[BHL

simply. "It was my interest in mechan­

algebraic topology" [Sch

ment of sheaves and their cohomol­

Leray had been interested in topology

tention behind them was different:

concerns a category of sheaves. The

even before the war, but as a tool

Leray aimed at studying, not only the

central problem, on which Voevodsky

rather than as an end in itself. The ho­

topology of a space, but the topology

made some recent inroads, is to find

motopy invariant now known as the

of a representation, i.e., topological in­

enough injective resolutions for coho­

Leray-Schauder degree was created in

variants for continuous maps. He took

mology to work. With the papers of Ko­

order to prove the existence of solu­

as starting point his notes on a course by E lie Cartan on differential forms,

daira and Spencer, and the Habilita­

tions to non-linear partial-differential equations. Such equations, particularly

published

1935 [Cart 1935]. He

sheaf cohomology crossed the French

in

those which stemmed from mathemat­

aimed

(which he persistently called homol­

sheaf cohomology to define hyperfunc­

1936, he published a

ogy) in a way similar to the de Rham

tions as generalised boundary values of

truly pioneering paper investigating

cohomology,

multiplicative

holomorphic functions, and investigated

the existence, uniqueness, and smooth­

structure. From his work with Schauder on

bundle. Sato's microfunctions are more powerful than Hormander's wave-front

ness of solutions of the initial-value problem

for

the

understand

with its

cohomology

1956],

ical physics, were at the centre of Leray's work. In

to

tionsschrift of Hirzebruch [Hirz

borders. Sato used complex analytic

microlocal analysis on the cotangent

three-dimensional

fixed-point theorems, he was used to

Navier-Stokes equations for incom­

the relative viewpoint. He considered

sets, which in turn were inspired by

pressible fluids. He showed, in partic­

mappings

Maslov. Later, Leray would devote a

ular, that non-stationary solutions for

the basic object. This was a lasting

whole book to the role of Planck's con­

smooth initial data remain smooth for

achievement. The Leray-Serre spectral

stant in mathematics, again in an at­ tempt to understand Maslov [Ler

between

two

spaces

as

1981).

a finite time only; beyond this, they

sequence of a filtration is still in gen­

may only be continued in a weak sense

eral use today. Grothendieck would

(giving rise to what are called weak so­

also stress the importance of the rela­

quences appeared first as a compli­

lutions nowadays). Leray called such

tive point of view in algebraic topology

cated set of relations among various

solutions turbulent, thereby suggest­

[Jack 2004].

Leray's

concept

o f spectral

se­

cohomologies of double complexes.

ing that the onset of turbulence is

Soon after his release, Leray found a

They allowed Leray to compute the co­

caused by the breakdown of smooth­

way to define cohomology with respect

homology of compact Lie groups and

ness. He certainly had good reasons

to sheaves, and introduced the spectral

flag manifolds. Serre used spectral se­

not to wish the Germans to learn of his

sequence of a continuous map, which

quences, already in their modern form,

work It is interesting to speculate

relates the cohomology of the domain

to determine the dimensions in which

what he would have done if he had

to that of the range and of the fibre. His

the higher homotopy groups of the

been given an opportunity to do scien­

original ideas, intended to be as general

n-sphere are not finite, namely n and

tific work for the Allies.

as

2n- 1. Massey made spectral sequences

his major interest" and started working

Frenchmen named Henri Cartan, Jean­

As it was, he "turned his minor into

possible,

enough,

were

however,

still not for

three

general young

on algebraic topology as an end in it­

Louis Koszul, and Jean-Pierre Serre.

self-Weil-style, as it were. He worked

They extended his concepts to obtain

1950, returned to

spectacular

analytic

ied the Cauchy problem, its connection

tion, avoiding contacts with German

spaces and algebraic geometry. In the

with multidimensional complex analy­

mathematicians.

some

late forties, the development became al­

sis, residue theory on complex mani­

reprints provided by Heinz Hopf, from

[ Gaz 2000). The two Fields medallists of 1954, Serre and Ko-

folds, and integral representations. Al­

from

neutral Switzerland, Leray was cut off

most breathless

to

Leray himself, after

partial-differential equations. He stud­

in great, but not total scientific isola­ Apart

applications

more easily accessible via the notion of exact couples.

gebraic topology became a tool again

© 2005 Springer Science+Business Media, Inc., Volume 27, Number 2, 2005

49

A U T H O II S

for Jean Leray. The interlude which had begun in the POW camp of Edel­ bach, as a kind of camouflage, was over. But generations of pure mathe­ maticians would exploit the ideas which had germinated in Oflag XVIIA.

Miller, J . Serrin, R. Siegmund-Schultze, A Yger, C. Houze!, and P. Malliavin. [Sch 1 990] Schmidt, M . : Hommes de science,

We thank Hofrat Dr. Andreas Kusternig for a wealth of information on Oflag XVIIA. Jean-Michel Kantor, Reinhard Siegmund-Schultze, and Han­ nelore Brandt provided considerable help in preparing this article.

sage, Birkhauser, Basel.

(Eke 1 999] Ekeland, I . : Jean Leray (1 9061 998), Nature 397, 482. [Gaz 2000] Gazette des mathematicians , Sup­ plement au no. 84, 1 -88 is entirely dedicated

XVIIA, Annates Scientifiques de Franche­

47, 350-359.

phier, reportage photographique clandestin

[Cart 1 935] Carlan, E . : La methode des reperes

sur Ia vie d'un camp de prisonniers francais,

mobiles, Ia theorie des groupes continus et

Oflag XVII A(Autriche)

/es espaces generalises, Notes written by J.

[Poll 1 989] Palleross, F.: 1938 Davor-Oanach:

Leray, Hermann, Paris. [Jack 2004] Jackson, A: As if summoned from the void, the life of Alexandre Grothendieck, No­

Horn-Krems. [CRAS 1 942] Leray, J . : Comptes rendus de

tices of the AMS 51 , 1 038-1 056, 1 1 96-1 2 1 2.

I'Academie des Sciences de Paris 214,

[Hirz 1 956] Hirzebruch, F.: Neue topologische

781 -783, 839-841 , 897-899, 938-940. [JMPA

1 945]

Leray,

J.:

Cours d'algebre

topologique enseigne en captivite, J. Math.

Kantor, Y . Choquet-Bruhat, J. Y. Chemin, H .

Pures Appl. 24, 95-1 67, 1 69-1 99, 201 -248.

THE MATHEMATICAL INTELUGENCER

Verlagsdruckerei, Berlin. [BHL 2000] Borel, A , Henkin, G . M . , and Lax,

P D . : Jean Leray (1 906-1 998), Notices AMS

Comte 3, 2 1 -24.

(Carr 1 954] Carre, M . : Defense de photogra­

to Jean Leray, and includes articles by J. M .

50

Munchen/Grafelfing. [Heim 1 942] Die a/te Heimat: Sudetendeutsche

Beitrage zur Zeitgeschichte des Waldviertels , REFERENCES

dem Oflag XVIIA, Nieder6sterreich Perspek­ tiven 3, 22-25.

[Fr 1 953] Frank, H . : lm Angesicht des Galgens,

Hermann, Paris. (Wei! 1 99 1 ] Wei!, A: Souvenirs d'apprentis­ [Ell 1 948] Ellenberger, F.: La geologie a I'Oflag

Acknowledgments

[Kus 2003] Kusternig, A: Grosse Flucht aus

Methoden in der algebraischen Geometrie,

Ergebnisse 9, Springer-Verlag, Berlin. [Ler 1 98 1 ] Leray, J . : Langrangian analysis and quantum mechanics, MIT Press, Mass.

CHRISTIAN BOYER

Some Notes on the M ag ic Sq uares of Sq uares Pro b em

Permettez-moi, Monsieur, que j e vous parle encore d'un probleme qui me parait fort curieux et digne de toute attention.

-Leonhard Euler, 1 770, sending his 4 X 4 magic square of squares to Joseph Lagrange.

Can a 3

X 3 magic square be constructed with nine distinct

1272

square numbers? This short question asked by Martin LaBar

[38] in 1984 became famous when Martin Gardner repub­ 1996 [25] [26] and offered $100 to the first per­

lished it in

son to construct such a square. Two years later, Gardner wrote

[28]:

�

�

1�

74

822

582 94 g72

L81. Three rows, three columns, and one diagonal have the same

So far no one has come forward with a "square of squares" -but no one has proved its impossibility either. If it exists, its numbers would be huge, perhaps beyond the reach of today's fastest computers.

magic sum 82

21609. But unfortunately the other diagonal has a

=

different sum 82

=

38307.

In the present article I add both old forgotten European works of the XVIIIth and XIXth centuries that I am proud

Today, this problem is not yet solved. Several other arti­

to revive (and to numerically complete for the first time

[ 10] [ 1 1 ] [ 12] [27] [29] [30] [49] [ 5 1 ] [52]. John P. Robertson [51]

of the very last months on the problem-and more gener­

cles in various magazines have been published

[9]) after years of oblivion, and very recent developments

showed that the problem i s equivalent to other mathe­

ally on multimagic squares, cubes, and hypercubes. And I

matical

have highlighted

problems

on

arithmetic

progressions,

on

Pythagorean right triangles, and on congruent numbers and elliptic curves

y2

=

x3 - n2x. Lee Sallows [52] dis­

cussed the subject in The Mathematical InteUigencer, pre­ senting the nice

(LSJ ) square, a near-solution with only

one bad sum.

52

THE MATHEMATICAL INTELLIGENCER © 2005 Springer Scierce+Business Media, Inc.

10 open subjects. An open invitation to

number-lovers! The magic square of squares problem is an important part of unsolved problem D 15 of Richard

K. Guy's Unsolved Problems in Number Theory [30], third edition, 2004, sum­ marizing the main published articles on this subject since

1984. I have organized my exposition around nine quota­ tions from Guy's text. 1.

"Martin LaBar asked for a proof or disproof that

a 3 X 3 magic square can be constructed from nine distinct integer squares."

He presents a numerical example (EL2), using (p, q, r, s) = (6, 5, 4, 2), with other interesting characteristics: •

•

Martin LaBar [38] is represented as the original prepounder of the problem in 1984: by Andrew Bremner [ 10], Martin Gardner [25], Richard Guy [29] [30], Landon Rabem [49], John Robertson [51]. But I have discovered that this prob­ lem was posed, more than one century before LaBar, by the French mathematician Edouard Lucas: in 1876, in the magazine Nouvelle Correspondance Mathematique [40], edited by the Belgian mathematician Eugene Catalan.

•

•

The three rows and the three columns have the same magic sum, which is an eighth power of an integer. The main diagonal, when its numbers are not squared, has a sum which is the fourth power of the same integer. The nine numbers, when they are not squared, have a sum which is a squared integer. The sum of the products of the rows is equal to the sum of the products of the columns.

Lucas does not remark that this last characteristic is also true for all 3 X 3 magic and semi-magic squares. He also does not remark that, with a good choice of (p, q, r, s), his family allows seven of the eight sums to agree, as I will show in part 4. But he proves that his family cannot pro­ vide a solution with the eight magic sums, adding that "cela ne prouve pas, il est vrai, que le probleme soit insoluble": "it's not a proof that the problem has no solution." See also below, part 4, part 5, and the supplement [9] for a complete numerical study of Lucas's family. Citing Legendre [39], Lucas mentions the 4 X 4 magic square of squares problem proposed by Leonhard Euler (see part 5), but seems to be unaware-like Legendre-that Euler had published in 1770 [ 18] this very similar family of 3 X 3 matrices, but with non-integer entries: A = rp2 + q2 - r2 - s'2)!u,

B = [2(qr + ps)]lu,

C = [2(qs - pr)]/u,

0 = [2(qr - ps)]lu,

E = (p2 - q2

F = [2(rs + pq)]lu,

Edouard Lucas (Amiens 1842-Paris 1891) stated the 3 X 3 magic

G = [2(qs + pr)]lu,

H = [2(rs - pq)]lu,

square of squares problem in 1876.

where u = p2 + q2 + r2 + s2 • Euler used directly his works on mechanics about the rotation of a solid body around a fixed point [ 19]. Joseph Lagrange, then later Arthur Cayley, worked on the same subject of physics, and used similar matrices. Euler announces that his solution has the 12 following characteristics:

In his article, Lucas particularly studies this parametric family (ELl) of semi-magic squares, "semi-magic" meaning that all the row sums and column sums are the same, but not the two diagonals:

EL1 . Lucas's 3 X 3 semi-magic square of squares family. The 3 rows and 3 columns have the same magic sum S2 = (p2 + q2 + r2 + s2)2.

e&2

"

7&2

1 &2

232

2SZ

.. 1 7

=

(6, 5, 4, 2) and moving

rows and columns. Or generated, for example, with (2, 6, 5, -4) and just inverting columns 2 and 3. S2 = 812 = 38• This square has other characteristics: 1 + 16 + 64

=

·

I = (p2 - q2 - r2 + s'2)1u,

A2 + B2 + C2 = 1

AB + DE + GH = O

02 + E2 + F2 = 1

AC + OF + Gl = 0

G2 + H2 + J 2 = 1

BC + EF + HI = 0

A2 + 02 + G2 = 1

AD + BE + CF = 0

82 + E2 + H2 = 1

AG + BH + Cl = 0

C2 + F2 + J 2 = 1

DG + EH + FI = 0

47157

28157

1 81&7

4157

23157

521157

32157

-44157

1 7157

G

LE1 . The smallest example published by Euler in 1 770, generated by (p, q, r, s)

=

(6, 4, 2, 1), solving 6 identities like A2 + B2 + C2

6 identities like AB + DE + GH

=

=

1 , and

0.

34, with 1 + 68 + 44 + 76 + 16 + 23 +

28 +41 + 64 192, and 12 682 442 + 762 162 232 + 282 · 412 542 12 . 7S2 . 282 + 682 . 162 . 412 + 442 . 232 . 642. =

r2 - s2)1u,

Euler gives four examples, his smallest being (LEl):

EL2. The example of a 3 X 3 semi-magic square of squares by E. Lu­ cas in 1876. Generated with (p, q, r, s)

+

·

·

·

·

=

We can very easily rewrite his result, removing signs and denominators and squaring the integers, and say that Euler

© 2005 Springer Science+ Business Media, Inc., Volume 27, Number 2, 2005

53

has found the first

X

3

3 semi-magic square of squares

(LElcb): 472

2fl

11'

4

'¥Jl

52' 72

44

LE1 cb. The smallest semi-magic square of squares, directly coming from (LE1). Magic sum S2

=

3249

=

572•

And even better: Euler has produced the smallest pos­ sible semi-magic example. There is no smaller example, even outside of this family. But because Euler never men­ tioned the problem to get (in his notation) the two further properties A2

+

E2

+

J2

=

1

C2 + E2 + G2 = 1 ,

we can say that the earliest known author to propose the problem of a 3 X

3 magic square of squares is Lucas in [41]. (ELl) again, but very briefly, without any comments [42] [43] [44]. Several years later, Lucas presents

2.

"Duncan Buell [1 2] searched for a 'magic hour­ glass' a+b+c a-c a-b a a -b-c a+b a+c with all seven entries squares, but found none with . < 25 . 1 024. Bremner [1 0] [29] found a magic square with seven square entries: (AB1) 2892 5652 3732 205 2

4252 52 72

222 1 2 1 "

It seems that the (ABl ) square was found independently by Lee Sallows

[29].

As well analyzed by Bremner [ 1 1 ] , it is easy to get a lot

of examples with six square entries. But it is strange to re­ mark that, excluding rotations, symmetries, and multiples of the

"Michael Schweitzer showed that any such square must have entries with at least 9 decimal digits"

Michael Schweitzer showed in

32'

380 7 2 1

3.

(ABl ) square, no other example has been found with

seven square entries, after long efforts of computing by dif­ ferent people . . . including me

[8] . . . .

And no example is known with eight square entries.

1996 [26) that if a 3

X

3

magic square of squares exists, the centre entry is bigger than

109, not all the entries. Because such a magic square

of squares needs to be at least a "magic hour-glass," the re­ sult of Duncan Buell done in

1998 [ 12] is far better: it im­ 3 X 3 magic square of squares exists, then its centre cell is > 25 1024 Martin Gardner was right in saying [28]: "If it exists, its ·

plies that if a

.

numbers would be huge." 4. "He [Michael Schweitzer] gives the following specimens in which only one diagonal fails. 1 272

462

582

22

1 1 32

942

742

822

972

(MS 1)

1 882

1 942

1 1 62

42

1 482

2542

2262

1 642

922

2822

29 1 2

1 742

62

2222

38 1 2

3392

2462

1 382

(MS2)

(MS3)

The magic totals are squares in each case: 1 472, Does this have to happen?"

2942, 44 1 2•

Lee Sallows also found in it is the

1996 [26] the first specimen (MSJ): (LSi) already presented in the introduction. It is the

smallest possible example in which only one diagonal fails. But I remark that it is simply a Lucas magic square! Apply with (p,

q, r, s)

=

(ELl) (1, 3, 4, 1 1), and you will get exactly the same

square. And it explains why the magic total is a square:

(p2

+

q2 + r 2

+

s2) 2

=

( 12

+

32

+

42

+

1 1 2) 2 = 1472.

The second specimen (MS2) also belongs to Lucas's fam­ ily! With

(p , q, r, s) = (2, 7, 4, 15), columns 2 and 3 being

inverted. It explains also why the magic total is a square:

(pz

+

q2

+

r z + s2) 2

The third specimen

=

(22

+

7z + 42 + 152) 2

=

2942.

(MS3) does not belong to Lucas's

family. The same magic sum of 441 2 is, however, possible, for example with

(1, 2, 6, 20), (1, 4, 10, 18), (1, 10, 12, 14), (2, 4, 14, 15), . . . but giving squares in which the two diag­

onals fail, instead of only one with this third Schweitzer ex­ ample. The solution

(3, 12, 12, 12) gives a square in which q = r s, the integers

only one diagonal fails, but because

=

are of course not all distinct. Bremner's first specimen (AB2), found before the above squares and published by Guy and Nowakowski in

1995 [29],

in which the two diagonals fail, belongs to Lucas's family.

AB2. This semi-magic square of squares by Andrew Bremner in 1 995, is a 3 X 3 Lucas square with (p, q, r, s) columns 2 and 3. S2

54

THE MATHEMAnCAL INnELLIGENCER

=

4225 = 652•

=

(2, 6, 4, 3) and inverting

Michael Schweitzer [26] was the first to fmd an example (MS4) in which again one diagonal fails but having a non­ square magic sum:

• •

pr + qs 0, a I c [ - d(pq + rs) - b(ps + qr)] I [ b(pq + rs) + d(ps + qr)]. =

=

The work o f Euler is linked to the theory o f quaternions 3'9

34951

29582

36422

2 1251

1 785'

2n!¥Z

2()582

3()05?

[2) [ 15] [36] [37], developed later in 1843 by William Hamil­ ton. In his (LE3) square, Euler reuses an identity that he found and sent to Christian Goldbach in

MS4. Example with a non-square magic sum 52 = 20966014, in which one diagonal fails.

1748 [2 1 ) :

(a2 + bz + c2 + d2)(p2 + q2 + r 2 + s2) = (ap + bq + cr + ds)2 + (aq - bp - cs + dr)2 Z + (ar + bs cp - dq) 2 + (as - br + cq - dp) . -

This identity also follows from the fact that the norm of the Because its magic sum is not a square, this example is of course not a member of the Lucas family.

12

1 82

382

1 12

212

chet, and Fermat. Using as a basis these partial results of Euler's, Lagrange published in

222

472

1 72

(AB3, with S2

332

1 32

=

2823)

432 � 32 31� As for the 3 X 3 magic square of squares problem in part 1, there is a forgotten work of an old and prestigious au­ thor on this subject. The first 4 X 4 magic square of squares was constructed by Leonhard Euler, in a letter [56] written in French that he sent in 1770 to Joseph Lagrange, not giv­

ing any method. This letter with the (LE2) square was known by Legendre

[39) in 1830 and by Lucas [40] in 1876,

but not the method.

41

fliP

2'¥1

1 7'

31

tiP

'Ill'

zt'

11

.,.

at

1754 [ 17) in a partial proof the sum ofat most four square

integers, an old co{\jecture announced by Diophantus, Ba­

"He [Andrew Bremner] [1 1 ] also gave the 4 X 4 magic square of squares: 232

Euler first used this identity in that every positive integer is

5.

372

product of two quaternions is the product of the norms.

371 .. 81

LE2. The first known magic square of squares, sent in 1 770 by Leon­ hard Euler to Joseph Lagrange. The 4 rows, 4 columns, and 2 diag­

1770 (55] the first complete

proof of this four square theorem, the same year as the let­ ter from Euler with the first

4 X 4 magic square of squares.

The magic square received by Lagrange is a member of

(a, b, c, d, p, q, r, s) = (5, 5, 9, 0, 6, 4, 2, - 3). In 1942, Gaston Benneton [ 1) [2) published another 4 X 4 square of squares using Euler's method. Its magic sum is 7150, a smaller sum than the Euler example. In 2005, we

this family of squares with

can now say that the smallest member of Euler's family producing distinct numbers is one not found by Euler, (GEl). And it is a member of a nice sub-family (CB2).

48"

zt'

at

18'

21

•

33'

322

1

3111

13'

422

222

�

44'

8'

CB1 . The smallest magic square of squares of Euler's family, gen­ erated by (2, 3, 5, 0, 1 , 2, 8, -4), giving 52 = 3230.

onals have the same sum 52 = 851 5.

Euler gave his method (LE3) at the St Petersburg Acad­ emy also in

1770 [ 18]:

CB2. Sub-family of Euler's magic squares of squares, 52 = 85(k2 + 29). k = 0, 1, 2 do not produce distinct numbers. k = 3 produces the (CB1) square. See [9] for another sub-family producing Euler's (LE2) and Benneton's squares. LE3. Euler's 4 X 4 magic square of squares family. Magic sum 52 = (a2 + b2 + c2 + d2)(p2 + q2

+

r2 + s2).

Its magic sum is smaller than in Euler's example (LE2), smaller than Benneton's square, but bigger than the Brem­

He needs two supplemental conditions in order to get the

ner example (AB3), which is not a member of Euler's fam­

two diagonals to sum to S2:

ily. I can confirm that the solution with the smallest sum

u

© 2005 Springer Science+ Business Media. Inc., Volume 27, N mber 2, 2005

55

Leonhard Euler (Basel 1 107-5t. Petersburg 1 783) sent in 1 770 a 4 X 4 magic square of squares to Joseph-Louis Lagrange (Euler's original letter: Bibliotheque de l'lnstitut de France, photo C. Boyer).

Euler's method published in Latin at St. Petersburg (Bibliotheque de I'Ecole Polytechnique, photos C. Boyer).

56

THE MATHEMATICAL INTELLIGENCER

is the Bremner sample (AB3) (and all its permutations). I can also add that it is impossible to construct another ex­ ample with a sum smaller than 3230: it means that the sec­ ond smallest solution of this problem is my square (CBI), coming from Euler's family. See the supplement [9] for a complete numerical study of Euler's family. Warning: both Lucas's and Euler's families often gen­ erate incorrect squares, because all their numbers are not always distinct. For example, the square (CB3) is smaller than the Bremner example, but it unfortunately contains the same number twice: a small game, will you quickly lo­ cate it?

CB3. A member of Euler's family, generated by {3, 2, 4 - 1 , 2, 8, -4), but unfortunately an incorrect magic square!

Note that with a = p, b q, c = -r and d -s, Euler's family of squares becomes (LE3cb ) It is a disguised and permuted version of Lucas's family (ELl) of 3 X 3 squares: =

=

.

CB5. The second smallest 5 X 5 magic square of squares. 82

=

1831.

Bimagic squares

Now, let us switch to bimagic squares: magic squares stay­ ing magic when their entries are squared-or, if you pre­ fer, magic squares of squares which stay magic when their entries are not squared. The previous examples in this ar­ ticle were not bimagic, because they do not stay magic when their entries are not squared. The first published bimagic square was an 8 X 8 square made by G. Pfeffer­ mann in 1890, and published in January 189 1 [4] [7] [45] [47] [62]. Amazed, Edouard Lucas immediately published in 1891 [42], just before his accidental death, a compliment to Pfeffermann for his achievement, together with a proof that a 3 X 3 magic square using distinct integers cannot be bimagic. John R. Hendricks published in 1998 [32] a different, long proof. I propose here a third-and far easier-proof of the impossibility, proving also the new result that even 3 X 3 semi bimagic squares are not pos­ sible. -

LE3cb. Transition from Euler's 4 X 4 to Lucas's 3 X 3 magic square of squares, generated by a

=

p, b

=

q, c

=

-r, d

=

-s.

And what about 5 X 5 magic squares of squares? They are also possible. Here are two examples, (CB4) (CB5). They seem to be the first published 5 X 5 magic squares of squares, and the smallest possible examples. Some related squares are given below (CB9) and in the supplement [9].

CB4. The smallest 5 X 5 magic square of squares. 82

=S2

=S1

CB6. A putative 3 X 3 semi-bimagic square.

=

1 375.

If such a square exists (CB6), then: Sl = a + b + c e=b+c-d

=

a+d+e

And: S2 z b + c2 b2 + c2 0 (d - b)(d - c)

= a2 + b2 + c2 = a2 + d2 + e2 = d2 + e2 d2 + (b + c - d? = 2d2 + 2bc - 2bd - 2cd = 0. =

This implies that d = b, or d = c. But a magic square has to use distinct numbers: without this requirement we would have bimagic, trimagic, . . . squares with, for ex­ ample, 1 in each cell! So a 3 X 3 semi-bimagic square can­ not exist, implying of course that a 3 X 3 bimagic square cannot exist.

© 2005 Springer Science+ Business Media, Inc., Volume 27, Number 2 . 2005

57

Now increment the size, searching for a 4 X 4 bimagic square. Cauchy worked in 1812 on a related subject. After partial proofs of Gauss (triangular case in 1796) and La­ grange (square case in 1 770, as already mentioned above), Cauchy was the first to prove completely what Fermat pro­ posed: every positive integer is a sum of at most three tri­ angular numbers, four square numbers, five pentagonal numbers, and n n-gonal numbers. . . . In his proof [ 14], Cauchy used the same system of two equations that we have to use in the search for a bimagic square: • •

k = t2 + u 2 + v 2 + w 2 s=t+u+v+w

It is not too difficult to find samples of semi-bimagic squares (4 bimagic rows, 4 bimagic columns), the smallest being (CB7). See also an example of a semi-bimagic square using only prime numbers in [9].

1

35

46

61

= 1 43

12

352

462

61 2

=

7063

37

71

13

22

= 1 43

372

71 2

1 32

222

=

7063

43

26

67

7

= 1 43 .....

432

262

672

72

62

11

17

53

= 1 43

622

1 12

1 72

532

= 1 43

=

1 43 = 1 43 = 1 43

=

7063

=

7063

=

7063

=

7063

nal. 51 = 1 43, 52 = 7063.

55

1 05

36

69

1 00

21

15

28

49

19

109

99

1

60

45

CBS. The smallest 4 X 4 semi-bimagic square with one magic diag­ onal. 51 = 205, 52 = 1 5427.

But in a 4 X 4 square having 4 bimagic rows and 4 bimagic columns, it seems very difficult-or impossible­ to get two simply magic diagonals; or one bimagic diago­ nal. And it is proved impossible to get two bimagic diago­ nals: answering puzzle 287, posed by Carlos Rivera and me in October 2004 [50], Dr. Luke Pebody (Trinity College, Cambridge, England) and Jean-Claude Rosa (Cluny, France) proved independently that a 4 X 4 bimagic square is impossible. It is unknown if a 5 X 5 magic square using distinct in­ tegers can be bimagic. The following (CB9) magic square has 5 bimagic rows and 5 bimagic columns. The two diag­ onals are magic but not bimagic: not yet a bimagic square, but a better result than the (CBS) square.

58

THE MATHEMATICAL INTELLIGENCER

44

16

34

35

1

12

38

41

8

24

40

7

10

36

47

13

14

32

4

28

11

45

The 6 X 6 magic square (GPJ) was published by G. Pfefferman in 1894 [46]. It has 6 bimagic rows and 6 bimagic columns. The two diagonals are magic, but not bimagic, as in (CB9). 6 X 6 magic squares with the same characteris­ tics were published by Huber in 1891 [35], Planck in 1931 [48], Planck modified by Lieubray also in 1931 [48], Venkat­ achalam Iyer in 1961 (RVIJ) [61 ] , Collison in 1992, page 147 of [31 ] .

CB7. The smallest 4 X 4 semi-bimagic square without magic diago­

9

20

onals. 51 = 1 20, 52 = 3970.

7063

It is also possible to get one magic diagonal. For example (CBS),

37

CB9. The smallest 5 X 5 semi-bimagic square with two magic diag­

= 7063 =

3

6

42

29

3

40

30

8

44

47

21

20

10

33

31

41

37

1

7

19

17

13

9

43

49

36

2

5

35

34

38

48

14

15

45

12

16

GP1. A 6 X 6 semi-bimagic square with two magic diagonals, pub­ lished by G. Pfeffermann in 1 894. 51 = 1 50, 52 = 5150.

4x+y

5x+6y

2y

6x+2y

x+6y

2x+y

5x+2y

y

4x+6y

2x+6y

6x+y

x+2y

6y

4x+ 2y

5x+y

x+y

2x+2y

6x+6y

0

4x+4y

5x+5y

x+5y

2x+4y

6x

5x+4y

5y

4x

2x

6x+5y

x+4y

4x+5y

5x

4y

6x+4y

X

2x+5y

RVI 1 . A 6 X 6 semi-bimagic square family with two magic diagonals, published by R. Venkatachalam lyer in 1 961. 51 = 18x

+

18y, 52 =

82x2 + 1 08xy + 82y2•

Open problem 3. What is the smallest bimagic square us­ ing distinct integers? Its size is unknown: 5 x 5, 6 x 6, or 7 x 7? My feeling is that 5 x 5 bimagic squares do not exist. Bimagic squares of izes 8 X 8 and above are al­ ready known, see part 7. Open problem 4. Construct a bimagic square using dis­ tinct prime numbers. [9) [50].

6.

"It's implicit in the work of Carmichael that there can be no 3 X 3 magic squares with entries which are cubes or are fourth powers"

The work of Euler implies already that there can be no 3 X 3 magic square with entries which are cubes. If z 3 is the number in the centre cell, then any line going through the centre should have x 3 + y 3 = 2z 3. Euler and Legendre [39] demonstrated that x 3 + y 3 = kz 3 is impossible with dis­ tinct integers, for k = 1, 2, 3, 4, 5. Adrien-Marie Legendre mistakenly announced that k 6 is also impossible: Edouard Lucas published the general solution for k = 6 in the Amer­ ican Journal of Mathematics Pure and Applied of J. J. Sylvester [41 ] , and gave the example 173 + 373 = 6 · 2 13. The equation x 3 + y 3 7z 3 has been known to be possible since Fermat, one of his examples being 43 + 53 7 33 . Legendre showed also that x 4 + y 4 2z 2 is impossible if x i= y. Because z 4 = (z 2)2 , this implies that there can be no 3 X 3 magic square with entries which are fourth pow­ ers. It's also implicit in the later work of Carmichael [ 13] that there can be no 3 X 3 magic square with entries which are cubes, or are fourth powers or 4k-th powers. Noam Elkies [26] points out that with Andrew Wiles's proof it can be shown that an + bn = 2cn has no solution for n greater than 2, and thus that there can be no 3 X 3 magic square with entries which are powers greater than 2. And as said in the D2 problem, "The Fermat problem," page 219 of Guy's book [30], "It follows from the work of Ribet via Mazur & Kamienny and Darmon & Merel that the equation xn + yn 2zn has no solution for n > 2 apart from the trivial x = y z." So, 3 X 3 magic squares of cubes are impossible. I think that 4 X 4 are also impossible with distinct positive inte­ gers. The 12 X 12 (WTl) trimagic square of part 7 below, when its numbers are cubed, is a magic square of cubes. If we accept negative integers, and using the interesting but obvious remark that n3 and ( - n3) are not equal (the rule in a magic square is to use "distinct" integers, and the trick is that they are distinct!), (CBJO) and (CEl l) are magic squares of cubes having a nuU magic sum. They seem to be the first published 4 X 4 and 5 X 5 magic squares of cubes. If you do not like the terminological trick I used, then Open problem 5 is for you! And the (CB12) square is a first step. =

=

=

·

=

=

=

Open problem 5. Construct the smallest possible magic square of cubes: 5a) using integers having different ab­ solute values, 5b) using only positive integers. Open problem 6. Construct a magic square of cubes of prime numbers [9]. 1 g3

(-3)3

(- 1 0)3

(- 1 8)3

(-42)3

213

283

353

423

(-21 )3

(-28)3

(-35)3

( - 1 9)3

33

1 03

183

CB10. A 4 X 4 magic square of cubes. 53 = 0.

113

( -20)3

1 23

1 33

1 43

(- 1 5)3

213

33

(- 1 0)3

(- 1 7)3

43

sa

(-5)3 1 73 (- 1 4)3

03 (-4)3 --1-

1 03

( - 3)3

(-2 1 )3

1 53

(- 1 3)3

( - 1 2)3

203

(- 1 1)3

CB1 1 . A 5 X 5 magic square of cubes. 53 = 0. A special property: four consecutive integers are used in the first row.

g3

473

543

643

963

233

973

63

483

(723

1 03

1 43

673

1013

423

1 1 03

363

213

33

283

403

703

983

1 83

383

CB12. The smallest 5 X 5 semi-magic square of cubes using posi­ tive integers. 53 = 1 408896.

7. "The following example [30] [34] , due to David Collison, is 'trimagic' in the sense that it is magic and stays so when you either square or cube the entries"

Collison's 16 X 16 trimagic square uses distinct but non­ consecutive integers. Bimagic and trimagic squares using consecutive integers, so more restricted than squares us­ ing non-consecutive integers, were published before this Collison square.

Wanting. In parts 1 to 6, we spoke about magic squares using distinct integers, but generally non-consecutive in­ tegers. ow, in parts 7 to 9, we speak about magic squares and cubes using consecutive integers. The first published n X n bimagic square using consec­ utive integers from 1 to n2 is an 8 X 8 square made by G. Pfeffermann in 1890, and published in January 1891 [45]. Various other bimagic squares are currently known with various sizes n ::=:: 8. Walter Trump and I showed in 2002 that an n X n bimagic square using consecutive integers is impossible for n < 8. The first published trimagic square, staying magic when you either square or cube the entries, is a 128 X 128 magic square made by Gaston Tarry in 1905 [59]. Later, smaller trimagic squares were found: 64 X 64 by General E. Caza­ las in 1933 [16], 32 X 32 by William H. Benson in 1976 [3], and 12 X 12 (WTl ) by Walter Trump in 2002 [60]. Even using consecutive integers, this trimagic square is smaller than Collison's square. Walter Trump and I showed in 2002 that an n X n trimagic square is impossible for n < 12. Nobody will ever construct a trimagic square smaller than Trump's.

© 2005 Springer Science+Business Media, I n c . , Volume 27, Number 2 , 2005

59

1

22

33

9

119

45

75

141

35

74

8

1 06

41

62

66

79

83

1 04

112

123

1 44

115

1 07

93

52

38

30

1 00

26

136

48

57

14

131

88

97

1 10

4

70

43

102

133

96

39

137

71

1--

140

101

12 49 1--11 24 42 60

1 22

76

1 42

86

55

27

95

1 35

- --

37

1 08

85

103

21

44

5

67

1 26

19

78

59

3

69

23

130

89

56

15

10

50

118

90

77

28

13

1 43

81

72

-

132

117

68

91

11

73

64

2

121

1 09

58

98

84

1 16

138

80

34

1 05

6

92

51

63

31

20

25

99

46 i113 32

-

134

--

54

-

-

36

24

1 29

7

29

61

47

87

1 27

18

53

1 39

40

111

65

1 28

17

1 20

1 25

1 14

82

94

16 -

--

-

WT1 . The first 12 X 12 trimagic square, constructed by Walter Trump in 2002. This square using consecutive integers from 1 to 144 is magic, and it is again magic when its numbers are squared or cubed. 51 = 870, 52 = 83810, and 53 = 9082800. It is impossible to construct a trimagic square of a smaller size (and using consecutive integers).

In 2001, I published the first known tetramagic and pen­ tamagic squares, staying magic when you square, cube, or

lines in his cube, but 64 remains an excellent result com­ pared to the needed 76 theoretical magic rows.

raise to the fourth or fifth power the entries, in an article

More than three centuries after Fermat's 4 X 4 X 4 cube,

of Pour La Science (the French edition of Scientific Amer­

Richard Schroeppel showed in 1972 [53] that a 4 X 4 X 4 per­

[4] after a joint work with Andre Viricel: 5 1 2 x 512

fect magic cube is impossible, and in 1976 [54] that, if a 5 X

ican)

tetra and 1024 X 1024 penta. Later, I constructed a smaller tetramagic square (256 X 256), and Li Wen a smaller pen­

if n

tamagic square (729 X 729). It is unknown

X

n tetra­
magic and pentamagic squares are possible for 12

5 X 5 perfect magic cube is possible,

then its centre cell is 63.

Walter Trump and I constructed in 2003 [6] [7] [63] the first known 5 X 5 X 5 perfect magic cube

(WT2CB13):

of

course, its centre cell is 63.

729. The first hexamagic square, magic

Bigger magic cubes of various sizes were previously

until the 6th power, was constructed by Pan Fengchu in

known, the first known perfect magic cube being a 7 X 7 X

2003 (4096 X 4096).

7 cube constructed in 1866 by Reverend Andrew H. Frost [20].

256 and 12


For more about all these multimagic squares, see refer­

The first published bimagic cube is a 25 X 25 X 25 cube

ences [4] [7] [ 16] [47] [57] [62 ] . Because they are at least

created in 2000 by John R. Hendricks [33]. Coincidence?

bimagic, all the multimagic squares are "magic squares of

bimagic cube of the same size was announced several years

squares."

before by David Collison Gust before his death, but unfor­

8.

news announced by John R. Hendricks himself in 1992 [31 ] .

Martin Gardner already asked this question in 1976 [22] [23]

cubes including the currently smallest known bimagic cube,

A

tunately never published) to the same John R. Hendricks:

"Has anyone constructed a 5 X 5 X 5 magic cube, or proved its impossibility?"

and again in 1988 [24]. Yes, a 5 X 5 X 5 magic cube has been constructed, re­ cently, in 2003. What should we mean here by "magic cube"? Richard Guy speaks about

X

X

n perfect magic cubes using consecu­ tive integers from 1 to n3, and having 3n2 + 6n + 4 magic lines: their n2 rows, n2 columns, n2 pillars, 3 2n plane diag­ onals, and 4 space diagonals. A standard magic cube ( non­ perfect) does not have its 3 2n plane diagonals magic. A 3 X 3 X 3 magic cube is possible, but a 3 X 3 X 3 per­ fect magic cube is impossible. In 1640, Fermat [58] sent to Mersenne a 4 X 4 X 4 nearly perfect magic cube (PFl) with n

n

·

=

In 2003 [5], I succeeded i n constructing various multimagic

a 16 X 16 X 16 cube, smaller than Hendricks's. I also con­ structed the first known perfect bimagic cubes, and the first known standard and perfect trimagic and tetramagic cubes. My smallest perfect bimagic cube is 32 X 32 X 32. I showed that

X

X

bimagic cubes are

n bimagic and perfect bimagic n < 8. It is unknown if n X n X n possible for 8 ::::: n < 16, and perfect

bimagic for 8

32.

n

n

cubes are impossible for :::::

n<

·

64 magic rows: Fermat mistakenly announced 72 magic

60

THE MATHEMATICAL INTELLIGENCER

Open problem 7. Construct the smallest possible magic cube of squares (the 16 x 16 X 16 bimagic cube, when its numbers

are

squared, is already

a

magic cube of squares).

/ ...r.J? / .!7 ...r 7 .- 7 ...r
r

/ 6".:7 / 6" / .T 1/ .:F.T / / ...r.T / -.T / - / ...?,&7 / / -.!7 / ..Z'.9 / ...rB' / 4IS" / [/ 4" / - / .:F.9 / .:r 17

I

/ 4I:SI" / ...:z:.r / ...r.t? I/ __. / / .Sf:.? / _,., / .J?-!7 / .2 ..9 / / .9'6" / ..?.:77 .szf7 -:5W7 / ..9 / .:F-5" / � / ..z:.? / /

I-

/
6U1' v

...r /

/ -...r / .23 / ....;;;?!? / -7 / .;...r / 4/CY / -- / .::;t.r 7

1/ - / ..2 / ..? / «r 1/

PF1 . Pierre de Fermat (Beaumont de Lomagnes 1601-Castres 1665) sent in 1640 a 4 X 4 X 4 nearly perfect magic cube to Mersenne. Among its 76 lines, 64 lines have the magic sum 51

=

130, but 12 lines have a different sum.

/ .2.!7 / ...r6" / - 7 ...r-7 ..9.:7 7 /.L.r .:F / .94" / - / ...r / ..9.T / / -<JC2 /�/ � / .2 / .T.!r / / 6"6"7 .?.::? 7 .2.T/...ra?/ - / / 6'".T / ...rB' /...:z:.r..?/...rPI6"/ .!7 7 / ..?...r / .T.T / .?-r / / .T.t? / / � / 6"- /..z.r.T 7 - 17 .r..Y"7 / .?.:7 /'...:z:.r B'/ � /..z:.? ..Y'712.?7 / .26'" / .?.9 / .R.? / - /1 -/ 1/'...:z:.rd"/ ...r..T / ...r- / .?:Y / ..9.!7 I/ / -..T / 6"...r / � / �liP7 4"6'" 7 /...r.t?.T/ 4/CY / � / ..Y'J v --7 / 4".9 / 6"4" -- / .:?..T / / .?.2 / ..9.? / - / 4CY / ..z19 / v -- / .!7.:7 / «r / 6:? 7 ..T..9 [7 / .J?...r / � /�/...r. �/ ...r.t? / / ...z:.2 / - / _,_ / B'..T1/..,r,e?,e?/ /...r �/ ..Y' /...r�7 4" / ..96" / /.L.r .JI'"/ .!7..T / .9 / dP / /101"7 / ..5"6'" /...z:.2.:7/ � / 11<9 / ..Y'-!7v /"..z:.? ...r/...r6'4"/ ..T / �!? / .!7..9 / � ...:z:.r / / .2..9 / .24" 7-Z.2?/...z:.2 / .£r / ...r.:F / ..c:r /�/ --7 / ..TB' / .5"- / .9:9 / .2- / ,6� / v .?6'" /'...:z:.r .t?/ -6'" / ....;;;?!? /...r.a: I/

WT2CB13. The first 5 x 5 x 5 perfect magic cube, constructed in 2003 by Walter Trump and Christian Boyer. Its 1 09 lines have the magic sum 51

=

315. The centre cell is the average cell value: 63. A 3 X 3 X 3 or 4 X 4 X 4 perfect magic cube is impossible.

© 2005 Springer Science+ Business Media, Inc., Volume 27, Number 2 , 2005

61

Open problem 8. Construct a bimagic cube smaller than 16 X 16 X 16.

to the 4th power, all the sums of these 201 millions of lines are equal to the same number: S4

Open problem 9. Construct a perlect bimagic cube mailer than 32 X 32 X 32.

=

1496577676620038940902162752365801 55584753888727040.

Similarly, the lines have the same sums, S2 and S3, when their numbers are squared and cubes, respectively. This cube is big:

My smallest perfect trimagic cube is a 256 X 256 X 256

•

So big that, if you built it, with cells of 2 em X 2 em X 2 em on each of which a number of 12 digits maximum

cube. I thank Yves Gallot, Renaud Lifchitz, Walter Trump, and Eric Weisstein, the author of the excellent MathWorld ency­

is engraved, you would get a cube bigger than Notre­

clopaedia

Dame de Paris.

http://mathworld.woifram.coml,

who each inde­

pendently checked this cube and confirmed its properties. My best multimagic cube is a 8192

X 8192

•

X 8192 per­

will need more than 17 years to check the whole cube.

fect tetramagic cube, staying perfectly magic when you square, cube, or raise to the fourth power the entries. The main challenge in magic squares or cubes is to have the maximum of characteristics in the minimum of space. That's why most of the open problems of this article are to fmd the

smaUest

squares or cubes having the prescribed

properties. A big object has no interest if it has poor char­ acteristics. But we fmd that each time we want to add char­ acteristics, i.e., a new multimagic degree, we are "forced" to work with a bigger object. I am unable, at least today, to get the perfect tetramagic characteristics in a smaller cube! This 8192 X 8192 X 8192 cube is monstrous. All its 201 million possible lines of 8192 numbers are tetra-magic: its 67, 108,864 lines, 67, 108,864 columns, 67, 108,864 pillars, 4 main diagonals, and 49,152 small diagonals. All the sums of these 201 million lines are equal to the same number S1

=

225 1 799813681 152. When its 550 billion numbers are raised

(CB14)

So big that, if you check 1000 numbers per second, you

It is very difficult to compute and to check such a huge cube. Eric Weisstein was able to check the 256th-order per­ fect trimagic cube, but he said that he and Mathematica were not able to check 5 12th-order or bigger cubes, mainly for memory reasons. I thank Yves Gallot, the author of the famous Proth pro­ gram used worldwide by searchers for big prime numbers, who checked this tetramagic cube and confirmed its prop­ erties. He wrote a nice piece of specific assembly and

C++

code to check the cube, different from my code: the com­ puting work behind this big cube is comparable to the com­ puting work behind a big prime number. One needs vast memory and multiprecision routines. Again in 2003, I constructed the first known multimagic hypercubes of dimension 4. For example, a 256 256

X 256 X X 256 trimagic hypercube, also checked by Yves Gallot.

About these magic and multimagic cubes and hyper­ cubes, see [5] [ 6] [7] [47] [62] [63].

9.

"Rich Schroeppel notes that the centre cell of a magic 95 is always the average cell value, and that a corollary is that there is no magic 96." This sentence needs an explanation: magic 95 and magic 96 mean perfect magic hypercubes of dimension 5 and 6, us­ ing consecutive integers. Schroeppel's demonstration referred to is similar to his demonstration in part 8: the centre cell of what he called a "magic 53"

(=

a perfect magic cube 5 X 5

X 5)-if it

exists-is the average cell value [54]. And a corollary is that there is no "magic 5 X 5).

54" (

=

perfect magic hypercube 5

X 5X

The current list of the conclusions of Richard Schroep­ pel are these: •

centre cell of magic 53 is average value � there is no magic

•

54.

centre cell of magic 74 is average value

�

there is no

magic 75. CB14. The 8192 X 8192 X 8192 perfect tetramagic cube compared

•

51 = 2251 799813681 152 52 = 825293359521335050119065600 53 = 340282366919700523424090353056775929856 S4 = 14965776766200389409021627523658015558475388872704{)

62

THE MATHEMATICAL INTELLIGENCER

centre cell of magic 95 is average value � there is no magic 96.

to the cathedral of Notre-Dame de Paris!

We can easily fill in the simple case: •

centre cell of magic 32 is average value � there is no magic 33

The general case would apparently be that for all k •

centre cell o f magic (2k + magic (2k + 1 )k+ Z.

1 )k +1 is average value

:2:

1,

�

no

[20] Andrew H . Frost, Invention of Magic Cubes, Quart. J. Math. 7 (1 866), 92-1 02 (21 ] P.-H. Fuss, Lettre C>N, Euler a Goldbach, Berlin 4 mai 1 748, Cor­ respondance mathematique et physique de quelques celebres geornetres du XV/1/eme siecle , St-Petersburg (1 843), 450-455

(reprint by Johnson Reprint Corporation, 1 968) (22] Martin Gardner, Mathematical Games: A Breakthrough in Magic Squares, and the First Perfect Magic Cube, Scientific American 234 (Jan. 1 976), 1 1 8-1 23 REFERENCES

[23] Martin Gardner, Mathematical Games: Some Elegant Brick-Pack­

(1 ] Gaston Benneton, Sur un probleme d ' Euler, Comptes-Rendus Hebdomadaires des Seances de I'Academie des Sciences

ing Problems, and a New Order-7 Perfect Magic Cube, Scientific American 234 (Feb. 1 976), 1 22-1 29

[24] Martin Gardner, Magic squares and cubes, Time Travel and Other

214(1 942), 459-461 . [2] Gaston Benneton, Arithmetique des Ouaternions, Bulletin de Ia So­

Mathematical Bewilderments , Freeman, New York, 1 988, 2 1 3-

225

ciete Mathematique de France 71 (1 943), 78-1 1 1 .

[3] William Benson and Oswald Jacoby, New recreations with magic

(25] Martin Gardner, The magic of 3

x

3, Quantum 6(1 996), n°3, 24-26

[26] Martin Gardner, The latest magic, Quantum 6(1 996), n°4, 60

squares , Dover, New York, 1 976, 84-92

(4] Christian Boyer, Les premiers carres tetra et pentamagiques, Pour

[27] Martin Gardner, An unusual magic square and a prize offer, CFF, no. 45, February 1 998, 8

La Science N°286, August 2001 , 98-1 02 .

[5] Christian Boyer, Les cubes magiques, Pour La Science N°31 1 ,

[28] Martin Gardner, A quarter-century of recreational mathematics, Scientific American 279 (August 1 998), 48-54

September 2003, 90-95. (6] Christian Boyer, Le plus petit cube magique parfait, La Recherche,

[29] Richard K. Guy & Richard J. Nowakowski, Monthly unsolved prob­ lems, American Math. Monthly 1 02(1 995), 921 -926; 1 04(1 997),

N°373, March 2004, 48-50. [7] Christian Boyer, Multimagic squares, cubes and hypercubes web

[30] Richard K. Guy, Problem D 1 5 - Numbers whose sums in pairs

site, www.multimagie.com/indexengl.htm [8] Christian Boyer, A search for 3

x

967-973; 1 05(1 998), 951 -954; 1 06(1 999), 959-962

3 magic squares having more

than six square integers among their nine distinct integers, preprint,

make squares, Unsolved Problems in Number Theory, Third edi­ tion, Springer, New-York, 2004, 268-271

September 2004

[31 ] John R. Hendricks, Towards the bimagic cube, The magic square

magic squares of squares problem" , downloadable from (7], 2005

(32] John R. Hendricks, Note on the bimagic square of order 3, J.

[9] Christian Boyer, Supplement to the article " Some notes on the [1 0] Andrew Bremner, On squares of squares, Acta Arithmetica,

course, self-published, 2nd edition, 1 992, 41 1 Recreational Mathematics 29(1 998), 265-267.

[33] John R. Hendricks, Bimagic cube of order 25, self-published, 2000

88(1 999), 289-297. (1 1 ] Andrew Bremner, On squares of squares II, Acta Arithmetica,

[34] John R. Hendricks, David M. Collison's trimagic square, Math. Teacher 95(2002), 406

99(2001 ), 289-308. (1 2] Duncan A. Buell, A search for a magic hourglass, preprint, 1 999 (1 3] Robert D. Carmichael, Impossibility of the equation x3 + y3 2mz3 , and On the equation ax4 + by4 cz2 , Diophantine Analysis , John =

=

Wiley and Sons, New-York, 1 91 5 , 67-72 and 77-79 (reprint by Dover Publications, New York, in 1 959 and 2004) [1 4] Augustin Cauchy, Demonstration complete du theoreme general de Fermat sur les nombres polygones, CEuvres completes, 11-6(1 887), 320-353 [1 5] Arthur Cayley, Recherche ulterieure sur les determinants gauches,

[35] A. Huber, Probleme 1 96 - Carre diabolique de 6 a deux degres avec diagonales a trois degres, Les Tablettes du Chercheur, Paris, April 1 st 1 892, 1 01 , and May 1 st, 1 892 , 1 39 [36] Adolf Hurwitz,

Ueber

die Zahlentheorie der Quaternionen,

Nachrichten von der Konig/. Gesellschaft der Wissenschaften zu Gottingen, 1 896, 31 3-340 [Reprint in Mathematische Werke von Adolf Hurwitz, Birkhauser, Basel,

2(1 963), 303-330.]

[37] Adolf Hurwitz, Vorlesungen uber die Zahlentheorie der Ouaternio­ nen, Verlag von Julius Springer, Berlin, 1 91 9, 6 1 -72

Journal fur die reine und angewandte Mathematik 50(1 855), 299-31 3

(38] Martin LaBar, Problem 270, College Mathematics J. 1 5(1 984), 69

(1 6] General Cazalas, Camas magiques au degre n, Hermann , Paris,

[39] Adrien-Marie Legendre, Theorie des Nombres, 3rd edition, Firmin­

1 934 (1 7] Leonhard Euler, Demonstratio theorematis Fermatiani omnem nu­ merum sive integrum sive fractum esse summam quatuor pau­ ciorumve quadratorum, Novi comrnentarii acaderniae scientiarurn Petropolitanae 5(1 754/5) 1 760, 1 3-58 (reprint in Euler Opera Omnia , 1-2, 338-372)

(1 8] Leonhard Euler, Problema algebraicum ob affectiones prorsus sin­

Didot, Paris, 2(1 830), 4-5, 9-1 1 , and 1 44-145 (reprint by Albert Blanchard, Paris, in 1 955) [40] Edouard Lucas, Sur un probleme d ' Euler relatif aux carres mag­ iques, Nouvelle Correspondance Mathematique 2(1 876), 97- 1 0 1 (41 ] Edouard Lucas, Sur !'analyse indeterminee du troisieme degre­ Demonstration de plusieurs theoremes de M. Sylvester, American Journal of Mathematics Pure and Applied 2(1 879), 1 78-1 85

gulares memorabile, Novi cornrnentarii academiae scientiarum

[42] Edouard Lucas, Sur le carre de 3 et sur les carres a deux degres,

Petropolitanae, 1 5(1 770) 1 771 , 75-106 (reprint in Euler Opera

Les Tablettes du Chercheur, March 1 st 1 891 , 7 (reprint in [44] and

Omnia , 1-6, 287-3 1 5)

[1 9] Leonhard Euler, De motu corporum circa punctum fixum mobil­

in www. multimagie. corn/Francais/Lucas. htm) (43] Edouard Lucas, Theorie des Nornbres , Gauthier-Villars, Paris,

ium, Opera posturna 2(1 862), 43-62 (reprint in Euler Opera Om­

1 (1 89 1 ) 1 29 (reprint by Albert Blanchard, Paris, in 1 958 and other

nia , 11-9, 431 -44 1 )

years) (reprint by Jacques Gabay, Paris, in 1 992)

© 2005 Springer Science +Business Med1a, Inc , Volume 27, Number 2 , 2005

63

[44] Edouard Lucas, Recreations Mathematiques, Gauthier-Villars,

A U T H O R

Paris, 4(1 894) 226 (reprint by Albert Blanchard, Paris, in 1 960 and other years) [45] G. Pfeffermann, Probleme 1 72 -Carre magique a deux degres, Les Tablettes du Chercheur, Paris, Jan 1 5th 1 89 1 , p. 6 and Feb

1 st 1 891 , 8 [46] G. Pfeffermann , Probleme 1 506- Carre magique de 6 a deux de­ gres (imparfait), Les Tablettes du Chercheur, Paris, March 1 5th 1 894, 76, and April 1 5th 1 894, 1 1 6 [4 7] Clifford A. Pickover, Updates and Breakthroughs, The Zen of Magic Squares, Circles, and Stars, second printing and first paperback

printing, Princeton University Press, Princeton, 2003, 395-401 CHRISTIAN BOYER

[48] Planck and E. Lieubray, Quelques carres magiques remarquables,

53, rue

Sphinx, Brussels, 1 (1 931 ), 42 and 1 35

De Mora

95880 Enghien les Bains

[49] Landon W. Rabem, Properties of magic squares of squares, Rose­

France

Hulman Institute of Technology Undergraduate Math Journal

e-mail: [email protected]

4(2003), N . 1 [50] Carlos Rivera, Puzzle 7 9 "The Chebrakov's Challenge", Puzzle 287

Christian Boyer was born near Bordeaux, and graduated from

"Multimagic prime squares, " and Puzzle 288 "Magic square of

two of the "grandes

(prime) squares", www.primepuzzles.net

eccles" of engi neering He has worked .

for Microsoft France, and more recently co-founded a suc­

[51 ] John P. Robertson, Magic squares of squares, Mathematics Mag­

cessful start-up in software. {He even offers advice to others

azine 69(1 996), n°4, 289-293

creating new companies.)

[52] Lee Sallows, The lost theorem, The Mathematical lntelligencer

Beside mathematical research and

popularizing of mathematics, he is a sports-car aficionado . He

19(1 997), n°4, 5 1 -54

is married, with three daughters.

[53] Richard Schroeppel, Item 50, HAKMEM Artificial intelligence Memo M.l. T. 239 (Feb. 29, 1 972)

[54] Richard Schroeppel, The center cell of a magic 53 is 63 (1 976), Villars, Paris, 2(1 894), pp. 1 86-1 94 (partial reprint of the letter at

www. rnultimagie. com/Eng/ish/Schroeppe/63. htm

[55] J . -A.

Serret,

Demonstration

d'un theoreme d 'arithmetique,

CEuvres de Lagrange, Gauthier-Villars, Paris, 3(1 869), 1 89-201

[56] J . A . Serre! and Gaston Darboux, Correspondance de Lagrange -

www.multimagie. com/Francais/Fermat.htm)

[59] Gaston Tarry, Le carre trirnagique de 1 28, Compte-Rendu de /'As­ sociation Fran9aise pour /'Avancement des Sciences, 34em e ses­

avec Euler, Lettre 25, Euler a Lagrange, Saint-Petersbourg , 9/20 mars 1 770, CEuvres de Lagrange, Gauthier-Villars,

Paris,

sion Cherbourg (1 905), 34-45 [60] Walter Trump, Story of the smallest trimagic square, January 2003,

1 4(1 892), 2 1 9-224 (reprint in Euleri Opera Omnia, IV-A-5, 477-482)

www. multimagie. com/Eng/ish/Tri 12Story. htm

[61 ] R. Venkatachalam lyer, A six-cell bimagic square, The Mathemat­ ics Student 29(1 961 ), 29-31

[57] Neil Sloane, Multimagic sequences A052457, A052458, A090037, A090653, A09231 2, A TT Research 's Online Encyclopaedia of

[62] Eri c Weisstein, Magic figures, MathWorld, http://mathworld. wolfram. comltopics/MagicFigures. html

Integer Sequences, www.research.att.com/ �njaslsequences

[58] Paul Tannery and Charles Henry, Lettre XXXV IIIb bis, Fermat a

[63] Eric

Mersenne, Toulouse, 1 avril 1 640, CEuvres de Fermat, Gauthier-

Weisstein,

Perfect

magic

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64

THE MATHEMATICAL INTELLIGENCER

cube,

MathWorld,

math world. wolfram. com/PerfectMagicCube.html

http:!/

MARTIN J. MOHLENKAMP AND LUCAS MONZON

Tri gonometri c dentiti es and Su ms of Se parab e Fu nctions

sin(z + 'Y - a) . sm( 'Y - a)

sin(y + {3- a) . . sm(x + y + z) = sm(x) . sm ({3 - a) sin(x + a - {3)

+

sin(a - {3)

sin(z + 'Y - {3) . sm(y) , . sm( 'Y - {3)

sin(x + a - y) sin(y + {3 - y)

+

sin(a - y)

sin(f3 - y)

(1)

cute. It illustrates a richness of structure that invites future study. Other mechanisms that allow representations of the form (2) with low separation rank are described in [2] .

sin(z)

hold for all values of x, y, and z? Motivation Modem computers have made commonplace many calcu­ lations that were impossible to imagine a few years ago. Still, when you face a problem with a high physical di­ mension, you immediately encounter the Curse of Dimen­ sionality [ 1 , p.94]. This curse is that the amount of com­ puting power that you need grows exponentially with the dimension. The simplest manifestation appears when you try to represent a function by its sample values on a grid. If a function of one variable requires N samples, then an analogous function of n variables will need a grid of N" samples. Thus, even relatively small problems in high di­ mensions are still unreasonably expensive.

A method has been proposed in [2] to address this prob­ lem, based on approximating a function by a sum of sepa­ rable functions: r

f(x l ,

·

·

. , xn)

=

L ¢{(xi) ¢�(x2) j �I

·

·

·

�

representation would require only r n N samples, so if the approximation can be made sufficiently accurate while keeping the separation rank r small, we can bypass the curse. We describe here a particular test of (2), when the "straight­ forward" approximation is exact but has very large separation rank. Although it may not be directly useful in applications, the result of this test is surprising, positive, and, we believe, This

For what value(s) of a, {3, 'Y does the equality

¢ (xn).

(2)

A Test Function Our test function is the sine of the sum of n variables, sin(�.f� 1 Xj), which is a wave oriented in the "diagonal" di­ rection in n-dimensional space. One could use complex ex­ ponentials to express it as the sum of two separable func­ tions, sin

( In ) xj

j� l

1

= ---;

n

.

1

n .

fl� l eur:J - ---; fl e -ur:J, 2t j� l

2t j

but in our test we allowed only real functions. You can use ordinary trigonometric identities to find such a representation. When n 2 we have =

sin(x + y) = sin(x) cos(y) + cos(x) sin(y),

(3)

which expresses sin(x + y) as a sum of two separable func­ tions. When n = 3 we have sin(x + y + z) = sin(x) cos(y) cos(z) + cos(x) cos(y) sin(z) + cos(x) sin(y) cos(z) - sin(x) sin(y) sin(z), (4)

This material is based in part on work supported by the National Science Foundation under grant DMS-9902365, and by University of Virginia subcontract MDA-97200-1-0016.

© 2005 Springer Sc1ence + Business Media, I n c. . Volume 27, Number 2, 2005

65

Figure 1. left: Graphical separated representation of sin(K + y + z) using the usual trigonometric identity (4). Each of the four rows gives the factors of a separable function. For example, the first row corresponds to sin(K) cos(y) cos(z). The separable functions from each row are then added. right: Graphical separated representation of sin(K + y + z) using (1) with

a = 0, {3

=

Tr/3, and y

= 2Tr/3. The amplitude has been

equidistributed. (The original illustration uses three colors, blue, red, and green, for the three curves in each graph.)

which uses four terms. The drawback to this approach is that, for

n

variables, the number of terms is

2n- l . This exponen­

tial growth in the number of terms negates the benefit of us­ ing the form

(2) . Indeed, if this really is the minimal number

of terms needed, then the entire approach is doomed.

We then asked what the minimal number of terms is, and our program replied shown in Figure

1

for all values ofA, B, s({3 - a) * 0). Proof

With the notation c(x)

metric identity

s(A)

(right). After some investigation, we de­

s(y)

(3)

Any representation of a function of a sum of n variables will have

�

- 1 free parameters, because one can include n shifts Xj Xj + ai and one linear constraint �J= 1 ai = 0 and have �J=l (Xj + aj) = �J= l Xj. The identity that we pre­ sent below in Theorem 2 has n 1 additional independent n

-

+

n

=

3,

( 1) holds for arbitrary a, {3, and y, as - {3) * 0, sin(a - y) * 0, and sin({3 - y) * 0.

Since these three parameters occur only as differences, only two of them are independent. One can introduce two additional parameters as phase shifts to make versions of

(1)

with different symmetries.

The Identity Lemma 1 The function s(x)

s(A + B) =

66

=

recognize as

satisfies the equation

s(A + a - {3)s(B) s(A)s(B + {3 - a) + s({3 - a) s(a - {3)

THE MATHEMATICAL INTELLIGENCER

(5)

s(B) (s(A)c(- y) + c(A)s( - y)) s ( - y) + c(A)s(B),

which we

s(A + B).

D

Theorem 2 Any junction s(x) that satisfies (5) also sat­

isfies

s

( � ) - �l ( ) L

j= l

Xj -

L

j=

s Xj

n n

s(xk + ak - aj) . s(ak - aj) k = l ,k ¥j

(6)

for aU choices of { llJ} such that s(ak - aj) * 0 for aU j * k. The proof is by induction and is given in the Appendix. We can generate a more general form by introducing

a/ s

sin(x)

par­

--

all terms cancel except for s(A)c(B)

it provides an answer for our opening

teaser. The identity long as sin(a

{3 - a,

Multiplying out and using that s(x) is odd and c(x) is even,

parameters, which play a structural role in our represen­ tation. When

cos(x) and y =

(s(B)c(y) + c(B)s( y))

termined the trigonometric identity that our program had un­ numerical, not symbolic, and so uncovered a trigonometric

=

to obtain

covered. What is most remarkable is that the program was identity without even knowing it was doing trigonometry!

and {3 such that s(a - {3) * 0 (and

tially expand the right-hand side using the usual trigono­

and produced graphs such as that

"n"

a,

Ct

Xj +

it

)

n

shifts

ai

=

n

n

I s(Xj + aj) =n k l,k¥j j=l

s(xk + ak + ak - aj) s(ak - aj)

By choosing different ways to satisfy the linear constraint

�J= l ai = 0, we can produce a variety of identities similar

to (6) without modifying the parameters a1, which are the

by setting y1

structural elements of our representation. Note that in the

identity, and then multiplying by exp(i�J= l x1)!2i.

set { ak - a1}Mj only n - 1 parameters are linearly inde­ pendent, say { a1 - ai1J=2·

Other Functions that Satisfy the Same Identity Because of Lemma 1 and Theorem 2, we know that sin(�J= l x1) is exactly separated with separation rank n. Moreover, this function is peculiar in that sin(·) is the only function used in the separated representation. We now consider the problem of finding other functions s(x) sat­ isfying (6). Since the general case (6) is equivalent to the

2 case, it is enough to describe all functions that sat­ isfy (5). n

=

Lemma

3 The function s(x)

=

x satisfies the identity (5).

This and the following lemma may be verified directly. Lemma 4

f(C D)

=

f(C)f(D ¢/8) + f(C8/¢)f(D) !( 8/¢) f( ¢/8)

is equivalent to (5) with the substitutions C exp(A), D exp(B), 8 = exp (a), ¢ exp(/3) , and s(x) = f(exp (x)). Sim­ ilarly, (6) is equivalent to =

=F 0, b,

and c

f

( Il

j= l

Yi

)

=

ate condition is that s(x) be meromorphic.

meromorphic function s(x) satisfies (5) if

and only if a exp(bx)x

or

s(x)

=

a exp(bx) sin(cx)

for some complex constants a =F 0, b, and c

=

I rl

j = l k= ! ,k #j

In analogy to Lemma 4, from the particular solutions f(x)

ln(x) and f(x)

=

=

1 - x to (8) we can generate other solu­

Remarks and Conclusions It is easy to extend our results to find similar identities for f(�J= l Xj), where f(x) could be cos(x), cos2(x), or sin2(x), for example.

We also tested the function of six variables sin(u + v + w) sin(x + y + z). Using ( 1) on each factor and then mul­ tiplying out yields a representation of the form (2) with 9 terms, but our program found a representation with 8

A survey on the problem of exact separated represen­

s (a + ak - a1) s(ak - aj)

When s(x) = sin(x), this result is presented in [4] . For a proof using Lagrangian interpolation, see [5, page 272]. The approach of [4] and [5], however, does not produce the gen­ eral results of Theorems 2 and 5. Conversely, our results can be used to derive only a few of the identities listed in

[5, Section 2.4.5.3].

out in Problem 4 on page 158, to find a minimal rank rep­ resentation for a separated representation is still an open problem. We believe that our Theorems 2 and 5 are an ex­ ample of such minimal representations. Lemma 1 can be proven geometrically, in a way similar to the geometric proof of the usual identity (3). We have not been able to find a geometric interpretation of (6).

Appendix: Proofs Proof of Theorem 2

The case n = 2 is Lemma 1 with A = x1, B = x2, a = a1, and {3 = a2. The proof will be by induction in n, so we as­ sume (6) has been proven for n 1. We will use (5) to sep­ -

The situation is different if we consider Milne's identity

[ 6, 3]

(8)

tations is the book [7] by Rassias and Sim8a. As they pointed

Under the same conditions as in Theorem 2,

s(a)

f(Yk8k181) , fC8k/8j)

to find the formula analogous to (6) for this case.

=F 0.

Extensions and Relationships with Other Identities If in Theorem 2 we set x1 = a for all j, we obtain the fol­ lowing corollary.

s (na)

k = l ,k #j

terms. After considerable effort, we have still not been able

The proof is given in the Appendix.

Corollary 6

I f(yj) Il

j= l

ization of Milne's identity.

=F 0.

(5), and then ask if we have missed any others. We only

Theorem 5 A

=

=

for constants a, b, and c. In this way we obtain a general­

wish to consider reasonably nice functions. The appropri­

arate out the variable Xn, then cancel like terms and reduce 1 case. the n case to the n -

1-

n

I1

j= l

n

Yi

=

I (1

j= l

-

yj)

n

I1

k = l,k #j

1 - Yk8k/81 1 - 8kJe1

First, expand the left-hand side of (6) using (5) with A

·

(7)

We can obtain another proof of this identity by setting s(x) = 1 - exp(x), ai In 8j, and Xj = In Yi in (6). Theorem 5 applies to this function because 1 - exp(x) = -2i exp( ) sin( ). Conversely, (6) for s(x) sin(x) can be obtained =

�

Simply note that the identity

axb f(XC)

Starting with our two basic functions sin(x) and x, we can use Lemma 4 to construct other functions that satisfy

=

A "multiplicative" version of the identities that we have discussed can be derived by generalizing this observation.

If s(x) satisfies (5), then so does

for all complex a

exp( - 2ia1) in Milne's

=

tions to (8), namely

a exp(bx)s(cx)

s(x)

exp( - 2ix1) and e1

=

=

�

r;: f Xj, B = Xn, (X = O'.n - 1 ,

and {3

=

O'.n to obtain

S(Xn + O'.n - O'.n- 1) s( O'.n - O'.n- 1) +S

(nfl J= l

Xj + O'.n - 1 - an

)

s(Xn) . a ns( l - an)

© 2005 Springer Sc1ence+Business Media, Inc., Volume 27, Number 2, 2005

=

(9)

67

On the right-hand side of (6), first separate off the j n term in the sum. When j i= n, we expand the k n term in the product using (5) with A = a, - a1, B = Xn, a = an - 1 , and {3 = an. Explicitly, the k = n term is =

=

s(Xn + a, - aj) s(an - aj) X

=

(

=

1 s(a, - aj)

S (a,, - aj)S(Xn + Cin - an- 1) s(an - Cin - 1)

s(xn + a, - Cin- 1) s(a, - Cin- 1)

+

(

+

S( Cin- 1 - aj)S(Xn) S ( Cin- 1 - a,)

s( Cin- 1 - aJ) s(an - aj)

)(

s(xn) s(an - 1 - an)

)

)

·

Note that the first term does not depend onj, and that when j = n - 1 the second term is absent. Combining these ex­ pansions, we can express the right-hand side of (6) as 1 s(xk + ak - aJ) s(xn + - Cin- 1) s(Xj) s(an Cin- 1) s(ak - aj) k=1,k"'J J=1

(I

(�

)

fY

�

n 1 S (Xk + ak - aj) S ( Cin - 1 - aj) n + L s(Xj) s(ak - aj) s(a n - aj) k= 1,k #J j= 1 2

)

Proof of Theorem 5, given Lemmas 7 and 8

We have already shown that these functions satisfy (5), so we need only show there are no more solutions. We now assume s(x) satisfies (5) and will deduce its properties. Using Lemma 7, we know that h(x) = exp( -bx)s(x) is an odd function. By Lemma 4, h(x) also satisfies (5). Then, by Lemma 8, h(x) is either ax or a sin(cx), so s(x) = a exp(bx)x or s(x) = a exp(bx) sin(x), which completes the proof. D The proofs of Lemmas 7 and 8 use the fact that s(O) = 0. By setting {3 - a = A in (5) and subtracting s(A + B) from both sides we obtain 0

=

s(O)s(B)

valid for all A such that s(-A) i= 0 and for all B. Choosing B such that s(B) i= 0 implies that s(O) 0. =

Proof of Lemma 7

We define the auxiliary meromorphic function

F(x) =

Now compare our expansions (9) and (10) of the two sides of (6). Using the induction hypothesis at n - 1, we can see that the first terms in (9) and (10) are equal, and so cancel. The remaining terms all have a factor of s(xn) in the numer­ ator and s(a,- 1 - a,) in the denominator, which we can also cancel. Thus we have reduced the proof to showing that n-1 Xj + Cin- 1 - an S J= l

(�

)

s(-A) '

s(x) s( -x) '

which cannot be identically zero, and show that it satisfies the functional equation

F(x + w) = F(x)F(w).

=

s(x)s(w) s(x + w)

+

s( - w)s( -x) s( - x - w)

)

_

.

(13)

Using (5) with A x, B = -x, a = -x, and {3 w, we con­ clude that the right-hand side of (13) is equal to s(O) = 0. D =

=

(

(12)

This functional equation is satisfied only by exponentials, so we can conclude that F(x) = exp(2bx) for some con­ stant b. Rewriting this condition in terms of s, we have exp( -bx)s(x) = -exp(bx)s( -x), which is what we are try­ ing to show. To show (12), we substitute in ( 1 1) and manipulate to form the equivalent equation 0

Now make the substitutions Xn - 1 = Xn - 1 + an- 1 - an and an - 1 = an and rearrange to obtain n-2 n -2 s(in- 1 + Un-1 - aj) X s I Xj + X- n- 1 = I s(xj) SC an- 1 aJ) j=l j=l

(11)

- -­

Proof of Lemma 8

Taking a derivative with respect to A in (5), using the fact that s is odd, and setting A = - a, B = a and {3 = - a, we obtain

s ' (O)s(2a) = 2s(a)s ' ( a). We recognize this equation as the n - 1 case of (6), which is true by the induction hypothesis. D The proof of Theorem 5 depends on two lemmas. Lemma 7 If a meromorphic function s(x) satisfies (5), then there exists a complex constant b such that exp( - bx)s(x) is an odd function.

Thus, s' (0) i= 0, and because of the invariance with respect to multiplication by constants, we can assume s ' (O) = 1. We have the system

{

s'(O) s(2a)

Since s(O) 0, we know that s is analytic around zero. We can write s(z) = kk=O akzZk + 1 and use the previous con­ ditions to obtain a recurrence for the sequence an, =

Lemma 8 An odd meromorphicfunction s(x) satisfies (5)

if and only if s(x) = ax

or

for some complex constants a 68

THE MATHEMATICAL INTELLIGENCER

s(x) i=

=

a sin(cx)

0 and c

i=

0.

{

a0 = 1 22n + 1 an = (2n

+

2)

k�=O an-kak.

(14)

A U T H O R S

MARTIN MOHLENKAMP

LUCAS MONZ6N

Department of MathematiCs

Department of Applied Mathematics

Ohio University

University of Colorado Boulder, CO 80309-0526

Athens, OH 45701 USA

USA

e-mail: [email protected]

e-mail: [email protected]

received his Ph.D. in 1 997 from Yale Univer­

Lucas Monz6n, following his undergraduate degree from the Uni­

sity. He spent a semester at MSRI in Berkeley and several years

versity of Buenos Aires, obtained his Ph.D. from Yale University.

Martin Mohlenkamp

at the U niversity of Colorado

in Boulder before moving to Ohio Uni­

Besides computational harmonic analysis, he enjoys theater, po­ etry, and the visual arts. His partner is Mariana lurcovich, an in­

versity. He works mainly in numerical analysis.

ternational consultant in public health.

The value of an for n > 1 is uniquely determined by the value of a1 , which is arbitrary. Setting A = 6a 1 we claim An (15) an = (2n + 1)! · When A 0 we have s(x) = x and when A =!= 0 we have s(x) sin(Ax) and the Lemma follows. We prove the claim by gen­ eralized induction on the variable n. Thus we assume (15) for 0 ::::; n ::::; N - 1, and show it for n = N. Using (14) with n = N, =

22N+ l aN

=

=

(2N + 2)

N

L

k �O

aN-kak

and thus

+-

N- 1

(

2N + 2

)

k2:.l 2k + 1 ' and the result follows because 2f� o (::�) = 22N+ l . aN = (2N + 1)! 22N l

2(2N + 2)

[ 1 ] Richard Bellman. Adaptive Control Processes: A Guided Tour.

Princeton University Press, Princeton, New Jersey, 1 96 1 . [2] Gregory Beylkin and Martin J. Mohlenkamp. Numerical operator cal­

culus in higher dimensions. Proc. Nat!. Acad. Sci. USA , 99(1 6): 1 0246-1 0251 , August 2002. University of Colorado, APPM preprint

August 2001 ;

http://www.pnas.org/cgi/content!abstract/

1 1 2329799v1 .

k�

1

REFERENCES

#476,

N- 1 AN-k = 2(2N + 2)aoaN + (2N + 2) (2(N - k) + 1)! (2k + 1)! ' AN

that the identity (6) is valid for s(x) = x, and so inspiring Theorem 5.

[3] Gaurav Bhatnagar. A short proof of an identity of Sylvester. Int. J. Math. Math. Sci. , 22(2):43 1 -435, 1 999. [4] F. Calogero. Remarkable matrices and trigonometric identities I I . Commun. Appl. Anal. , 3(2) :267-270, 1 999. [5] F. Calogero. Classical many-body problems amenable to exact treatments. Lecture Notes in Physics, monographs m66. Springer­

Verlag, 2001 . D

Acknowledgments

Thanks to Gregory Beylkin for leading us to separate sine in the first place. Thanks to Richard Askey for pointing out

[6] S. C. Milne. A q-analog of the Gauss summation theorem for hy­

pergeometric series in u (n). Adv. in Math. , 72( 1 ) : 59-- 1 3 1 , 1 988.

[7] Themistocles M . Rassias and Jaromir

S msa.

Finite sums decom­

positions in mathematical analysis. Pure and Applied Mathematics.

John Wiley & Sons Ltd. , Chichester, 1 995.

© 2005 Springer Science+ Business Media, Inc., Volume 27. Number 2 , 2005

69

GERALD L. ALEXANDERSON AND LEONARD F. KLOSINS

Mathemati c i ans and O d Books hose interested in book collecting, mathematical books in particular, have observed remarkable phenomenon over the past few years: the prices of rare mathematical m terials have no t been driven downward by the recent recession and the bursting of t. dot-com bubble. While prices for fine art definitely slumped and prices in other are'

of book collecting have declined or held steady, reflecting the state of the economy, prices of books in mathematics and science, particularly at the high end, have risen, even after extraordinary price increases in the 1980s and 1990s. In 1982 we wrote a short article on collecting rare math­ ematics books [1). Recently we learned that the article was scheduled to go into a mathematical anthology and, when we reread it, we found that it was very out of date, not only the prices but other aspects as well. We decided more could be said on the subject. Here we'll be talking mainly about first editions of mathematical classics, unless otherwise noted. Probably the most extraordinary price increases in mod­ em times appeared in the famous auction of the Haskell F. Norman collection at Christie's (New York) in 1998. Nor­ man had been a psychiatrist in San Francisco and collected rare books of high quality in science and medicine. His son, Jeremy Norman, the prominent San Francisco rare book dealer, upon his father's death, put the collection up for auc­ tion, and Christie's, aware that the collection would attract lots of attention, took some of the best pieces in the sale out on the road to show to collectors in Milan, Paris, Lon­ don, New York, Chicago, San Jose (!), and Tokyo, among other cities. The sale included many items that had been ac­ quired some years ago and are now seldom seen on the mar­ ket. There were unique items like a first edition of Euler's VoUstdndige Anleitung zur Algebra (1770) that Euler had presented to Lagrange, with an inscription-very desirable to a knowledgeable collector. In the 1980s one could find a copy of this book for about $1000; this very special copy in

70

THE MATHEMATICAL INTELLIGENCER © 2005 Springer Science + Business Media. Inc.

the Norman sale went for $19,550. The whole Norman c lection (632 lots) sold for almost $8,000,000. After the Norman sale the conventional wisdom arne collectors and dealers seemed to be that the high pri<

Fig. 1 . The cover of the Christie's catalogue for the Norman s; decorated with the title page of Newton's Principia of 1687.

m o Ujidn�ige

� n le tt u n g

,. 1 g''t t> o n

6 t a

� r n. 2to n�orb � u l er.

•

·•

��·�����������·�

srbnufr �19

e5 t. '}> c t u6 & t1 r g. 'll:cob. tlf 'll:hff•nfcll• rm

Cn' .l'i•9f·

mo.

Fig. 2. The cover page of Euler's Algebra, with the inscription to Lagrange. The writing is almost certainly that of an assistant since, by the time this was published, Euler was almost totally blind.

could have been a local maximum and resulted from two

cachet, and the best copies, just like great paintings, come

factors: (1) Norman was a well-known collector, and

with a long list of the names of previous, and preferably

(2) Christie's created hype surrounding the sale by arrang­

prominent, owners.

ing the world tour of the principal books. Provenance is

The surprise is that prices have held at the high level

important. Coming from a famous collection gives a book

and are actually increasing. If prices were driven up by the

© 2005 Springer Science+ Business Media, Inc., Volume 27. Number 2, 2005

71

Norman sale and by a lot of dot-com millionaires with an interest in mathematics and science, why have the prices held after many of those dot-com fortunes vanished? We don't have an answer. As we write this, the huge and important Macclesfield sale is going on at Sotheby's in London. This extensive li­ brary in the history of science was housed in Shirbum Cas­ tle near Oxford, and, according to the catalogue, encapsu­ lated "almost everything published on the astronomical and mathematical sciences up to about 1750: from Regiomon­ tanus and Peurbach to Copernicus and his contemporaries, and from there through Tycho Brahe, Giordano Bruno, Gilbert, Galileo and Hevelius to Newton, Leibnitz, Whiston and Euler. . . . There are many books famous in the annals of science, such as the copy of Copernicus's De revolu­ tionibus . . . which Owen Gingerich saw in situ and on a well-chosen day (as he relates in The Book Nobody Read, he went on a Thursday which was, according to Lady Mac­ clesfield, 'the only day we have a cook, and we can invite you to lunch') . . . . The genesis of this remarkable collec­ tion . . . lies in the biographies and interests of two men, . . . John Collins, the mathematician [and friend of Newton] whose collection was acquired by William Jones about twenty-five years after his death, and Jones himself, who bequeathed his library to the second Earl [of Macclesfield]."

The first Macclesfield sale of mathematics books took place June 10, 2004, and covered only the books by authors with names beginning with A-C. (A second sale in the se­ ries was to take place November 4, 2004.) The first 251 lots sold in June brought in almost $6.5 million, and that's for only three letters of the alphabet! Prices in the sale so far indicate that they are not only holding their own but in­ creasing. The Copernicus of 1543 went for $1.2 million. It was a nice copy, heavily annotated by the astronomer John Greaves. Still, for most people that's a lot of money to spend on a book (In 1964 Ernst Weil in London listed the 1543 Copernicus in his catalogue for $2,380. In the Honeyman sale of 1978, there were two copies-one sold for $35,000, the other for $ 19,000.) This book and the great works of Galileo, the first edition of Newton's Principia of 1687, the 1482 Ratdolt Euclid (the first printed Euclid, produced roughly 40 years after Gutenberg's invention of moveable type), and other such rarities now go for very high prices; few copies remain in private hands. As with any other items of interest to collectors, prices fluctuate over time, moving upward with overall inflation, and both upward and downward depending on shifting fashion. Fermat's 1670 edition of Bachet's Diophantus (this edition was supervised by Fermat's son and published shortly after Fermat's death) contains the printed version

O B S E R V A T I O D O M I N I P E T R I D E F E R M A T. J'/,.,.

••l<m ;, "••s ••h•s , •111 f"•Jr•t•fuJr•u• "' ""'' '!••Jr4111J11Urlfl•l s•••r•lltiT """•• "' ;,,,;,.,. .,,,. , •• J,.,,.,,. p•�
C6 J,.,

v

tk•••J1r•ll•"'"'

Q.Y A: S T l O

flu

•l•/·

l X.

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R

• •' •

+

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•

1!f!l¥W Yo,icacibus r 6 . & lie

•

1

N.'l czit

\U' A: S T ! O

I X.

:!T

n ,. •>• «, ,; -,.o-. • Mi- .. �:� . ,. .....,..,... (( H.rl'ittr,w �· ·"-�· f.., A w.,.,..., ,; T'l�)..,, " :, .n ,... .._, � �· .... r.. '"' � '"'"'�i ·on;�+!
E

Fig. 3. The page from the 1670 edition of Sachet's Diophantus, with the famous remark by Fermat on having a miraculous proof but inade­ quate space to write it in the margin (under "Observatio Domini Petri de Fermat").

72

THE MATHEMATICAL INTELLIGENCER

of what Fermat supposedly wrote in the margin about hav­ ing a proof but not having enough room in the margin to write it down. Not long after Andrew Wiles's 1995 proof of "Fermat's Last Theorem," a copy of the Diophantus ap­ peared in a London dealer's catalogue for $9,075. Then in the Norman sale in 1998 it sold for $25,300 (after an esti­ mate of $2,000-$3,000 by the auction house!), and later a New York dealer had a copy listed at $45,000. One of the authors of this article bought a copy in San Francisco in 1965 for $484. At a June 2004 auction in Paris a copy sold for $31 ,020. The increase is far beyond the rate of inflation. But, of course, not every book enjoys the stimulus of a widely publicized proof after 350 years, one that made "Nova" and the front page of the New York Times. Perhaps another book belongs in the category above: Gauss's Disquisitiones Arithmeticae of 1801, a later book than the Euclid and the Newton but one of considerable mathematical interest and rarity. One of us bought a copy in 1967 from the dean of West Coast book dealers, Warren Howell in San Francisco, for $ 1,350. In 2003 a special copy that had been owned by C. J. Doppler was listed by the London dealer, Bernard Quaritch, for $60,800, but a plain, ordinary copy sold at a recent Christie's sale in Paris for $49,350. It is perhaps not surprising that this, probably the greatest of Gauss's works, should have this amazing in­ crease in selling price, but a copy of his Theoria motus cor­ porum coelestium, a less well-known book, was purchased from Howell for $190 in 1963 and a London dealer listed a copy in 2001 for $6,800. So even lesser works by great math­ ematicians have shown a significant increase. What can explain prices like this? After all, the text of many of these works is available in modem editions or on the Internet. The splendid 1847 edition of Oliver Byrne's Euclid in brilliant colors printed by woodblock and still available on the market for something like $7,500 (one of us owns a copy bought in a London bookshop in 1960 for less than $ 13) can now be seen on the Internet at http://www.sunsite.ubc.ca/DigitalMathArchive/Euclidlbym e.html. It's a beautiful book either way, but looking at pages on the screen is not quite the same experience as holding a copy in your hand and flipping through the pages. Byrne, incidentally, seems to have had time on his hands; he is identified on the title page as "Surveyor of Her Majesty's Settlements in the Falkland Islands." Here are a few examples of the increases over the years for some important books in mathematics: •

•

•

•

the editio princeps of the works of Archimedes of 1544: Honeyman sale of 1978, $ 1,280; W. P. Watson in London, $ 1 12,000 in 2002; Jacques Bernoulli's Ars Conjectandi: Honeyman sale, $900 in 1978; Norman sale, $ 12,650 in 1998; W. P. Wat­ son, $21,000 in 2004; Jean Bernoulli's Opera Omnia: Quaritch, $75 in 1962; the same dealer, $7,200 in 1997; Euler's Introductio in Analysin Injinitorum: Zeitlin & Ver Brugge in Los Angeles, $75 in 1961; Jonathan Hill in New York, $ 13,500 in 2000;

•

•

•

Galileo's Dialogo of 1632: E. Weil, $294 in 1958; Zeitlin & Ver Brugge, $5,500 in 1978; Norman sale, $27,600 in 1998; Christie's in Paris, $49,761 in 2004; Lagrange's Mechanique Analitique: Joseph Rubinstein in Berkeley, $250 in 1973; Norman sale, $ 13,800 in 1998; Watson, $20,000 in 2002; Newton's 1687 Principia: E. Weil, $490 in 1948; Zeitlin & Ver Brugge, $15,000 in 1978; Norman sale, $321,500 in 1998; W. P. Watson, $356,000 in 2004.

Even the third edition of the Newton ( 1 726), of interest because it was the last edition published in Newton's life­ time, was recently listed at $35,000 by a Los Angeles dealer. One of us paid $236 for a copy in 1969. Why does a person want to collect old and rare mathe­ matics books? Why indeed? They are often in Latin, a language not understood by many these days, and they are often filled with notation unfamiliar to modem mathemati­ cians. But there is a certain thrill for someone who loves mathematics to pull a volume off the shelf and see, for ex­ ample, the first appearance in print of Euler's formula for polyhedra in "Elementa doctrinre solidorum," in the Novi Commentarii Academiae Scientiarum Imperialis Petro­ politanae (1758), 109-60, or the first appearance of the dif­ ferential calculus as most of us know it, Leibniz's "Nova methodus pro maximis et minimis," in the Acta Eruditorum of Leipzig for 1684. Sometimes one gets lucky and comes across something totally unexpected, like a short manuscript we bought from a New York dealer that was a report signed by Laplace and Legendre and sent to the Academie des Sci­ ences, Paris, saying in French, but here translated: "We have examined at the request of the Academy the description of a vessel capable of traveling under water at a determined depth for whatever amount of time, and diving to any depth for a certain amount of time. The means that the author pro­ posed did not seem to us to be practicable or worthy of the attention of the Academy." So much for the idea of a sub­ marine. On another occasion, a letter turned up, signed by Gauss. There was nothing so unusual about that except that it was in English. At first glance that might appear suspi­ cious, but perhaps less so when one realizes that many of Gauss's descendants ended up in the United States, some re­ putedly because they couldn't get along with him. Potential collectors should be concerned about the pos­ sibility of buying a fake. Obviously this is a problem in the art world, where skilled and successful forgers often have fooled even the museum experts. Autographs and manu­ scripts are also susceptible to forgery. Books have been rel­ atively safe, at least until the prices went so high, because the amount of work involved in typesetting with antique typefaces and printing on paper of the correct period, would be prohibitively time-consuming and expensive. Still, a collector has to be careful. In 1976 one of us bought a copy of Gauss's doctoral dissertation, published in Helm­ stadt in 1799 and containing an important result, the first generally accepted proof of the Fundamental Theorem of Algebra. The book was purchased for $ 1,000 from a dealer of impeccable reputation in San Francisco, acquired from

© 2005 Springer Sc1ence+ Business Media, Inc., Volume 27, Number 2, 2005

73

A P X I M H � OY � T 0'1

:l Y P /1. K OV % I or,

A R C H i d E D I S PHI L 0

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(Sorum C:ualogum ucrf� �a rfont

E V T O C II l �

..A

E. 0 $ D B ,\\

C ..A L O N I T<.Alt

A. 1\ C U I

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.

brosCommcnt1n":�,ir<m Gr.tcc�uon ', nunqu:un �rca =fa.

L 1.

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B c.A $ I L l!. c.A1!,

1tM1111ts Hr:rullg/NS cxnulifol/, An.

1-1 .0 X L 1 J l 1 .

Fig. 4. The title page of the first printed collection of the works of Archimedes, in Greek and Latin (1 544) .

a collection ostensibly brought by a refugee family from Europe when fleeing from the Nazis in the 1930s. It was only some years later that it was discovered that this col­ lection had been stolen, book by book, from the John Crerar Library in Chicago by a person who was to become well known as one of the cleverest book thieves of the cen­ tury. So the Gauss dissertation-really quite rare, the pre­ vious sale of a copy having taken place in 1928-was "re­ quested" by the U.S. Attorney in Chicago as evidence in the trial of the alleged thief. There it stayed for several years, during which time the remainder of the Crerar Library was acquired by the University of Chicago. Finally, at the end of the trial (the suspect was convicted), a letter came from

74

THE MATHEMATICAL INTELLIGENCER

the University asking whether they should return the $1,000 or the book The choice was clear: the book There was no copy of this book in the Honeyman or Norman sales, and it's unlikely there will be one in the Macclesfield sale, for the book came along about fifty years late for that collec­ tion. It is certainly not a common book, so it is difficult to guess what copies would sell for today. But $1,000 did seem like a bargain. A New York dealer subsequently listed a copy in his catalogue at $39,500 in 1997. Occasionally one finds interesting copies of books that have been owned by other mathematicians. One or the other of us has acquired over time Cayley's annotated copy of Rie­ mann's Gesammelte Mathematische Werke; Dedekind's own

M.ENSIS OCTOBRIS

.A,jM DC LXXXIV.

467

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w....:- .:.

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yy QJ!_oadSif-.o hoc probt not&ndum, cum in calculo pro littta fubllituiturfimphcittr <JUI dilftrcntialis, ftrvari q uldcm eadem C.gnJ, & pro + z fcribi + dz, pro -z fc:ribi - d z , u t cx addition< & fubrra­ tuone paulo ante pofito appartt ; fed quando ad extgcfin V>lorum vcnitur, feu cum confidcra.tur ipfius z ctlatio ad z, rune :appartrrj an nlor ipGusd z fit quantitas al!irmativa, an nihilo minor feu ncgativo: quodpollcnuscomlit, tunc t>ngcns Z E ducitur a pvndo Z non vcr­ fils A, fed in partes conrr.uias feu infr> X,id tl\ tunc cum ipfzordinm• • deereNnnJ

t �rAJ..� .

Fig. 6. A report by Laplace and Legendre recommending against fur­ ther investigations by the Academie des Sciences on an idea for a submarine.

Fig. 5. The first page of Leibniz's article of 1 684 on differential cal­ culus. Note the addition, subtraction, product, and quotient rules for differentials in the second paragraph.

copies of the works of Dirichlet; Darboux's copy of Gauss's Werke (only nine volumes, because the remaining volumes appeared after Darboux's death, after which he stopped col­ lecting-to borrow a phrase from H. W. Lenstra, Jr.); Hardy's copy of Waring's Meditationes Algebraicae (the volume with the first appearance of Wilson's theorem, and of special in­ terest because of the contributions of Hardy and Littlewood to the solution of Waring's problem); a manuscript by Fatio de Duillier on alchemy, with "corrections" in longhand by his close friend Isaac Newton; Charles Babbage's copy (pre­ sented to him by the translator) of the four volumes of Nathaniel Bowditch's translation into English of Laplace's Traite de Mecanique Celeste, where, because Laplace's work was so unclear, Bowditch would give a few lines of Laplace with the rest of the page devoted to notes in small type to explain what Laplace had just said; and offprints of George Green's that he had presented to Jacobi. These last offprints, bought from the Honeyman collection, were valuable to Mary Carmell in her biography of Green, in showing that he had been more in touch with continental mathematicians than had been previously thought. These things are out there to be collected. A person just has to read auction house and dealers' catalogues or Web sites assiduously. Manuscript materials are, in general,

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© 2005 Springer Science+- Business Media. Inc .. Volume 27, Number 2. 2005

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harder to find than good books. When asked about the pos­ sibility of getting a couple of manuscripts from the Stanitz collection when that came up for auction at Sotheby's in 1984, the dealer who would be doing the bidding warned that there was little chance of getting them. "The institutions will be out in force bidding for those"-one a manuscript of a paper on elliptic functions by Jacobi, the other a collection of manuscripts by Cauchy written when he was a student. We were assured that it would be much easier to get books in the sale. As it turns out, we got no books but got both of the manuscripts, and at a good price. Not every manuscript attracts the sort of attention the Archimedes palimpsest did in 1998 at Christie's. It sold for $2 million. That price was predictable, but generally at auctions a person just never knows how things will turn out. It can be exciting. The discoverer of Vitamin E, Herbert McLean Evans at the University of California, Berkeley, led the way for sci­ ence collectors in 1934 when he issued a small booklet list­ ing the books that he thought were the "epochal achieve­ ments in the history of science." He built seven great science collections himself and is generally accepted as the pioneer in science collecting, which then received another boost when the Grolier Club (an organization of book col­ lectors in New York) published a handsome volume in 1964 called One Hundred Books Famous in Science, by Harri­ son Horblit, another great science collector. (Even this modem book, that just lists books, had gone up in price from $50 to $750 when it appeared in a recent catalogue of a Los Angeles dealer.)

76

THE MATHEMATICAL INTELLIGENCER

Some who may know rather little about mathematics can find for themselves a little bit of immortality by assembling a great collection in the history of mathematics or science. Who would remember the name Robert Honeyman today had he not accumulated an enormous book collection? Hon­ eyman was an engineer, a graduate of Lehigh University, to which he gave his great literature library and his extraordi­ nary Darwin collection in the 1950s. His Western Americana collection went to the Bancroft Library at the University of California, Berkeley. For some reason his science collection went on the auction block at Sotheby's Parke Bernet in Lon­ don in 1978, and, for some of us, it's a good thing it did: we enjoy seeing on our bookshelves the telltale bright red mo­ rocco slipcases that Honeyman commissioned for his books. This auction may have stimulated the first major increase in prices for mathematics and science books, for it put on the market many books previously not of great interest to a large group of collectors. There were some young dealers who were looking for stock and who moved into the science area: Roger Gaskell, Jonathan Hill, Michael Phelps, W. P. Watson, and Jeff Weber. Prior to that, the principal dealers in science were Bernard Quaritch, Dawson's of Pall Mall, Ernst Well, and R. D. Gurney in London; Alain Brieux, Paris; Rosenkilde & Bag­ ger, Copenhagen; E. Offenbacher, H. P. Kraus, and Lathrop C. Harper in New York; Jacob Zeitlin, the Rootenbergs, and Harry Levinson in Los Angeles; and Warren R. Howell (John Howell-Books) and Jeremy Norman in San Francisco. Only five of the firms in this second list survive: Quaritch, Daw-

son's in the form of Pickering & Chatto, Brieux, B. & L. Rootenberg, and Jeremy Norman. Once when scouring Lon­ don for a rare book in mathematics and having no luck, we were told that we really should be looking in Los Angeles. The legendary Jake Zeitlin there was a pioneer in making scientific books attractive to collectors. He, with E. T. Bell, advised William Marshall Bullitt, a prominent Louisville, Ken­ tucky, lawyer and U. S. Solicitor General under William Howard Taft, on building an extraordinary collection of rare mathematics books now in the library of the University of Louisville. Bullitt was prompted to build a collection by a parlor game suggested by his friend, G. H. Hardy. He was a good friend of mathematics and was instrumental, along with John von Neumann, in arranging for Charles Loewner's first appointment in the United States at Louisville. Bullitt may have heard the answer to our question of why prices of math­ ematics books continue to rise: the most renowned of Amer­ ican book dealers, A. S. W. Rosenbach of Philadelphia, ad­ vised him that "science and mathematics books [are] likely to prove more valuable as investments than books on Napoleon or Henry Clay." For more information on this fas­ cinating collector see [2]. Like Warren Howell, Jake Zeitlin was a scholar. A visit to either of these booksellers was like going to a private club. It involved a move into their private offices, where the good things were kept in safes, and one could settle down for an hour of good conversation about books, han­ dle some important pieces, and often meet science histo­ rians of note. We remember meeting Stillman Drake for the first time in Zeitlin's office. Haskell Norman writes glow­ ingly in his introduction to the catalogue of his collection about his Saturday afternoon visits to Warren Howell's of­ fice where he would often find Herbert Evans. The group of science dealers, experts, and collectors was small in those days and everyone knew each other. Someone wanting to see the great books and manu­ scripts in mathematics, but unable to afford them, can view them at some of the great libraries of the world. For a de­ scription of some of the European libraries with important mathematical holdings-the British Library, the Vatican Li­ brary, the Bodleian at Oxford, and a great California col­ lection, the Huntington near Los Angeles, see [3]. Other im­ portant collections are at the Bundy Library of the Dibner Institute at MIT, the Institut Mittag-Leffler near Stockholm, the Linda Hall Library in Kansas City, MO, the Thomas Fisher Library at the University of Toronto, the Lownes Col­ lection at Brown University, the Bullitt collection men­ tioned above, the DeGolyer Collection at the University of Oklahoma, and the David Eugene Smith Collection at Co­ lumbia University, among others. These collections were rarely assembled book by book by the institutions; they were usually gifts from generous patrons. How does one get started in this highly entertaining and rewarding (but often expensive) hobby? It would help to have been born many years ago and to have started col­ lecting in the 1950s at the latest. But that's hard to arrange. Given the market these days, most have to give up any as­ pirations to acquire a 1543 Copernicus or a major Newton

or even a major Gauss. It's probably best to pick a subdis­ cipline, perhaps one's own specialty, and look for the best available in the field that one can afford. Then there are the books slightly out of the mainstream: books on mathemat­ ics and music (there are affordable works by Descartes, Euler, d'Alembert), or books on art and perspective, say. These are really quite cheap, as long as a person is not try­ ing for something like Pacioli's great work with the Leonardo drawings of the polyhedra. Some things get to be expensive not for their mathematical content but for their association with someone well-known, in this case, Leonardo. Perhaps mathematics and poetry could be pur­ sued. D. E. Smith wrote on this subject. The French poet Paul Valery was more or less a mathematician. Sylvester wrote poetry, rather bad poetry, actually. Cajori says in his A History of Mathematics (Macmillan, 1919) that: "At the reading, at the Peabody Institute in Baltimore, of his Ros­ alind poem, consisting of about 400 lines all rhyming with 'Rosalind,' he [Sylvester] first read all his explanatory foot­ notes, so as not to interrupt the poem; these took one hour and a-half. Then he read the poem itself to the remnant of his audience." Warren Howell in 1977 had a copy of Sylvester's Spring's Debut, a Town ldylVin Two Centuries of Continuous Verse, printed "for Private Circulation Only" in 1880. It too has each line ending in the same sound. This

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Fig. 9. The title page of Euler's treatise on music of 1 739.

© 2005 Springer Science+Business Media, Inc., Volume 27. Number 2. 2005

77

slim volume sold for $80, the price of a good dinner now and a very good dinner then. That the price was not higher may be surprising, since it is a presentation copy with an inscription by Sylvester to "Professor Child" and with hand­ written corrections by Sylvester. But it perhaps shares the fate of Hamilton's Lectures on Quaternions, very com­ monly inscribed by Hamilton. They didn't sell, so the au­ thor had to give them away-ours was presented to Sir George Biddell Airy, onetime English astronomer royal. If a collector's resources are limited (and whose are not?) another way of building an interesting collection is to acquire reprints. They're often presented by the authors with interesting inscriptions. For example, Mark Kac, knowing of P6lya's work on the problem, "can you hear the shape of a drum?", sent P6lya a reprint of his own paper on the subject and inscribed it "From another drummer, Mark Kac." A person can pick up things like this when col­ leagues retire and vacate their offices. Dealers who carry such things are Malcolm Kottler at Scientia, Arlington, MA; Ray Giordano at Antiquarian Scientist, Southampton, MA; Elgen Books, Rockville Centre, NY; and Jeff Weber, Los Angeles. Weber's catalogues are filled with interesting and modestly priced books in mathematics. How does a person find such things, besides reading cat­ alogues and Web sites? One haunts bookstores, and there are still lots of wonderful bookstores around. Fewer seem to be able to afford street-level shops anymore, so one may have to climb a few stairs. But they're there, and the deal­ ers are usually eager to educate the interested novice. A Cecil Court bookseller in London once gave very good ad-

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vice that we've both tried to follow: "It's never the things you buy that you regret, only the things you don't buy!" A few years back, a beginning collector of science wrote in the magazine Biblio that he visited the Quaritch shop in London and asked about rare books in science, adding rather apologetically that he realized, of course, that he would never be able to afford a Copernicus or a Galileo Di­ alogo. The attendant asked gently whether he would like to look at their copies. The days of being able to walk into a shop and browse through a first edition of Copernicus are probably over. Even a shop like Quaritch, at those Olympian heights, probably would not keep a copy in stock. With the recent Owen Gingerich census of copies of Coper­ nicus, we are aware of the location of every known copy of both the first and second editions. New York and other major cities like Leipzig, Amster­ dam, Paris, and Milan, have book fairs. California has a huge annual fair, held in alternate years in San Francisco and Los Angeles. These are good ways of visiting dealers from all over the world. The last fair in San Francisco had almost 250 dealers present. And they often bring their best things to fairs, particularly if they know there are interested collectors for specific material in the area. Auctions are tricky. In person one can get carried away bidding and end up paying more than is prudent. Absentee bidders can submit a bid to the auction house directly or have a dealer bid for them for a small fee. The dealer or the dealer's agent can check in advance on condition, col­ lation, and other details to fill in gaps in the catalogue de­ scription. At auctions, though, one must always keep in

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Fig. 1 0. Sylvester's Spring's Debut, both the title page and a page of verse (corrected by Sylvester).

78

THE MATHEMATICAL INTELLIGENCER

mind that the buyer pays the house an additional fee, the premium on hammer price. So the outlay can end up con­ siderably higher than the bid. On the other hand one can get bargains if one is shrewd and informed. And you know that if you're bidding against a dealer, that person is going to sell the book for a much higher price in the shop. Auc­ tion houses to check out are Sotheby's in London, New York, and Paris (sothebys.com); Christie's in those same cities (christies.com); Swann's in New York (swanngal­ leries.com); Bonham's and Buttertield's in San Francisco and London (buttertields.com); the Dorotheum in Vienna (dorotheum.com); Tajan in Paris (tajan.com); Reiss und Sohn in Konigstein, Germany (reiss-sohn.de ); Zisska & Kist­ ner in Munich (zisska.de); and Pacific Book Auctions in San Francisco (pbagalleries.com). Can one make a lot of money investing in rare books? Probably not. Like the stock market, prices fluctuate; and just because mathematics books have been "hot" over the past few decades, there's no guarantee they'll stay that way. Over the long term, art has largely kept up with Standard & Poor's, but art, books, stamps, and other fields of col­ lecting pose special problems. When a person wants to dis­ pense with a collection, it's not so easy. An auction house will sell the books but will charge the seller (as well as the buyers) a percentage. And dealers in rare books, as a rule of thumb, mark books up by 10(}0;\J over what they paid for them. So don't expect to get more than 50% of the current retail price. Most dealers do not have the wherewithal to buy a whole collection, so they will often agree to take

something on consignment and then charge the seller a cer­ tain percentage of the sale price. By contrast, selling stocks or bonds is relatively easy. But as an expert pointed out re­ cently in the New York Times, when stock prices collapse, the investor is left with a stack of worthless paper, whereas the wine collector can at least drink the wine and a book collector can ef\ioy the books. So collecting is not a way to get rich. A collector has to like the books themselves, and the discoveries made in col­ lecting them. Further, if you have a good collection, rare book librarians at major institutions will be very nice to you if they find out what you have. If the collection is good enough, you may get your name attached to a library col­ lection. That's immortality of a sort. Here's to good hunting! Acknowledgement

The authors wish to express their gratitude to Ellen Hef­ felfinger, librarian and bibliographer of the American In­ stitute of Mathematics, for her extraordinarily helpful and perceptive comments on an earlier draft of this article. REFERENCES

[ 1 ) G. L. Alexanderson and L. F. Klosi ns ki, On the value of mathemat­ ics (books) . Mathematics Magazine 55 (1 982), pp. 98-1 03.

[2) R. M. Davitt, William Marshall Bullitt and his amazing mathematical collection, Mathematical lntelligencer 1 1 :4 (1 989), pp. 26-33.

[3) S. I. B. Gray, A mathematics treasure in California , Mathematical ln­ telligencer 20:2 (1 998), pp. 4 1 -46.

A U T H O R S

GERALD L. ALEXANDERSON

LEONARD F. KLOSINSKI

Department of Mathematics & Computer Science

Department of Mathematics & Computer Science

Santa Clara University

Santa Clara, University

Santa Clara, CA 95053-0290

Santa Clara, CA 95053-0290

USA

USA

e-mail: [email protected]

e-mail: [email protected]

Gerald L. Alexanderson has been collecting rare mathematics books, Lewis Carroll works, and press books since 1 961 . He was

Leonard F. Klosinski collects sculpture, drawings, maps, and paint­ ings, along with books if they're old enough, preferably incunab­

chair of his department at Santa Clara University for 35 years and

ula. He has taught mathematics and computer science at Santa

has been a member of the faculty there since 1 958. He has served

Clara University since 1 964 and has directed the William Lowell

as Secretary and President of the Mathematical Association of

Putnam Mathematical Competition since 1 975, longer than any

America and most recently received the MAA's Haimo Award for

previous director. In 2001 he won the MAA's Haimo Award for Dis­

teaching and the Yu-Gin Gung and Dr. Charles Y. Hu Award for

tinguished College or University Teaching of Mathematics. He has

Distinguished Service in 2005. He does not like to travel.

traveled to all seven continents.

© 2005 Springer Science +Business Media. Inc . Volume 27. Number 2. 2005

79

i;l§iil§ldJ

Osmo Peko nen , Editor

I

Using the Mathematics Literature edited by Kristine

K.

Fowler

NEW YORK, MARCEL DEKKER, 2004. 389 PP., US $165, ISBN 0-8247-5035-7

REVIEWED BY J. PARKER LADWIG AND E. BRUCE WILLIAMS

Feel like writing a review for The Mathematical Intelligencer? You are welcome to submit an unsolicited review of a book of your choice; or, if you would welcome being assigned a book to review, please write us, telling us your expertise and your predilections.

Column Editor: Osmo Pekonen, Ago ra Center, University of Jyvaskyla, Jyvaskyla, 40351 Finland e-mail: [email protected]

80

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T

his book deals with the basic tools and skills needed in the mathematical laboratory." It is written not only for librarians, but more im­ portantly for undergraduates doing mathematical research, for graduate students, and for faculty exploring new areas. This is the 66th volume of Dekker's Books in Library and Information Sci­ ence. It is divided into two parts, "Tools and Strategies," and "Recom­ mended Reading by Subject," and it has two indexes: author and subject. The first place a seasoned mathematician is likely to browse is Part II, "Recom­ mended Readings by Subject," just to check the list of resources for his or her subject. The subjects included are:

enough information to locate and even order the resource, and often a one- or two-sentence description. The empha­ sis is on books, but key journals and on-line resources are also indicated. The chapters generally contain an introduction, a section on general sources, and then sections for the ma­ jor subdivisions of a field. Sections of­ ten refer to general texts, and then further refine the subdivision. For ex­ ample, the chapter on topology by Alan Hatcher contains a section on intro­ ductory books, then sections on al­ gebraic topology, manifold theory, low-dimensional topology, history, and other resources. The section on mani­ fold theory, for example, discusses differential topology, piecewise-linear topology, topological manifolds, and surgery theory. Here are a few additional sugges­ tions for the chapter on topology:

History of mathematics Number theory Combinatorics Abstract algebra Algebraic and differential geometry Real and complex analysis Differential equations Topology Probability theory and stochastic processes Numerical analysis Mathematical biology Mathematics education NOTE: We understand from the editor that contributors for other subjects (like mathematical logic) would have been welcome.

II. Algebraic Topology A. Introduction I. Madsen and J. Tornehave. From calculus to cohomol­ ogy: de Rham cohomology and characteristic classes. Cambridge: Cambridge Uni­ versity Press, 1997. Chern classes are constructed via curvature forms and via pure algebraic topology. This is done after the reader is given the necessary background in topology and de Rahm the­ ory. III. Manifold Theory A. Differential Topology V. Guillemin and A. Pollack. Differential topology. Engle­ wood Cliffs, N.J.: Prentice­ Hall, Inc. , 1974. This is a very readable book. The main topics are trans­ versality, intersection the­ ory, and differential forms.

Each subject is given a chapter writ­ ten by a mathematician and/or a math­ ematics librarian. Each entry gives

T. Brocker and K. Janich. Introduction to differential topology. Translated from the

THE MATHEMATICAL INTELLIGENCER © 2005 Springer Sc1ence + Business Media, Inc.

German by C. Thomas and M. Thomas. Cambridge: Cam­ bridge University Press, 1982 This is an excellent, concise introduction to basic mater­ ial in algebraic topology. Dundas, Bjorn, Differential topology, http://www.mi.uib. no/�dundas/ This is a very reader-friendly expansion of the Brocker­ Janich book. D. Surgery theory W. Luck. A basic introduc­ tion to surgery theory, http:// wwwmath. uni-muenster. de/index.en.html The main topics are s-cobor­ dism theorem, Whitehead tor­ sion, surgery exact sequence, the Farrell-Jones cor\iecture, and computations of K- and L-groups.

one just beginning research in mathe­ matics. Because this work is primarily arranged by discipline, it offers a dif­ ferent perspective than Nancy D. An­ derson and Lois M. Pausch, editors, A Guide to Library Service in Mathe­ matics (Greenwich, CT: JAI Press, Inc.), 1993. One might also consult the $65 book by Martha Tucker and Nancy An­ derson, Guide to Information Sources in Mathematics and Statistics (West­ port, CT: Libraries Unlimited), 2004. The book's major drawback is its price-42¢ per page vs. 19¢ for Tucker and Anderson. However, it is still an important addition to your library's collection, a relevant resource for un­ dergraduate and graduate student ad­ visors, and perhaps a gift for the new librarian who will be working with your department. Mathematics Library University of Notre Dame

Part I, "Tools and Strategies," con­ tains three chapters. The first is a very interesting one on the culture of math­ ematics. For undergraduates who are thinking about advanced study (or for friends and family who are puzzled about what a mathematician does), this is a concise and even elegant overview. "Tools" continues with chapters on "Finding Mathematics Information" and on "Searching the Research Liter­ ature." Both chapters are written by experienced mathematics librarians and answer questions asked by those learning and studying mathematics. "Finding Mathematics Information" contains sixteen sections (too many to enumerate)-two of our favorites are "Locating Definitions and Basic Expla­ nations" and "Finding or Verifying Quo­ tations and Anecdotes. " As with Part II, each entry contains complete biblio­ graphic information with a one- or two­ sentence abstract. "Searching the Research Literature" contains five sections: introduction, strategies, finding journal articles us­ ing indexes, finding papers on the Web, and obtaining the resources found. This chapter is more of a discussion than a list of resources, but like the chapter on "Finding Mathematics In­ formation" would be helpful for some-

Notre Dame, IN 46556-5641 USA e-mail: ladwig. 1 @nd.edu Mathematics Department University of Notre Dame Notre Dame, IN 46556-5641 USA e-mail: [email protected]

Mathematics in Nature. Modeling Patterns in the Natural World by John A. Adam PRINCETON UNIVERSITY PRESS, 2004. 354 PP., ISBN 0-691 - 1 1 429-3, US $39.50

REVIEWED BY THOMAS GARRITY

O

nce in the early evening in high school, on the flat desolate plains of the Texas panhandle, I was on the back of a horse with a girl that I was trying to impress. As we rode along, she pointed out to me the beauty of the sunset, which was spectacular. It so happened that earlier that day I had read about the underlying mathematics behind sunsets. Of course I had to tell

her my newly acquired knowledge, in part to impress my riding partner but also to share my newfound apprecia­ tion and awe of the sunset. Unfortu­ nately, she looked at me not with any sort of admiration but with contemp­ tuous disdain. I realized that I had ru­ ined any chance of a romantic moment. Still, even in high school, I could not fathom how understanding the under­ lying science and mathematics of a physical phenomenon could at all de­ tract from its immediate aesthetic, if not sensual, appeal. Surely insight into causes could only add to the delight of our experience. While I suspect that that girl of long ago would still find my viewpoint a bit ridiculous, I am confi­ dent that John Adam would whole­ heartedly agree with me. In fact, the point of his Mathematics in Nature: Modeling Patterns in the Natural World is that the underlying math can add immeasurably to our delight in merely walking about and looking at the world around us. In the first chap­ ter, in talking about this very issue, Adam states, "I have always found, for example, that my appreciation for a rainbow is greatly enhanced by my un­ derstanding of the mathematics and physics that undergird it. . . . " More generally, Adam states in the first chapter, "The idea for this book was driven by a fascination on my part for the way in which so many of the beautiful phenomena observable in the natural realm around us can be de­ scribed in mathematical terms. . . . " He describes how much he er\ioys observ­ ing the world as he walks to work each day, and he wants to share this joy. Thus his goal is to foster in the reader the skills needed to model the phe­ nomena in nature that we encounter in our day-to-day lives. He wants to ex­ plain how a few simple principles can be used to understand many seemingly disparate physical phenomena. It is not surprising that Adam's ini­ tial stab at modeling involves Fermi problems. This is another name for tra­ ditional "back of the envelope" calcu­ lations. An example of a Fermi prob­ lem would be to approximate how many Snickers bars are eaten each year in, say, Chicago. The goal is to use a few pieces of numerical information

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(in the above you would need to have a rough guess as to the population of Chicago). You win with these problems if you can get an answer that is accu­ rate up to an order of magnitude. As Adam states, this chapter owes a lot to Paulos's Innumeracy and Harte's Con­ sider a Spherical Cow. In a similar vein, in the next chapter he turns to the basics of scaling, or in other words, dimensional analysis. Here he discusses how to make quick estimates of sizes of area, volumes, etc. For example, this chapter would pro­ vide the reader with the ability to see that a mouse will shiver in the cold far more quickly than an elephant. The rea­ soning is as follows. The amount of heat in an animal should be propor­ tional to its volume. The amount of heat that an animal loses, on the other hand, will be proportional to its surface area, as it is only through our surfaces that we can lose heat. Crudely, volume goes up by radius cubed while surface area goes up by radius squared. The mouse shivers because its ratio of surface area to volume is far larger than the corre­ sponding ratio for the pachyderm. (This also suggests why Harte titled his book Consider a Spherical Cow.) The rest of the book develops mod­ els for specific physical phenomena, from the meandering of rivers to the formation of hurricanes. You will be able to understand the clouds in the sky as well as the dunes at the beach. After spending some time on waves (both linear and nonlinear), you will be able to see how a bird flies. Adam closes with dispersion relations, em­ phasizing the now standard explana­ tion for how a leopard gets its spots. As can be imagined, there is a lot more that is discussed. In fact, with an ad­ mitted bit of exaggeration, the author claims, " . . . if you can see it outside, and a human didn't make it, it's proba­ bly described here!" There is the question of who is the intended audience. For example, sup­ pose I have a neighbor who readily ad­ mits to not liking math but who also asks me for a book that will show the importance of math in everyday life. I would recommend a book like Innu­ meracy by Paulos. Mathematics in Nature requires greater mathematical

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maturity. This is not quite what the au­ thor intends; as he states in the pref­ ace, "Anyone interested in the beauty of nature, regardless of mathematical background will also (I trust) ef\ioy much of this book. Although the math­ ematical level ranges across a broad swathe, from 'applied arithmetic' to partial-differential equations, there is a measure of nonmathematical discus­ sion of the basic science behind the equations that I hope will also appeal to many others who might wish to ig­ nore the equations (but not at their peril). Thus those who have no formal mathematical background will find much of value in the descriptive mate­ rial contained here." I am skeptical. While certainly anyone would profit from paging briefly through the book, I have a difficult time imagining any­ one spending serious time reading the text who was not at the least a fairly strong undergraduate math or hard sci­ ence major. It would take some effort for any of us to work out the details be­ hind the equations in the book. This minor fault is no doubt due to the author's enthusiasm in trying to convey to others his awe at the math­ ematics of the world. Few readers of The Intelligencer will have these prob­ lems. For this audience, the book can be both dipped into to get a feel of the underlying math behind nature and also studied with paper and pen. Both approaches will be ef\ioyable. I liked this book a lot. Department of Mathematics and Statistics Williams College Williamstown, MA 01 267 USA e-mail: [email protected]

Modeling Differential Equations in Biology by Clifford Henry Taubes UPPER SADDLE RIVER, NEW JERSEY, PRENTICE-HALL, 2000. 479 PAGES, $58.99, ISBN: 0 1 301 73258

REVIEWED BY THOMAS HILLEN

S

hall we expose undergraduate stu­ dents to original research articles?

Taubes answers this question with a boldface YES. In this introductory textbook for Mathematical Biology, he presents a large number of original re­ search articles which are interwoven with the main text. Most of these arti­ cles are from Nature or Science, and the collection shows an amazing vari­ ety over many important parts of Biology. Before I comment more on the pa­ pers in this book, let me talk about the real contents. The text deals with ordi­ nary and partial differential equations applied to biological systems. The level of exposition is very basic, directed mainly to students in biology. A student in mathematics will not be satisfied with the exposition, because many de­ tails are omitted in favor of "rule-of­ thumb" statements. The text introduces the biology student to modeling with 0 DEs (ordinary differential equations) and PDEs (partial-differential equa­ tions). Indeed, one strength of the book is the painless treatment of PDEs. If needed, a student can continue in the study of differential equations using a more mathematical textbook. I mentioned the research articles of this textbook already, and you might get the impression that they are dis­ cussed in detail and that the corre­ sponding models are analyzed with the methods just introduced. But this is not the case, which I consider a weak part of the book. The research articles merely parallel what has been done in the corresponding sections. Some of them are only loosely, or not at all, con­ nected to the contents of the section at hand. For example, Chapter 2, on "Ex­ ponential Growth," concludes with four research articles: first an article on HIV virus load, which goes way over the head of a beginning student; sec­ ond, the discussion of right-handed and left-handed snail shells, which is en­ tertaining and also understandable on the undergraduate level; third, an arti­ cle on gene control of kidney develop­ ment-way too difficult; finally, one answer to the symmetry problem of the snail shells. The set-up of Chapter 2 forms a good example of the overall text. Very basic mathematics is followed by about four research articles of mixed

level. This has the advantage that an in­ structor can choose articles depending on the level of the students. Another example is the choice of pa­ pers concerned with reaction-diffusion equations and pattern formation in Chapter 18. We all know that the Tur­ ing mechanism creates wonderful pat­ terns, but it is still unclear if this mech­ anism is responsible for animal skin patterns, for example. Taubes's selec­ tion of papers shows the controversy quite nicely. An initial publication on fish-pattern is opposed by a second ar­ ticle, which then is commented on by the authors of the first article. This leaves the true impression that this dis­ cussion is still open. Other topics of the text: ODEs, phase-plane analysis, linearization, vector-matrix notation, advection, dif­ fusion, separation of variables, reaction­ diffusion equations, pattern formation, traveling waves, periodic solutions, fast and slow dynamics, and chaos. Although the text is not suitable for a course in mathematics, the enormous number of well-chosen references makes it a useful addition for the shelf of a generally interested researcher. As Taubes says in his preface, his "goal is to introduce to future experimental bi­ ologists some potentially useful tools and modes of thought." Department of Mathematical and Statistical Sciences University of Alberta Edmonton, Alberta T6G 2G1 Canada e-mail: [email protected]

Probability Theory: The Logic of Science by E. T. Jaynes CAMBRIDGE, CAMBRIDGE UNIVERSITY PRESS, 2003. 727 PP., $65.00, HARDBACK ISBN 0-52 1 -59271-2

The Fundamentals of Risk Measurement by Chris Marrison BOSTON, McGRAW-HILL, 2002. 4 1 5 PP. $44.95 HARDBACK ISBN 0-07-1 38627-0

The Elements of Statistical Learning: Data M ining, Inference and Prediction by Trevor Hastie, Robert Tibshirani, and Jerome Friedman NEW YORK, SPRINGER-VERLAG, 200 1 . 533 PP. $82.95 HARDBACK ISBN 0-387-95284-5

REVIEWED BY JAMES FRANKLIN

A

standard view of probability and statistics centres on distributions and hypothesis testing. To solve a real problem, say in the spread of disease, one chooses a "model," a distribution or process that is believed from tradi­ tion or intuition to be appropriate to the class of problems in question. One uses data to estimate the parameters of the model, and then delivers the re­ sulting exactly specified model to the customer for use in prediction and classification. As a gateway to these mysteries, the combinatorics of dice and coins are recommended; the ener­ getic youth who invest heavily in the calculation of relative frequencies will be inclined to protect their investment through faith in the frequentist philos­ ophy that probabilities are all really rel­ ative frequencies. Those with a taste for foundational questions are referred to measure theory, an excursion from which few return. That picture, standardised by Fisher and Neyman in the 1930s, has proved in many ways remarkably serviceable. It is especially reasonable where it is known that the data are generated by a physical process that conforms to the model. It is not so useful where the data is a large and little-understood mess, as is typical in, for example, in­ surance data being investigated for fraud. Nor is it suitable where one has several speculations about possible models and wishes to compare them, or where the data is sparse and there is a need to argue about prior knowl­ edge. It is also weak philosophically, in failing to explain why information on relative frequencies should be rele­ vant to belief revision and decision­ making.

Like the Incredible Hulk, statistics has burst out of its constricting gar­ ments in several directions. In the foundational direction, Bayesians, es­ pecially those of an objectivist stamp like E. T. Jaynes, have reconnected sta­ tistics with inference under uncertainty, or rational degree of belief on non-con­ clusive evidence. In the direction of en­ gagement with the large and messy data sets thrown up by the computer revolution, the disciplines of data min­ ing and risk measurement, represented by the books of Hastie et al. and Mar­ rison, have developed data analysis and tools well outside the traditional boundaries. The essence of Jaynes's position is that (some) probability is logic, a rela­ tion of partial implication between ev­ idence and conclusion. According to this point of view, statistical inference is in the same line of business as "proof beyond reasonable doubt" in law and the evaluation of scientific hypotheses in the light of experimental evidence. Just as "all ravens are black and this is a raven" makes it logically certain that this is black, so "99% of ravens are black and this is a raven" makes it log­ ically highly probable that this is black (in the absence of further relevant ev­ idence). That is why the results of drug trials give rational confidence in the ef­ fects of drugs. Galileo and Kepler used the language of objective probability about the way evidence supported their theories, and in the last hundred years a number of books have filled out the theory of logical probability­ Keynes's Treatise on Probability (the great work of his early years, before he went on to easier pickings in econom­ ics), D. C. Williams's The Ground ofIn­ duction, George P6lya's Mathematics and Plausible Reasoning, and now E. T. Jaynes's posthumous master­ piece, Probability Theory: The Logic of Science. Jaynes's school are called "objective Bayesians" or "maxent Bayesians," to distinguish them not only from frequentists but from "subjective Bayesians," who think that any degrees of belief are allowable, provided they are consistent (that is, obey the axioms of probability such as that the proba­ bility of a proposition and its negation

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must sum to one). The objectivists em­ phasise, on the contrary, that one's de­ gree of belief ought to conform to the degree to which one's evidence does logically support the conclusion. In asking why evidential support should satisfy the axioms of probability the­ ory, objectivists have been much im­ pressed by the proof of R. T. Cox (American Journal of Physics, 1946) that any assignment of numbers to the relation of support between proposi­ tions which satisfies very minimal and natural logical requirements must obey the standard axioms of conditional probability. They have been corre­ spondingly unimpressed by supposed paradoxes of logical probability that purport to demonstrate that one can­ not consistently assign initial probabil­ ities. In some of his most entertaining pages, Jaynes exposes these "para­ doxes" as exercises in pretending not to know what everyone really does know. His reliance on symmetry prin­ ciples to assign initial probabilities shows its worth, however, well beyond such philosophical polemics. In in­ verse problems like image reconstruc­ tion, where the data grossly under­ determines the answer, it is essential to assign initial probabilities as non­ dogmatically as possible, in order to give maximum room for the data to speak and point towards the truth. Jaynes's maximum entropy formalism allows that to be done. In the business world, there is the same need as in science to learn from data and make true predictions. But among other forces driving the expan­ sion of commercial statistics are the new compliance regimes in banking and accounting. Following a number of corporate scandals and unexpected collapses, the world governing bodies in banking and accounting have de­ cided on standards that include, among other things, risk measure­ ment. The Basel II standard in banking says, in effect, that banks may use any sophisticated statistical methodology to measure their overall risk position (in order to determine the necessary reserves), provided they disclose their methods to their national banking reg­ ulator (the Federal Reserve in the U.S.,

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the Bank of England in the U.K.). Mar­ rison's book is an excellently written introduction to the standard ideas in the field. It avoids the unnecessary el­ ements in usual statistics courses and goes immediately to the most applica­ ble concepts. These include the "value­ at-risk" formalism, which measures the loss such that worse losses occur exactly 1% (say) of the time, and the concepts needed for precision in han­ dling rare losses, such as heavy-tailed distributions and correlations be­ tween the losses of different financial instruments. It is significant, for ex­ ample, that foreign exchange rate changes resemble a random walk, but are heavy-tailed, are heteroskedastic

Mathematicians , pure and appl ied , t h i n k there i s someth i ng wei rd ly d ifferent about statistics . (variable in "volatility," that is, stan­ dard deviation), and have some ten­ dency to revert to the mean. It is per­ haps surprising to learn that credit ratings are intended to mean absolute probabilities-a AAA rating means one chance in 10,000 of failure within a year; naturally it is hard to ground so small a probability in data, so one pre­ sumes that credit rating agencies will need to use priors and qualitative evi­ dence (a euphemism for market ru­ mours?) in the style of Jaynes. Marri­ son's insights into how bank risk teams really work is enlivened by oc­ casional dry humour: in pointing out that profits from risky trades need to be discounted, he adds, "Convincing traders that their bonuses should be reduced according to Allocated Capi­ tal X Hr is left as an exercise for the reader." In accounting, the forthcoming IFRS (International Financial Reporting Standards) compliance standard will

play a role similar to Basel II in bank­ ing, in enforcing higher standards of mathematical competence. It will be necessary to price options reasonably in the interests of their truthful display on balance sheets, for example. The book for accountants corresponding to Marrison's appears not yet to be writ­ ten, so there may be a gap in the mar­ ket for an ambitious textbook writer who would like to become very rich very quickly. If there is a dispute in statistics as heated as that between frequentists and Bayesians, it is that between tra­ ditional statisticians and data miners. Data mining, with its roots in the neural networks and decision trees developed by computer scientists in the 1980s, is a collection of methods aiming to un­ derstand and make money from the massive data sets being collected by supermarket scanners, weather buoys, intelligence satellites, and so on. "Drink from the firehose of data," says the science journalist M. Mitchell Wal­ drop. It is not easy-and especially not with the model-based methods devel­ oped by twentieth-century statisticians for small and expensive data sets. With a large data set, there is a need for very flexible forms to model the possibly complicated structure of the data, but also for appropriate methods of smoothing so that one does not "over­ fit," that is, learn the idiosyncracies of the particular data set in a way that will not generalise to other sets of the same kind. Are specialists in data mining (or "analytics" as they now often prefer) pioneers of new and exciting statistical methodologies, or dangerous cowboys lacking elementary knowledge of sta­ tistical models? Those who enjoy vig­ orous intellectual debate will want to read data miner Leo Breiman's pug­ nacious "Statistical modelling: the two cultures," Statistical Science 16 (2001), 199-2 19, with a marvellously supercilious reply on behalf of the tra­ ditionalists by Sir David Cox. As an in­ troduction to the field for practitioners in the business world, Michael Berry's Mastering Data Mining (New York, Wiley, 2000) is often recommended, but for mathematicians interested in understanding the field, Hastie et al.'s

Elements ofStatistical Learning is the ideal introduction. Assuming basic sta­ tistical concepts and an ability to read formulas, it runs through the methods of supervised learning (that is, gener­ alisation from data) that have come from many sources: neural networks, kernel smoothing, smoothed splines, nearest-neighbour techniques, logistic regression and newer techniques like bagging and boosting. The unified treatment and illustration with well­ chosen (and well-graphed) real data­ sets makes for efficient understanding of the whole field. It is possible to appreciate how different methods are really attempting the same task-for example, that classification trees de­ veloped by computer scientists to suit their discrete mindset are really per­ forming non-linear regression. But the differences between methods are well laid out too: the table on p. 313 com­ pares the methods with respect to such crucial qualities as scalability to large data sets, robustness to outliers, han­ dling of missing values, and inter­ pretability. The less-tamed territory of unsupervised learning, such as cluster analysis, is also well covered. One topic of current interest missing is the attempt to infer causes from data, but, as is clear from Richard Neapolitan's Learning Bayesian Networks (Har­ low, Prentice Hall, 2004), that theory is still in a primitive state. Spatial statis­ tics and text mining are not covered ei­ ther; they too await readable textbooks of their own. Mathematicians, pure and applied, think there is something weirdly dif­ ferent about statistics. They are right. It is not part of combinatorics or mea­ sure theory but an alien science with its own modes of thinking. Inference is essential to it, so it is, as Jaynes says, more a form of (non-deductive) logic. And, unlike mathematics, it does have a nice line in colourful polemic.

School of Mathematics University of New South Wales Sydney 2052 Australia e-mail: [email protected]

Mathematics Across Cultures Helaine Selin and Ubiratan D'Ambrosio, editors KLUWER ACADEMIC PUBLISHERS, 2000, 479 PAGES HARDBOUND, ISBN 0·7923·648 1 ·3, € 1 95.50 PAPERBACK, ISBN 1 ·4020·0260·2, €63.00

REVIEWED BY HELENE BELLOSTA

T

his book is meant as a supplement to the Encyclopaedia of the His­ tory of Science, Technology and Med­ icine in Non-Western Cultures (Kluwer Academic Publishers, 1997) and is aimed at a more scholarly audience; the aim is to explore the same topics in greater depth. The book is divided into two parts: the authors of the six essays in the first section try to define the field of ethno­ mathematics and to make a general study of the connection between math­ ematics and culture as well as the vari­ ability of the concept of rationality, while the second part is devoted to the description of fifteen individual cul­ tures and their mathematics in various "non Euro-American" areas: the Middle East, America (native cultures), the Pa­ cific and Australia, Africa, and the Far East. If the intention behind the book-to rehabilitate the so-called non-Western cultures and to denounce the damag­ ing effects of cultural imperialism and eurocentrism, the consequence of which is a certain contemptuous dis­ regard for these cultures-is highly laudable, this enterprise is not entirely free from danger. The main difficulty is defining and naming the field of study: how should we divide sciences into Western and non-Western, or Euro­ pean and non-European? The criterion is not geographical but cultural (H. Selin, p. v), for the studies in this book deal with mathematics in the Far and Middle East, as well as mathematics in Aboriginal, Amerindian, or African so­ cieties. Should we, as some authors do, speak of "non-modem" or "traditional" sciences, even though this mixes up different eras, from the 3rd millennium Be to today? Should we then group these sciences together under the

heading "ethno-sciences"? But what then are the criteria that include sci­ ence in Mesopotamia or ancient Egypt in ethno-sciences, but exclude Greek science, although all the authors in Greco-Hellenistic Antiquity regard sci­ ence in ancient Egypt as the origin of Greek science? Why should Arab math­ ematics, the heir to Greek mathemat­ ics, whose contribution is essential to understand the constitution of classical mathematics in 17th-century Europe, be included in ethno-mathematics? This book seems to make a rather strange division. On one side we have ethno-sciences, bringing together sci­ ences as different as science in ancient China and science in present-day Abo­ riginal societies, these being viewed as sciences of unusual societies, the pe­ culiarities of which, together with their incommunicability, some papers dili­ gently stress; and on the other, by de­ fault, Greek science, European science from the Renaissance to nowadays as well as science in the USA, would be left as non-ethnic sciences (white sci­ ence versus colored science?). If we continue to follow the unspoken logic of this division, these sciences should then show the opposite qualities and be a contrario universal. We should not be surprised then to find here and there in some papers hasty judgments and worn-out commonplaces on these "dif­ ferent" civilizations, which could be de­ fined as the eurocentrism the editors intended to stigmatize: "The transfor­ mation of the word science as a dis­ tinct rationality valued above magic is uniquely European" (H. Selin, p. vi) or "the development of this concept of ra­ tionality (i.e., European's 1 7th century) was not universal. For example, it was not paralleled in Islamic society where men were denied rational agency; they were held to lack the capacity to change nature or to understand it. Knowledge was instead to be derived from traditional authority" (D. Tum­ bull, Rationality and the disunity of the sciences, p. 47). One of the authors (R. Eglash, Anthropological perspec­ tives on ethnomathematics) is clearly conscious of the difficulty of defining what ethno-mathematics or non-West­ em mathematics actually are, and also

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aware of the disparity in level and con­ tents of what the editors group under this heading. He therefore suggests distinguishing between mathematics of "non-Western empire civilisations" (India, China, Japan, Arab world)­ which could be compared to "West­ em" mathematics-and mathematics of "indigenous societies." However, this leaves us to wonder what consti­ tutes an "indigenous society." Would it not be more epistemologically conve­ nient to try to distinguish between mathematics and what could be called proto-mathematics, and to try to spec­ ify the criteria that would enable us to define each of them (criteria of ratio­ nality, modes of proof, degree of ab­ straction, nature of the problems solved . . . )? We could try to define mathematics as a human activity the purpose of which is to solve abstract problems dealing with number and magnitude or with geometrical figures, with methods having some degree of exactness and generality; moreover, this activity is ex­ plicitly bound, by those who practise it (geometers, mathematicians . . . ) and name it (geometry, wasan . . . ), to some criteria of rationality (even if they could differ) and to the necessity of some type of proof (whatever it is, should it be the constraining form of euclidean formalism, or a mere justifi­ cation of calculus procedures and al­ gorithms). Proto-mathematics would then be a contrario a set of calculus techniques and geometrical construc­ tions, directly born of practical neces­ sities or esthetical concerns, and aimed at solving problems arising in a some­ how organized society (the knowledge of these techniques and geometrical constructions is not restricted to a dis­ tinct social class, and those who know them and use them do not bother about the necessity of proof). Let us note that this implies no disrespect for proto­ mathematics; furthermore the opposi­ tion between mathematics and proto­ mathematics is not so clear-cut as this brief account suggests, and more often it will be possible to distinguish in proto-mathematics the forms of more sophisticated future mathematical the­ ories.

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Part One Leigh N. Wood , Communicating

uct essentially dependent on commu­ nity and tradition.

mathematics across culture and time

The author intends to stress the diffi­ culties inherent in communicating mathematics across cultures and time periods. Mostly relying on erratic and superficial knowledge, the author jumps haphazardly from Babylonian tablets to the counting system in Ocea­ nia and Papua New Guinea, from Greece to China . . . Historians of sci­ ence will certainly be pleased to learn that "Leonardo Fibonacci . . . [solved] cubic equations" (p. 10) (sic).

Helen Veran, Logics and mathematics: challenges arising in working across cultures

This paper combines the author's per­ sonal experience teaching mathemat­ ics in Australian Aboriginal and Niger­ ian classrooms (see below) with a criticism of colonialism and general considerations on the relativity of logic and rationality. It concludes that "the foundationist mode of ontology/episte­ mology is a moral project of European colonising" (p. 72) (sic).

Ron Eglash, Anthropological perspectives on ethnomathematics

This paper is the only one to mention the difficulties in defining ethno- or non-Western mathematics and to bring up the ideological dangers of this en­ deavour together with its a-historiciz­ ing effect. In answer to some papers, like Helen Veran's, he remarks judi­ ciously that "what goes under the name of multicultural mathematics is too of­ ten a cheap shortcut that merely re­ places Dick and Jane counting marbles with Tatuk and Esteban counting co­ conuts" (p. 20). Edwin J. Van Kley, East and West

A brief survey of what European sci­ entific development owes to China. Conjectures the author puts forward as if they were facts, e.g. , "Simon Stevin whose sailing chariot probably was in­ spired by description of similar Chi­ nese devices . . . also introduced deci­ mal fractions and a method of calculating an equally tempered musi­ cal scale both of which might also have been inspired by Chinese example" (p. 32). His conclusion is as condescend­ ing as it is naive: "Obviously Asians are as able to master its intricacies [i.e., in­ tricacies of the rational scientific in­ dustrialized culture] as are Europeans and Americans" (p. 34)! David Turnbull, Rationality and the

Ubiratan d'Ambrosio, A historical proposal for non Western mathematics

This paper focuses on the social, politi­ cal, and cultural factors in the dynamics of the transfer and production of math­ ematical knowledge in the colonies, as well as on the recognition of non-Eu­ ropean forms of mathematics extant or buried in the colonial process. Part Two

The chapters in the second part describe individual cultures and their mathemat­ ics. They vary greatly in the fields of study and in the level of the discussion. Eleanor Robson, The uses of mathematics in ancient Iraq,

600o-600

BC

The author, who deplores the fact that the history of mathematics is too often written by mathematicians or mathe­ matics historians and intends to rem­ edy that, makes "a first approximation to a description of Mesopotamian nu­ merate quantitative or patterned ap­ proaches to the past, present, and fu­ ture; to the built environment and the agricultural landscape and to the nat­ ural and supernatural worlds" (p. 95). James Ritter, Egyptian mathematics

A conscientious, well-documented study of Egyptian mathematics by one of the specialists in the field.

disunity of the sciences

In this paper the author expounds the thesis of "post modem" relativism: that the concept of rationality is relative, and hence science is but a social prod-

Jacques Sesiano, Islamic mathematics

Following the inner logic of this book, i.e., the division of mathematics into

two distinct fields, Western and non­ Western, the author stresses that Mesopotamian mathematics would be the ancestor to Arab mathematics. However, he gives no evidence of how it was transmitted across the centuries, and carries on to suggest, still with no proof, that algebra originated in India. The anachronistic use of the word al­ gebra-Indian algebra "dealing with the practical needs of daily life and trade," geometrical algebra of the an­ cient Greeks-leads the author to un­ derrate the radical novelty and impor­ tance of al-Khwarizmi's Algebra, which for him is "probably not very original" (p. 143). This preconceived idea con­ cerning the supposed lack of original­ ity of Arab mathematics, backed up by some allusions to the so-called con­ formism of mathematicians ("a refer­ ence to the ancients had almost the weight of a formal proof' (p. 139), is belied by all the scientific literature. This portrayal of Arab mathematics, in addition to its many errors of perspec­ tive-importance given to minor au­ thors or to secondary problems and words, and misunderstanding of the novelty of others-leaves aside entire chapters of these mathematics (infini­ tesimal mathematics, geometrical transformations, spherical geometry . . . ). No wonder that the bibliography given at the end of the paper is quite outdated; it ignores almost all the fun­ damental publications of the last 20 years, both specialized studies and en­ cyclopedic works. Tzvi Langermann and Shai Simonson, The Hebrew mathematical tradition

In this paper Tzvi Langermann investi­ gates numerical speculation and num­ ber theory (mainly based on the works of Abraham ibn Ezra ( 1092-1 167), a central figure in Hebrew mathematics) and Hebrew contributions to geometry (mainly translations from Arabic to He­ brew of Euclid's works), while Shai Si­ monson focuses on algebra. As with Sesiano, the anachronistic use of the word "algebra," applied to Babylonian algorithms as well as to the so-called geometrical algebra of the an­ cient Greeks, leads Shai Simonson to paint a quick and rather summary pic-

ture of this chapter of the history of mathematics, mixing up epochs and traditions: "Throughout these 3000 years, the Greeks, Indians, Chinese, Muslims, Hebrews and Christians seem to have done no more than present their own versions of solutions to linear and quadratic equations, which were well known to the Babylonians" (p. 174). Si­ monson then studies the contributions of Abraham ibn Ezra and Levy ben Ger­ shon to algebra. In doing so, he de­ scribes Levy's method for square and cube root extraction, in great detail. However, this probably owes less to Babylonian or Chinese origins (spanning centuries and distances with no textual evidence of such transmission), than to the classical method set out by al­ Khwarizmi (IXth century) in his arith­ metrical treatise (translated several times into Latin in Spain during the XI­ Ith century), or by al-Karaji in his al-Kafi fi al-hisab (end Xth beginning Xlth cen­ tury) for the square root, and by al­ Samaw'al (died 1 1 74) for the cube roots. As for ibn Ezra, if he knew the approx­ imation formula Vx2 ± A = x ± _:'\_ . Z,r , � r +A (not v x- :c:: A = '--i-' as wrongly mdicated by the author), it seems more likely that he borrowed it from al­ Khwarizmi than from the Babylonians or Greeks . Thomas E. G isl dorf, Inca mathematics

This paper, which ambitiously aims to describe "the development of Inca mathematics in order to give a cultural perspective and to emphasize the in­ terdependence of mathematics and non-mathematical factors" (p. 189), shows how difficult it is to study a dead civilization which has left no other written traces than decorative patterns bearing geometrical motives or quipus (devices formed by knotted strings). If the needs of this central­ ized society, as presented by the au­ thor (water control, civil and agricul­ tural engineering, astronomy, and time-keeping) seem to have been gen­ erally the same as in other civiliza­ tions and surely imply some mathe­ matical knowledge, the lack of written testimonies or oral transmission nonetheless makes it difficult to re-

constitute. This leads the author to put forward as fact second-hand hy­ potheses which have been taken from different authors and are quite often contradictory. Michael P. Closs, Mesoamerican mathematics

A precise, well-documented paper in which the author studies in detail the numeration and calendar systems in the Olmec, Zapotec, Epi-Olmec, Maya, Mixtec, and Aztec civilizations. Daniel Clark Orey, The ethno­ mathematics of the Sioux tipi and cone, Walter S. Sizer, Traditional mathematics in Pacific cultures, Paulus Gerdes, On mathematical ideas in cultural traditions of central and southern Africa

Three papers in the same vein. Reha­ bilitating the culture of the Sioux Indi­ ans after they have been slaughtered is certainly a worthy project, but I cannot see how the fact that Clark Orey de­ termines the height of a tipi and the center of gravity of its base, or gives a (false) calculation of his own of its lat­ eral area, helps preserve the beauty of this civilization. If the observation of the straight spears of New Guinea war­ riors leads Walter S. Sizer to assert that "most cultures display considerable geometric understanding through their manufactured products" (p. 260), the discovery of ancient lunar calendars makes Paulus Gerdes wonder "who but a woman keeping track of her cycles would need a lunar calendar?" (p. 3 14) before concluding that "women were undoubtedly the first mathematicians" (p. 314)! Helen Veran, Aboriginal Australian mathematics: disparate mathematics of land ownership and Accounting mathematics in West Africa: some stories of Yoruba number

In the first paper, H. Veran studies the subtle and unusual links the Aborig­ ines have forged from time imme­ morial with their native land, their despoilment at the hands of the colonizers, their continuing conflict with the Federal Government, as well as the kinship systems in their society.

© 2005 Springer Science+ Business Media, Inc., Volume 27, Number 2, 2005

87

This study, combined with the axiom "politics, logic and mathematics are in­ separable" and the constant use of analogy, leads the author to somewhat

man Singh's The Proof, with the superb

The Parrot's Theorem

opening scene of Wiles's reflections. In a more esoteric vein, an opera about

by Denis Guedj

the eleventh-century Islamic mathe­

strange conclusions concerning the

NEW YORK, THOMAS DUNNE, 2001 . 352 PP., US$24.95,

mathematics of a society in which

ISBN 0-31 -228955-6

premiered in Qatar. Fictional mathe­ maticians have also enjoyed some at­

there seems to be no use of numbers, such as "land is a matrix of vectors" (p. 308). The second paper is a lexical

The Fractal Murders

study of the numeration system of

by Mark Cohen

tention, in, for example, Tom Stop­ pard's Arcadia or his screen play for Enigma; Matt Damon and Ben Af­ fleck's Good Will Hunting; or Daniel Aronofsky's chaotic Pi.

Yoruba (Nigeria). BOULDER CO, MUDDY GAP PRESS, 2002. 282 PP., US$25.00, ISBN 0-97 - 1 89860-X

Mathematicians

Jean Claude Martzloff, Chinese mathematical astronomy

A serious and documented study of

The Da Vinci Code

Chinese mathematical astronomy by

by Dan Brown

dabbled in mathematics from time to time, and his nemesis Moriarty was a mathematician as well as a master crook. Michael Innes's

ISBN 0-38-550420-9

T. K. Puttaswamy, The mathematical

ematics, marred both by numerous

The Curious Incident of the Dog in the Night-Time

mistakes in mathematical formulas

by Mark Haddon

A brief survey of ancient Indian math­

and geometrical figures, and by ety­ mologies of pure fantasy.

Jochi Shigeru, The dawn of Wasan

idence is a classic,

Weight of the Ev­

Desmond Cory's Pro­

fessor Dobie stories date back a few years, and Scott Turow's first huge suc­ cess,

Presumed Innocent,

featured a

mathematician. Not all mathematicians in mysteries are fictitious. A very un­

NEW YORK, DOUBLEDAY, 2003. 240 PP., US$22.95,

pleasant John Wallis figures promi­

ISBN 0-38-550945-6

nently and Newton makes a cameo ap­ pearance in lain Pears's wonderfully

(Japanese mathematics}

Leaning towards Infinity

This is an introduction to the history of

by Sue Woolfe

atmospheric

gerpost.

An Instance of the Fin­ Dark Matter

In Philip Kerr's

Newton himself is the detective, albeit

Japanese mathematics, with a very comprehensive bibliography.

mathematics

Not that this is new. Sherlock Holmes

NEW YORK, DOUBLEDAY, 2003. 454 PP., US$24.95,

mathematicians

and

have also invaded the mystery genre.

one of the specialists of the field.

accomplishments of ancient Indian

matician Ibn Sina (Avicenna) recently

NEW YORK, FARRAR STRAUS & GIROUX, 1 997. 393 PP., US$24.95, ISBN 0-57- 1 1 9905-4

in his role as warden of the Royal Mint. And Escher's mathematics, if not the artist himself, appears in Jane Langton's

Kim Soo Hwan, Development of materials for ethno-mathematlcs in Korea

The author illustrates his interest in

The Escher Twist. A cadre of more con­

REVIEWED BY MARY W. GRAY

T

temporary mathematicians is thinly dis­ he past few years have seen a pro­

guised in Maths d

liferation

got Bruyere, a long-time administrator

of portrayals

of real

school and everyday mathematics by

mathematicians

the description of some everyday tra­

Sylvia Nasar's A Beautiful Mind; Hugh

in

popular

media.

mort, written by Mar­

at IHES (lnstitut des Hautes Etudes Sci­ entifiques de Bures-sur-Yvette).

ditional objects to which one can ap­

Whitemore's

ply mathematics.

Hoffman's charming rendition of the

women in mathematics appears higher

life of Erdos,

than in real life; of the mysteries re­

Centre d'Histoire des Sciences et des

Numbers;

Philosophies Arabes et Medievales

Breaking the Code;

Paul

The Man Who Loved Only Proof

and David Auburn's

come to mind as contributing to the

In

fiction

the

representation

of

viewed here, only

The Curious Inci­ dent of the Dog in the Night-Time has

CNRS UMR 7062

caricature of mathematicians as pecu­

no woman mathematician involved in

7 rue Guy M6quet

liar, to say the least-especially if one

the plot. (Yes, the parrot is female !).

94801 Villejuif Cedex

thinks

to

Presumed Innocent is in this tradition,

France

Nasar's story. Rumors have circulated

as are the Laurie King books featuring

e-mail: [email protected]

that a film about the Unabomber is in

Sherlock

the works, and perhaps a version of

later wife, the mathematics student

After Math by Miriam Webster

of what

Hollywood

did

Holmes's

apprentice

and

Ahmed Chalabi's adventures will in­

Mary Russell; P. M. Carlson's series

clude his brief fling as a mathemati­

with statistician Maggie Ryan;

cian. His subsequent shady political ca­

dia;

and

Arca­

Proof

reer may have some echoes of the

Various theories for the increased

tactics of Newton in Carl Djerassi's re­

interest in and popularization of math­

WAYNE, PA, ZINKA PRESS, 1 997. 280 PP., US$1 2.95,

cent play

Television has

ematics in the media have been put

ISBN 0-96-47 1 7 1 1 -5

been somewhat kinder, producing Si-

forward. Perhaps for many authors it

88

THE MATHEMATICAL INTELLIGENCER

Calculus.

is the strangeness of mathematics that

Austin environmental community as an

overlook the propensity of the state of

makes mathematicians attractive sub­

advocate for the use of bicycles and as

Texas to impose the death penalty.

jects, although the writers may feel

a city council candidate, claims to fol­

It is fair to say that some readers

compelled to exaggerate our peculiar­

low the revelatory maxim of Anthony

must have liked the book better than I

ities. For others-and perhaps for their

Trollope, "The author scorns to con­

did. A recent site visit to Amazon.com

readers-the notion that mathemati­

ceal from the reader any secret which

showed copies being offered for as

cians are, after all, very weird if not ac­

is known to himself. " However well

much as

tually deranged, may justify their math

this may have worked for Trollope, in

If Babich's primary goal was to

phobia. And yet often the books, plays,

her case it takes out the mystery, leav­

write a murder mystery, Denis Guedj's

and

and

ing very little except for the atmo­

in

mathematicians written by non-mathe­

sphere of a university mathematics de­

been to make the history of mathe­

maticians are wonderfully sensitive to

partment.

films

about

mathematics

us and our work. The heightened attention to mathe­

$296.

The Parrot's Theorem seems to have

matics palatable for the masses. Or, as

The author generally has a good feel

Simon Singh put it in a review in

The

for this life, but contrary to her char­

Guardian: The Parrot's Theorem

has

matics in the media has done nothing

acterization, it is NOT the case that all

a "different objective from most other

for the recruitment of students, at least

statisticians prefer their chocolate in

fiction-to smuggle mathematical con­

in the U.S. and the U.K. It has been con­

the form of M&Ms when it comes to

cepts into the mind of the unsuspect­

jectured that the spurt in applications

eating, no matter what else they may

ing reader by wrapping the maths in­

to law schools in the U.S. can be traced

use them for. My colleagues have a

side

back to television's LA

strong preference for Godiva, but then

succeeded fairly well,

0. J.

Law

or to the

outcome of the trial depends on statis­ tive recruitment device. Although we do not want our field overcrowded, it would be nice to have more students in undergraduate courses because they love mathematics and not because it is

We

market, a mathematical recluse in the

al l the pri n c i pal g uard ians of the

called Max, through whom the reader

sacred doctri ne

tory, from Thales to amicable numbers

bookseller, and an amazing deaf child is introduced to

be male seems a

orem and the Goldbach Conjecture.

Uncle Petros and the Goldbach Conjecture intro­

(Apostolos Doxiadis's

duces another recent fictional claimant to the proof of the latter.) Perhaps the

bit n egative .

so many more lawyers than mathe­

5000 years of math his­

to claimed proof of Fermat's Last The­

over the cent u ries

a requirement for graduation. Are there

frequent references to problems un­ solvable by use of a compass and ruler

maticians because we are so litigious, or is it the other way around? Or can it

has

Amazon jungle, a delightful Parisian

homes tuned in to the trial. Maybe an

tical evidence would be a more effec­

Guedj

although the

have a real parrot bought in a Paris

. letting nearly

cians in the latter coincided, it is re­ ported, with peak usage of water in the

and Order in which the

engaging plot."

"mystery" is pretty transparent.

Simpson trial. Unfortunately the

appearance of the competing statisti­

episode of Law

an

none of us regularly wear Birkenstocks

(instead of straight edge) and a few

be because faking it is so much easier

nor find it necessary to run our hands

other inaccuracies are a result of mis­

in law than in mathematics? Of course,

over our bodies to determine whether

translation. My main quarrel is with his

lawyers from Fermat through Cayley

or not we are wearing clothes, other

attribution

and Sylvester did produce significant

characteristics of mathematicians, ac­

thought of as the Problem of Dido to

of what

I

have

always

mathematics, but the day of the ama­

cording to Babich. Nor can I agree with

Hassan

teur mathematician may be past, al­

her

is

Khayam. A nice touch is the link he

though several of the books reviewed

about rules." Doesn't that just perpet­

sees between the origins of mathemat­

here do feature such anomalies.

uate another stereotype?

ics and tragedy in ancient Greece.

Magic realism in the hands of a mas­

assertion

Ajler Math

that "mathematics

also suffers from an

Sabbah,

a friend

of Omar

The Parrot's Theorem would

be an

ter like Gabriel Garcia Marquez is, well,

overabundance of characters, few of

excellent text for a general education

magical. The same cannot be said for

them well developed; from excessive

course in the history of mathematics,

reliance on quotes from Shakespeare,

although those with little mathematics

the supernatural

After Math

adven­

tures of the dead mathematician Ray

Trollope,

Raymond

background might find some passages

Bellweather and equally dead graduate

Chandler, Wittgenstein, Gogol, Goethe,

tough going. Also, some of the more

Byron,

Heine,

student Glen Vesper, as they combine

and others; and from coy asides from

colorful legends he describes cannot

forces with two colleagues still among

the author to the reader. At least the

be taken too seriously. Although the

the living to expose their murderer.

murdering mathematician is not crazy,

book was a best-seller in France, it is

Not that there was ever much doubt

just venal. However, his complacency

unlikely to repeat its success in the

about who was the murderer. Author

as he contemplates a long imprison­

United States. The mathematics is too

Amy Babich (Miriam Webster is a pseu­

ment, remarking that good mathemat­

simple or too familiar to mathemati­

donym), a mathematician known in the

ics has been done in prison, seems to

cians (of whom there are in any case

© 2005 Springer Sc1ence+ Business Media, Inc., Volume 27, Number 2, 2005

89

too few to confer best-seller status)

A symbologist and a cryptographer

has charming simplicity appealing to a

and too inaccessible and digressive for

set out to solve the murder in the Lou­

broader audience. The "detective," in

others. Guedj's writing does combine

vre of its chief curator, and to halt at­

whose voice the book is written, is fif­

insight and appreciation with a joy he

tempts to suppress entirely the preser­

teen-year-old Christopher Boone, who

would like his readers to share, but his

vation of the

"sacred feminine" in

suffers from Asperger's syndrome, a

combination of whimsy, mathematics,

religion. Although making the cryptog­

symptom of which is an inability to fig­

and mystery doesn't quite achieve the

rapher a woman (and not a first for

ure out what is going on in the minds

proper balance.

Brown, as his earlier

and emotions of others--not that every­

My favorite among the books re­

Digital Fortress

introduced Susan Fletcher as the head

one does not have occasional problems

of code-breaking at the National Secu­

doing so. Christopher's curious incident

where I think the author has achieved

rity Agency) may be a gesture for fem­

is less benign than Sherlock's, as the

a great balance between mathematics

inism, letting nearly all the principal

dog is impaled on a garden fork Find­

and mystery. A Heidegger-obsessed for­

guardians of the sacred doctrine over

ing out who did this leads to scary rev­

mer Judge Advocate General lawyer

the centuries be male seems a bit neg­

elations

turned private investigator and a Uni­

ative. Granted, it's in keeping with

and, crucial to Christopher himself, al­

versity

mathematician

what we know about the leaders of

most causes him to miss out on a math­

team up to solve the mystery, enlivened

most religions, not to mention those in

ematics exam to which he was looking

by a fairly accurate lay description of

influential positions of any kind.

forward with great joy.

viewed here is

of

The

Fractal Murders,

Colorado

about

Christopher's

family,

fractals. In fact, the book has Mandel­

It cannot be said that the reader's

The narrative could have become

brot's endorsement. The mysterious

understanding or appreciation of math­

patronizing or pathos-filled, but Had­

deaths of three mathematicians who

ematics is enhanced by reading The Da Vinci Code, but for sheer escapism the

tone, making his hero endearing in

were working on applications of Man­

don manages to achieve just the right

delbrot's trading time theorem trigger

book is hard to beat. Yes, Brown im­

spite of his quirks. Some of these, like

the investigation. The lives of academic

plies that the Golden Ratio is a ratio­

his giving each chapter a prime num­

mathematicians are quite well cap­

nal number, and the Da Vinci Code is

ber, are endearing, but others, such as

tured, even though the investigator be­

not much of a code, but this is a thriller,

his refusal to be touched, are less so.

trays a lack of understanding of the eco­

not a textbook Criticisms seem to be

Readers are left to interpret for them­

nomics of mathematics textbooks by

felt to be necessary by those who

selves words and actions that are a

being shocked by a

$44.95

price tag he

considers excessive.

probably not only secretly enjoy the

mystery to Christopher, whose view of

book but identify with the protago­

the world is totally literal. Although

The monetary value of fractals is

nists. Apparently these critics need to

Christopher loves mathematics, there

also central to the plot of Robert God­

display their superior knowledge, ig­

is not a lot of it in the book (except for

dard's

featuring one of

noring Brown's skill in using various

some interesting bits about prime num­

my favorite fictional creations, a female

devices to make the book exciting

bers and the Monty Hall problem). For

mathematician who is described as be­

reading. They remind one of those in

that matter, there is very little mystery,

ing at the lAS at Princeton contempo­

the United States who felt obliged to

but the book is brilliantly written.

raneously with Einstein, Godel, Man­

display their credentials by panning

Whether it is an accurate portrayal of

delbrot, and von Neumann. Both of

Michael Moore's

as if

autism, I cannot say, but it feels au­

these books are helped by great loca­

it were someone's dissertation, not a

thentic. Unfortunately, the book may

tions, ranging from Nederland,

Col­

skillful propaganda piece with some

feed the myth that anyone with As­

orado, to a remote English village. (OK.

basic truths and a few exaggerations.

perger's syndrome is a mathematical

I confess that any book that uses Ne­

The originality and factual basis of

genius. On the other hand, it certainly

braska as even a minor locale has a lot

Brown's theories may be in question,

has led to an increased interest in, and

going for it as far as I am concerned;

but he has written an absorbing mys­

one would hope understanding of, the

we Nebraskans do not see much about

tery. As such, it is a page-turning suc­

condition, a recent indication of which

our home state in print, fact or fiction.)

cess.

is a wonderful compendium, Ian Stu­

Out of the Sun,

Fahrenheit 9/1 1 ,

They can be recommended as good dis­

Another best-seller is Mark Haddon's

tractions on the long plane rides and se­

The

art-Hamilton's An Asperger Dictio­ nary of Everyday Expressions, ex­

curity delays faced by peripatetic math­

Curious Incident of the Dog in the Night-Time, borrowing from Conan

ematicians. Cohen is planning a series

Doyle's "Silver Blaze." (Inspector Gre­

horns" is not just the tactic of a des­

featuring the same main characters, so

gory: "Is there any point to which you

perate matador. Maybe Christopher's

watch for their next appearance.

would wish to draw my attention?"

enthusiasm will even inspire others

Holmes: "To the curious incident of the

with an eagerness to take mathematics exams!

The Da Vinci Code has topped best­ seller lists and created a lot of contro­

dog in the night-time." Inspector: "The

versy with theologians, historians, and

dog did nothing in the night-time."

mathematicians,

Holmes: "That was the curious inci­

plaining why "taking the bull by the

least eight books have been written de­

dent."] Originally classified as a book

Finally, there is Leaning towards In­ finity by Sue Woolfe, who has also writ­ ten a book, The Secret Cure, about

bunking various aspects.

for young people,

Asperger's. There is not a lot of mathe-

90

among

THE MATHEMATICAL INTELLIGENCER

others.

At

The Curious Incident

matics in Leaning towards Infinity, and

ematician would go so far as to bare her

the only interesting mystecy is how it

breasts to get attention when presenting

won an award in its author's homeland

her results to an indifferent male-domi­

--, The Dobie Paradox, New York: St.

Martins Press, 1 994. Matt Damon and Ben Affleck, Good Will Hunt­

of Australia. No doubt the prize com­

nated audience. (Maybe I have yet to see

mittee had few mathematically inclined

true desperation!) In any case, both

Carl Djerassi and David Pinner, Newton 's Dark­

members. A complicated plot begins

Juanita and Frances go crazy and the

ness: Two Dramatic Views , London: Imper­

with Ramant\ian's

1913

letter to Hardy

revolutionary mathematics never gets a

ing, film, Miramax Films, 1 997.

ial College Press, 2003.

of

chance. That the "mathematics" is vague

Apostolos Doxiadis, Uncle Petros & Gold­

women: Juanita Montrose, the amateur

and improbable is understandable, but

bach's Conjecture, New York: Bloomsbury

mathematician who is supposed to be

in general the book is difficult to wade

another Ramanujan; Frances her daugh­

through, with the different narratives be­

ter, who was advised not to study math­

coming confused.

and

involves

three

generations

ematics as it might be injurious to her

azine, 1 892.

It seems on the evidence here that

health, and so ended up teaching litera­

compelling mysteries are best written

ture, more suitable for a woman; and

by someone other than a mathemati­

Frances's daughter Hypatia, who is or­

cian, although Leaning

ganizing the book her mother wrote. Nei­

ity

ther Hypatia nor Woolfe admits to know­

guarantee success. Ultimately, it would

ing anything about mathematics,

an

assertion the reader can easily believe.

towards Infin­

shows that ignorance does not

be interesting to know how many read­ ers previously uninterested in mathe­

Juanita's creativity began, we are

matics are motivated to explore more

told, with the understanding of Zeno's

deeply concepts from the best of these

paradoxes at age nine, and Frances fell

books such as prime numbers, growth

in love with the stocy that Einstein's wife was

responsible

for

e

=

mc2,

giving

equations, probability, the golden ratio, cryptography, and fractals. REFERENCES

of Sonya Kovalevskaia is the idea of

Daniel Aronofsky, Pi, film, Artisan Entertain­

effect. After many unsatisfactory years in her chosen field, Frances responds to an ad soliciting entries for a context for

ment, 1 998. David Auburn, Proof, London: Faber & Faber, 2001 . Dan Brown, Digital Fortress , New York: St.

Editeur 2002 (originally published as Dis-mol

Montrose

numbers,

apparently

con­

ceived by Woolfe as something like transfinite numbers that will turn the mathematical

world

upside

down.

Frances's alleged geometrical intuition pushes the concept beyond the first of these new numbers; she sees herself em­ ulating Tartaglia in winning the contest and changing the world. There are two themes in the book that strike a realistic chord. The first is the difficulty of combining mathemati­ cal research with primacy responsibility for child care. The second is the atmos­ phere of an international mathematics conference. With regard to the first, the Montrose women are more or less fail­ ures, although their frustration is well portrayed. Much about the second rings true, but I doubt that any woman math-

lishers, 2002. Laurie R. King, The Beekeeper's Apprentice, New York: St. Martins Minotaur, 1 994. -- , A Monstrous Regiment of Women,

New York: St. Martins Press, 1 995. , A Letter of Mary, New York: St. Mar­

qui tu aimes, je te dirais tu ha1s, 1 990).

P. M. Carlson, Audition for Murder, New York: , Murder is Academic, New York: Avon , Murder is Pathological, New York: , Murder Unrenovated, New York: Ban­ , Rehearsal for Murder, New York: Ban­ , Murder in the Dog Days , New York: , Murder Misread, New York, Double­

Gate Films, 2004. Sylvia Nasar, A Beautiful Mind, New York: Si­ lain Pears, An Instance of the Fingerpost, New Simon Singh, The Proof, video, WGBH, 2003. 1 995. 2001 . Everyday

Expressions,

London:

Jessica

Scott Turow, Presumed Innocent, New York: Farrar Straus & Giroux, 1 987.

day Books, 1 990. --

N H : Thomas T. Beeler Publisher, 2003. Michael Moore, Fahrenheit 911 1 , film, Lions

Kingsley Publishers, 2004.

Crimeline, 1 990. --

, The Game, New York: Bantam, 2004.

lan Stuart-Hamilton, An Asperger Dictionary of

tam Books, 1 988. --

--

Tom Stoppard, Enigma , film, Miramax Films,

tam Books, 1 987. --

, Justice Hall, New York: Bantam, 2002.

Tom Stoppard, Arcadia , New York: Doubleday,

Avon Books, 1 986. --

--

York: Putnam Publishing Group, 1 998.

Books, 1 985. --

, 0 Jerusalem, New York: Bantam,

1 999.

mon & Schuster, 1 998.

Avon Books, 1 985. --

, The Moor, New York: St. Martins

Press, 1 998.

Jane Langton, The Escher Twist, Rollinsford

Martin's Press, 1 998. Margot Bruyere, Maths a mort, Paris: Aleas­

velopment of her mother's work on

New York: Dodd Mead, 1 943. Philip Kerr, Dark Matter, New York: Crown Pub­

--

"radical" new ideas about mathematics,

winning contribution is the further de­

bers , Westport CT: Hyperion Press, 1 998.

Michael Innes, The Weight of the Evidence,

--

the reward to be an invitation to present the ideas at a conference in Athens. Her

Henry Holt & Company, 1 997. Paul Hoffman, The Man Who Loved Only Num­

tins Press, 1 997.

Woolfe employs. Borrowed from the life mathematical notes used as wall paper,

Robert Goddard, Out of the Sun , New York:

--

some idea of the level of sophistication

although obviously not to the same good

USA, 2000. Arthur Conan Doyle, Silver Blaze, Strand Mag­

, Bad Blood, New York: Doubleday

Sue Woolfe, The Secret Cure , Sydney: Pica­ dor, 2003.

Books, 1 991 . Desmond Cory, The Catalyst, New York: St. Martins Press, 1 991 . --

Department of Mathematics and Statistics

, The Strange Attractor, New York:

MacMillan, 1 992. --

Washington DC 2001 6-8050

, The Mask ofZeus , New York: St. Mar­

tins Press, 1 993.

American University USA e-mail: [email protected]

© 2005 Springer Science+Business Media, Inc., Volume 27, Number 2, 2005

91

Adventures in Group Theory: Rubik's Cube, Merlin's Machine & Other Mathematical Toys by David Joyner BALTIMORE AND LONDON, THE JOHNS HOPKINS UNIVERSITY PRESS, 2002. 280 PP. US $69.95 HARDCOVER ISBN 0-801 8-6945-0 US $22.95 PAPERBACK ISBN 0-8018-6947-1

REVIEWED BY GERHARD BETSCH

E

rno Rubik was born in Budapest. He studied architecture and design and is at present a professor of interior design at the Academy of Applied Arts and Design in his hometown, Bu­ dapest. In the mid-seventies he patented a cube-shaped mechanical puzzle that has since captured the imagination of millions of people worldwide. By 1982, "Rubik's cube" was a household term, and an entry in the Oxford English Dictionary. More than 100 million cubes have been sold world-wide. The more senior readers of The Intelli­ gencer will remember the time around 1980, when playing with a Rubik's cube was almost endemic. It was clear from the beginning that this toy could and should be consid­ ered in a systematic, or rather mathe­ matical way, with the tools of permu­ tation group theory. And very soon papers and books on "cubic" problems appeared in print. On the other hand, Rubik's cube and similar games or toys seemed to be a good opportunity, a good starting point for an introduction to group theory. The book under review is another attempt at an introduction to permuta­ tion group theory starting from fasci­ nating applications in "everyday life." To quote the author: "All the abstract algebra needed to understand the mathematics behind Rubik's cube, Lights out, and many other games, is developed here . . . . This book began as some lecture notes designed to teach discrete mathematics and group theory to students who, though cer­ tainly capable of learning the material,

92

THE MATHEMATICAL INTELLIGENCER

had more immediate pressures in their lives than the long-term discipline re­ quired to struggle with the abstract concepts involved." Joyner's book owes much to the monographs by Chr. Bandelow [ 1 ] and D. Singmaster [2]. A sketch of the contents may be in­ structive. Chapters 1 and 2 deal with elemen­ tary mathematical concepts: Some logic, sets, functions, relations; matrices and determinants; some elementary facts from combinatorics. Chapter 3 is devoted to permutations, with a fine introduction to "permutations and bell ringing," or campanology, which is the art and study of ringing cathedral bells by permuting "rounds" of bells. Cf. the murder mystery solved by Lord Peter Wimsey, in Dorothy Sayers's novel The Nine Tailors. This is fol­ lowed by a chapter on permutation puzzles, which also provides a first contact with the 2 X 2 and the 3 X 3 Rubik's cube. Chapter 5 presents some elemen­ tary group theory: the quaternion group, the finite cyclic groups, the di­ hedral and symmetric groups; the con­ cepts of subgroups, cosets, conjuga­ tion, actions and orbits. Chapter 6 expounds the mathematical structure in some mathematical models of some of the "Merlin's Magic Family" of games. The following chapter is de­ voted to graphs, and graphical inter­ pretation of permutation groups (Cay­ ley graphs). Chapters 8-10 provide more group theory: the Platonic solids and their symmetry groups; factor groups, direct and semidirect products, wreath prod­ ucts; and the machinery of generators and relations, as well as the presenta­ tion problem. Chapter 1 1 is a highlight of the book: the mathematical description of (legal) moves of the 3 X 3 cube, the cube group, and moves of order two. To this point the book has ad­ dressed the "general reader," or the "ordinary student." Chapters 12-14 are more advanced and address readers who have some knowledge of abstract algebra (say, undergraduates majoring in mathematics). A few key words from

these chapters: finite projective linear groups, Mathieu groups, Golay codes. The "cubic" enthusiast will be quite happy with the final presentation, Chap­ ter 15, which provides concrete solution strategies for Rubik's cube, the Master­ ball, the Skewb, and a few others. Most theorems in this book are given with proof. The author provides stimulating exercises, denoted as "pon­ derables." REFERENCES

[ 1 ] Christoph Bandelow: Inside Rubik 's cube and beyond, Birkhauser, Boston, 1 980.

[2] D. Singmaster: Notes on Rubik's magic cube, Enslow, Berkeley Heights, N.J., 1 981 .

Furtbrunnen 1 7 D 71 093 Wei! im Schonbuch Germany e-mail: Gerhard. [email protected]

Origami Design Secrets: Mathematical Methods for an Ancient Art by Robert J. Lang WELLESLEY, MA, A K. PETERS. LTD. 2003, ISBN 1 -5688 1 - 1 94-2 594 PP. US $48.00

REVIEWED BY THOMAS C. HULL

D

uring the past 35 years or so the art of origami has been in a ren­ aissance. The word origami is Japan­ ese and reflects their historical prac­ tice, some say dating back to the 1600s, of folding paper into representational and abstract shapes. But modern origami, in the sense of what one finds when looking at a typical origami in­ struction book, only dates back to the 1940s and 1950s. That was when a handful of individuals, namely the ma­ gician Robert Harbin of England, Lil­ lian Oppenheimer (the "grandmother" of origami in the USA) of New York, and paper-folding master Akira Yoshizawa of Japan, took it upon them­ selves to communicate and popularize this art to the rest of the world. As a result, the first mass-marketed origami instruction books began to appear, and

interest in the art slowly began to grow. Technical advances in folding tech­ nique were created. Most notably, Uchiyama from Japan attempted to classify origami "bases" from which many models can be derived, and the method of "box pleating" was invented by folders like Neal Elias and Max Hulme. This led to more complex origami model design, but nonetheless there were respected paper folders in the 1970s making statements like, "It is impossible to make a grasshopper us­ ing one sheet of square paper" ([5], p. 131). The general impression seemed to be that there were definite limits on what one could do with origami. This all changed when American John Montroll published his first book, Origami for the Enthusiast, in 1979 [6]. There it was, at the end of the book, the most complex origami model pub­ lished up to that time: a grasshopper with six legs, abdomen, thorax, wings, head, and two antennae, all folded from a single square of paper with no cuts. Furthermore, his methods used no box pleating, which was the most complex technique devised at the time. Rather, Montroll invented methods that were merely natural generaliza­ tions of the classic bases (fish base, bird base, and frog base) used by the ancient Japanese.

John Montroll's grasshopper

Needless to say, Montroll's work amazed everyone. Soon numerous pa­ per-folders were mimicking Montroll's techniques and causing an explosion in the level of complexity seen in origami. This was when Robert Lang, then a CalTech grad student, emerged with ever-more-challenging models, like a cuckoo clock (complete with cuckoo that pops out when the pendulum is pulled) from a rectangular piece of paper. And in Japan physicist Jun Maekawa was creating models like his devil (complete with eyes, nose, tongue, ears, horns, tail, legs, and arms with five fmgers on each hand), which took Man­ troll's innovations even further. Such models were breaking away from the concept of general "bases" and instead developing ways to collapse the paper that were individual for the subject at hand. Furthermore, some of these new models (Maekawa's devil among them) were so elegant and exhibited such eco­ nomic use of the paper that one sus-

pected, as with the classic Japanese crane and frog models, they were some­ how meant to be folded. Concurrent with this was the emer­ gence of an increased interest in the mathematics of origami. In fact, many origami enthusiasts at the time (Lang, Montroll, Maekawa, and Kawasaki, just to name a few) were either mathemati­ cians, physicists, or engineers by day. It is perhaps not surprising that people with scientific interests would be drawn to the challenge of origami design, where the rules of economy and the physical restraints of the paper must be overcome, making each model design a puzzle. Some of these same origamists began observing mathematical patterns among the creases in the models they were folding. Thus emerged the so­ called Maekawa's Theorem, which states that the difference between the number of mountain (convex) and val­ ley (concave) creases at a flat-foldable vertex in an origami crease pattern is al­ ways two, and Kawasaki's Theorem, which states that the alternating sum of the consecutive angles between creases in a flat-foldable vertex is always zero (see [1] and [2] for more details). The French mathematician Jacques Justin, who was also an origamist, discovered these same basic results at the same time, the 1980s. Mathematicians were

Illustration of the tree algorithm using one of Meguro's bugs. (Not all creases are shown.)

© 2005 Springer Science+Business Media, Inc., Volume 27, Number 2. 2005

93

paying attention to the on-going origami revolution and initiating origami-math research. As origami entered the 1990s, paper­ folders were beginning to approach complex design with more sophisti­ cated techniques. In fact, Fumiaki Kawahata, Toshiyuki Meguro, and Jun Maekawa in Japan and Robert Lang in the United States independently dis­ covered a connection between origami design and circle-packing. The idea is that when designing something like an origami insect, one needs to extract many appendages, like legs, wings, and antennae, from the square piece of pa­ per. Sometimes these appendages can be made from the comers or edges of the square. For example, the classic Japanese crane makes appendages for the head and tail from two opposite comers of the square and the wings from the other pair of comers. But a more complex subject might require appendages to be created from the in­ terior of the paper. In all these cases, however, we can think of the ap­ pendage as being like an umbrella, that is, a stick made from pleats radiating from a point. Folded up, we would have our appendage. Unfolded, we would see that the area of paper de­ voted to this appendage would ap­ proximate a disk. Thus when planning where appendages should be extracted from the paper, we can try arranging circles, possibly of different sizes for different-sized appendages, on the square. Our only requirement would be that the circles not overlap and that the center of each circle be contained in the interior or the boundary of the square. Finding an optimal arrange­ ment of such circles, where they might be arranged symmetrically for ease of folding or be as large as possible for ef­ ficiency of the design, would be the first step toward developing a crease pattern for a base that would have all the appendages we need for our origami insect. This is, of course, easier said than done. But it does suggest that an actual algorithm for origami design might be possible. Further, mathematical tools like Maekawa and Kawasaki's Theo­ rems among others, had emerged by the 1990s to help form crease patterns

94

THE MATHEMATICAL INTELLIGENCER

that would actually fold up. By the mid90s origamists were writing about such algorithms in detail. The Japanese de­ signers called their algorithm the bun­ ski, or molecule method, while Robert Lang called his the tree method of origami design. But their approaches were similar: start by drawing a stick figure, a tree graph, of your subject with weights on the edges to indicate their relative sizes. Then each leaf node with its edge weight l determines a cir­ cle of radius l that must be packed on the square. Find a way to arrange the circles, making them reasonably large and, for convenience, positioning them along axes of symmetry of the square, if possible. Then connect some of the centers of neighboring circles with creases to begin an outline of the crease pattern. This will result in various poly­ gons drawn on the square with comers at the centers of the circles. These polygons can then be collapsed by in­ serting more creases-the molecules, in the Japanese terminology-into each polygon. For example, a triangle can easily be collapsed by making creases on the angle bisectors. Collapsing each polygon should result in the paper transforming into an object whose out­ line resembles the tree graph with which we started. An origamist can then, with skill, make this look like the original subject. But there were many gaps in this al­ gorithm. How does one find an optimal arrangement of circles? What if the tree graph has lots of vertices which are not leaf nodes? And origamists had not then discovered ways of collapsing all possible polygons properly. Robert Lang spent a lot of time in the 1990s and beginning of the 21st century working on these details. The fruits of his labor were compiled into the in­ creasingly complex versions of his TreeMaker computer program, which al­ lowed a user to draw a stick figure, weight the edges, and then watch as the program used a constrained optimiza­ tion algorithm to fmd an optimal circle packing. After Lang discovered his uni­ versal molecule for collapsing any poly­ gon, his program was able to go on to produce a full crease pattern, which can then be printed out and folded. The book under review, Origami

Design Secrets, is Lang's magnum opus, collecting all this work in printed form. But this is not a mathematics text, nor is it a strict technical manual. Lang chose to strike a balance between a book that describes origami design algorithmically and one that appeals to the origami community. Thus, 151 of the 585 pages in the book are devoted to folding instructions for a variety of origami models. But even for non-ex­ perts at folding, these diagrams serve to illustrate the sheer complexity that can be found in modem origami. Nearly 30 of the diagram pages are de­ voted to one model: Lang's tour-de­ force Black Forest Cuckoo Clock (an insanely more elaborate version of his earlier Cuckoo Clock), which has leg­ endary status among origamists. And there are simpler models, like his Koi Fish and Snail, which illustrate the de­ sign techniques on a smaller scale. The book is written in a simple style that will seem elementary to most math­ ematicians, and will only disappoint people who are already deeply familiar with Lang's techniques. For mathemati­ cians and origamists alike, Lang's ex­ pository approach introduces the reader to technical aspects of folding and the mathematical models with clarity and good humor. Details of the underlying mathematics are sprinkled throughout, and those wishing for full details of the optimization side of the tree algorithm will fmd them in the last chapter. Two other features make this book especially valuable. One is the sheer abundance of gorgeous figures. This alone makes the book suitable for un­ dergraduates to understand (although Lang's clear writing will suit them too) and is a boon for readers struggling to visualize the tricks of paper-folding. The other is that Lang was not sat­ isfied with only describing his tree al­ gorithm. He also includes, with histor­ ical references, numerous other design techniques that are part of an origami creator's toolbox. These include graft­ ing multiple crease patterns together, splitting crease patterns to insert more details, box pleating, and even finer el­ ements of the tree theory like rivers and stub points. Being as complete as possible in this regard makes Origami Design Secrets a literal encyclopedia of

paper-folding methods. It also offers

field of

computational origami.

One

or even try their own hand at the com­

never-before-seen continuity and clas­

of the now-classic results in this field is

sification of all that has been discov­

the fold-and-cut theorem, which states

ered by origami designers over the past

that by folding a piece of paper one can

REFERENCES

30 years. Further, Lang had the foresight

remove with a single cut of a scissors

[ 1 ] T. Hull, The combinatorics of flat folds: a

to describe box pleating, which for over

any planar embedding of a graph (with

survey, Origam1'J: Proceedings of the Third

a decade has been considered by many

its interior). (See [3]) The proof of this

International Meeting of Origami Science,

plexity of origami model design.

as an old-school design technique from

result actually uses a method similar in

Mathematics, and Education , T. Hull ed. ,

the 1970s, in a more modem design con­

concept to Lang's universal molecule.

A . K . Peters (2002), 29-38.

text. This is especially helpful for those

Furthermore, one of the authors of this

who want to understand the models of

result is MacArthur Award winner Erik

the Connoisseur, Japan Publications, 1 987.

Satoshi

a young Japanese

Demaine of MIT, who has done more

[3] E. Demaine, M . Demaine, and A. Lubiw,

folder who for the past several years has

than anyone to galvanize this field. It is

"Folding and One Straight Cut Suffice, " Pro­

been redefining people's conception of

not

that

ceedings of the 1 Oth Annual ACM-SIAM

Kamiya,

an

exaggeration

to

say

[2] K. Kasahara and T. Takahama, Origami for

ultra-complex origami (see [4]). Kamiya

Origami Design Secrets

is required

Symposium on Discrete Algorithms (SODA

uses

reading for mathematicans interested

'99}, Baltimore, Maryland, January 1 7-1 9,

box pleating in ways no one

thought was possible. Lang's book is current with such developments. The book also points out open prob­ lems in this field. The biggest is that the tree algorithm only produces

uniaxial

bases, that is, bases whose flaps (ap­

in computational origami.

1 999, 891-892.

Those wishing to learn about math­ ematical methods in paper-folding will have a frustrating time sources.

Aside

from

finding re­

a proceedings

book from a 2001 conference (see the

pendages) all lie along a single axis, or

reference for [ 1 ] ) on origami in math,

line. There are a number of origami

science, and education, there are no

[4] S. Kamiya, Web page: http://www.asahi­ net.or.jp/�qr7s-kmy/ [5] T. Kawai, Origami, Hoikusha Publishing Co. , 1 970. [6] J. Montroll, Origami for the Enthusiast, Dover, 1 979.

bases without this property, and in­

detailed texts or monographs in the

corporating them into Lang's algorithm

field.

is a wide-open area for study.

huge void in that regard. It is highly rec­

Merrimack College

ommended for mathematicians

North Andover, MA 01 845

In fact, the problems dealt with in

Origami Design Secrets

Origami Design Secrets

fills a and

Thomas Hull Department of Mathematics

have grown

students alike who want to view, ex­

USA

over the past decade into the emerging

plore, wrestle with open problems in,

e-mail: [email protected]

Acknowledgment We acknowledge with thanks that Lyon P. Robinson of Sydney contributed the photographs in the article about the Sydney Opera House in vol. 26 (2004), no. 4, 48-52.

© 2005 Springer SCience+ Business Media, Inc., Volume 27, Number 2, 2005

95

'"-J@ij,j.Mg.h.i§i

Robin Wilson

The Philamath's Alpha bet-H

]

and geometrical optics, and revolu­

known for his important contributions

tionised algebra with his investigations

to number theory-particularly, trigono­

into non-commutative systems. This

metric sums and Waring's problem­

stamp commemorates his discovery of

and to the study of several complex

quaternions.

variables.

Heisenberg: Werner Heisenberg (1901-

Revolution of 1966-1976, he travelled

Throughout

the

Cultural

1976) took an algebraic approach to

widely through China, lecturing on in­

quantum mechanics, realising that quan­

dustrial mathematics to audiences of

tities such as position, momentum, and

up to 100,000 factory workers.

Halley: Edmond Halley ( 1656-1742) is

energy can be represented by infinite

Huygens: The scientific contributions

primarily remembered for the comet

matrices. Using the fact that matrix mul­

of Christian Huygens (1629-1695) were

whose return he predicted and which

tiplication is non-commutative, Heisen­

many and varied. He expounded the

is named after him. Having observed

berg deduced his 'uncertainty principle',

wave theory of light, hypothesised that

the comet in 1682, he predicted its re­

that it is theoretically impossible to de­

Saturn has a ring, and invented the pen­

turn in late 1758 or early 1 759, and its

termine the position and the momentum

dulum clock and spiral watch spring. In

of an electron at the same time.

mathematics he wrote the first formal

Isaac Newton's theory of gravitation.

Hipparchus: The first trigonometrical

probability text, introducing the con­

Earlier, Halley had been influential in

approach to astronomy was provided

cept of ' expectation', and analysed such

persuading Newton to develop this the­

by Hipparchus of Bithynia (190-120

curves as the cycloid and the catenary.

ory and publish its conclusions in the

BC), possibly the greatest astronomi­

appearance

did

much

to

vindicate

Principia Mathematica.

cal observer of antiquity. Sometimes

Hamilton: William Rowan Hamilton

called 'the father of trigonometry', he

(1805-1865) was a child prodigy who

discovered

discovered an error in Laplace's trea­

equinoxes, produced the first known

the

precession

of

the

tise on celestial mechanics while still a

star catalogue, and constructed a 'table

teenager and became Astronomer Royal

of chords' yielding the sines of angles.

of Ireland while a student. He did im­

Hua Loo-Keng: Hua Loo-Keng [Hua

portant theoretical work in mechanics

Luogeng]

( 1 9 1 0-1985)

is most well

Hipparchus

Hamilton's quatemions Hua Loo-Keng

�p · �q � h

Halley

Htiun�hz Uns
Please send all submissions to the Stamp Corner Editor, Robin Wilson, Faculty of Mathematics,

The Open University, Milton Keynes, MK7 6AA, England e-mail: [email protected]

96

Heisenberg

THE MATHEMATICAL INTELLIGENCER © 2005 Spnnger Science+Bus1ness Media, Inc

Huygens