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.
Comment on
1r
Is Wrong!
In his note in the Mathematical Intel l ig en cer vol. 23(2001) no. 3, 7-8, Bob Palais gives his reasons why he thinks the symbol 1r should have been be stowed on 27T. It is of interest to note that Albert Eagle in his book The El liptic Functions As They Should Be (Galloway and Porter, Can1bridge, 1958), thinks 7T should have been be stowed on 7T/2. His reasons are given in his preface, and I quote. There are one or two other impor tant innovations I have made which have nothing to do with elliptic functions and the first of which may cause an outcry at first; but its im mense convenience must surely soon be realized. Most mathemati cians must have often wished that there was a single symbol that could be written instead of +7T or 7T/2. When one thinks that for the ellip tic functions we have the conve nient symbols of K and K' for the two quarter periods, it is really too absurd that for the circular func tions, which are employed millions of times as often as the elliptic func tions, we have no symbol for the quarter period at all, and have to ex press it as "half the half period." The letter 7T should have been used to denote but the meaning of 1r can obviously not be changed now. But the Greek letter T, with its one leg instead of two, so closely re sembles 7T cut in two that I have ap propriated that letter exclusively for +7T. Consequently I ask my readers every time they see it to say to them selves "half-pi" or "pi by two" in stead of "tau" or "taw." If a shorter name is desired for it what could be more appropriate to imply half-pi than "hi"? How convenient it is to write T as the upper limit of a trig integral instead of And how ,
+'IT,
f7T!
convenient it is to say "the integral of sin "I:J dI:J from nought to hi" since the integrand is zero at the lower limit, and the upper limit is its high point! It is natural that the practical man, measuring the diameter and the circumference of a cylinder, should want a symbol for the ratio of the two lengths. But a pure math ematician, noting that a diameter of a circle divides the circumference into two halves, would think it more reasonable to introduce a symbol for the ratio of half the circumfer ence to the diameter. And he, per haps rather surprisingly, would be showing better common sense about the matter than the practical man did! Seriously, who can want to have e -!7T or e - 7TI2 printed instead of e-7? Or who won't much prefer to write T than 7T for the upper limit of a trig integral? Those who peruse the numerous formulae in my book will, I think come to see that the finding of a sin gle Greek letter T to stand for +7T was a necessity that was forced upon me. How immensely nicer books on Fourier's Theorem would look with it! (pp. ix-x).
+
Murray S. Klamkin Mathematics Department University of Alberta Edmonton, Alberta T6G 2G1 Canada e-mail: mklamkin@math. ualberta.ca
. . . and again
•
.
.
I agree with Bob Palais's 7TOUS Opinion, although some might think it 27TOus. Charles W. McCutchen Camp Asulykit
Lake Placid, NY 1 2946-9600 USA
© 2002 SPRINGER-VERLAG
NEW YORK. VOLUME 24,
NUMBER 2. 2002
3
R. MICHAEL RANGE
Extension Phenomena in Multidimensional Complex Analysis: Correction of the Historical Record
omplex analysis in several variables is not a mere extension of the familim· theory in the complex plane. Novel extension behavior is found, with deep consequences for the nature of complex analytic functions. This article is a brief introduction to some of these fas cinating phenomena, both local and global. In it, I also take the opportunity to correct major inaccuracies in the histor ical record, as it has appeared in much of the pertinent lit erature since the mid-1950s. In particular, I wish to put on record important, yet generally overlooked or forgotten con tributions by Francesco Severi, Hellmuth Kneser, and Gae tano Fichera. Thus this article may have much to say to his torians, as well as to experts in multidimensional theory. Preliminaries
To set the stage, I briefly review the central concepts. Com plex Euclidean space IC" = [z = (zb . .. , Zn): ZJ E IC} is a vector space of dimension n over IC. The familiar identifica tion of IC with IR2 extends to a natural identification of IC" with IR2" , thereby giving immediate meaning on IC" to all con cepts familiar from multivariable real analysis. A complex valued C1 (i.e., continuously differentiable)1 functionf: U--'> IC on the open set U C IC" is holmnorphic on U iff is holo morphic in each variable separately, that is, if it satisfies the Cauchy-Riemann equations iJf/iJZj = 0 on U in each variable Zj (j = 1, . . . , n) separately.2 The space of holomorphic func1
It is standard that the
4
Hartogs's Theorem
We begin with the following astonishing result discovered by F. Hartogs [Hartogs 1906]. It is fair to say that this dis covery marks the birth of multidimensional complex analy sis as a new independent area of research.
Theorem 1. Suppose D c IC" is open and bounded, with connected boundary iJD. If n � 2, then every fE O(iJD) (i.e., holomo1phic on some open neighborhood of iJD) has a holomorphic extension to D. The crux of the proof is contained in the following sim ple special case.
Lemma 2. Suppose K is compact and n � 2. Then every bounded f E CJ(IC"\K) is constant on the unbounded com pon ent of IC"\K.
C1 hypothesis can be replaced by continuity alone. Much deeper is a result of F. Hartogs that complex differentiability in each variable sepa
a/a�= 112(iilox1 + i a/ay;). for j
rately already implies continuity in all variables jointly. 2Recall that
tions on U is denoted by O(U). Iteration of the one-variable Cauchy integral formula for discs yields an analogous for mula on products of discs (so-called polydiscs), which read ily leads to other standard properties of holomorphic func tions J, such as the fact that such functions are in C"(U), and have local representations by power series.
=
1,
. . .
, n.
THE MATHEMATICAL INTELLIGENCER © 2002 SPRINGER-VERLAG NEW YORK
side. In fact, Hartogs's first example of simultaneous ana lytic extension arose in the following context.
2 Lemma 3. Consider H = {(z, w) E C :lzl < 1, 1/2 < l wl 1 2, l wi < 1}. Then evmy fE C0(H) has a holo < 1} U liz I< / morphic extension to the bidisc P = {(z, w): lzl < 1, lwl < 1}.
1""
E
Figure 1 . Lemma 2: The complex line {(z, w'): z the ball of radius R.
C} does not meet
Clearly this Lemma includes the obvious generalization of Liouville's theorem (K = 0) to n variables. Thanks to the ex tra variable, one easily obtains the stronger statement given here, as follows: ChooseR so large that K C {z: lzl < R}. Then t for fixed w' E p- (here we use n � 2!) with lw'l > R, the complex line {(Zt. w') : z 1 E C) does not meet K, so that by the one-variable Liouville theorem,fCz t. w') is constant in z1. Interchanging variables, it follows that fis constant in each of the other variables as well, whenever the remaining vari ables are fixed with sufficiently large modulus. It follows that jis constant outside a large ball, and hence, by the identity theorem, on the unbounded component of C"\K.• Given the Lemma, the idea of the proof of Hartogs's the orem is now very simple, at least conceptually. Choose domains D1 and D2 with (piecewise) C1 boundaries, with D1 cc Dcc D2, so that fE fJ(D2\Dt) and C"\D1 is con nected. In case n = 1, the Cauchy integral formula gives f(z) = rCz) - F(z) for z E D2\Dt. where j+(z) =
�
2
Lv2 ��df,
f-(z)
=
� ��df.
2
Lv1
Clearly r E C:J(D2), FE V(C\Dt), and limzl_,"' f-(z) = 0. In higher dimensions, a corresponding integral formula (the Bochner-Martinelli formula) yields the same decomposi tion with analogous properties. (I will discuss this more in detail below.) Hencej-is bounded outside a large ball, so if n � 2, the Lemma implies thatj-is constant there, and hence must be 0. The identity theorem implies thatj- ""'0 on C"\Dt. and hence f=j+ on D2 \ D1• So j+ does indeed extendjto D2.• Hartogs's argument in 1906 was actually quite different. First of all, there was no appropriate higher-dimensional integral representation formula known at that time, which would have yielded the decomposition f= j+ - f-in ar bitrary dimensions. More significantly, Hartogs's point of view was directed to the i nterior of D, and not to the out-
-------- ----
lwl
H
lf.J.
0
Jzl
Figure 2. Lemma 3: Representation of H in the (JzJ, JwJ)- plane (the
absolute plane).
<
The proof is, again, surprisingly simple. Choose 1/2 < 1, and consider
F(z, w)
=
1 f(z, ;;)d?; . . u.• 2 m '�=r . !>r
-
J
r
_
It follows readily by basic results that F is holomorphic on 1 ,f(z,·) W =liz I< 1, lwl < r}. Now for fixed z, with lzl < !2 is holomorphic on the whole disc lwl < 1, so F(z, w) = f(z, w) by the Cauchy integral formula. Hence F =f on H n W, and F defines indeed a holomorphic extension off• The geometric setting in which this latter argument can be applied is quite flexible. Via deformations and identity theorem, it can be adapted to obtain global versions. Bar togs's 1906 proof of Theorem 1 was based on these ideas. The proof of Theorem 1 presented here arose from ideas discovered much later by R. Fueter [Fueter 1939). Fueter had developed a function theory for quatemions, and proved a corresponding Cauchy integral formula. Relating quatemions to two complex variables, he was able to de duce the fundamental decomposition f= j+ - f- when n = 2, and complete the proof of Hartogs's theorem along the lines given above. 3 Shortly thereafter, Fueter's papers provided the inspiration for E. Martinelli [Martinelli 1942/43] and, independently, S. Bochner [Bochner 1943], to replace Fueter's quatemion integral formula by the integral formula they had discovered around that time (the now
3Fueter also handled the case of more than 2 variables by utilizing a function theory for more general hypercomplex variables [Fueter 1941 /42].
VOLUME 24. NUMBER 2, 2002
5
formulation, it is not hardly anyone out id the circle of experts in e\ eral omplex variables took n ti . This condition be cam quite a bit more widely known in the 1960s, after L. Ehr npr is rectiscovered it (in ( ) in c mlS of the existence
Rez
functions.5 Given thi
holomorphi
urprising that
·
of olutions
with compact �upport of th "
u
auchy-Riemann equations au
inhomogeneo':!_S
f. whenever the given a cl sed (0, I)-form/had compact support [Ehrenpreis 1961]. Ehrenpreis's proof of Hartogs's Ill r m became the stan dard proof for generations of analysts, especially after it gained wide publicity by its inclu ion in L. Honnander's classi treatise [Hommnder 1966]. It hould be mentioned that Ehrenpreis proof was an ·elementary by-product of d p inve tigations of extension ph nomena for solutions of gen -ral systems of partial.ffiff r -ntial operators.6 =.
Tangential Cauchy-Riemann Equations
w
There is a version of Hartogs's theorem that involves
tions detined
·plane
trinsic differential
Figure 3. The "tomSito can principle'': A function holomorphic in
neighbomood of the oper�ed side of 1he whole
can.
can
op.
a
elrter�ds holomorphicalfy to the Ina
formul !Martirlelli 1938], cit.J).-!Martinelli and Bochner were thus able
to r cast Fi.Ieter's arguments completely within standard compl x analysis and thereby compl t the proof of Har togs' lh or· ru as discussed her . It hould be noted that Bo hner's approach was more g neral. He applied Fueter's argun1ents to Green's fonnula in real analysis, thereby ob
taining
a
Hartogs-type extension result for certain har
monic functions in a neighborhood of the bonndary i:JD whi h atisfy an additional partial-differential equation of a p cial typ , such as for exampl the Cauchy-Riemann equations for n ;::: 2. Specializing to the complex case, Bo hner then obtained essentially the same proof as M ar tinelli. I hall return to Bochner's fundamental 1943 paper,
lat ron. Bv n
utiliz
thougll
d th
n
Fueter, Martinelli,
and Bochner
� 2 variant of Liouville'
de
facto no
theorem, there is
i.ndi ation that lhey fuUy r:-ecogniz d the
particu
C" \D. After researchers turm d lh ir attention to more abstra t paces, uc.h as com pi x manifolds, the relevant condition was made explicit by J. P. S rr [Serre 1953]. True to th time , Sen·e stated tlus onditi.on as HJ(X,C) = 0 (n 5: 2), that 1s, the vanish ing of the first cohomology group with mpact .supports of th pace X with coefficients in th sheaf (') of germs of lar prop rty of the
complement
r:-elevant
on
the
func
boundary, which satisfy an in
known as the "tangential
condition,
auchy-Riemao.n equations."
·Lassical Bochner-Martinelli
[Bo hner
only
Theorem 4. Let D c Cr1 be bounded, ·with connected dif ferentiable boundary :JD ojcla C1• upposefE C1(JD) atisfie dJ l\dz1A ... 1\d.z., = 0 n iJD. If n � 2, theu f agr e with the bou11dmy t•alue of a lwtomorphic junc tion on D, i.e. there e.rists FE D) n G(D) with F =I on aD.
This theor�m was first proved
[Severi 1931) in case the boundary
in 1931
by Francesco Seven
aD an.d the
given
func
real-analytic. In 1936 H Umuth Kneser [Kneser 1936] the theorem in c� in case the boundary is of class Cl and stri tly pseudoconvex. 7 The general differentiab!.e v rsi.on without any convexity assumptions was proved b Ga tano Fichera [!Fich ra 19571, who required tile bound ary to b of class C1 +� any £ > 0. A few years thereafter, tion a.r
pwv- d
Martin lli obtained a different simpl· r proof, valid w1der the
[Mrutin Jli 1961]. nJortunately, these pionee1ing inv stigations have been large�y overlooked in the lit-raturc. Since the mid l960s, this result has quite univ rsally been referred to as minimal hypothesis stated here
Bochn r's theorem, with refer nc
per [op. cit.]. k y
id
Jffi
at the core
I shall discuss the
of these contributions and attemp \.he confusion in the publish d r ord. holomorphic in a neighb rho d U of iJD, then
as
clear up
to S. Bodmer's 1943 pa·
In the rest of thi arti l
to
j df= > _· - on U, )�l iJZjdZ n
(J
and hen the hypothesis in the theorem is satisfied. So thi theor m indeed generalizes Hartogs's theorem. The
'Note that t.1artin li's paper is from 938. while the tootnote co 111e fiJSI page or Bocl1ner·s 1943 paper indicates that Bochner had lectured on the formula ·as early Wimer 1940/41: naware of Maltl ·s ear6er paper 6a:red on the pvblshed rl;lCQI"d, tlleor"9 IS lh1.1s no Q\.leSiion about M
6
"I"J.IE MATJ.I5MA11CAL INTEUIGENCE�
novel feature in Theorem 4 is that f and its differential df live intrinsically on the boundary. Obviously one can al ways extend f to a differentiable function in a neighbor hood of aD; in that case the differential df appearing in the theorem must be understood as the pullback t*(df) to the boundary. The hypothesis can be reformulated more explicitly in terms of the tangential Cauchy-Riemann equations, as fol lows. Suppose aD is described as [z: r(z) = OJ, where r is a C1 function with dr =I= 0 on aD. A differential operator
n
(i.e., a tangent vector to e of type (0, 1)) is tangential to the boundary at p E aD if
ar ) ar-:=(p) = o . i
dZj
Such tangential Cauchy-Riemann operators act on C1 func tions on the boundary. It involves just routine arguments to verify that df 1\ dz 1 1\ . . . . /\ d z 11 = 0 at p E ciD if and only if L(f ) = 0 for all vectors L of type (0, 1) which are tangential to aD at p. In modem terminology, this is ex pressed as abj = 0, and a C1 function! on an open subset W c aD which satisfies df 1\ dz 1 1\ . . . . 1\ dz, = 0 on W is called a CR ( = Cauchy-Riemann) function on W. Note that in case n = 1 every f E C1(aD) is trivially a CR function. Functions on hypersurfaces in IC", or on submanifolds of higher codimension, which satisfy the tangential Cauchy Riemann equations, were first introduced by W. Wirtinger [Wirtinger 1926], who invented the calculus of the differ ential operators a/az1 and a/aZj , which turned out to be so remarkably useful in multidimensional complex analysis. Such functions arise naturally as the restrictions of holo morphic functions to a submanifold M, or as differentiable boundary-values on aD of holomorphic functions on D. Starting from the intrinsic system of tangential CR equa tions on M, Wirtinger launched the program to develop an existence theory for solutions of such systems, viewing it as an extension of Riemann's investigations of the analo gous question in the (real) two-dimensional case. Wirtinger ran into formidable new difficulties, and his investigations were mainly limited to formal considerations. Still, even though the CR theory literature hardly mentions Wirtinger any longer, it is fair to say that Wirtinger's 1926 paper ini tiated CR function theory, which began to blossom toward full maturity only 30 years later. The Real-Analytic Case
The first substantial result in CR function theory is proba bly Severi's 1931 proof of Theorem 4 in the real analytic case.I shall discuss his arguments in some detail, as they involve a beautiful and transparent application of the "metoda dell'estensione dal reate al complesso," introduced and used successfully by Severi in related investigations. The heart of the matter is the following local extension re sult, which had of course been known in dimension 1 since "antiquity. "
Proposition 5. A.sswn e M is a real-analytic hypersu r face in an open set in IC", and f: M � IC is a real-ana lytic CR function. Given P EM, there e.rist a neighbor hood U of P and FE O(U) w i th F = f on U n M.
Proof As a warm-up, let us consider Severi's argument in case n = 1 first. (The result is essentially trivial in this case: if M C IR, just replace x by z in the Taylor series off; the general case follows by applying a holomorphic coor dinate change.) We thus assume that M is a real-analytic arc through P = 0, which-without loss of generality-can be assumed tangential to the .r-axis at 0. Hence M can be described near 0 by ( r(.r,y ) = y - g(:r) = 0}, where g is a real-analytic function with dg(O) = 0. Replacing .r = (z + z)/2 andy = (z - z)/2 i in the power series representation of r, one obtains a power series in two complex variables r* (z,w), i.e., a holomorphic function in (z, w), which agrees with r(x,y) for w = z. Clearly ar*/an' =I= 0, and hence the implicit function theorem implies that r*(z,w) = 0 defines a holomorphic function w = h(z), such that r*(z,h(z)) 0. Analogously, the given real-analytic function f on M (no other restriction when n = 1) leads to a holomorphic func tion f*(z, w) with f*(z , Z ) =f(x, y). It follows that F(z ) = f*(z ,h(z )) is a holomorphic function of z near 0, which on M-i.e., at points z where h(z) = z-agrees withf Now consider the generalization of this argument to higher dimensions. The one-dimensional argument yields an extension F which is holomorphic in the normal direc tion, and the main point is to make sure that the tangential CR equations on M will be satisfied for the extension as well. We may again assume that near 0 one has M = [ r(z) = Yn - g(x,yb . . . Yn - d = 0}. We may then extendjfrom M to a function on a full neighborhood of 0, which is inde pendent of y, . A frame for the tangential vector fields of type (0,1) is given by =
ar a ar a Li = ---=- --= - --= - --=az, az1 az1 az"
U=
1,
...,n-
1).
Notice that by the special fom1 of r, the coefficients of this frame are independent of y11, and therefore the tangential CR equations Ld = 0, which, by hypothesis, hold on M i.e., when Yn = g(:r, Yb . .. Yn-1)-hold in a full neighbor hood of 0 as well. As in case n = 1, one obtains a holo morphic function r*(z,w) in 2n variables, which agrees with r(z ) if w 1 = Z1, . . . . , W 11 = Z11 • The equation r*(z,w) = 0 can be solved for Wn = h(z 1 , z,, w 1 , . . . , W11 - 1 ), where h is holomorphic in 2n - 1 variables. It follows that M = [z = (zb . . . , Zn-b z, ): z, = h( z , z1 , . .. , z,_I)J. Analo gously, the extended real-analytic function! can be written as f(z ) = f*(z , Z ), where f*(z , w ) is holomorphic in (z , w ) near 0 in IC2". Then F(z ) = f*(z , zb . . . , Z11-1 , h(z1 , ... z,, z1 , . .. , Z n-d) agrees withf(z) on M.Note that F is holo morphic in z,. Now for j = 1, . . . , n - 1, by the chain rule, aF!aZj = aj/aZj + aj/azn all/au] . Similarly, by applying a/aZj to r*(z, z', h(z , z')) = 0, one obtains ar!aZj + ar/az11 ahlillL) = 0. Upon solving for ilh /auj and substituting, it fol lows that .
•
.
VOLUME 24, NUMBER 2. 2002
7
aF az;
_
_jf_ az;
_
U n aD which has property (A) can be extended continu ously to a junction holomorphic on V n D.
( ar )-1
a j i!!:_ az, az; a z,,
( ar )
= --:=dz,
So F is also holomorphic in
Just as in the real-analytic case, Kneser obtains the fol
-1LJf = 0.
lowing version of Theorem
... , Zn-1·• The proof of the global Theorem 4 in the real-analytic Zt,
case is now an immediate consequence of Hartogs's theo rem.In fact, the local holomorphic extensions of f given by
to a holomorphic function F in a neighborhood
U of aD.
the first part patch together-via the identity theorem If
n � 2, Hartogs's theorem extends F to a holomorphic
function on D.•
Kneser's Extension Theorem
4.
Corollary 8. Suppose D is a strictly Levi pseudoconvex domain with connected boundary. Then evay continu ous junction f E C(aD) with prol!!!. rty (A) on aD extends to a holomorphic junction FE C(D) n fJ(D) 1vith F = f on
aD.
proof. Suffice it to say that one may assume that aD is
This is not the place to present the details of Kneser's
strictly convex at P, and that the extension F is defmed for
z E V n D by an integral formula
The local extension result fails in general for differentiable
CR functions. Yet it is a most remarkable higher-dimen sional phenomenon, that under suitable local convexity conditions on the hypersurface, local
CR functions extend
holomorphically to at least one side! This was discovered
1936 by Hellmuth Kneser. Kneser considered strictly (Levi) pseudoconvex hypersurfaces in IC2. This somewhat technical condition showed its importance in 1912 when first in
E. E. Levi discovered its central role in the characteriza tion of the natural boundaries of regions of holomorphy. The more general version known as
pseudoconvexity is a
concept peculiar to higher dimensions, which even today
is the object of deep investigations. All that's relevant here
is that there is a simple quadratic local holomorphic change
of coordinates which turns a strictly pseudoconvex hyper surface (locally) into a strictly convex one, i.e., with a defin ing function whose real Hessian is positive-definite on tangent vectors. In fact, this property-well known to all practitioners of several complex variables-seems to have been proved and used explicitly for the first time by Kneser. It also is remarkable that Kneser replaced Wirtinger's dif ferential condition, the tangential
CR equations, by a more
general integral condition, as follows.
Definition 6. A continuous junction f is said to have property (A) on the hypersurjace M C IC2 if the integral .f(z)dz11\dz2 = Ojor any 2-dimensional smface B C M which bounds a 3-dimensional region in M.
fs
It follows immediately from Stokes's theorem that a holomorphic function f on an open set
M c U. A
U enjoys
property
fJ(D) extends continuously to the bound ary aD, thenjhas property (A) on aD. Kneser also shows that for fE C1(M) property (A) is equivalent to f being a (A) on any hypersurface
shows that ifjE
continuity argument
CR function.
Kneser's main theorem then is the following [Kneser
1936).
Theorem 7. Assume that PE aD, and that there is a neighborhood U of P such that U n aD is strictly Levi pseudoconvex (this requires in particular that aD is of class C2 near P). Then there exists a neighborhood V C C U of P such that every continuous junction f on
8
THE MATHEMATICAL INTELLIGENCER
where B caD n U is a suitable closed surface, and l{"l, tPl are complex affme linear functions which depend holo
t1CzJ · t2CzJ
morphically on
z, with
z,
and which vanish simultaneously only at
i=
0 on
B.
Property
(A) is used to prove
B and t1Czl, t2Czl, subject to appropriate restrictions. The proof that limz--- q F(z) = f(q) for q E V n aD involves specific choices of B and explicit estimations. that this definition is independent of the choices of
Kneser's pioneering work should have been the catalyst for further investigations of
CR functions, but it appears to
have remained largely unnoticed. Interest in those years was focused mainly on the principal global problems on domains of holomorphy as they had been formulated, for example, in the influential Behnke-Thullen Ergebnisbericht [Behnke and Thullen
1934]. The war severely disrupted sci
entific communications in subsequent years. Times were very different 20 years later, when Theorem 7 and its corollary were rediscovered-for CR functions of class C1-by Hans Lewy [Lewy 1956]. Lewy made reference to Severi's 1931 result, and most likely he was also famil iar with Wirtinger's 1926 work; in fact, he formulated the hypothesis on fin terms of Wirtinger's differential condi tion. Yet he seems to have been completely unaware of Kneser's
1936 result. Lewy's paper appeared at a time when
researchers in partial-differential equations-P. Garabe dian, L. Nirenberg,
D. Spencer, and others-had begun in
earnest to look at overdetermined systems, and in particu lar at the several-variable version of the Cauchy-Riemann
1956 paper, H. Lewy [Lewy 1957) used the tangential Cauchy-Riemarm operator on the
equations. Shortly after the boundary of the ball in ferential equation in
IC2 to produce a smooth linear dif
3 real variables without any local so
lutions. (More accurately, Lewy used the unbounded ver sion of the ball, i.e., the Siegel upper half space, whose boundary is the well-known Heisenberg group.) This phe nomenon, completely unexpected at that time, highlighted the need to investigate tangential
CR equations, and Lewy's
work gained immediate wide recognition. After Lewy's pa pers,
CR function theory grew at an ever-accelerating pace,
eventually developing into a major research area at the in-
Knese r can be
found
in a commemorative article written
on the occasion of his lOOth birthday [Betsch and H o fmann
1998). The bibliography in that paper does include Kneser's
paper, although the authors seem unaware of its signifi cance. Certainly Kneser was still very active mathemati
cally in the late 1950s,
and remained keenly interested in new mathematical developments. Thus it is quite likely that he did learn of Lewy's work, especially since Kneser and Lewy surely had known each other in earlier days. Kneser earned his doctorate in 1921 with David Hilbert in Gottin gen, and then was Richard Courant's assistant during the time that Lewy was one of Courant's doctoral stud en ts. Of course, Courant and Le,vy, along with many others, were forced to emigrate, and it is not known whether Kneser had any contacts with Le\vy in later year.s. Perhaps the mystery is rooted in Hellmuth Kneser's per sonality. Colleagues and students describe him as a modest and generou� man, especially in his relations with yoWlger mathematicians, and it is reported that on other occasions Kneser tactfully remained silent when younger mathemati cians rediscovered with much effort results that Kneser him self had obtained long before [Betsch and Hofmann, op.
and K H.
Figure 4. Hellmuth Kneser working outdoors at the Oberwolfach In· stitute in 1952. He served as Director of the Institute in 1958.
Hofmann,
personal communication).
cit.,
The Global CR Extension Theorem
of the local CR extension the have sparked interest in the corresponding global question; it seems likely that meanwhile the global problem had remained in the back of Severi's mind from the time he proved the real-analytic ver Publication of Lewy's proof
orem in
the differentiable
case may
tersection of several complex variables and partial-differ
sion. In particular, Severt mentioned the problem to Fichera
nom enon discovered by Kneser and rediscovered by Lewy,
Fichera's investigations.
ential equatio ns. In particular, the local CR
extension phe
inspired numerous investigations and generalizations. The definitive version for real
CR extension (at
C2
hypersurtaces
M states
that
least to one side) is possible near a point
P EM if and o n ly if M does not contain any germ of a com plex hypersmface through P [Trepreau 1986]. Not only Lewy,
but in fact j ust about everyone else at that
time and in the years thereafter, was unaware of Kneser's
remarkable
1936 paper. In spite of Hellmuth Kneser's stature
[Fichera 1957, p. 707], providing the stimulus that led to Fichera approached the pro blem from the perspective
of real analysis. The presumed extension is n ecessarily the known unique solution of the (harmonic) Dirichlet prob
lem for the given boundary values. Thus one is led to search for conditions on the boundary which, via Stokes's theo
rem, would force the harm onic extension to be holomor
phic.
Pursuing this idea,
Fichera was able to
identify suc h
a necessary and sufficient condition for functions f whose
harmonic extensi on had fmite
Straube. Subsequently I discovered that Kneser's result is
Dirichlet integral; it was ex terms of moment conditions dw 0 for all (11, n - 2)-forms w with polyn omial coefficients. In case f E C1, Fichera showed by integration by parts that this lat
sequent publications (see, for example ( Wells 1974}), Wells omits any further reference to Kneser, and he credits the lo cal CR extension result only to Hans Lewy.9 The question arises of why Kneser himself did not men tion or remind anyone, including students and coUeagues in Tubingen, of his 1936 work Personal details about
contrast of Theorem 4, Fichera's proof does not make use of Hartogs's theorem, but instead yields yet an other proof of that result as welL Inspired by Fichera's result, Martinelli (MartineiJi 19tH] discovered that his earlier integral formula proof of Bar togs's theorem could be strengthened to produce a simple direct proof of TI1eorem 4. The additional ingredient re-
as a leadin g mathematician of his times, and in spite of the
growing interest in
CR function theory,
Kneser's
remained �forgotten." I fll'St learned of it in
mentioned in
a
1936 pap r
1999 from E.
survey article by G. Fichera [Fichera 1986,
p. 69], and also in a paper by R. 0. Wells [Wells 1968} (so far the earliest post-war reference I know.)8 However, in sub
pressed in
j,,0f
=
ter condition is equivalent to the classical Wirtinger condi
tion; i.e.,/satisfies the tangential CR equations. In
to the earlier proofs
Btn a footnote (p. 69, op cil.). Fichera credits E. L. Stout with telling him about Kneser"s paper. Stout ni tum. panted me toward Wells's papers. 9Asked about this matter. Wells replied in June 2000: "I don't have any reason for having deleted the reference later, except that I was probably cmcentrating on other things.·
VGLlJM� 24. NUMBER 2. 2002
9
quired was a generalization of the classical "jump" formula for the one-variable Cauchy kernel to the Bochner-Mar tinelli integral kernel KBM· Without getting into the techni cal details, let me just mention that this kernel KBM is a nat ural generalization of the one-dimensional Cauchy kernel d�/21Ti(� - z). Even though KBlvi (?, z) is no longer holo morphic in z in dimension greater than 1, it still shares im portant features with the Cauchy kernel. For example, both Martinelli and Bochner knew and used in their respective proofs of Hartogs's theorem in 1942/43, that F(z)
=
Ji1D fWKBM (?, z)
defines a holomorphic function on IC" \ aD whenever f is a holomorphic function in some neighborhood of aD. Mar tinelli now noticed that this conclusion remained valid iff satisfied only the tangential Cauchy-Riemann equations. He then replaced the argument of Fueter by proving the following generalization of a classical one-dimensional re sult. Lemma 9.
some F-
=
Suppose aD is of class C1 andf E Lip" (aD) fo r 0 (e.g., f E C1(aD)). Then both F+ = FID and FIC" \ D extend continuously to an, and a >
f(z)
=
F + (z) - F - (z) for z
E aD.
As in the proof of Hartogs's theorem given above, The orem 4 is now an immediate consequence of this Lemma. In fact, iff is a CR function, then F+ and F- are both holo morphic, and since limz�x F- (z) = 0, Lemma 2 implies that F - """ 0 when n � 2. Sof = F+ on aD, and the proof of The orem 4 is complete. (By refining these ideas, one can show that the holomorphic extension F is even in C1(D). More generally, if aD is of class Ck and f E Ck(aD), then F E Ck(D). This seems to have been noticed first in 1975 [Cirka 1975], [Harvey and Lawson 1975]; a simplified version of their proof is given in [Range 1986].) Bochner's 1 943 Paper
I already mentioned that since the mid-1960s the global CR extension theorem, under varying hypothesis, has been widely attributed to S. Bochner. Among the many exam ples are [Hom1ander 1966], [Andreotti and Hill 1972], [Wells 1974], [Cirka 1975], [Harvey and Lawson 1975], [Polking and Wells 1975] [Weinstock 1976], [Boggess 199 1 ] , [Ja cobowitz 1995]. However, a careful reading of Bochner's 1943 paper, which is their reference, reveals that Bochner did not prove any such result, nor did he ever indicate that he had been contemplating any generalization of Hartogs's theorem in that direction. The published record suggests that in 1943 Bochner was completely unaware of the idea of tangential Cauchy-Rie mann equations, and, in particular, of the earlier work of Wirtinger, Severi, and Kneser. Instead, the fornmlation and proof of Bochner's main Theorem 4 (p. 659, op. cit. ) clearly shows that he was looking for conditions that would imply the extension of harmonic functions defined in an open neighborhood B of a sum B of simplices, e.g. , in case B 10
THE MATHEMATICAL INTELLIGENCER
equals the boundary aD. In the application to the complex case immediately thereafter (Theorem 5, p. 660), a casual reading of the hypothesis an analytic function of several
complex variables defined i n the connected boundary
might tempt one to think of CR functions, but there is sim ply no evidence to back up such an interpretation. Not only was Bochner just using the standard formulation that a function is holomorphic in a point (or in some set) if it is holomorphic in some open neighborhood of that point (or set), but more importantly, Bochner's proof (i.e., the proof of Theorem 4 in his paper) does not make sense unless the function f to be extended is known to be harmonic in an open neighborhood of aD. In Theorems 6-9 thereafter, Bochner specializes to IC" and derives from Green's fommla what is now known as the Bochner-Martinelli formula. The hypothesis on the function f still states explicitly an a nalytic function in B (i.e., an open neighborhood of B) (pp. 662-663). Bochner explicitly proves Theorem 8, to the effect that F(z) = KBJl,J(?, z) is holomorphic in z outside of aD (see the remark made earlier), but for the other results, in particu lar for the main extension Theorem 9 (p. 663), no proofs are given. Instead, these are viewed as analogues of the cor responding theorems proved earlier for harmonic func tions, and hence it would seem that Bochner simply had in mind the obvious modifications of the proofs of the former theorems. Bochner's later papers (for example [Bochner 1954]) confirm that he had not been thinking at all about any generalizations of Hartogs's theorem to the setting of CR functions. The function to be extended is explicitly as sumed to be an analytic solution of an elliptic differential operator on a neighborhood of the bou ndmy (op. cit., p. 3, and Theorem 5, p. 7, 8). I conclude that the known relevant published record 1943-1954 does not support any attribution of the CR ex tension theorem to Bochner. With hindsight, it is of course true that the conclusions of Theorem 9 in Bochner's 1943 paper remain correct if the functionfis only assumed to be a CR function on aD. How ever, the proof in that case, as completed by Martinelli in 1961 (op. cit), does not just follow by "forgetting" the addi tional hypothesis. The jump formula now does require a new proof involving arguments quite a bit more delicate than those used by Bochner in 1943. If this tenuous connection is all that is available to justify naming the CR extension theorem after Bochner, equal credit should then be given to Martinelli, whose 1942 proof of Hartogs's theorem is defacto identical to the proof of Bochner's Theorem 9. However, given the earlier contributions of Severi and Kneser, the lack of any evidence that Martinelli and Bochner were even re motely aware in the 1940s of the concept of intrinsic tan gential Cauchy-Riemann equations, and lastly Fichera's first complete published proof of the general case, any such at tribution would misrepresent the historical record.
JaDf(0
How the Confusion Arose and Spread
Having reached this point, it remains to clarify the origins of the erroneous attribution. After reviewing the literature
and consulting with key mathematicians close to the mat ter, it appears that the attribution to Bochner resulted from unfortunate misunderstandings. The earliest reference linking Bochner to the CR exten sion theorem seems to occur in a paper by J. J. Kohn and H. Rossi [Kohn and Rossi 1965], who state in their intro duction, "Bochner's proof of this theorem [i.e. , the classi cal Ha rtogs Extension theorem, ref is to Bochner's 1948 paper] shows thatf can be so extended under the weaker hypothesis that it satisfies only the equations [i.e. , the ta n gential Cauchy-Riem a n n equa tions] on bM . . . " (op. cit., p. 45 1 ] . Since no precise hypotheses are given, the mean ing of this statement is not clear. The best I can extract from Bochner's proof is the conclusion that given f har monic in a neighborhood of aD, iff satisfies the tangential CR equations on aD, then the extension given by Bochner's Theorem 4 is indeed holomorphic on D. After more than 35 years, Kohn and Rossi do not recall whether they had anything else in mind. They were aware of Lewy's local CR extension theorem in the strictly pseudoconvex case, and the corresponding global question seemed an obvious and natural problem. In the cited paper, Kohn and Rossi proved the CR extension theorem (in the e x category) and its gen eralization to ab-closed forms on complex manifolds, as suming at least one positive eigenvalue for the Levi fom1 everywhere on ?_D, by using Kohn's recent deep regularity results for the ii-Neuman problem. Strictly speaking, the rather limited statement of Kohn and Rossi quoted above does not add up to an attribution of the CR extension the orem to Bochner, yet it could easily be so interpreted, es pecially if no references to earlier work on global CR ex tension were known. Around that time Honnander had also been investigat ing the Cauchy-Riemann equations. He was familiar with Ehrenpreis's recent proof of the Hartogs theorem, and also with Lewy's local result. From the perspective of partial differential equations, it was clear to Hom1ander that the proper setting for these results involved the local and/or global Cauchy problem for the Cauchy-Riemann operator a with zero boundary values on hypersurfaces. In particu lar, in his 1964 lectures at Stanford University, he adapted Ehrenpreis's proof to obtain proofs (under not quite opti mal differentiability hypotheses) of both the local and the global CR extension theorem (the latter one seemingly a new result). These proofs were then included in his mono graph, published in 1966. Attempting to recollect events from over 30 years ago, Hormander believes that he talked about his proof of the global CR extension theorem to Kohn, who was also visiting Stanford at that time, and that Kohn suggested to him that he should mention Bochner's work in this context. Honnander followed this suggestion, and in his book attributed the result to Bochner without ha\-ing personally checked Bochner's 1943 paper. 10 In the absence of any knowledge of the work of Severi, Kneser, and Fichera, Hormander could very well have been cred-
1 0This account is based on a porsonal communication by
L. Hbrmander in Fall
1997,
ited with the proof of the global CR extension theorem, so Hommnder's decision to defer to Bochner-inaccurate, as it turned out-reflects a sense of fairness and generosity. Later authors appear to have simply accepted Homlan der's account. Conclusion
Even though Bochner should not be credited with the proof of any version of the CR extension theorem, his 1943 pa per remains a landmark in the history of the Hartogs ex tension phenomenon. His vision to enlarge his horizon from holomorphic functions to certain ham1onic functions set the stage for further generalizations by himself (for exam ple [Bochner 1954]) as well as for Ehrenpreis's investiga tions on related problems for solutions of more general el liptic partial-differential operators. In closing, it should be pointed out that Bochner's 1943 paper, in an ironic twist, includes an important result for which Bochner did not receive any credit until recently [Range 1986, p. 188] . Bochner proved the solution of a on polydiscs (for (0, I)-forms in the real-analytic case, which was the case of interest to him), via the Cauchy transform with paran1eters in dimension one, and by induction on the number of differentials dZj appearing in tile given form (The orem 1 1 , op. cit., p. 665). This result, with essentially the same proof, 10 years later becan1e widely known as the Dol beault-Grothendieck Lemma. But this is another story. . . . Acknowledgments
I am grateful to E. Martinelli, now deceased, for telling me about the work of Severi and Fichera, and of his own proof of the global CR extension theorem. While preparing the new printing of [Range 1986] , I set out to correct the record, and I thank R. Gunning, L. Hormander, J. J. Kohn, and H. Rossi for their assistance. I became aware of Hellmuth Kneser's 1936 paper only in 1999, unfortunately after pub lication of the corrected 2nd printing. Instead of waiting for a 3rd printing, I thought that a separate historical account might be of interest to a wider audience. I thank E. Straube for pointing out this important historic contribution, as well as G. Betsch and K. H. Hofmann for sharing their 1998 preprint and other personal details about Kneser. REFERENCES
[Andreotti and Hill 1 972] Andreotti, A., and Hill, C . D . : E. E. Levi con vexity and the Hans Lewy problem, Part
1: Reduction to vanishing
theorems. Ann. Scuola Norm. Sup. Pisa 26 (1 972), 325-363.
[Behnke and Thullen 1 934] Behnke, H . , and Thullen, P . : Theorie der Funktionen mehrerer komplexer Veranderlichen. Springer-Verlag, Berlin 1 934, 2nd. ed. 1 970 [Betsch and Hofmann 1 998] Betsch, G., and Hofmann, K. H . : Hellmuth Kneser: Pers6nlichkeit, Werk und Wirkung. Preprint Tech. Univ. Darmstadt, 1 998. [Bochner 1 943] Bochner, S. : Analy1ic and meromorphic continuation by means of Green's formula. Ann. of Math. (2) 44 ( 1 943), 652-673.
and was confirmed in another communication in July 2000.
VOLUME 24, NUMBER 2, 2002
11
lntegralformel bei Funktionen mehrerer Veranderlichen. Sitzungsber.
AUTHO R
Kongl . Bayer. Akad. Wissen 36 (1 906), 223-242.
[HaNey and Lawson 1 975] Harvey, R. . and Lawson, B.: On boundaries of complex analytic varieties I. Ann. of Math. (2) 1 02 {1 975). 223-290. {Hormander 1 966] H6rmander, L.: An Introduction to Complex Analy· .
sis in Several Variables. Van Nostrand, Princeton 1966. 3rd. ed.
North-Hol land Pub!. Co., Amsterdam-New York (1 990). [Jacobowitz 1 995) J acobowitz, H . : Real hypersurfaces and complex analysis. Notices Amer. Math. Soc. 42 (1 995) 1 480- 1 4 88. [Kneser 1 936] Kneser. H . : Die Randwerte einer analytischen Funktion
z:weier Veranderlichen. Monats. f. Math.
[Kohn and
of Math.
(2)
81 ( 1 965). 451 -47 2 .
[Lewy 1 956) Lewy, H . : On the local character of the solutions of an
Departme11t of Mathematics and Statist�
atypical linear differential equat ion i n three variables and a related the·
State University of New York at Albany
Alb6rly, NY 1 2222 USA
[email protected] .edu
Germany and raised in M ilano ,
Phys. 43 (1 936), 364-380.
hol omorphic functions from the boundary of a complex manifold. Ann.
R. MICHAEL RANGE
Born in
u.
Rossi 1965] Kohn. J. J . , and Rossi , H.: On the extension of
Italy, R. Michael Range
earned his Dip/om in Mathematik in Got1ingen, whe re lectures
orem for regular fui1ctions of two
(21
G4 (1 956), 51 4-522.
[Lewy 1 957]
complex variables. Ann. of Math.
Lewy, H. : An example of a
smooth
lial equation wi thout sollJI:ion. Ann. of Math.
linear
{2) 66
partial differen·
( 1 957), 1 55- 1 58.
of Hans Grauert got him hooked on Multidimensional Com
[ Martinell i 1 938] Martinelli. E. : Alcuni teoremi integral i per
1 97 1 . He has held academic positions at Yale University and
[Martinel l i
plex Analysis. A Fulbright Exchange Fellowship brought him
to the United States and UCLA, where he received a Ph.D. in
at the University of Washington, as wel l
as
research positions
at various institllles. A frequent visitor abroad. Range is fluent
le funzioni
analitiche di piu variabili complesse. Mem. della R. Accad. d'ftalia 9
(1 938), 269-283 .
per un
1 942/43)
Mart inell i. E.: Sopra una dimostrazione di A. Fueter
teorema di Hartogs. Comm. Math. He/v.
340-349.
1 5 (1 942/43),
in five langu ages, a skill he has often used In lectures in North
[Martinel li 1 96 1 ] Martinelli, E . : Sutla determinazione di
son, he got into ice climbing and alpine mountaineering. He
[Potking and Wells 1 975] Polking, J . . and Wells, R. 0.: Hyperfunction
America and Europe . Range loves mcuntains and Is
an
avid
downhill skrer. A few years ago, inspired and guided by his and his wife Sandrina have three grown cl1i ldre n . He is shown hP..re
on the summit of the Monch (4099 m.), with the Jungfrau
in the background.
una
fun.::ione ana
lltica dl piu varlabili complesse in un campo, asseg natane Ia traccia sulla frontlera. Ann. Matern. Pura e Appl. 55 ( 1 96 1 ) , 1 9 1 -202.
boundary values and a general ized Bochner-Hartogs 'theorem. Proc.
Symp. Pure Math. 30. 1 87-1 94. Amer. Math. Soc., Providence. Rl, 1 977.
(Range 1 986] Range, R. M. : Holomorphic
functions and Integral Rep
resentations in Several Com plex Variables. Springer-Verlag, New York
[Bochner 1 954] 8ochner.
S.: Green's formula and analytic continuation.
1 986, 2nd. corrected printing ( 1 998).
[Serre 1 953] Serre. J. P . : Quelques problemas globaux re lat lfs aux var iates de Stein. Coli. Plus. Var., Bruxel les.
i Differential Equations. ed . L. In: Contributions to the Theory of Partal
(Severi 1 931 ] Severi. F . : Risol uzione
Riemann Complex. CRC Press . Boca
[Struppa 1 988] Struppa, D.: The first ei ghty years of Hartogs' Theorem.
Bers et al . . Ann. of Math. Studies 33. Princeton Univ. Press. 1 954.
[Boggess 1 991 J Boggess. A.: CR ManifOlds and the Tangential Cauchy
Raton.
FL. 1 99 1 .
[Cirka 1 975] Cirka. E . M . : Analytic representation of CR·functions. Math.
USSR Sbomik 27
(1 975).
526-553.
[Ehren preis 1 961 ) Ehrenpreis.
L.: A new proof and an extension of Har
togs' theorem. Bull. Amer. Math. Soc. 67 (1 9 6 1 ). 507-509.
[Fich era 1 95 7) Fichera. G . : Caratterizazione della traccia. sulla frontiera di un campo. di una fu nzlone analitica di
piu variabili complesse. Rend.
Accad. Naz Uncei VIII. 23 (1 957). 706-71 5 .
1 953.
57-68.
generale del problema di Dirichlet
per le funZiOni blarmoniche. Rend. Reale Accad. Uncei 23 ( 1 931).
7 95-804,
Seminari di Geometria 1 987-88. Univ. Bologna. I taly 1 988, 1 27-209.
[Trepreau 1 986]
Trepreau.
J . IM.:
Sur le prolongement holomorphe des
fonctiones CR defin1es sur une hypersurtace reelle de classe C2 dans
C". Invent. Math. 83 ( 1 986), 583-592.
{Weinstock
1976)
Weinstock. B. : Cont i nuo us boundary values of holo
morphic functions on Kah le r domains. Can. J. Math. 28 (1 976),
5 1 3-522 .
[Fichera 1 986] Fichera. G.: Unification of glObal and local existence the
[Wells 1 968] Wells. R. 0., Jr .: Holomorphic hulls and holornorphic con
[Fueter 1 939) Fueter. R.: Ober einen Hartogs' schen Satz. Comm. Math.
[Wells 1974f Wells. R. 0. . Jr.: Function theory on differentiable sub·
orem s for holomorphic functions of severat complex variables. Atli
Ace. Lincei Mem. tis. . S. VII I . 18 ( 1 986) . 6 1 -83. Helv. 1 2 ( 1 939), 75-80.
[Fueter 1 94 1 I42) Fueter. R . : Ober einen H artog s · schen Satz in der The erie der analytischen Funktionen von n komplexen Variablen . Comm.
Math. Helv. 14 ( 1 94 1 /42), 394--400.
[Hartogs 1 906) Hartogs. F.: Einige Folgerungen aus der Cauchyschen 12
T>-IE MATI-lEMATOCAL NTELLIGEI-ICE�
vexity of differentiable submanifolds. Trans. Amer. Math. Soc. 132
( 1 968). 245-262.
manifolds. In: Contributions to
Analysis.
al., Academic Press. New York (1 974).
('Mrtinger
407-44 1 , eel. L. Ahlfors
et
1926) Wirtinger, W.: Zur formalen Theorie der Funktionen
von mehreren
357-375.
komplexen
Veranderlichen. Math. Ann.
97 ( 1 92 6).
M.effli·i§u@hl£11fuiui§ i§4fii.J •ifi
This column is a place for those bits of contagious mathematics that travel from person to person in the commun ity, because they are so elegant, suprising, or appealing that one has an urge to pass them on. Cont;·ibutors are most welcome.
M i c h ae l Kleber and Ravi Vaki l , E d i t o rs
Four, twenty· four, t .
.
.
Michael Kleber
E
nglish is blessed with the word 'four,' which stands out from the other, more humdrum digits because it is, in fact, four letters long. There aren't any more number names like this, which is good: if there were too many, we'd be using a system in which the lengths of the names grow linearly in the value of the number. As a big fan of decimal notation, and of logarithmic use of resources in general, I'm happy keeping things short. On a long drive to Manhattan, Kansas ("The Little Apple"), Rick Char-
Rick reported that in the margins of his notes for selected math lectures he ruled out all numbers up to around five million (!). Impatient sort that I am, I set about teaching my computer how to nan1e numbers, and it promptly reported on 84,672. The way I was taught, that's 'eighty four thousand six hundred sev enty two,' which gives 6 X 4 X 8 X 3 X 7 X 7 x 3 = 84,672. Did Rick miss one? No: his name for the same number ends 'and seventy two,' so the product would be off by a factor of three. The Chicago Manual of Style is implicitly on my side: "One hundred ten men and 103 women will receive advanced degrees this quarter," says the 14th edition, and the editors kindly inform me that the 15th ed. (forthcoming, in 2003) will ex plicitly note that 'and' properly does not appear. Rick, who is sufficiently Canadian that he thinks the alphabet
Cal l a n u m ber fortuitous if it is
equal to the p rod uct of the lengths of the words in its nam e .
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
trand pointed out to me that the next step is to appreciate 'twenty-four': the two words have six and four letters, re spectively, and 6 X 4 is, indeed, 24. So call a number fortuitous if it is lucky enough to be equal to the prod uct of the lengths of the words in its name. If you're feeling industrious, or are trapped in a long car ride, you can search for the next one. It only takes a little work to rule out everything under a hundred. The words in number names are short enough that, aside from seventy-whatever, we can restrict ourselves to numbers with factoriza tions involving only 2, 3, and 5, and poor 5 has its odd multiples ruled out right away, since they end in a four-let ter word. With such tricks you can quickly pare down the numbers to check, even as the range gets larger;
ends with 'zed,' wonders idly whether there might be a question of American versus British usage. I, being American, decline to acknowledge that other us ages exist. In either case, there are still larger fortuitous numbers. If we eschew 'and' entirely, as I do, the next few are 1,852,200, then 829,785,600, then 20,910,597,120, which is around twenty 'billion' (not 'thousand million'; see above comment on being American), then 92,2 15,733,299,200. If you like the word 'and,' and put it after 'hundred' whenever there's something else in that block of 3 digits, then the computer eventually comes across 333,396,000 and 23,337,720,000 and 19,516,557,312,000 and 56,458,612,224,000 and after that 98,802,571,392,000. (If you follow nei ther of these rules, perhaps your name
© 2002 SPRINGER-VERLAG NEW YORK. VOLUME 24, NUMBER 2. 2002
13
for 207,446,400 uses the word 'and' ex properly, this results in only two more actly once.) These exhaust the exam fortuitous nun1bers, 107,8272 and ples under one 'quadrillion, ' when sud 5,3661,7156,6080, below 102-1. denly you need to consider numbers But seriously, . . . with 1 1 as a prime factor as well. The English examples, that is. If you believe Rutherford's physics/ Thanks to Mme. Eisler in seventh stan1p-collecting dichotomy, it's high grade, I am equipped to observe that time for some of the former, and there French is not endowed with an ana is a natural question to ask: do we ex logue of 'four': under the map taking n pect this fortuitous coincidence to oc to the length of its name, the small cur infinitely often? This is meaningful numbers fall into the cycle trois -> only if we have a system for giving cinq -> quatre -> six. But cinquante (let's say) English names to arbitrarily quatre gives 9 X 6 = 54, and the ves large numbers; Conway and Guy's tigial quatre-vingt dLr gives 6 X 5 X Book of Numbers proposes one such. 3 = 90, the best reason I've ever heard Unless we plan a rigorous analysis, for not switching to nonante. The next I suspect it doesn't much matter what example is 2560, deux m ille cinq cent the system is, provided that number soixante, and based on the unreliable names behave about as we expect. hypothesis that I remember how to Which is to say, as a generic N gets construct number names correctly, these three are the only ones below 109, whose name I shall omit-I won't attempt to deal with the mille m il lion I m illiard I billion mess. While we're playing polyglot, other large enough to be worthy of capital alphabets are extra fun: in Hebrew, letter status, its name should grow to l':li t-:: is again four letters, and Dmitry around log N words and consist of sec Kleinbock tells me that in Russian, 54 tions that look something like 'five is miTb,Il;eCHT 'leThipe, the same fortu hundred eighty-nine supergazillion' itous word lengths as in French! Chi the heart of the system is a way to con nese is fascinating: each "word" in the struct arbitrarily large zillion names. nan1e is a single character, which Mark Those zillions will have to get longer, Zeren recommends dealing with by eventually growing as something like counting strokes, making 1, 2, and 3 log log N in any sensible scheme. into fortuitous digits. The name of a So let f be the map taking a natural large number is read off from its deci number to the product of the lengths mal fom1, grouping digits by fours, so of the words in its name, which has that 1234,5678 is (1 1000 2 100 3 10 4) fixed points at four, twenty-four, etc. 10,000 (5 1000 6 100 7 10 8); a new name At least one in five words in the name is introduced for each power of ten of any N are zillion names, mostly long thousand. l But any power of 10 whose ones, sofwill generally grow as around N coefficient is zero is dropped, and there (log log N)10g , better known as JVIog log log N_ is a special 13-stroke character that ap It seems f(N) eventually pears-! just got Bong Lian to explain grows much faster than N does. Have it to me, I hope I get it right-any tin1e we shown thatfhas only finitely many this drop creates an internal gap in the fixed points? sequences of 10" for n = 3, 2, 1, 0, or Hardly. The generic N may have a n = . . . , 16, 12, 8, 4, 0. If I understand name of log N words, but the nan1e
Other al phabets are extra fu n .
'The numbers of strokes in the characters for 1 /2/3/4/5/617/8!9 are 1 /2/3/5/4/4/2/2/2. respectively. For 1 0" with they are 1 2/1 5;6/8/9.
14
THE MATHEMATICAL INTELLIGENCER
length ignores any part of Ns decimal expansion which is a long string of ze ros. We can readily construct copious examples of N in the same ballpark as j(N), where a ballpark is around three orders of magnitude: given any f(N) »N, write N in decimal, and start inserting zeros, three at a time, just af ter the first few digits. This multiplies N by around 103, but has approxi mately no effect on f(N)-it merely changes one zillion name for the next larger one, which is only a few char acters different in length. We can eas ily puff N up with zeros until it is close to j(N). More generally, a large fixed-point f(N) = N could appear as long as one out of every log log log N of the blocks of three digits in N is ',000,' which certainly happens in abundance. But such an N would also need to have its largest prime factor on the order of log log N, the length of the longest words that appear in its name. If we assume having enough zeros and hav ing small enough prime factors are in dependent, my gross estimate is that there are infinitely many N that do both. But if we then think off as tak ing on random values with the right sized prime factors, the probability that any one of these candidate N is a fixed point of f seems to fall off so quickly that I'd reluctantly guess the total for tuity of the natural numbers is finite. Perhaps f might have infinitely many cycles (like the period five 168 -> 525 -> 672 -> 441 -> 420 -> 168, or the one with period four beginning at 1 , 170,993,438,720)? I can't think of how to begin to estimate this, which may be just as well. Perhaps it's time to stop.
Department of Mathematics Brandeis University Waltham, MA 02454 USA e-mail:
[email protected]
n
= 1 !213 they are 2/6/3, and with
n
= 4.'8/1 2/1 6120
PETER A. LOEB
A Lost Th eo re m of Ca c u u s •
n setting up the Riema nn in tegral for an application, hmo does one determine that the in tegrand has been correctly chosen ? Why, for example, is f(x) tl.x a good ap proximation in calcula ti ng the a rea under the graph off bu t tl.x a bad approxima tion jor the graph 's length? Decades ago, ma thematicians used Duhamel 's prin-
ciple to answer the question (see [5], [4], or [2], page 35); they later used a substitute by Bliss [1] when applicable. These theorems are now known to only a few in the old est generation of mathematicians. In modem texts, a rig orous treatment of the integral uses upper and lower sums for both the theory and its applications, and most instruc tors are unaware that this method fails as a test for the in tegrand in simple problems that arise even in the first course. I became acutely aware of this inadequacy when, as a visitor at an Ivy League university, I was required over my strong objections to ask the following question of my stu dents in a combined calculus final: Set up and fully justify the integral for the force of water on a circular window of radius R in the wall of a swimming pool, where the water level is just at the top of the window. The students in the course had learned only the method of upper and lower sums, so they could correctly solve the problem for the top half of the window, but not for the bottom half (see Ex ample 2 below). The difficulty that occurs with this test of integrands is somewhat subtle. If a quantity Q is equal to the integral of a function!, then every upper sum off is larger than Q and every lower sum off is smaller than Q. On the other hand, even with some applications occurring at the most ele mentary level, it is not possible to know a priol'i that up per and lower sums bound Q. One knows this only after
showing in some other way that the integral off equals Q. Consider, for example, the area between the graphs of the functions g(x) = 1 + a:· 2 and h(x) = 2.r2 on [0, 1]. While for a small �.r > 0, the maximum of g(.r) - h(.r) on [0, �x] oc curs at 0, no rectangle of height and width .ll· contains the region between the graphs over [0, �x], so it is not clear a pri o ri that 1 ir is larger than the area of that region. Of course there are several methods to justify the integral needed here (see Example 1), but even for this simple ex ample the "universal" method of upper and lower sums fails, and Bliss's theorem also fails, as a test for the inte grand. What is my answer? In calculating a quantity Q as the integral of a continuous function! on an interval [a, b], one approximates on the ith interval of a given partition a part. Q; of Q by the ith tem1 of the Riemann sum off The re sulting errors e; must be small enough so that their sum goes to zero as the partition is refined. We shall have an swered the opening question when we say how small each e; should be. The answer given below in Theorem 2 is a special case of Duhamel's principle derived from Keisler's [3] Infinite Sum Theorem. Keisler's result uses infinitesimal numbers. It says that for a partition by infinitesimal inter vals, each error e; should be an infinitesimal times the length of the corresponding interval. I replace infinitesimals here with functions having limit 0 at 0. To make the answer useful, I must also challenge the widely held belief that uni-
1
·
© 2002 SPRINGER-VERLAG NEW YORK, VOLUME 24, NUMBER 2. 2002
15
form continuity cannot be introduced at the elementary level; it can indeed be introduced there with the relatively easy, equivalent condition given below in Theorem 1. As in most introductory courses, I have implicitly as sumed so far that a desired quantity Q exists as a number and the problem is to correctly calculate it with an integral. This point of view is appropriate for even advanced appli cations of calculus when physical principles and elemen tary considerations define a quantity and one needs to check the correctness of the value given by a particular in tegral. Theorem 2 provides a general method for validating such a calculation. While in more fom1al settings, quanti ties to be considered must first be defined, this is usually accompanied with some justification. Throughout the lit erature one can fmd arguments doing this that approximate parts Q; of a not-yet-de fined quantity Q. As in the area example discussed above, however, the mean ing of any part of Q may be no clearer than that of the whole. The simple mod ification of Duhamel's method for justifying an in tegral definition given in Theorem 3 does not refer to individual pieces of a quantity Q, but rather to intervals J; in which such num bers, if they exist, can arguably be found. Whether one takes the formal or informal approach to setting up an integral, errors can and do occur. Because no one can lmow to what use students will eventually put their training, they should be prepared for totally new applica tions. It seems strange, therefore, that order-of-magnitude checks on the validity of an integral are not included in cur rent calculus texts. These texts are, after all, replete with methods for numerically evaluating integrals and for check ing the accuracy of those evaluations. At the same time, many Computer programs can be used instead to evaluate any integral at lea.•';t numerically, and often symbolically. What these computer programs cannot do is tell the user if the answer is correct for the initial problem. Given the decreased need to evaluate integrals by hand, error esti mates to justify the choice of integrands should take a prominent place in the training of future generations.
ferentiation. Keisler notes that if a < c < b, then using val ues of !l.r; that evenly divide [a, c], one can show that the integral over [a, b] is the sum of the integrals over [a, c] and
[c, b).
We shall want to use the following result. It is equiva lent to the fact that a continuous function! on a closed and bounded interval is unifomlly continuous. The reader can easily establish this equivalence, and instructors can prove the result itself to beginning students for the case that the function f has a bounded derivative. THEORE!It
1 (MAXIMUM CHANGE THEOREM). Let f be a contin uous function on [a, b]. Given .:U· > 0 and the con·e sponding partition of [a, b), let M; and m ; denote the max imum and min imum values of the function f in the ith interval [X; - 1 > x;]. Set £j(!::.).r :
Com puter p rog rams can be
(M;
=
-
maximum, ,;; ; ,;; " lim.l.r o
m;). Then Er(!l.r) 0.
used to eval uate any i ntegral ;
_
=
With this result, one can easily show that the inte gral of a continuous func tion exists and has the usual properties. The map ping f- E1, defined for each continuous j, is nec essary for this development. I also use such functions in formulating and then applying the following simple form of Duhamel's principle.
what they can not do i s tell the user if the answer is correct
Q;
for the i n itial problem .
Duhamel's Principle and Uniform Continuity
Given a continuous function! on an interval [a, b], I follow H. J. Keisler [3] in forming Riemann sums. Each &r > 0 cor responds to a unique partition with n subintervals, where n is the first integer such that a + n&r :2: b: The partition end points are X; = a + i!::..x for 0 :5 i :5 n - 1, and Xn = b. We let &r; denote the length of the ith subinterval of the partition; of course !::u:; = &r except for the last subinter val, which may be shorter than !::..r. Always evaluating at the left, the Riemann sum is 1-;'= 1 f(x; - 1 ) !l.r;, and the in tegral is just the limit as !::..r - 0. It is not necessary to in troduce a limit concept different from the one used for dif16
THE MATHEMATICAL INTELLIGENCER
THEOREM 2 (ERROR-SUM PRINCIPLE). Let f be a continuous function on an interval [a, b]. A quanti.ty Q is equal to the integral f(x)dx if there is a. junction E of t:u· with the following"properties: For each positive !::..r and co/re sponding partiti.on of [a, b ], the number Q can be written as a sum Q = 1.;'= 1 Q; in such a way that for each i be tween 1 a nd n,
r
I Q; - f(x; - 1 ) ·
and lim.l.l·
_,
0
E(oi.r)
= 0.
t::...·r;l
:5 E(t:.x) · !l.r;,
PROOF. The quantity Q is the limit as !l..r - 0 of the cor responding Riemann sums, since for any given !l.x,
I Q - i f(X; - 1) !l..'t·;l i i
=
:5
1
s
i
"
L =
;
1
=
1
J Q; - f(X; - 1 ) .
E(t:u·) .:l:x; ·
=
!::..r; J
E(!::..r)
·
(b - a). D
It should be clear to anyone familiar with the historical literature that the above fommlation of Duhamel's princi ple is more appropriate for beginners than the original from 1856 [2], or Osgood's 1903 revision [4], or any later formu lation using Landau's little o notation. Although it may not be suitable for beginning students, we can modify Duhamel's principle as stated by Osgood [4] and Taylor [5] so that while, as in those results, the total
quantity Q is not assumed
to
exist,
now the
emphasis
is on
the intervals in which its parts might be found. The point
of the modification is to remove any ass umption that we
are given at the outset a real-valued map i
Qi for each partition of {a, b ]. '!'his yields the following version of Duhamel's principle that can be used to justify the use of a particular integral to define a quantity. �
THEOREM 3. Let f be a continuous ju nction on an interval [a, b], and let E be a nonnegative fu nction of tl.t· with lim.u- - o+ £(fix) = 0. Associate 1vith each .l:c > 0 a nd each pa rtition interval Cri - b xd an i n ten;al Ji(;l.c) co n ta in ingf(;ri - l)ill·i a n d having length at most E(�J:) · �.r. Then fo r anu choice of poi n ts qi E Ji(.l.r), 1 $ i s n , set ting Q(.l.r) :'= � qi we have lim .l.i· --" 0 + Q(Q.r) f(x)d.r. .
PROOF. Set S(Ax) : = .u· $ 1 ,
IQ(�l:)
-
t j(..r)d.:rl a
s
:s:
i
I l-i= H t ftX.:- 1) �i -
f
1
''
lqi - .1\..r; - J ) ;;U·11
E(il..r) · (b =
[''
+ l
- a) ·
+
=
r
_((.r)rui . When +
·cr
S(�:r)
S(.ll) _. 0.
D
A corollary of Theorem 2 is the 1914 result of G. A Bliss [ l ] that tests the appropriateness of an integrand for a given application when the integrand is a product of two contin uous functions g and ll-. The test requires that when l; is the ith interval of the partition of [a, b] generated by a .::l.r > 0 and Q1 is the corresponding part of the desired quantity Q, then there should exist a point (s;, t;) in f; X 11 such that Q; g(s;)h(tL)J:r;. Alternatively, one can use Theorem 3 to show that for any choice of(s,, I;) in /; x 1;, lim..l.,- - 0 + !i'-1 g(s;)h(t;)!!..x1 g(J')h(.x)d.r. In either case, a proof sets E(!l.r) Eh(ar)"- mruqa,bJ lfll + E9(tb.:) mruqa,bJ lh l .
not show that the integrand is c orrect by using upper
lower
sums; a simple
and
is given in the introductory
example
paragraphs in this article. On the other hand, our more gen eral methods can be directly
Yi
applied
and z1 be the maximum values of y
respectively,
in
the interval
to this problem. Let
[.:ri- " xi), and let Ui and � be the g(:r) and
=
z =
h(x),
minimum values. With the partition interval [.r;- 1 1 xiJ, associate the inteiVal .J;(w) = [� - z;) · ari, Cii; - �a · AxiJ. This interval contains both the ith tenn of the Riemann swn and, assuming it exists, the corresponding part of the area. Now we can apply either Theorem 2 or The orem 3, since the length of .J;(;l.r) is at most [Eg(�.r) + En(.ix)] a.t·i and lim..1.x - o [Eg(�r) + Eh(ax)] = 0. corresponding
·
ExAMPLE 2. Consider a window of height 1 in the wall of a swimming pool with the surface of the water at the top of the window. The window is symmetric about a vertical axis and at depth .x has width sin 7T'..t·. Let w denote the weight density of the water. We want to show that the water force
F on the window
is given by the i ntegral j
1
w · .r · sin m: ti:r.
Upper and Lower sums work to justify the integral for the
top half of
the
II
window, but they do not justify the integral
for the bottom half. There,
length
as
the pressure
increases, the
of horizontal strips decreases. Bliss's theorem can
A U T H O R
=
=
r
The Fundamental Theorem of Calculus is also a corol =
·
lary of the Error-Sum Principle: Given an antiderivative Y
=
continuous function ] on [a,b] and a positive Ax, there is in each subinterval [X;-1, x;] a point c; such that .1 Y1 : = F(r;) - F(,r,. _J) j(c.) .l..r;; the desired quantity is F(b) F(a) 1: ;'= 1 llY;, and for each i, the absolute value of the error is I J(c;) - j(X; - l) l · .1.1.·; s E,r(O:r) f:J.x;. (This inequality also shows that Riemann sums need not be eval uated at the left end points of partition inteJVals.) Nate that the Riemann stm'\ !�'= L f(x; t).l.t-; is the sum of differen tials 1:�=t d Y1• As tl.l· goes to 0, the difference between the sum !;'�1 � Y1 of the actual changes in Y and the sum '2:�'= 1 d Y1 of the approximations to those changes along tan F(:.r) of a
=
-
·
PETER A, I.OEB
=
Depar1ment of Mathematics
·
University of Illinois Urball6 , IL 61 801
_
gent lines goes to 0. This helps explairl the integral notation. Examples
of the Riemann integral, there are elementary problems for
upper and lower sums cannot validate the choice of an integrand.
be
directly applied to
EXAJ..IPLE: 1. Let g and h be continuous functions with g(:L·) :2: h(x) for all .r in [o,b]. By translating graphs and subtract ing ·the resulting areas, one can show that the area between the graph of g and the graph of h .is given by the integral
f (g(x)
-
h(:r)) d.x.
Peter Loob received his Ph. D. degree u nder H. L Royden at Stanford in 1 964. His seventy publications in real
especially wi1h
I conclude with two examples showing that in applications which
USA
e-mail;
[email protected]
Without such manipulations, one can-
measures
in potential
theory and
analysis deal
with Robin
son's no nstand3rd analysis. He addressed the 1 983 lnterna
t iooal Congress on the formaHon of standard measure spaces on
nonstandard
models
that
al low
infinite
stochastic
processes to be treated with the combinatorial tools available
for finite processes. Loeb has taught introductory Cillculus
classes from both 1he nonstandard and the standard view
point. His other interests- electronics. photograpf<w. and oom
puters-.are a natural outgrowth of h is belief that he wlm dies
with the most toys wins.
VOLUME 24. NUMBER 2 . 2002
17
be used for this part of the problem, but I shall directly ap ply Theorem 2 using the fact that force exists as a physi cal quantity and is additive. Integrating on the interval [ U2, 1 ], we have for each il..r > 0 and each subinterval [.r;- t, .:ri], bounds on the corresponding fo rce F1 given by The ith term
of the Riemann sum is also between these be justified using ei ther an argument invoMng pressure or one involving the work that would be done by gravity in bringing water down from the surface to push the part of the window from .1.·;- 1 to Xi out a distance h.) The correctness of the above inte gral now follows from the error-srnn principle using the fWlc tion E(�) = w · 1J.· + w Esrn ,l.:U:), because for each i , th.e difference between the bounds is oll·; multiplied by bounds. ('The bounds on the force can
·
·w(x;
sin 7rX; - t - .:l'i - 1 sin ru·; - 1
+ .r; - L sin 11X; - 1 - J:; - 1 sin
For these problems, as indeed for most
encountered in a calculus course, there are check the correctness of the
application. Students
18
'ff.t';)
::::::
E(iir_).
of the problems special ways to
choice of an integral for each should, however, be given a general
11-<E MATJ-;EMAT!CAL NlHLIG.NCffi
principle that they can remember and apply in their future work Such a general pdnciple is the order-of-magnitude test given in Theorems 2 and 3. With the increasing power of computers to evaluate integrals both numericall y and symbolically, once they are correctly set up, such a test should again play a major role in the calculus curriculum. Acknowledgments
I am indebted to Daniel Grayson and the referee for helpful suggestions. REFERENCES
[I ] G. A- Bliss. A substitute for Duhamel's theorem, Annals of Mathe·
121 [31
matics (2) 1 6( 1 914- 1 5). 45-49.
J. M. c. Duhamel. Elements de Cafcul lnfinitesimal, Paris.
1 856.
H. J. KeisJer. Elementary Calculus, An tnflflftesimal Approach, Prindle, Weber & Schmidt. Boston. 1 986.
[41 W. F. Osgood. The integral as 1he lim it of a sum. and a theorem of Duhamel's, Annals of Mathematics.
(2) 4( 1 903). 1 61-1 78.
[51 A. E. Taylor. Advanced Calculus. Ginn & Company. Boston, 1 955: third editiOn coauthored with w. Robert Mann, John Wiley and Sons. New YorK 1 983.
'
M a t h e m a t i c a l ly B e nt
Col i n Adams.
ditor
saddlebag. Suddenly out of nowhere,
Homotopy on the Range Colin " The p roof is in the pudding.
Opening a copy of The Mathematical l ntelligencer you
unea.t.'ily, "J.Yhat is
may ask yourself this anyway-a
mathematical journal, or u•hatr Or
you rnay ask, "Where "Who am
tion
is at
open
am /?" Or even
I?" This sense of disorienta
its most acute urhen you
f() Colin Adams's column.
Relax.
Breathe regularly. It's
mathematical, it's a humor tolunm,
and it may even be harmless.
Adams
c "Oh,
ookie, these pancakes taste like " COW p ies .
ah
wouldn't use that
twice, " guffawed
Cookie with
trick
a twin
kle in his eye. His dirt-crusted face split
Sioux in Sioux City.
homology'?"
I ask d.
�oh, Like most everybody else, I
picked it up on th e range.
My first cat
fore you !mow it, I was
I spent eight years learning their ways. Mostly
with 'em,
homotopy-theoretic, completely alge--
braic, no topology. They looked at
copy of Whitehead's Elenumts
We
ing longhorns. First day of the drive, the
foreman rode over and hande
motopy Theory.
of Ho
'I think you'll be need�
ing this,' he said. He was sure
about thal Hard to
right believe, but back
a contravariant functor."
sympathy.
I was a real greenhorn. Nights around the eampfi.re, we'd talk spectral sequences, and I can't believe some of the dumb things I would say. TI1at
crew'd be in stitches, making fun of my interpretation of the Eil enberg-Zilber
I felt like such a fool . Butt of every joke. Hey, tenderfoot, they'd say, what does the Alex:ander-Vlhitney diagonal product tell you about the re chain map.
lationship between the slant and cup
start laughing, answer it. But I
products? They'd just
knowing
I couldn't
gotta tell you. By the time we hit Kansas, I !mew my cohomology. And it's a darn good thing I did. I wouldn't be alive to tell this story v:ithout it." "How's that?" I asked.
"Well, we'd just forded the Arkansas
River,
and were bedding down for the
night. I was plu m b tuckered
thinking,
out.
ot
I'd left my Whitehead in my
be
lhing with the
things different from the
"Yep,
e-mai l:
[email protected]
to talking about cohom ology, and
tle drive was out of Austin, Texas. Herd
I shook my .head in
Williamstown, MA 0 1 267 USA
talk at that
"So, Cookie, where'd you learn your co
of pancake around
my plate, debating where to drop it.
and
Department of Mathematics. Bronfman
Didn't you give a conference on Algebraic K-Theory over in Mason City?' " ''By gum, it was him, and was he
you Squatting Deer?
tluilled I remembered his talk. We got
I pushed a piece
between a relative Alexander presheaf
Science Center. Williams College,
' ·· r said, 'Hey, wait a second. Aren t
into a tooth-deprived grin.
then, I didn't understand the difference
Column editor's address: Colin Adams,
lndians swooped down on us. I think they were Sioux from Sioux City, Iowa, branch campus of Iowa State. They wiped us out. I was the last cowboy alive. One of them had me by the hair with his hatchet raised. I was a goner for sure, but I realized in that instant that I recognized him.
"And
way we do.�
then what happened'?"
"Well, one year, there
musta gone
without a result.
a
was a drought. good two months
Everyone was ex
tremely depressed. So we headed west
for a conference in Cheye!Ule, on Stiefel Whitney classes.
It wasn't
our kind of
stuff but we were desperate. And that's where I fell in love."
"Yeah?" I said. "You found true love?"
"Yep, I \\'as head over heels. Stiefei
Whitney classes just blew me away.
So
me and the Indians, we parted ways.
When the conference ended , I drifted
from town to town, going wherever
there was a seminar on. fiber bundles.
Did odd jobs here or there , refereed a paper for Topology, wrote an expository piece for the Bulletin. Fore you know it,
I had built up a little reputation. Outfits
can always use
someone
Stiefel-Whitney. I got a lot
who knows
of work But
you know, then the winds, they changed
People started getting interested in
invariants. So now, I'm making flapjacks.� quite a tale, Cookie. WeU, I
Seiberg-Witten
lucky to be "That's
best be climbing in my bed roll and working on those Seiberg-Witten in
vrui.ants. Thanks for the
grub. " "Sure thing, Pardner, don't let the
Be·tti bugs bite. "
0 2002 SPRINGER-VERLAG NEW YOOK. VO..lJME 24 NUMBER 2. 2002
19
BEATRICE LUMPKIN
M ath e m ati cs U sed Egypti an Co nstru cti o n an d Boo kkee p i n g
m·
historians of ma thematics, it is fortu nate tha t the ancient Egyptians believed
"you ca,n take i t with you . " Much of our knowledge of mathema tical applications in ancient Egypt comes from inscriptions on walls of tombs and rolls of papyrus buried w ith the dead. In 1 860, a precious object was taken from the grave of Neferhotep, who was a scribe of the royal establishment, c 1700 BCE [ 1 1 , 51]. The object looked anything but precious, more like a bun dle of dirty rags. Once unrolled to its length of 7 1/2 m, it showed hieratic writing and numerals written in two different hands. It is now known as Papyru s Bulaq 18. The beginning and part of the middle of the papyrus are missing. What remains tells a vivid story of provisioning the Royal Court of Egypt, 3, 700 years ago. Neferhotep, according to the papyrus, was the Head Scribe in charge of disbursements of foodstuffs for the Court. From the capital at It Tawy near present-day Cairo, Neferhotep had traveled south to Thebes with Pharaoh Sobekhotep of the Thirteenth Dynasty and his ret inue. Many Egyptologists have written about this interesting papyrus, including Alexander Scharff in 1922 [ 1 1 ] and An thony Spalinger in 1985 [ 12]. Papyrus Bulaq 18 was an ac count book containing daily balance sheets of income and expenditures for the Pharaoh's court during their travels. At the end of each day, bookkeepers totaled receipts and expenditures. Then total expenditures were subtracted from total receipts. Any surpluses were forwarded to the
20
THE MATHEMATICAL INTELLIGENCER © 2002 SPRINGER-VERU\G NEW YORK
next day's balance sheet. On some days, additional outlays were required that could not be covered by the regular daily income of the Pharaoh's traveling court. These added quan tities were shown as amounts "due from the temple of Amun" in the relevant income colunms [ 12, 189-190]. Amounts due could be promptly collected because the tem ple of Amun was located in Thebes, the site of the royal visit. Papyrus Bulaq 18 was recorded about the same period as the major surviving mathematical papyri of ancient Egypt. The longest is known as the Rhind Mathematical Papyrus, written c 1650 BCE by the scribe Ah'mose and copied from a work that was 200 years older [6, 45]. An other important mathematical text was written c 1890 BCE and is now known as the Moscow Mathematical Papyrus [2, 20]. However, the scribes of the extant mathematical pa pyri did not face the same challenge that confronted the scribes of Bulaq 18. To complete the balance sheets, it was necessary to express zero remainders. For many foodstuffs recorded in the daily balance sheets, the total disbursed equals the total received; sub traction left zero remainders. The example below, adapted
from Spalinger's translation, was dated Day 1 of the third month of Akhet (inundation season), year 3 of the reign of King Sobekhotep [ 12, 187] . Only vegetarian items appear on this balance sheet; distribution of meat was given else where in the accounts [ 1 1, 5 1-68, 1**-24**] . Spalinger believes that the second column for bread ( i nw) is for special state needs beyond the usual allotments and is reflected in an increase in the "Due today" line [ 12, 193]. The two columns each for beer, dates and vegetables differ by package size. Calculations were mostly correct, Spalinger writes, but 2 jugs of beer that remained on day 27 were not carried over to day 28. He doubts that this was an error, instead "quite possibly due to a personal appropriation of little consequence" [12, 191 ] .
59]. Sir Alan Gardiner also studied Papyrus Boulaq 18. In the excerpt below, Gardiner translated nfr "perhaps as symbol for zero." In the dictionary section of his Egyptian Grammar, he gave "zero" as the meaning of nfr(w) (5, 574), and referred the reader to section 35 1 for a discussion from which we excerpt: [5, 266] nj'r with the meaning of a negative word. Besides its senses "good," "beautiful," "happy" the adjective nfr has sometimes the signification "finished," "at an end"; compare the related nouns nfrw "lack," nfrw [written differently, BL] "end room," and njryt "end" in the compound preposition neftyt-r "down to," lit. "end to"; perhaps also nfr as symbol for "zero."
The balance sheet was
written i n hieratic scri pt , i n
The balance sheet shown in the accompanying tables was written in hieratic script, including hieratic numerals. For the 4 zero remainders in the bottom row, the scribe entered nfr, using the trilateral glyph [5, 574]. Vowels were not shown. On the balance sheet, Spalinger translated nfr as "0." Scharff, on the other hand, retained the nfr hieroglyph to express zero remain ders in his translation (Fig. 1). He described the remain ders as das glatte Aufgehen, the flat or even outcome [ 1 1,
�
Day 1, Month
Gardiner then gave an example of a Middle Kingdom proverb that he translated as "Ye shall offer to me with what is in your hands; if there chance to be nothing in your hands, ye shall say with your mouths." In this proverb, nfr has the meaning of "nothing."
c l u d i n g h ieratic n u merals .
Zero Remainders
Zero Level
About 1000 years before Papyrus Bulaq 18 was written, nfno was already used as a label at construction sites. Con struction of large tombs and pyramids required deep foun-
3 Bread
Income
Beer
(lrt
(!pt
1 680
1 35
2
1
200
2
1 00
10
to Palace Pharaoh
inw
des jugs
varied
Dates
(!nw
Vegetables sacks
52
bundles
200
Day 30 balance Temple of Amun Due 938
90
7
938
237
9
1
52
625
60
2
1
52
630
61
50
525
38
50
Today
IN
7
TOTAL 1 980
7
200
Expenditures
to Palace, Pharaoh
1 00
Workhouse Officials Workhouse ordinary Officials, harem
310
35
290
22
600
216
5
7
Officials, citizens TOTAL OUT Remainder
1 780
200
The balance sheet illustrated in Figure
338 1.
21
7
2
1
0
52
0
7
0
200
0
VOLUME 24, NUMBER 2. 2002
21
- -- - -
----- --
@)':xxvu 2.. � x� vm � �· (�@�@J-r�� J!mJcii )Ao �Jqo i @ � \w1r�y�� e 3 � 1-� � -Qj}i.�P- f8 ��� -;7.' ii �� � ! �I 1i li 1 .p,� - CR 135" z 1 52, },q-g't ao. � �1 � Q� l£.r-1a "' :7: �- 16if1 2, s 1 e � A � 7'f0 ��� '7:'il!I 'T �� 0 zw 1� 1rrob ] E i +� � , � �� 1 Q�Q�� '2.) 'l 938 9cr ? \ x � )\Jr �. '-t 1\. JLj C> � '?E... 1 rlih 04 Q 1 93 0" � zqa5t ..
0
c::::>
c::>
'"
�o Q
d
0
.......
I:::J
0
0
& ��
9
-+-
d!!.
-. <>
I l l
,a
0
�
I l l ,.._..
-:;!u I:J 1¥ � 0 �� rlih --;; � 0 0 � � C'J -A A1>.D 0 � ...-A Q R 10'" , D .o � -a o -yqq1 � � -ii � c;:::l ..,....A C"J--6 ,__ .,;....Q c::::l �
-Ji
,._ , "
rt � ·t-:>1 � �'it »
--D
43
I ll
I
,......
..
I l l
"'
��>I ,....._ A']:
A
6 0
0 0
� -,t ,� • �t.lJ e-t I
�
0
�
b�U 525
11-
"'
0
&1
2/tcr
310"'
1'1-itr
z rru A�. -1b � � �� � � \Q.a �f�� 1'ia-n;:: � ll20"
-1 s
'1�+1) t
b25"
" ' A I ,.,o
A� � IJI}tj � � � � � 1t � t� n Q �
,..,.�d-4
93K 23t 9
-
b(ft
o
3 31
36!T 1 i
1 1
52
'l
1fT(T Sq-
38
5q"
2�
3S 5"
216 �
--
2,1
-
.t
1
-
t
Si
\
t
�
b
2(1'�
t
Figure 1 . Hieroglyphic transcription of balance sheet for day 1 month 3 of Akhet. Scharff, Zeitschrift fiir Agyptische Sprache und
A/tertumskund. 57, (1 922), 12**.
dations. To refer to an exact level, ancient Egyptian archi tects created a system of vertical coordinates. These coor dinates were directed numbers, given as the distance above or the distance below nfno. Here, rifrw was used as the zero reference level and was usually placed at pavement height. Such an example can be seen at Meidum, Egypt, in the foundation trench of Mastaba 17, a massive tomb with base 52.5 X 105.0 m (Fig. 2). Start ing at ground level and going down to the bedrock, thir teen horizontal lines were drawn on the wall of the trench. The lines were spaced 1 cubit apart. A vertical line crossed the horizontal lines and some intersection points were labeled. Labels that can still be read include nfnv for ground or zero level, then 5 lines lower, "5 cubits below nf;w" and 3 lines further down, "8 cubits below nfno. " This grid was also the background for a slanted line that showed the desired inclination for the walls of the tomb.
The measure of the inclination was the Egyptian sqd. It was the inverse of the slope ratio commonly used today, the tangent of the angle of inclination. Sqd was usually given as the number of palms of horizontal setback for a rise of 1 cubit. The height of the rise could be read from the la beled horizontal guidelines. A cubit stick could have measured the run. For the inclination line shown at Mastaba 17, the sqd was fixed at 1 I 4 [ 1 , 12). Sqd examples make up problems 56-60 of the Rhind Mathematical Papyrus [3, 96-99]. The njr1v label for zero level can still be seen at some pyramid construction sites, including Meidum and Khufu's huge pyramid at Giza in the outskirts of Cairo. Series of horizontal lines drawn with red ochre on the stone blocks are still 'visible at the site. Egyptologists believe these hor izontal lines were used as leveling guidelines to help the masons keep each course of stones even. As at Meidum Mastaba 17, nfrw labeled a horizontal line at or near the
The nfrw label for zero level can be seen at some
pyram id construction s ites .
22
THE MATHEMATICAL INTELLIGENCER
Egyptologists Borchardt and Petrie, who had studied Old Kingdom pyramids [9, 76-77] :
pavement level of the pyramid. Course heights were given as the number of cubits either above or below nfnc [ 1 , 15-18]. The use o f directed numbers as vertical coordinates to locate leveling lines was well known to early Egyptologists. In 193 1 , George Reisner described the leveling lines at the Mycerinus (Menkure) pyramid at Giza, built c. 2600 BCE. He gave the following list, including terms collected earlier by
,
•
. .
•
..
.,�\\ !
.-'1 .
..
'\ I
.
, '•
,•
�
--{
,
y .•
. I.
�••
.· . ,_,· ·
, ....
_/
•
,
_..
.. i(.
..
,
I
·
·
·
/
•
.
i
.-'
....
•
}·'
,.
j./· i
·
.
•
/
.·
,...
,.· ·
.·· ·
· ·
·
y -
�'
/
� �··
.
\
.·· �
_
·
···
!
. .
.
,
.
,
·
/
··
..
/,
/_./
- �- -· - .. ·
.
...
.
.-'
//
·
·· - - -
_ ..
·
·
·
·
-
--
-
-
-·
:
= :
iI
--
.. �..;t.,• I J
-- - - -- - - - - -
-t.
- - - �. .· - -
�••_:·•' � �
I i:D L' Q:-:b :._�.:;,;;. .. --+ ----------1-----j�: : I " ! .. � •
, I
•
� ·
•••
,.. .. , _, ..
I --··:·---,-
· --
'
I I
I f
--
-
· ·
-·
,'
- - - -- -
' I
-
- -- ·
·- - ·
I
I
- · ·
--
-
I --·
-- -
-- - · --
- - -- -- -
- --- - - - - -
- --
..
:� <> f & �
-
-
1
r
--:---· -----· r ··------·-----r·----- - =-.r :.:-
I
I I
!
+-
:
·
l
-
·
,t-
·
- · · - · - -- -
·
;
I
-
-
..
,·
1
,
.
.. . . I
. .
:
.
�
J:
I
...
. .
I
-- - - - - - - -- - -
l
I
. .
.
_ _ _ _ _ _ __ _ _ _ _ _ _ _
. .. .
I
- - -
.
-
. ·
L
:
r
,
- - - - - - - - -- - - -
I
i
/· ·· ·
·
•
, - - -- - - - - - - · · - - -
��-····
. ... \
' :
I
/ - / · ·1 :- - , ,.....,.. \-·· '
. , ·· _
..
· ···
..
.
• .··
··�
· . ··
: // .....r . I
• • •
�j' /
,
.
.
,,./
__
:·
- - - - - -- -- - - - - - - - - - -
. ......-
. // ..··
_
,.·
· \v·/.
•. . •",
:
.
.
,
,,
·
...
, ··
..-;_�/ L-"' ....j ·
�
,
- , ·
.
-
---
·
···tr-------
-
· - - -
-·
-r<-= L ..'
-
-
- -
-
•
.
..
.
· . .·
·
·
..
. •
-
- ·-
----
·
--
• • • ••
/
\.···· . . , I /; ,/ ..·· ' �··· ) . ,.�- - -
·
.
-
·
.•·
.··
i t' �, \
·'
/.
· · ·· · · · · · ·
nfnv, nfnt.'
n
hr
- - · ·· · ·
.. . / 1 . . I ·"'· . . ..-/ _//,{..,/... 1 ·� · ....
/
.
..
· _, ·
�r nfnv
-
��
zero zero line above zero below zero
njr'1o
n
m!i. �zr
,. I
// /
. _ . . ....
...
,./' ./
tp
m
� -- -------
,- -
__,
nfrw
· -
-
- -
- --
· -
-
�
] 'I
- - -----·-----·-
_ _ _ __ _ _ ___ _
-1·
_ _ __ __
!
-
. .
·--- - - - - - --
_ _
__
/
Figure 2. Construction guidelines at Mastaba 17 at Meidum. Adapted from D. Arnold, Building in Egypt, 1 2. © 1 991 by Dieter-Arnold. Used
by permission of Oxford University Press, Inc.
VOLUME 24. NUMBER 2, 2002
23
L.,.
...
--
•••
{(
L'1 1
tt � (tc
Li II
,,
...
,,.
Figure 3. Photograph and tracing of architect's diQilram of Thi'rd Dynasty. Gunn, Annates du Senrke des An·tiquites de I'Egypte 26. 1926, 1 �7-202. �otograph p. 277, tracing, p. 1 98.
24
THE MATHEMATICAL IN'IELUGENCCR
In an
1865
Richard Lepsius de of horizontal and vertical guideli nes at a number of other pyramids. He showed that the spacing be classic of metrology,
scribed systems
tween the vertical llnes
was
often
an
A U T H O R
integral number of cu
bits. Lepsius's study of these guidelines helped hlm fix the
length of the royal cubit at 0.525 m [8, l l-13[ .
Rectangular Coordinates The record sh ows that ancient Egyptian bui lders had ex tensive experience with horizontal and vertical guidelines. Egyptian artists also used grids of squares to maintain pro portion in art and to tra.n:sfer artwork, square by square [ 1 0, IJ. To con.sider this experience as the precursor to the use of �oordinates would be purely speculative. However, an architect's plan found at Djoser's pyramid complex (c. 2700) at Saqqara, does give numbers that appear to be co ordinates for point:;; on the plan. The plan was inscribed in red ink on a l imestone flake or ostracon (also ostrakon). It shows a curve and 5 equally spaced vertical segments extending down from the curve (Fig. 3). Next to each segment is a hi eroglyphic numera l giving lengths in cubits, palms. and fingers. The equivalent lengths in fingers, re adi n g from left to right are 98, 95, 84. 68, and 4 1 [7, 1 97-202]. o horizontal spac i ng for th es e segments is given. Egyptologist Batiscombe Gunn, who studied the plan, thought "the very fact that the distance is not specified makes it most probable that it is to be understood as one cubit, an implied unit foun d elsewhere" [7, 200, n. l ] . G u nn used the given values as vertical coordinates and standard horizontal spacing 1 cu bit apmt. When he connected the poin ts with a smooth curve, he got a close match to the curve shown on the
BEATRICE
ChiC<�go, IL 60049
7 1 23 S.
USA
,
,
ostracon. The ostracon was found next to a solid, sa.ddl.e-backed structure. The reconstructed cmve at the top of this struc ttue matched the curve on the d ia gram within a few cen timeters. Clarke and Engelbach agree d with Gunn's analy sis. They wrote, "It seems very likely indeed that the diagram on the ostrakon was i ntended to setve as a guid e
to the masons in constructing the saddle-back. If this
true,
it shows
dinates was
that the system of drawing a understood" 14, 53].
be
curve by coor
book th e development of rela
These examples from ancient construction an d
keeping practices indicate that
concepts, such as recognition of zero as a quantity and the metricizing of space, has a long history, going back at l east 4,700 years in ancient Egyptian mathe tively modern
matical
applications.
RII'IAINCIS
1.
2.
Arn old , D. Building in Egypt. Oxford U niversity Press, New York, 1991 .
Boyer. C. B. A History of Mathematics, revised by U. C. Merzbach.
Wiley. New York. 1 991 . orig. 1 968.
Crandon A-venve LUMPKIN
e-mail;
[email protected]
After years
as
factory worker, union organizer, and technical
writer, Beatrice lumpkii1 found her true role
as a
mathemat
ics teacher, She was at Malcolm X College of the Chicago
City CoHeges from 1 967 to her retirement in 1 982. Student
demand for Black histcry awakened her interest in history of
mathematics. especially of ancient Egypt She has published numerous books and articles on this subject, including Alge
bra Activities from Many Cultures (J . Wis on Walch . 1 997).
3. Chace. A. B. The Rhind MatflemaUcal Papyrus. National Counci l of Teachers of Mathematics iNCTM). Reston, VA. 1 986. orig . 1 927-2fl.
4. Clarke. S. and R. Engelbach. Anclent Egypti<Jn Construction and Architecture. Dover . New York. 1 990, orig. 1 930.
5. Gardiner. A. H. Egyptian Grammar. Griffith Institute. Oxford. 1 978.
6. Gillings. R. J. Mathematics in the Time of the Pharaohs. MIT Press. Cambridge. 1 972 , rep rin ted Dover. New York, 1 982.
7 . Gunn. B. "An Architect's Diagram of the Third
Dynasty" in Annales I'Egypte (Cairo) 26, 1 926. Tile l ime stone fl ake containing this diagram was listed in the Cairo M useum Journal d!Entree. number 50036. du Service des Antiquites de
8. Lepsius. R. The Ancient Egyptian Cubit and its Subdivision. Eng
l ish translation by J. Degreef from the 1 865 original, Die Aft Agyp
tische Elle und lhre Eintheilung. Edited by M. St. John. Museum Bookshop Publications. London. 2000.
9. Reisner. G. A. Mycerinus: the Temples of the Third Pyram1d at Giza.
Harvard University Press, Cambridge. 1 93 1 .
1 0. Robins. G. Proportion and Style in Ancient Egyptian Art. of Texas. Austin, 1 994.
1 1 . Schartt , A. "Ein Rechnungsbuch des K6nig lichen
Dynas
ie
University
Holes aus der 1 3 . Agyptische
(Papyrus Boulaq Nr. 1 8),'' i n Zeitschrift fOr
Sprache und A/tertumskunde 51, 1 922.
12. Spalinger. A. "Notes on
the Day Summary Account of P. Bu laq
18
and the Intradepartmental Transfers , " in Studien Zur A/Ui.gyptischen
Kultur 1 2. 1 985.
VOLUM£ 24. MJMBER 2. 2002
25
i,i,fflj.t§p@ih%1ifJ.ipipijj.iihfj
Mathematical Activity in Mozambique Paulus Gerdes
This column is a forum for discussion
of mathematical communities throughout the world, and through all
M arj o r i e Senec h a l , E d i t o r
M
ozambique lies on the southeast em coast of Africa. The country has an area equal to that of France and Great Britain together. From the inte rior of Africa, Bantu agriculturalists immigrated into the region from the early centuries AD onwards, encoun tering small groups of Klwi and San hunter-gatherers. The coastal zones were visited by seafarers from Arabia, Persia, India, and Indonesia, and after the late fifteenth century, from Portu gal and other European countries as well. The name "Mozambique" seems to derive from Mussa ibn Bique, the name of the ruler of a coastal island, which was mistakenly thought by Vasco da Gama in 1498 to be the name of the region. For several centuries Mozambique suffered from the slave trade to islands in the Indian Ocean and to Brazil. Mozambique's borders were laid
include "schools" ofmathematics,
l i beration stru g g l e , Mozam b i q ue
circles of correspondence,
became i nd ependent .
mathematical societies, student organizations, and informal communities of cardinality greater than one. What we say about the communities is just as unrestricted. We welcome contributions from mathematicians of all kinds and in all places, and also from scientists, historians, anthropologists, and others.
Please send all submissions to the Mathematical Communities Editor,
Marjorie Senechal, Department
of Mathematics, Smith College, Northampton, MA
01 063 USA
e-mail:
[email protected]
26
mixed population. There are eighteen indigenous languages. The official lan guage, Portuguese, is the mother tongue of only about 6% of the population; but school education is conducted mostly in Portuguese. About one third of Mozam bicans are Muslim; about one third, Christian. The country is predominantly rural; subsistence farms are maintained primarily by women, with cattle-raising, fishing, and hunting traditionally male acti\
I n 1 97 5 , after 1 1 years of
time. Our definition of "mathematical community" is the broadest. We
I
down at the Berlin Conference of 1885. They did not reflect the cultural and linguistic realities, but only served to "legitimize" Portuguese conquests. In 1975, after 1 1 years of liberation strug gle, Mozambique became independent. The school system was little devel oped, illiteracy was 95%, and though there was a colonial university in Lourenc;o Marques (today's Maputo), most of the professors and students left after Independence. Only one mathematician educated at the colo nial university, Ana Maria Pires de Car valho, stayed on, and only two mathe matics students remained. Today Mozambique has around 17 million people, mostly Bantu, with mi norities of Arab, Chinese, European, and Indian origin, and a substantial
THE MATHEMATICAL INTELLIGENCER © 2002 SPRINGER-VERLAG NEW YORK
and physics, or of mathematics and bi ology. The establishment of programs in mathematics education and mathe matics was supported by mathemati cians coming from countries including France, Guinea Conakry, Netherlands, and especially the then Gem1an Dem ocratic Republic and Soviet Union. Ini tially mathematics teachers were trained in Maputo at the former colo nial university, renamed in 1976 Uni versidade Eduardo Mondlane (UEM), after the late cultural anthropologist and first president of Mozambique's liberation movement. From 1986 on wards teacher education for secondary schools took place at the Instituto Su perior Pedag6gico, later transformed into the Universidade Pedag6gica (UP), which today has campuses in Ma-
puto (southern region), Beira (central), and Nampula (northern). Publications
Since Independence, tens of mathe matics books have been published in Mozambique: school texts, university manuals, and books with research re sults. Jan Draisma, who had a mathe matics degree from Amsterdam and who before Independence had taught in the secondary school of the Mozam bican liberation movement in Baga mayo, Tanzania, coordinated the de velopment of mathematics books for primary schools. A team of Soviet and Mozambican mathematicians, coordi nated by Evgenil Shcherbakov and Ana Maria Pires de Carvalho, wrote mathe matics books for secondary school. Joao Carlos Beirao wrote a series of calculus textbooks, the first volumes with a Soviet collaborator, Rodeon Aleksandrov, who later died in Angola. In the 1980s came Michail Evdokimov's
Analytic Geometry and Linear Alge bra and Igor Mozolevskii's handbook of calculus problems. Danielle Huillet from France and Achim Friedrich from the German Democratic Republic pub lished a booklet for future teachers on how to prove theorems. Ida Alvar inho* 1 and Sergei Vodopyanov wrote a manual on Euclidean geometry. In the 1990s, a younger generation of Mozambicans, among them Abdul carimo Ismael* and Ismael Nheze, * wrote experimental books for sec ondary schools. Marcos Cherinda* contributed to a book on famous theo rems of geometry. A couple of sec ondary-level mathematics exercise books were written by Zeferino Mar tins* together with Raul Carvalho, for mer director of the Teacher Education College of Settibal, Portugal. During the 1980s the Faculty of Ed ucation of UEM published a mathe matics education journal called TLANU. For the last decade, the De partment of Mathematics, Statistics, and Computer Science of UEM has published a research bulletin. Several Mozambican mathemati cians have published abroad. The Uni-
Figure 1 .
versity of Perm, Russia, recently pub lished the doctoral thesis of Manuel Alves on Second-Order Singular Func tional Differential Equations; the same university awarded a Ph.D. to Paulo Munembe for his research in the same field. I will mention below sev eral publications in my own area of eth nomathematics. Olympiads
During the 1980s Mathematical Olympiads took place in Mozambique. Many thousands of secondary school students participated. The journal TLANU published booklets with prob lems from the Olympiads. Besides pop ularizing the subject, the Olympiads helped in spotting talent. For example, Esselina Macome,* the highest-ranked girl in the first Olympiad (1980), is to day Deputy Dean of the Faculty of Sci ences at UEM and is completing a doc toral thesis in computer science. Abdulcarimo Ismael, * the highest ranked boy in the same competition, received his Ph.D. from the University of Witwatersrand, South Africa, in 2002, with a thesis on Probabilistic As
pects of a Manca1a Board Game from the North of Mozambique. Both re ceived Master's degrees from German institutions. The Olympiads were sus pended because of the war but have been restarted as of 2001. During the 2000 International Mathematics Year, for the first time a team of four Mozam-
bicans took part in the Pan-African Mathematics Olympiad in Cape Town, South Africa. Higher Education in Mathematics
The former Faculty of Mathematics of UEM, now the Department of Mathe matics, Statistics, and Computer Sci ence, gives service courses in other faculties of the UEM; there is research in applied mathematics, especially func tional differential equations. There is a plan to offer B.Sc. and M.Sc. programs in mathematics. The UEM's Faculty of Education recently introduced an M.Ed. in mathematics and science education. The private Instituto Superior de Cil�ncia e Technologia de Mo<;ambique, created in 1996, also offers computer science. The Catholic University very recently opened a department of math ematics education in Nampula. The UP now has mathematics de partments at all three campuses and of fers B. Ed. and M.Ed. programs in math ematics education, directed toward middle-school and high-school teach ing. At the Beira campus, the UP also trains specialists in mathematics edu cation for primary schools. The gradu ates of this unique program go on to become teacher-trainers, curriculum developers, and textbook writers. Research in mathematics education is conducted at both UP and UEM, and UP also has programs in the history of
1 An asterisk after a name means the person was a student of mine at some time in the 1 970s or 1 980s.
VOLUME 24, NUMBER 2. 2002
27
research symposium took place in 1991 in Maputo. A special SAMSA sympo sium on mathematics in primary schools was held in 1993 in Beira. Var ious Mozambicans have been officers of the Association. Similarly, Mozambicans are active in the continent-wide African Mathe matical Union (AMU). Since 1986, the newsletter of its Commission on the History of Mathematics in Africa has been published in Maputo. Among those active on AMU commissions are Esselina Macome,* Jan Draisma, Marcelino Lufs,* and myself. Ethnomathematics, a Personal Note
Figure 2.
mathematics and in ethnomathemat ics, of which I will say more below. Broader Role of Mathematicians
Mathematicians in Mozambique appear on a wider stage. The "nestor" of Mozambican mathematicians, Joao Carlos Beirao, educated in Portugal, taught before Independence at the Uni versity of Angola. He becan1e in 1978 Academic Director of the UEM, and was later appointed Deputy Minister of Education. The former mathematics teacher Zeferino Martins* held the san1e post in 1994-1999, and in 2000 be came Deputy Executive Secretary of the Community of Portuguese Speak ing Countries. (This Community com prises Angola, Brazil, Cape Verde Is lands, Guinea Bissau, Mozambique, Portugal, and Sao Tome and Principe; East Timor is expected to join.) My own posts have included Rector of the Universidade Pedag6gica 1989-1996. Mario Getimane* and Ida Alvar inho* for several years were, respec tively, the scientific and pedagogic di rectors of UEM. Daniel Soares* is Principal of the Beira campus of the UP. Generosa Cossa, a UEM mathe matician trained in the Soviet Union, coordinates a program for the introduction of computers in secondary schools. She has been city councilor of Maputo for social affairs and educa tion, and is the co-founder and chair2This appeared in German
woman of the Mozambican Associa tion of Female University Lecturers. Mathematician Jan Draisma re cently became Dean of the Faculty of Education at Nampula campus of the Catholic University. Joao Loureiro* is President of the National Institute of Statistics. Several mathematics educators oc cupy key positions in the Ministry of Ed ucation. Sarifa Magide Fagilde* is Na tional Director of Secondary Education; Simao Mucavele* is Director of the Na tional Institute for the Development of Education; Ismael Nheze* is National Director for Pedagogical Support; and Agostinho Coetzee* is National Director for Primary School Teacher Education. In short, mathematicians and mathe matics educators are overrepresented among administrators. Thus Mozambique, which did not suffer a brain drain to foreign countries after 1975, experiences an analogous diversion of people with mathematical training away from teaching and ap plied mathematics into other func tions. They are also recruited by banks, insurance companies, and computing and statistical organizations.
When mathematics teacher education began at the Universidade Eduardo Mondlane in 1977, it confronted a wide spread antipathy to mathematics. Most students regarded mathematics as a useless subject imposed by the colo nialists to block Mozambicans' progress in school. An immediate response by the lecturers was to give the subject "Mathematics applied in daily life" an expanded share in the program. This did not address the image of mathe matics as alien to Mozambique. Did mathematics have no roots in African cultures? Stimulated by this challenge, I pro duced in 1978 a research proposal on the "traditional empirical mathemati cal knowledge of the Bantu popula tions in Mozambique," which ten years later became the Mozambique Ethno mathematics Research Project (MERP). My study of culture and the origins of geometric thought was concluded in 1985. 2
SAMSA and AMU
Mozambicans played an active role in the creation in 1980 of the Southern African Mathematical Sciences Associ ation (SAMSA). The biennial SAMSA
Figure 3.
(Ethnogeometrie, Franzbecker Verlag, Hildesheim, 1 99 1 ) and in Portuguese (Universidade Federal de Parana, Curitiba, Brazil . 1 991 , 200 1 ).
A shorter English-language version, with a preface by the late Dirk Struik, is forthcoming from MEP Press, Minneapolis, MN USA.
28
THE MArHEMATICAL INTELLIGENCER
tartin,g in the late 1980s a number young, eager Mozambican hav
f
join d
train d
MERP-people
in
Mozambique who did graduate tudy in
Democrati R ·publ ic, Ismael,* Marc herloda,* a nd Daniel Soare . * I re· · f n d
lik
bdulcarimo
in
ticular
SofaJa
and
p rovinces. The e scholaJ
Zamb zia
also
ha
parti ipat d in collective studies, lik
one on nu meration systems in Mozam bique, published in 1993.
A third generation in MERP is com· of some of our fom1er stud en· UP. For in tance, Salimo Saide"' did
pos d at
fi ld work among Yao women in t he the country, analyzing th
nonh of
geom ·try of their pottery decoratiOJ Evari
aile*
analyzed basket w
ing among the Changana in t he
_:J
av
outh.
Gildo Bulafo* did field work am ong
Tonga women to understand the g o
m tric and aritJunebc und rp in nings of
their b autiful handbags: � Abilio Ma
papa* studied the geometri
of
hi ldr n who make miniature wir
cars. th
and Bu l afo are lectur rs at
ail
ira
ampus
of the
P. Said
and
working in custom and at comm rcial bank A b ook l et Explo
1apapa are a
thinking
mlioll ·
in
Eth nomath ema t i.c.tS
and
Effmo cience i n Mozambiq·u e ( 1 994)
introduc
s
generation.
lat
th
work of
the younger
Much of my own research sin -e th '
1980s has been on mathematical el
em nls of the pictograms drawn by tra·
d i tional tory-tellers of the Tchokwe in
eastern Angola. This has appeared
in
books uch as Une lmdition geom 't riq'Lie en Ajlique (l'Hanuattan, Paris, 1995) L·unda Geometry (MERP Ma puto, ! 996), Geametricai Rec�rea liOiti oj Afr ica-Luso na (l'Hanuattan, Paris,
1997), Eth nomathem atik um Beispiel , . Sana Geomelrie ( pektrum erlag, Heidelberg 199 ) . A chapter of my Geomehyfrom Aj�"ica (MAA, Washing ton, DC, 1999) is an introdu tion to the ubj ct, and further wol'k .in progress ( e Figs. 1-3). Most scientific work in Mozambiqtle depends on foreign financing. The wedish government agen y SAREC has been the principal financial sup· porter of "capacity building� r s arch. It generously support d MERP for twelve years. This has now dried up, as they feel that "capacity was v ry sue-
A U T H O R
d
·
fully built up." Th
MERP, the
matics
formed donors.
c
u
sor
PAULUS GERJ)ES
MozwtbtCan Elhn:llltla hemallCS Research Centre
to
C.P. 91 5
Mozambi an Eth nornathe Research Centre (MERC),
Mdputo
Mozarnl;Jique
in 1999, is se king alternative
e·ma11.
[email protected]
Paulus Gerdes has been
lar>e
The major challenge aft r lndep n d nee in 1975 as to build up a Mozam from
coloniali ·m
impetus to this
gave
has
Mozambique now
dozen of pro
mathematician
and
d reds of mathematics teachers. The obstacles
hun
these were reviewed In the The Math
ematical /l'ltellgencar. vol. 23 (2001), no. 2, 65-68 He Is currently Chair man ci the African Malhlrnafical Unbn
v · re.
wer
The .ountry has suffe r d fr m floods (end
of the 1970s, and 20 0 and 200 1 ); from
former
' hite
Rhodesian
regime.
By any
countries in the world. Now living in pea
ar
to
truggling
trengthen this
to
mmunity and
drain. We have
of specialists
avoid re
m asure,
1 992, we
l idate
and
mathematical futur
brain
tart the training
education at all levels, drawing as
E hnomathamalics. and
Science and Cultural
poorest
wi1thin Mozambique. We quality of mathemat
have to raise the
ics
in con
young
on
President
of the lntamatlonal Association for
uth African
Mozambique is stiJI one of th
AfriCa (AMUCHMA), Pres
ident of ll'e lntema lona l Study GroJp
1980s
an inswTection
early 1990s from
supported by the former
apartheid
ema ICS 10
minority
r'gime ( 1 970s); and during the and
CommiAA n on 'he His
by the
drought ( 1 980s); from atta k
arld he Unwersidade
1 976. Ho was visiting professor at t'le
other count1ies was important to it.
fessional
unlver.>rty
UnNcroity of Georgia (USA) 1996-1998. Among his books are Geometry from Africa (MathomaticaJ Association of America, 1 090) and Le Cercle et le Carre (I'Harmattan. Paris, 2000);
great
and h. lp from
eff01t,
proTessor
Pedagogics in Mommbq.Je sirce
bican mathematical community. Liber
atjon
a
of mathematics al the Eduardo Mord·
Challenges
much
as p
ematical roo
ciety 19
of
ibl .
W
on
to carry on th
by th
Friends
Diversity.
indigenous math·
n ed an active so
work done in the
Mozambican Association
of
Mathematics.
We
we l
come t h e coop ration and friendship
of colleagues at"ound
the world.
3S. Salde. "On the geomet.y of pottery docoratlon by Yao women." in: P. Gerdes, Women, Art and Goometty in Southern Africa, Africa Word Press. Trenton NJ, 1998. •p, Gerdes and G. Bulafo. SipatSt: Technology. M and Geometly in lnhambane. MERP. Maputo, 1994. [The translation 1n o English was Clone by Arthur owell. a pro· f S\lOI' lrom Rutgers in the SA woo was a VISitlng proles501 01 UP.)
MARIO MARKUS AND ERIC GOLES
C i cad as S h owi n g U p After a Pri m e N u m be r of Years
•
~
n 1 9 70, while Bob Dylan 1oas receiving an honorary degree from Princeton Un iversity, a plague of millions of cicadas poured over the event. This i nspired Dylan clearly not an entomologist-to write his song "The Day of the Locusts " [ 1]. Exactly 1 7 years before, a cicada plague had appeared in the same place, a nd the same thing
happened in 1987. This did not astonish entomologists, who know well these periodic insects and could confidently an nounce to the public the next plague in 2004. Other cicadas of the same genus appear punctually every 13 years; others, every 7 years [2-5]. They accumulate up to 300 in sects/m2 over many hectares: a unique accumulation of an imal biomass. The cicadas stay in this crowd for a few weeks-mating, laying eggs, and dying. Then the next gen eration spends the rest of their time as larvae feeding on roots below the ground. Even larger biological cycles are known (e.g., 60 years for the seeding of bamboos [6]), but the particular feature of the cicadas is that their periods are prime numbers. Bi ologists have been explaining, for centuries, these prime periods with the following reasoning: suppose that the ci cadas had a period of 12 years; then they could be eaten by predators appearing every 1, 2, 3, 4, 6, and 12 years; a mutant to a 13-year cycle has fewer possible enemies and would thus be selected. This reasoning, however, contains a weak point: nobody has ever seen the relevant predators. To excuse their absence, it has been conjectured [7] that the enemies of the cicadas (say, fungi feeding on eggs or adults) had nowhere to go but to extinction after the mu tant cicadas were selected. Shaky though the hypothesis is,
30
THE MATHEMATICAL INTELLIGEN CER © 2002 SPRINGER-VERLAG NEW YORK
we thought it might yield an entertaining model. In the pre sent work, we present an algorithm simulating this evolu tionary scenario, assuming that the predating fungi really existed. As a curiosity, we can put our biological algorithm to work (assuming inordinately large cycle-lengths) as a generator of very large primes. Fitness, Mutations, and Selection
We assume a fungus (predator) of period F interacting with a cicada of period C. We assign a momentary fitness cbc(t) of the cicadas in a year t as follows: it is zero if cicadas are not present, it is - 1 if both fungi and cicadas are present, and it is + 1 if the cicadas are present but the fungi are not. Note that we punish cicadas that appear and meet fungi ( cbc (t) = - 1), as compared to the case of non-emergence of the cicadas (cbc(t) = 0); we do this because emergence uses up metabolic resources due to metamorphosis, mating, and death; these resources are lost if the cicada eggs are eaten up by the fungi, while they are preserved if the cicadas stay as larvae below the ground. The momentary fungus fitness ¢.r(t) is defined analogously as for the prey, but with oppo site signs. The fitness F1, resp. Fe. is defmed by the sum over the ¢.r(t), resp. cbc(t), t 0, . . . , FC, divided by the number of fungus, resp. cicada, generations. (Note that this yields =
(.)
0
Instability of Non-Prime Prey Cycles
10
/
8
C=E=21 47483647
Ci s .Q �
4
2
50
1 00
150
Figure 1 . Generation of a large prime number (E).
t
an average valid for t � oo, because the process is periodic with period FC). We divide by the number of generations because we found that not doing so would favor short gen eration times, instead of properly timed generations (prime cycle lengths). This is important if one considers that each generation uses considerable metabolic energy, as men tioned above, and these expenses should be minimized in the long run. Assuming that cicada larvae can live safely for a long time below the ground, our model favors infrequent emergences, as long as the cicadas are safe when they do appear for mating. A similar reasoning applies to the fun gus; such a predator can survive for a long tin1e as spores, and our model favors infrequent appearances, as long as they get nourishment when they appear. Nevertheless, this model does not cause divergence of cicada cycles, because these cycles eventually get locked into a prime number as we will show below-bringing evolution to a stop. Let lcm (F, C): least common multiple, gcd(F, C): great est common divisor of F and C. In FC years, the fungi ap pear C times, both cicadas and fungi appear FC!lcm(F, C) times; thus fungi without prey appear C - FC/lcm(F, C) times. Considering that gcd(F, C) · lcm(F, C) = FC, we thus obtain the fungus fitness Fj(F, C) = 2gcd(F, C)/C - 1. Anal ogously, one obtains the cicada fitness Fe(F, C) = 1 2gcd(F, C)/F. We compare a cicada mutating to a cycle C' with the resident cicada (cycle length C), at constant F. Analo gously, we compare mutant cycles F' with resident cycles F, at constant C. A mutant cicada (resp. fungus) replaces the resident if and only if Fe' > Fe, resp. F.t > F.t' Thus, in the case of fitness equality, the resident and not the mu tant is selected. Here and below, we assume that all inter acting populations are synchronized, thus being all present at t = 0. Proposition
We will show the following: if we allow random mutations, which can be of any size as long as they lead to mutants obeying 1 < F < C (condition K), then a sequence of such mutations will finally lock the cicadas into a stable prime period C. To be more precise, we will show that if C is not a prime then there exists a sequence of mutations that will change C, and if C is prime then no mutation will change it. Note that condition K means that all cycles remain above the main diagonal C = F on the C-F-plane.
Let us assume that C = Cv is not a prime; Fr(F, Cv) has the maximum value 2 gcd(FM, CN)/CN - 1 at the fungus period FM = gd(CN) (gd(x): greatest divisor of .r, excluding x). A sequence of random mutations keeping C = CN constant will eventually lead to FM. However, (FM, CN) is abandoned if mutations lead to CFM, eN ::'::: 1). In fact, gcd(F!If, CN) = F,v, implying that Fc(FM, CN) = - 1; gcd(FM, CN ::'::: 1) can not be equal to FM (the reason is: (CN ::'::: 1)/FM = Cv!FM ::'::: 1/Fp,J, the first term being an integer, but the second not, so that FM is not a divisor of eN ::'::: 1) and gcd(F!If, eN ::'::: 1) can certainly not be larger than FM; thus gcd(FM, eN ::'::: 1) < FM; this implies that Fc(FM, CN ::'::: 1) > - 1 = Fe(FM, CN). Thus, we have shown that there exists a sequence of mutations such that cicadas with a non-prime cycle C = CN are ex tinguished. (Note: (FM, CN) may also be abandoned by mu tations larger than CN ::'::: 1). Stability of Prime Prey Cycles
Assume that C = Cp is a prime; by virtue of condition K, any F is relatively prime to Cp; therefore gcd(F, Cp) = 1, so that starting from (F, Cp), there exist no fungus mutants that are fitter than a resident fungus. On the other hand, for any F, gcd(F, C') 2: 1, where C' is a cicada mutant, as compared to gcd(F, Cp) = 1, so that Fc(F, C') ::; Fe(F, Cp), i.e., no prey mutant is fitter than a resident. In conclusion, any initial random choice of (F, C) and mutations fulfilling condition K will lead and lock to a prime C after a suffi ciently large number of mutations. Note that we cannot loosen condition K, because the points (jCp, Cp), where Cp is prime and j = 1, 2, 3, . . . , are unstable with respect to prey mutations. This can be shown easily: gcd(jCp, Cp) = Cp, while gcd(jCp, Cp- k) with k = 1, 2, 3, . . . cannot be larger than Cp- k; thus Fc(jCp, Cp) < Fc(jCp, Cp- k). This means that convergence to cicadas with period Cp is not possible if mutations to the points
,--- - - - - - - - - - - - - - ���--
18
c
17 16 15
14
13 12
L------------------------
11
10
�--�--+---�--�---+----r---�
1
2
3
4
5
6
7
Figure 2. Evolution of cicadas (C) and fungi (F).
F
a
9
VOLUME 24, NUMBER 2 , 2002
31
(j
p,
p) are permitted. These points are rul
dition K.
Generation of Very
Large
d out by con
Primes
i!l!nd of Blo,1ogit;al Cycles
pr ess, w assumed that mutation and selection of F alternate in successive time tep with mutation and selection of C. In order to get very lar primes, we allow for mutations with unbiologically lar e ycle changes. As an exampl , w onsidered muta tions of th siz 10'1, where 11 is randomly and equally cho sen ln fO, 6]; a r suit i..s shown in Pigur· 1 , starting at F 3, C = 9 and leading to the Eui r-paime E given in the ftgur .. oming back to earth, we now considec a biologically realistic process. For chis we ha'0 to Lake into account that condition K is not plallSible in li ing nature. In fact, this condiLion would mean that the r strtctions for cicada mu tations depend on those for fungus mutations, and vice =
v r a. Such dependence is avoided by r stric:ting mutations
U2, U2
2
within
K on the C-F�plan,
d fin d by:
2 s F :s: number main diagonal C =
:s: C :s L, where L is an arbitrary even
Larger than 6. This square lies above the
F (condition K is fulfilled), and it contains the point that de tabil izes non-prime cicada cy I , nru1nely (F 1, CN). be caus l < £';,, gd(CN) ::::: L/2. Figure 2 shows an evolu tionazy process, \vi:thin such a square (dash d lines; L 18), locking at the prime cicada cycl C 1 7. =
=
=
are
faster
methods for the det
Th
than
the method pre
ti n of large primes which nted
here (see e.g. [ )). work, however, is lhe biological rationale underlying tit prime-generating al gorithm. Our algorithm dis:pla.ys th merging of two seen t ingly u_nr lated subjects: numb r th ory and population biology. Fina!Jy, we hope you do not find our explanation for plim r productive cycles too convincing, for it says noth ing aboul why many species hav non-prime cycles. arc
For mimicking an evolutionary
to a squar
Concluding Remariks
There
remarkable feature of the p11escnt
Acknowledgments W thank the Deutsche Forschung gemeitn:scb.aft and F 1 AP ( hile) for financial upport. Also, we thank Oliver S hulz and David Bressoud for fmitful interactions. References
[1 ] Dylan, B. . "New Morning"; Columbia Records; PC30290. [2) May, R. M., Nature 1978, 277 . 347-349. 131 Karban. R. , Nature 1 980, 287, 32�27. 14] Krttsky, G . . Nature 1 989, 344, 288. [5] Simon, C .. Martin, A . Nature 1 989. 34 1 , 288-289. [6] Bamboo ReseaJch in Asia essard. G .. Chouinard. A.. Eds.), [7)
Dev. Res. Center: Orawa. 1 980.
aye!,
M ..
Dybas,
H..S. EvolutiOn 1 966. 20. 466-505.
181 Rlbenbolm , P. The New Book or Prime Number Records, Springer·
Verlag:
New York. 1 991 .
AUT H O R S
MARIO MARKUS
�-Pianck-Jnstitw fOr Mdekulare
Otto-Hahn·Strasse
11
D-44227 Donmund Germany
ERIC QOLE8
Culter for
Mathen teal Modei!L"'g Univ� de Ohl!e
Pt1ysioi0Q'E1
Mario Markus was born in C11i e. He carne to Germany when received
pmsent he heads
a
his PhD In Physics at Heidelberg At
research group at the Max Planck lnsti
tut In Dortmund and is also a Professor at the
Unrtersity. He
cdlects Rorran coins and produces much-exhlbted computer graoncs: bu mainly. he 'Mites and Pt.1btshes poetry.
32
1 1-lo MATI
Caail e 1 70 3 Santiago
o-ma il: markus®mpi dortmund.mpg.da
he was 20 aM
tnt
ChHe
�ric Goles got
e-mail:
[email protected]
e
PhD in Computc;lr Science and another PhD
in Mathematics. both at Grenoble, France. Ha ls Pr-ofessor at
lhP University of Chie 2fld also PrASident of the National Re search Council of ducted
a
Chile (CONICYT}
Since 1996 he has con
OOPJiar soentllc TV program. He has two ctt ldren.
He colects fossils in the Chilean desert.
MARK MCCLURE
D i g raph Se f - S 1 m 1 ar Sets and Aperiod ic Ti 1 ngs
elf-similarity is a concept often associated 1oithfractal geometry. There are many i n teresting self-similar sets in the plane which would not generally be con sidered fractal, hmoever, a s they a re of dimension two (a1though thei r bound a ries might be fractal) . Such sets expan d the possibilities for tilings of the plane. Furthermore, a generalization of self-similarity called di graph self-similarity provides a way to construct aperi odic tilings. In this paper, I begin by looking at the connection be tween self-similarity and tiling. I also describe how self-sim ilar sets are detem1ined by iterated function systems. I then consider the generalization of self-similarity, digraph self similarity, and examine how it is related to Penrose kite and dart tilings. Finally, I describe digraph iterated func tion systems and use such a system to construct a visually interesting aperiodic tiling based on the Penrose pentacles. Self-Similarity and Tiling
A set that is composed of several scaled images of itself may be thought of as self-sim ilar as described in [5] and [6]. (I will write a more mathematical definition shortly.) A square is a simple example of a self-similar set in the plane, being composed of four copies of itself scaled by the fac tor 1h Each of these four copies is in tum composed of smaller copies. At any of the scales, we generate an obvi-
ous checkerboard tiling of the plane. A tiling is simply a family of closed sets that cover the plane and intersect only in their boundaries. A tremendous amount of information on tilings may be found in [8] . The connection with self-similarity suggests the possi bility of introducing fracticality into the picture. Figure illustrates this with a set called the terdragon, which is com posed of three copies of itself, all scaled by the factors A tiling using the terdragon is illustrated in Figure 2. Sets with fractal boundaries that tile the plane are called "fractiles" in [5]. General techniques for creating self-simi lar tiles are discussed in [2] and [4] . A more explicit description of self-similarity is in terms of an iterated ju nction system, or IFS. If r is a positive real number, a similarity with ratio r is a function j : IR/2 � IR/2 such that IJCx) - f(y) l = rfx" Yl for all x, y E IR/2 • If r < 1 , the similarity is called contractive. An iterated function sys tem is a finite collection of contractive similarities
1
1/V3.
-
J )"' lfi
i = l·
© 2002 SPRINGER-VERLAG NEW YORK, VOLUME 24. NUMBER 2, 2002
33
.
;.2. (:r ) = . ( .)
;J .:r
J
Figure
For any IFS,
E
closed, bounded
"'
=
U flE) t�l
overlapping only in boundaries. The set E i s called the i n varia n f' set of thE> I FS ,
and sets constructed i n this manner
self-sim ila 1·.
Iterated fnnction systems are easily described
using ma trix representations. Any contractive similarity may be ex pressed in the form 4i + 6, where A is a matrix and 6 is a translation vector. A rotation about the origin through an gle (} may be represented using the matrix
..
(U)
R
Multiplying the
2
I
lt2
.r +
- 1/(2\
.r +
and Aperiodic Tllings
Figure 3 illustrates a generalization of self-similarity called
there is a unique non-empty,
are also called
1r
Digraph SeH·Similarity
1 . The terdragon.
subset E of IR2 with
"""
R( ) .. ( J) ) I/2 ) R( 11') - ( 11(2v3) \73 - 6 I \ ·3
= (cos
8
- si.n
sm 8
matrix by
r
cos
(})
B .
achieves the desired contrac
tivity factor. For example, the
following
defines the IFS for the terdragon.
Figure 2. A tiling using terdragons.
list of functions
digraph self-si m i la rity, also described
a
concept
introduced
in [ 10]
and
in [5). The tenn m i.red se{{-si m ila1·ity was
Lntroduced in [ 1 ] to describe the same idea. I will use the temtinology of [5]. Digraph self-similarity is exhibited by a collection of sets, each of which is composed of scaled :im
In Figure 3, for example, is composed of two copies of itself, to gether with one copy of the type B triangle. The type B tri angle is c-omposed of one copy of itself, together with one copy of the type A triangle. The scaling factor for all im ages is 1/T, wh ere T is the golden ratio. As with self-similar sets, the basic decomposition may be iterated to obtain tihngs. In Figure 4, we see the fowih step in the decomposition. It is a bit more difficult to see how the digraph self-similar scheme can be used to tile the plane. That is, how do we use higher levels of the decom position to extend partial ti.lings defined by a lower level ages
chosen from the collection.
the type A triangle
of the
decomposition? Note that the construction implies
that copies of the lower-[e.,·el decomposition '�ill be
in the higher lined in bold
le\rels. For example, in
a
Figure 4 I
fonnd out
haw
copy of the leve l-�:me configuration inside
Type A triangle
Type
8 triangle
r
Figure
3.
A digraph pair of triangles.
th� level-four configuration. We may extend the level--one configuration by dilating the level-four configuration by the factor il, then rotating and shifting until the outlined por tion exactly coincides with the level-one confi.guration from Figure 3. A collection of digraph self-similar sets can be described using a directed-graph iterated j11nction system, or di-
Figure 4. Generating a tiling with the
graph IF'S. A digraph IFS consists of a directed muJtigraph G, together with a contractive similarity fe from �2 to �2 associated with each edge of G. A directed multigraph con sists of a fmite set V of vertices and a finite set E of di rected edges between vertices. Given two vertices, 11 and v, denote the set of all edges from u to v by Em•· Associ ated -witl1 a digraph IF'S is a unique set of non-empty, closed,
ctigraph triangles. VOLUME 24. N.JMBER 2.
2002
35
Figure
5.
The digraph IFS for the triangles.
bounded sets u
EV
Kv, one for each Kll
=
v E V, such that for every
U
cc. 1 -, ft:::.. Eru
The
J.(Kr) .
Kite
Sets constructed using a digraph IFS are said to exhibit d i
graph self-similarity.
The digraph IF S for the type A and B triangles is shown
in Figure 5 . There are two edges from node A to itself and
one edge from node A to node
B, because the type
A tri
angle is composed of two copies of itself, together with one
B triangle. Similarly, there is one edge from B to itself and one edge from node B to node A, be cause the type B triangle is composed of one copy of itself, together with one copy of the type A triangle. The labels copy of the type
node
on the edges correspond to the following similarities map
angle.
ping one triangle to part of another (perhaps the same) tri
The Dart
- --_;1 R(37T) - (1) .5 .' ,.,/p,.) ) 1 (47T)(- 1 1 ) ( cos(� --_;R 5 sine,.,/,)
a1(:r)
=
a2(.f)
=
_
bz(:r)
=
X+
O 0
0
,f +
- R( 47T) ( cos(,., sm. /;,5))) 1
r
- ---za .
_
X
+
e rr;
Note that these similarities involve reflections as well as rotations. The need to include reflections will become clear in the next section. T
Penrose Tilings
In 1 973 and 1 97 4, Roger Penrose discovered several fami
lies of sets that tile the plane aperiodically and (if certain
matching conditions are enforced) only aperiodically . In troductions to Penrose tilings may be found in [7] and chap
ter 10 of [ 8 ) . The most well-known of these are tilings by kites and darts. It turns out that this type of tiling is closely
related to the digraph self-similar triangular tilings of the T
previous section.
Figure 6 illustrates the kite and dart. The dotted lines in
.1 triangles and B triangles. The filled and
dicate that the kite is the union of two type the dart is the union of two type Figure
6.
Penrose's kite and dart.
36 THE MATHEMATICAL INTELUGENCER
untilled disks at the vertices are used to enforce a match ing condition. When tiling the plane with kites and darts,
Figure
7. An alteration of the digraph triangles.
we demand that fille d disks meet filled disks and unfilled disks meet unfilled disks. This matching condition guaran tees that any tiling by kites and darts will be aperiodic; i.e., no translation of the tiling maps each tile to another tile.
Figure 7 shows how we may
generate a tiling by kites and
darts using the digraph self-similar set strategy. The top two
triangles of
Figure
Figure
7 are simply modifications of the trian-
gles from Figure 3, where
I have deleted an edge
from each
triangle and marked the vertices to match the markings of
the kite and dart. The bottom portion of
Figure 7 shows the moditied triangles. Note that the functions a2 and b1 from the previous section in volve reflections to get deleted edges to line up. Figure 8 shows the tiling after four steps in the iteration. the action of the digraph IFS on
8. Generating a tiling by kites and darts.
VOLUME 2J. NLJMBER 2. 2002
37
2
Figure 9. Penrose's pentacles.
38
THE MATHEMATIC-"L NTELLIGENCEA
0
1
Figure 1 0. Part of
a
tiling by pentacles.
Pen·rose Pentacles and Fractal Bo·undarles
can use digraph lf· i milarity to generate ap riodi Liling by sets \\dth frac tal boundari s. One such example, a modification of the kH and da:rt tilings, was published in [ 1 ) . I present a dif fer nt su h Uling here. General tech.niques for construct ing tilin Ys based on digraph self-similarity are discussed in [9]. Ju t as with strictly self-similar
odi
ur aper;octic fi·acliles will b
, w
-
bas d on the flrst aperi
·et or Wes disc overed by P · nro e.
the pen tacle
are desc rib d in
hav
length one. and the angles ar
"'/5. Th
lab I
indicate
These tiles, called chapter 10 of [8f. 9. All of tile sides
[ 1 2] and
The p ul!
al!1 int ger multiples of
a matchlng ondlt ion, which again
ass ur
ap
riQ_!Iicity.
The
�ge·
lab I d 0
!!lUSt
fit agai n l
edge labeled 0, 1 against l, and 2 against 2. Note that the lhr
p
ntagons are congruent but hav
different matching
cond!Lions. A portion of a tiling by p ntacles is shown in figur
1 0.
In the analysis of tile p ntacles in .
tile
bling
are assembled into "pat he
tM original tiles,
patchwork
12. Th
edg
b tw en any two \ erti
·
upper
in Lhe
Figure 1 1 . This Figure
digraph shown in
\iated by collapsing all into one and labeling i
with a list of funclioru;. For xa!inp! th
clion 10.3 of [8] . the
approximately resem
as i l lu trated in
trongly suggests th digraph has been abbr
"
th
star-like patch in
left of F'iglll'e 1 1 con i of on copy of the tar upper left of Figu!'e 9', five ,copies of the half star in ,
VOLUME 24.
�M6t:""A 2. 200!
39
the upper right of Figure 9, and fiv·e copies of the pentagon the lower Left of F'i.gure 9. Again, the labels conespond to a list of functions. For example, the function s1 is de in
fined by
where R( fJ) represents the matrix that rotates
through the digraph has 42 functions in all. The digraph IFS shown in Figure 12 Lmiquely detem1ine six sets that could also be used to tile the plane. These sets have a fractal boundary and are sh0\..'11 in Figure 13. We see how they fit together to form a set of digraph self-sim ilar sets in Figure 14. Part of a tiling by these sets is shovm in Figure 15. It should be mentioned that while not explicitly con structed, the fractalized pentacle tiling described here is al luded to in [3). The Wings described in [3] ar·e by fractal ized �·ersions of the kites and darts and another type based on Penrose rhombs. One strength of the tilings in [3) is that the fractal nature of the tiles enforces matching rules cor responding exactly to the matching rules stated by Penrose. The tiles described here do not have this pmpe1ty. For ex angle f). The
ample, the fraccalized. diamond has twofold rotational sym
metry, while Penrose's original diamond (with matching
rules) does not. Figure 1 1 . Penrose's patches of pentacles. St
Figure 1 2. The digraph IFS for th e fractal boundary pentacles. 40
11-!t MAll-i�MATlCAL 1!-ITEUJGENCER
Implementation Notes All of the images in this paper were generated by the DigraphFractals Ma thematica package wtitten by the au thor and described in [11]. All code for these images and more examples are available at the autl1or's web page
http:l/www.uncaedul-mcmcclur/mathemat icaGraphics/ DigraphFractiles.
REFERENCE$
[1 ] Bandt, C . .
(2] [3]
·Self-similar sets. Il l. Constructions with sofic systems."
Monalsh. Math. 108
(1989),
no.
:2-3. 89- 1 02.
Bandt, C., ·'Self-similar sets . V. Integer matrices and fractal tilings
of IR".' Proc. Amer. Math. Soc. 1 1 2
( 1 99 1 ) ,
no.
:2, 549-562.
Bandt. C. and Gummelt, P., "Fractal Penrose tilings. I. Collstruc
tioo and matching rules . " Aequa/iones Math 53 295-307.
(1 997),
[4] Darst. R . . Palagallo, J., and Pri�e. T., "Fractal tilings
(5]
Mathematics Magazine. 71
(1 998). no .
1,
1 2-23.
no.
3,
in the plane."
G. A. Edgar, Measure, Topology, and Fractal Geometry. Springer
Verlag, New York, 1 990 .
[6] K. J.
Falconer, Fractaf Geometry: Mathematical Foundations and
Applications. John Wiley and Sons, West Sussex,
UK, 1990.
Figure 13. Versions of the pentacles with fractal boundaries. A U T H O R
MARK MCCLURE
Department of Mathematics UNG-Asheville
Ashewle, N C 28804
e-mail:
[email protected]
Mar!< McClure is an Assistant Professor of Mathematics at the
Univ�ersity of
North Carolina at Asheville. He recewed his Ph.D. University under the directiOn of Gerald Edgar.
at Ohro State
His primary research interest is the study of fractal dimensions
of various sets arising in analysis. He also loves hiking and
biking in the mountains of western North caro�na.
Figure 14. F'rtti ng 1he pieces together.
VOLUME 24. MJMElEF! :l. ;lOOZ
41
Figure 15. Part of a tiling using the pentacles wittt fractal
[7]
boundaries.
Gardner, M . . Penrose Tiles to Trapdoor Ciphers. W. H. Freema11,
· New York, 1 989.
directed constructions, " Trans. Arner. Math. Soc. 309 {1 988),
[8] Grunbaum. B. and Shepard, G. C . , Tilings and Patterns. W. H.
[1 1 ]
[9] Kenyon,
[12)
Freeman, New York. 1 987.
)1 0)
42
R. , "The Construction of Self-Similar Tilirtgs." Geom.
Funct: Anal. 6
(1 996),
no. 3, pp. 4 7 1 -488.
Mauldin. R. D. and Williams . S. C . , " Hausdorff dimension in graph
TH� MATHEMAnCAL •r.ITElLIGeiCEfl
2. pp, 81 1 -829.
no.
Mcaure, M . . "Directed-graph iterated function systems. " Mathe
matics in Education and Research 9 Penrose, R., "Pentaplexity:
plane. · Mathffi!attcaf
37.
a
(2000) .
no.
2.
class of non periodic mings of the
lntelligencer
2 (1 979/80), no.
1,
pp. 32-
Dedicated
Commemorative Windows in the Hall of Gonville The and Caius College, Cam bridge, England
to JOHN VENN Fellow 1857-1.923 President 1903-28 R. A.
a nd
FISHER
Fellow 1 920-26 1943-62 President 1956-59 two windows, designed by
Maria McClafferty following my
proposals. 1989.
were installed in
August
The upper window shows a Venn Di agram for three sets, ftrst used in set theory by John Venn in 1880 and made popular by its inclusion in his book Symbolic Logic the following year. Each circle represents a set; each pair of circles overlaps, and hence the three ways in which three sets can intersect in pairs are represented; and fmally all three circles overlap in the centre, rep resenting the single way in which three sets can triply intersect.
A. W. F. Edwards
Does your hometown have
any
math.enwtical tourist attraelions such as statu.�s, plaques, ,graves, the cafi where thefamous conjecture uoas made, the d�sk where the famous initials
are scratched, birthplaces, houses, or
memorial.5? Have you
encountered
a mathematical sight on
lf so,
we
column
invite you
a picture,
f.()
your tra vels?
submit to this
a descriptio n of its
matltema.t'ical si.qnificauce, and either
a
map
or directions so
may follou
tha.t others
in your tracks.
Please send all submissions to
Mathematical Tourist Editor.
Dirk
Huytebrouck,
Aartshertogstraat 42,
8400 Oostende. Belgium
e�mail: dirk. huylebrouck@ping. be
C«nmemorative windows in the Hall of Gonville and Caius College, Cambridge, England.
e :2oo2 SPRoNCEI'I- vER..AG NE'W YOAK. liOLUME 24, NUMBEFI 2. 2002
43
The colours of the intersecting sets have been accurately portrayed by us ing the technique of plating, the glass being actually two and three layers thick where the sets overlap. The three basic colours are identical to three of the colours in the lower, Fisher, win dow, and the colours of the three pair wise overlaps will be seen also to match colours in the lower window.
wheat for yield, given a square field for the experiment. Each colour corre sponds to a strain of wheat, and it will be seen that each occurs just once in each row and once in each column, but that in other respects the distribution of the colours is arbitrary. If there should be a change in the fertility of the soil across the field, such a Latin square pattern ensures that it will not
distinguish the "colours." It is not known why Fisher chose this particu lar square for the dust jacket. Maria Ulatowska McClafferty, the artist, was born and educated in War saw. She studied at Warsaw Univer sity and in the studios of Stanislaw Szczepanski and Adam Niemczyc. She emigrated to the U.K. via Rajasthan. Major exhibitions of her work as a
Each colour corresponds to a strain of wheat , a n d each occ u rs once i n each row and once i n each col u m n . Only the red of the lower window is missing from the upper one. The cen tral colour, formed by triple plating, is of course practically black. The lower window, commemorating R. A. Fisher, depicts a 7 X 7 Latin square. It was erected as a contribution to the celebration of the centenary of his birth (17 February 1890), and the actual design is taken from the dust jacket of his famous book The Design of Experiments, published in 1935. The best way to visualize the use of such a design is to imagine the prob lem of testing seven different strains of
44
THE MATHEMATICAL INTELLIGENCER
benefit any particular strain of wheat, whilst at the same time the pattern pro vides a degree of arbitrariness which, according to Fisher's theory of the de sign of experiments, enables a statisti cal analysis of the yield differences to be undertaken. Such designs have had a great in fluence on experimentation through out science, and not just in agriculture. Latin squares are so called because when they were first studied by the great Swiss mathematician Leonhard Euler he labelled the squares with Ro man, as opposed to Greek, capitals, to
painter-printmaker have been held in India, Poland, Sweden, Australia, and Canada, as well as the U.K. In 1981 she turned to decorative glass as a medium. Major works are installed in palaces in Abu Dhabi, Al Ain, and Dubai, and at home the Great Rose Window of Alexandra Palace is her work.
A. W. F. Edwards Gonville and Caius College Cambridge CB2 1 TA U.K. e-mail: awfe@cam . ac . u k
lj.i%,1,fflj.l§,pih¥11i.)IIQ?Ji
D i r k H uy l e b r o u c k , Ed itor
Plagiary in the E Renaissance Kim Williams
Does your hometown have any mathematical tourist attractions such
I
ven the smallest towns of central Tuscany are dotted with monu ments that attest to the cultural wealth of fifteenth-century Italy. One such town not only contains monuments, but was the birthplace of two mathe maticians who are "monuments" in themselves. The town is Sansepolcro, or, as it was known 500 years ago, Borgo San Sepolcro; the mathemati cians are Piero della Francesca (Fig. 1) and Luca Pacioli (Fig. 2). Piero's skill as a mathematician has been admirably laid out in the Mathematical Intelli gencer by Mark Peterson [26], who also touched on the controversy that sur rounds Luca's publication, under his own name, of Piero's work. Reading Mark's article, I was sur prised at the plagiary charge against Luca. The Franciscan's portrait adorns the cover of one of the books in my li brary; looking at it, I doubted he could be so guilty. Here, I said to myself, is a case not only for the Mathematical Tourist, but for the Mathematical Pri vate Detective!
Borgo San Sepolcro was founded by pilgrim saints Egidio and Arcano, who, on their return from the Holy Land at the end of the tenth century, built a chapel to house relics they had re trieved from the Holy Sepulchre (hence the name). The town grew up around the chapel, which has long since disappeared. Today Sansepolcro is a thriving, pleasant town of some 20,000 inhabitants. It lies in the upper Tiber valley, on the winding road (S73) that connects Arezzo in Tuscany to Urbino in Umbria. The historic center, surrounded by high walls and entered through formal arches, retains much of its Renaissance ambience. Thanks to its being off the beaten tourist path, it is uncrowded and somehow private. Fifty years ago, Kenneth Clark wrote of it, "Borgo San Sepolcro . . . presents today very much the same aspect as it did in Piero's time, and everyone who has visited it has felt its affinity with his work" [3, p. 2].
The Scene of the Crime:
T h e Victim: Piero della Francesca
Borgo San Sepolcro
Piero della Francesca (also known as Piero Franceschi or Piero dei Fran ceschi) was born in Borgo San Sepol cro in the early years of the fifteenth century, perhaps in 1406 but possibly as late as 1412. His family were well to-do merchants. He might have be come a successful businessman, as did his brothers Marco and Antonio, had he not become one of the most signif icant painters of Renaissance Italy. Al though it would have been customary for the son of a merchant to study com mercial mathematics in an abacus school, Piero studied in Sansepolcro's municipal grammar school; here he probably encountered the geometry of Euclid for the first time [ 18, p.624]. Piero lived most of his first forty years in Sansepolcro, until more and more prestigious commissions for paintings and frescoes called him to the courts of princes like Malatesta in Rimini,
Sansepolcro originated as a Roman
castmm. Popular legend has it that
as statues, plaques, graves, the caje where the famous conjecture was made, the desk where the famous initials are scratched, birthp laces, houses, or memorials? Have you encountered
a mathematical sight on your travels? If so, we invite you to submit to this mathematical significance, and either column a picture, a description of its 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]
Two men born in the same town in the sanw century, pursuing the same math ematical interests: Was Luca an ac complished mathematician, or a fraud ulent plagiarist? Did a scheming Franciscan aching to leave his mark on posterity mean to consign Piero to ob scurity as a mathematician, though highly praising him as an artist? I de cided to visit Sansepolcro, to bring Piero and Luca to life and provide a his torical context from which to reflect on plagiary in the Renaissance. Besides, Sansepolcro offers a rich itinerary for the mathematical tourist: a fme mu nicipal art museum, the house of Piero and the monastery where Luca lived, two public parks with statues of the protagonists of this story, and com memorative plaques dedicated to each. The mostly regular, orthographic layout of the streets suggests that
© 2002 SPRINGER-VERLAG NEW YORK, VOLUME 24, NUMBER 2, 2002
45
When
Piero
Piero's father
died in 14-64,
b£>came head. of his family and
began a project
to reconstruct the fam
ily house, now known as the Casa Piero
della Francesca (Fig. 3), located at Via Aggiunti 71. He is credited with having done the
archltectural design
A
on
in the 1 470s, w hich reflects
himself,
the archi
tecture of the Ducal Palace in Urbina.
(Fig. 4) reads: plaque
the fa<;:ade of
In
the house
honor
Of' PlERO DELLA F'llANCESCA in the :fifteenth century
[ the] sovereign
painter
from whose mastery Perugino learned the rna.IVels of art
and Italy the geometric principles perspective
of
Here where her great son lived and where at 82 years he ascended
Figure 1.
[this
Pi&ro della Francttsca.
Este in Ferrara, and Montefel tro in
Urbina. Still, he continued to accept
cormnissions from local
patrons in Sansepolcro, although he sometimes stipulated enormous lead times in the contracts for them, reflecting the fact that they were done in spare moments. Piero's ties with Sansepolcro were
so great
that tradition
claimed he de
picted his birthplace in the background of his St. Jerome and a
Disciple,
now
in the Galiena dell'Accaclemia in Venice,
painted in
Resurrection,
the Poli.Uico della Miseri and S. Giuliano. It also contains a full-length portrait of Pi.ero painted by Santi di Tito around 1630. He is depicted as a young man, wearing the formal robes appropriate to me positions of honor that he held in Sansepolcro. Next to the portrait is a case displaying a fac simile of the archive conta.iniJlg the reg oon/.i.a,
istration of Piero's death.
about 1450. The affection that Piero felt for Sansepolcro was re turned; he was a most respected citi zen of the town. From 1477 on, he was a member of the city counciL He was priore, a very honorable position, of the charitable Confraternita di S. Bar tolomeo and was appointed an accoun tant to oversee city finances, evidence of his skills in commercial mathematics. He was also, of course, the town's most fa mous artist Most of the frescoes he did
in Borgo are lost today. Fortunately for the matbematicaJ tourist,
the Museo
Civico1 contains three of his works, the 'The Museo Civico, Via Aggiullll 65. is open seven ooys a week trorn 930 to 1 300 and 1 400 to 1800. Telephone + 39-0575-732218. e-mail museoclvioo@tacmet .it
46
lHE MA�MAfiCAL
INTELLIGENCER
Figure 2. Luca Pacioli.
[by a]
into the
memorial ]
1 876
grateful and
It was
heavens
is placed
in
the year
reverential
countlJI'.
while living in this house that
P.iero ¥.'Tote his mathematical treatises.
decorated the house with his these are now :mostly lost (a detached fresco from the house depicting a young Hercules i.s now housed in the Isabella Stewart Gardner Museum in Boston) Fiero's house is today lhe home of the FonHe also
own frescoes, tho ugh
..
Piero was buried on 12 October 1492 (the day Columbus discovered America) in the family tomb in the chapel of S. Leonardo in the cloister of a church known as the Badia, as he had requested in his will. The Badia no
longer exists, having been absorbed into the Cathedral. There � no precise evidence of where Piero's tomb is.
Parco Piero della Francesco, a pub lic park across the street from the Fon dazione, contains a statue sculpted by Arnalda Zocchi in 1892 on the occasion of the 400th anniversary of Piero's death (Fig. 6).
Piero's Art and His Mathematles Piero's paintings and frescoes, which have an undefinable crystalline quality,
hold a special fascination and enduring
Figure 3. Piero's hous� at Via Aggiunti,
eredlte
now
71 , Sansepolcro, the
populartty today. ar(:l\itectural ttesign of which is
the home of the Fondazione
Piero
della Francesca.
Some
art historians
attribute the special character of his
works to the serene expressions on the ·faces of the characters in the paintings, others to their peculiar quality of ap pearing frozen in a particular moment. The more mathematically inclined crit ics cite the influence of geometry on Piero's art. There are at least three as pects
of his painting that have earned the reputation as the most mathe. matical of painters: the use of tmderly him
ing geometric constructions to govern
his compositions3; the use of perspec-
Figure 4. The plaque commemorating Plero della Francesca. dazione Piero della Francesca,2 a study
cenler and collection of documenta
tion related to Piero. The Fondazione also
contains a small research library (Fig. 5), where the matl1ematical tourist can browse books on Piero, and look at
the anastatic and facsimile works of Piero's and Luca's mathematical trea
tises
the
[6-8, 22-25], as I did while doing
reseat·ch for this article.
In the last years of his life, Piero went blind, although he continued his intellect.ual activities with the help of a readet and. a secretary. Picturing him being read to, meditating on what he heard, and dictating h.is thoughts to a sec;:retary is a happier picture than that depicted in the memoirs of one who re called, as a child of 10, leading Piero by the hand around. the Borgo.
2"fhe F{)ndazione Piero della Frarcesca is normal\,' open to the public from 91 5 to 1 400 Moncay through Fn· day. I recommend caling �gnora 5eretla Magn�ni for an appo1ntrnent. -t39 · 1)575-7404 1 1 . or e-mail
[email protected]. 3For studoes that point out tne underlying geometry in Piero's paintings. ;;ee Cha�es Bouleau. La geometria segrela dei pirtori. Milan. Electa. 1 995: 1 01 - 1 1 5: Rooert Lawl()(. sacred Geometry: Philosophy and PractiCe. Lordon, Tnames ano Hudson. 1:'192. p. 63; Orests Piero della Francesen, Milan. Rizzoij, 1 96 7 : 1 06.
Del
Buono and
P�erluigi
oe vecchi. L "opera compifJta rJi
Figure 5. The research library of the Fon
dazione Piero della Francesca, Inside
Piero's
house.
VOLUME 2d. NUI\I8EI'1 2. :!002
47
1458, Piero
was
commissioned by the
magistrates of Bargo San Sepol cro, the Conserva tod, to paint a Re:s W'"I"ect ion to adorn the meeting room in which public audiences were held. Moved by 1480 (that is, during Piero's lifetime) to its present location in what was at that time the Palazzo MWlici pale, this splendid fresco was an im portant symbol for the town (the Holy Sepulchre). In the Resu rrection, Christ's high
head is at apex of an
the
isosceles triangle,
other angles of which lie in the
lower comers of the painting. (Piero
also included a self-portrait in
the
fresco; the sleeping soldier on the left,
propped against Christ's tomb with his
head back, is identified with the artist.)
As a mathematician, Piero produced
Uu·ee treatises,
De prospectiva p in gendi [7], Tra ttato d 'abaco [6], and De corpmibus regula ribus [8]5, all three probably written between 1480 and 1 490, while Piero was in residence in
Sansepolcro
and already more than
70
years old (though Tm tt(lto d 'abaco is
sometimes dated as early as
1450 [ 18, 635]). De pTospecliva p i ngendi was dedicated to artists, as its title On
p.
pai nting in perspecti. z:e, implies. But, ,
as
Mark Peterson has shown, Piero did
n't limit himself to simply laying out
techniques for pain ting , but treated
perspective as a mathematical tool,
de E uropean theorem in geometry after Fibonacci [26, p. 34). Piero's s econd treatise, the Trattato d 'aoaco, deals v.ri.th commercial mat h veloping t he first
ematics and the gauging of volumes. I
have mentioned that Piero came from a mercantile family, and
that
his skills
in commercial mathematics were used
in his Figure 6. Kim William$ wtth the statue of Piero (photograph by Mark Reynolds). tive constructions to create the illusion
of pictotial space4; the translation of all
of his forms, both architectural and fig
ural,
into resemblances of reguJar geo
metric solids.
An example of Piero"s use of un
derlying geometric constructions to
govern his compositions
fresco
of
is found in the
the Resurrection in the
Museo Civico in Sansepol cro. In about
capacity as a controller of fi
nances i n Borgo San Sepolcro .
One
of
the greatest skills of the merchant was
the ability to gauge volume, in order to
judge quantities and
costs [ 1, p. 86-89. 1 Piero's interest in gauging volumes is evident in the volumes represented in his paintings. One example is the Drea m ojConstan tine, part of the fres coes in the Cathedral of Arezzo that il
4 For studies of the unde�yfng perspective constructl0f1s. see Laura Gealli and Lt.x:iano Fortunati. "The Ragelfation of Christ by Piero dela Francesca: a Study of its Perspective" in The llisval Mmd, Michele Emmer, ed . . Cambridge MA, MrT Press, 1994:207-2 13; Marilyn Aronberg Lavin. Piero della Francesca ·s Baptism of ChriSt. New Haven and London. Yale University Press, 1 972: Rudolf Wittkower and BAR. Carter. "The Perspective of Piero della Francesca's Rage!lation," Jouma/ of the Warburg and Covrtauld Institutes XVI (1 953):292-302 . 5 Piero·s work was publishe
THE t..I All-l!OMATlCAL INTELL!GENCER
ross,
lustrate th Legend of th True
in which the pavilion wh re tine
sleeps is depicted as cyJind r and cone.
tion of
Ba:xiilldaH points out . . . there i
onstan
a combina As Michael
a continuity b tw en the
and perspecti
is the medium of in-
inuation. Pi,ero's
therefore
painting
pp.
ial proport i nalir:y and lu id
!Borgo after his
those used
to produ
-
by
the pictor
solidity
that strike us as so remarkable t o
day. Pi ro' De abaco is th token of this continuity. The status of these
skins in his ociety was an encour .agement o the painter to assert them playfuUy in his pi ture . we
can see, he did. It was for co nspic uous skill his patron paid him [ 1 , p.
Paciolo
Paciuolo, and sometimes as Lu
hometown)
with Piero in aJmo t Pier
town,
remained
Luca
clo
a
or
di
c .ntrd.Sts
very way.
y tied to hi
Where home
left Bargo as a young man
and lived a peripat
·
tic life. Whcr Piero respected, honored dtizert, Luca was diffic ult to ge along with a.nd in vo h;. d ln frequent quarrels with his fel low Franciscan fliars. Where Pi ro, in was a
additi n to being on of the most im portant artists of th R naissance, made original contributions to mathematics [4, 24 ] Luca was mainly a compiler, ev n
stooping, as we shaii see, to
Luca'
The Perpetrator: Luca Pacloll
Luca Pacioli (also known as
mercial p ople and
o.f an
1 42-143].
mathemat ical skills us d by com
the paint
een
as the i Uustration of one
kind of geom try by mean
other [ 10,
is
ample figure, clothed in hls Fran
ciscan rob s, will b
plagi.ru:y.
almost as familiar
to most readers as his mathematical
works. The most famous portrait of him, painted by Jacopo dei Darbari in 1495 and o rved m the Museo di Capcxti monte, in 'ap1es, has me to repre nt th ar hetypal Renaissance mathemati cian. Luca even includ d his own por trait in lh printed edition of the great est of his books (Fig. 7). Luca \ as born in Sorgo San Sepol cro in 1445. His par nts probably died
101-1 02] .
Piem's i.ntere t in gauging volumes to his int rest in the
is closely related
solids. His th.ird treatise was
De
cor
pol'ibus regula1·ibus, in which he decrtbes the
onstructions for th
the center of the
solids
plagiarism con tr'Oversy involving Luca). In tum, his in
(at
terest in solids appears to be closely re
lated to his interest in perspective, .for in
order to repres
nt
objects correctJy in a
perspecti e construction, th y must be
made somehow to conform. to a more or less regular solid. In De pmspect im pin
gendi,
for instance, Piero tak
sideration th that has no
s into con
human head. an
straight lines that
object
can dis
appear toward vanishing p ints. If tile head can be abstracted and treated as imilar to a regular solid, th n it can be made to confomt to a perspect ive contruction. KeiUl th Clark, comparing one of Piero . drawings of a head in perspectiv �ith one of his frescoes, recognized "the deep reserve of hu maJJity with which. Piero in hls finest
work could cover the Euclid an frame
[3, p.54]. Thus, though apparently
work of his fomlS
three treatises deal with Uu·ee
subjects,
that is
the
practical mathe
matics of vol u me, the
and
Piero's
separate
regular sotids, the geometry of perspective, each
relates to IJle ol.he;r for
wrote:
Piero insinuates
bodies'] g
as
R bin Evans
their !the regular ometry into v rything,
F191ure 7. Luca Pacloli
as
depicted in hts Summa.
VOWME �4. "UMBER 2. 2002
49
as
when Luca was a child,
raised in
a
he was
foster family. In 1 464, at
about the age of 19, he went to l ive i n
the house o f Sig. Rompiaci, a Venetian children. He attended. lessons given by
merchant, as a tutor for tl1e master's Domenico Bragadino, Ven ice
s
'
public
busi
lecturer in mat hematics. He also ac companied Rompiaci on many
ness trips, acquiring great skill in com
mercial mathematics along the way.
His first mathematical treatise was written for his young pupils in this pe
riod.
Luca donned the robes of the
Fran in 1470, at the age of 25. In doing so, he joined the ranks of other friars who distinguished themselves in mathematics (among them, John Pecham, 1230-1292, author of Perspec tiva com munis, an important work on optics, and Robert Grosseteste, 1 1681253, appl ied mathematician and trans lator of Nicomachus and D ionysi us the Areopagite). It was from then on that ciscans
the
Chiesa cti
San Francesco (Fig. 8) in
Borgo San Sepolcro when he
became
i nvo lved in a major conflict with the general minister of the Franciscan or der. On 29 June the general minister is sued an order forbidding Luca to teach mad1ematics to young laymen , threat ening him with excommunication if he disobeyed. Disobey he did, and on 3 August the general m inister issued an order to the abbot of the Borgo San Se polcro monastery not to receive Luca. But eventually Luca was forgiven. In May 1492 he was allowed to reenter the
monastery, and in March 1493 he was once again allowed to preach during Lent. Later, in 1509, his fellow friars
privileges by the Pope , especially a large were
outraged when Luca was awarded
expense account The rift healed only when Luca agreed to renounce these
privileges [ 13] .
In 1496 Luca was
called to the Sforza in Milan as public lecturer in mathematics. There he met Leonardo da Vinci, with whom he would remain in contact until the last years of his life. While in Milan, in 1498, Luca ""Tote De divina proporthme, ded icated to l i suoi ca rissirni di.scipvli . del B01yo San Sepulch ro (his dearest disciples of Borgo San Sepolcro) lv5 is court of
Ludovico
il Moro
.
.
well known, it was Leonardo who sup .
plied
the 59 illustrations of the solids
Luca was known as "fra Luca " (fra is an abbreviation
significant
Franciscan
that
of jrate-friar). It is join the
Luca chose to for
order,
Franciscans
were among the few religious orders
tha t operated outside the
con text. As a Franciscan,
monastetic Luca was
free to travel, and he taught matics in many cities: Perugia,
mathe
Venice,
Florence, Rome, N aples, Bologna, Mi
lan, Pisa, Paris, and Zara (present day
Zadar in Dalmatia, then under the do m inio n of Venice).
Luca's choice to join an order may
but young
indicate not only re ligious fervor
also personal ambition. To a
man with no family connections and no
wealth, belonging to a religious order
secured a degree of otherwise could not simply
enough
plivilege t hat he have enjoyed. But
belonging to the order was not to satisfy
his
aspirations to
greatness. He tried to buy a cardinal
ship,
"offering ftrSt 30,000 ducats, then
40,000, to .Pope Alessandro VI Borgia to make him a
cardinal" [25,
p. 79] !
Apparently, Luca was som eone you
either loved or hated. Franciscans take
a vow
of happiness when they join the hard to sow discord among
order; it 's
them, but Luca managed to do so at least twice. In 1 49 1 , he was resident in a Franciscan monastery attached to 50
THE t.AATHEIMTICAL INTELLIGHiC£R
Figure 8. The monastery and church of San Francesco in Sansepolcro, where Luca lived when
in resldence in Sorgo, with the statue of Luca placed in front.
for
Luca's
De
di vina
propm ·tione
(imagine having Leonardo illustrate your book!). In Luca's unpublished manuscript De viribus quantitatis (On the Force of Mathematics) [25], he
praises Leonardo's unsurpassed mas tery in
drawing the geometric solids. Forced to flee Milan in 1499, he and Leonardo lived in Florence for the next few years, during which time Luca as sumed teaching positions in both Bologna and Pisa. Luca specified in his will that he be buried in the Franciscan church in whatever city he died in. He probably died in Sansepolcro in 1 5 1 7, and is
probably
buried in the church of San Francesco in Sansepolcro, next to which is the monastery where he lived, but there is no tomb for him in the church. and no documents record his
burial there.
In
the piazza in front of
the Franciscan chun::h and monastery
stands his statue. tlle back of which be, for mathematicians, perhaps
will
more interesting than the front. lt indi cates what Luca is for:
his treatise
on
best remembered the divina propo·r
tione and his geometrical methods for constructing the capital letters of tl:le
alphabet (Fig. 9). In the loggia of the Palazzo dei Laudi is a plaque dedicated to Luca Pacioli (F'ig. 10), which says: To Luca Pacioli
Who was friend and consultant to Leonardo da Vinci
and Leon Battista Alberti who first
the
gave to algebra
Language and structure of science and whose was the great discovery of applying it to geometcy He discovered the double accounting gave
of commerce
to works of mathematics
principles and norms invariant
in lucubration for posterity
The people of San Sepolcro
Upon the initiative of the
of workers
society
ashamed of 320 years of oblivion
to the great fellow citizen placed [this memorial in] 1878.
Luca's Mathematics
Luca's
prime
mathematics.
occupation
was teaching In this capacity, he wrote
Figure 9. The back of the base of the statue of LUCll, depicting the "divln� proportione" and the gowmetric constn.1ction for the capital A from luc;a's ;�lphabet.
one
didactical
treatise in Perugia a ritm etica e algebm , a manuscript copy of which sur
(Trattato 1478,
di
vives in the Vatican Library), and oth ers
in Venice and
Za.ra, both now lost.
In addition, Luca delighted in collect
ing puzzles and other mathematical
amusements, compiling them in a work titled De ludis or Schij( moia
Va nqu isher),
(Bo redom
also lost. He displayed
his skills in com mercial and practical
mathematics in
viribus
his
unpublished De
quantitatis [25).
But it was his printed works that
earned him his place in the history
mathematics. In 1494, two
of
years after
Piero della Fran cesca's death, Luca's
Su mma
de
prop(!l"fio n i
A 1'ithm etica
et
geomelria
prop(wtio-nalita
was published in
Venice
by Paganino
dei Paganini (it was reprinted The
Sum ma
(24]
in 1523).
is a compendium of four
fields of mathematics:
arithmetic, alge-
bra, Euclidean geometry, and double entry bookkeeping. The importance of the Su mnw can hardly be overesti mated. It became the starting point for
of mathematics in the Ren relegating the manuscripts that preceded it to obscurity. Though the study aissance,
the authors of the publicabons that fol lowed on the heels of the S1l mma
pointed out errors in Luca's text,
macy was unchallenged.
its pri
ln 1 509, Luca's second major publi
cation, De divi-no
pmportione (22,
23 1 ,
was issued. Written in 14-96, while Luca
was in Milan
with
Leonardo. it. was
published in a combined ed ition two
other works,
chitettu.ra and a treatise on regular solids,
with
the Trattato dell 'm·
the five with the unwieldy name
Libel/us in Ires partiales tr-actatus di visus qu inque cmpm ·iu m regola riwn
(Book divided i n to three parts on ji l..'e regu lm- bodies, hereinafter VOllJM" 2.!. NuMB£R 2. 2002
the re51
sistant [2, p. l07], but there is no proof of this. The age difference of some 40 years might seem to support the idea that Piero needed a collaboration with
Luca, but is it after all so impossible
that a man of 70 could and would ded icate himself with such great effect to studies of mathematics? It may be that after a busy life devo ted to his art, ad vanced age finally gave Piero the time for mathematics! Piero himself provides a
clue, writing in the dedicatory letter of
De
corporibus
re!Julmibus to Duke
GuidobaJdo Montefeltro that he under
took this work in his old age so that "his wits migh t not go torpid with dis use" [20, p.488; 3, p.53]. Each m an created, i n h is own way, a testimonial to the other. Piero's trib ute to Luca appears as a cameo portrait in the Madon na and Child with Sa ints und Angels Adored by Fede1·igo da Montefellro (called the Pala di Brera), painted around 1475 and now in the Pinacoteca di Brera in Milan. Kenneth Clark identified it with this comment: . . . St. Peter Martyr, who looks over the shoulder of St. Francis to the right is a portrait of Luca Pacioli, the fa mous mathematician, who was a native of Bargo and a friend, though ulti mately a treacherous friend, to Piero" [3, p.49 j. Luca's lestimonial to Fiero ap "
.
.
.
pears in the dedication to the Summa,
where he cal ls Piero, famously,
the painting of our times. " Further along in the Summa , on page 68 verso, he credits Piero with having written a " worthy" book on perspective in which he speaks "estimably of paint ing. setting forth in an estimable way the technique and illustration of the method." However, Luca always "monarch of
Figure 10. The plaque at the loggia of the Palano dei Laudi commemorating Luca.
fen·ed to as
Libe/l.us).
The treatise on
architecture was no doubt the result of Luca's time spent in Rome
in 1471 with Leon Battista Alberti, author of De re aed.ificatoria (Ten Boor.:s on A rclz i tec l u re) . The Libellus, we now know, is Luca's translation into the
vernacular
of Piero's Latin De corpo>·ibus regu lmibus. Luca's thi rd book, his able
translation from Latin into the vernac ular of
Euclid's Elemen ts, was also
published in 1 509 in 52
Venice.
11-E I>!ATHCMATICAL INTEI.LJS,.NCER
Plero and Luca
There is no documentaiy evidence to show what
kind
of
relationship Piero
della Francesca and Luca Pacioli had,
whether a friendship., a master-and-pupil
relationship, or even a correspondence
between professionals. Piero knew Luca,
for they were both present in the court of
Montefeltro in Urbina in th e period
1 472-1474.
when
It
has been suggested that
Piero was
losing his sight in his
later years, it was Luca
who was his as-
praises Piero as a painter, or at most as
an
author on painterly subjects, but
not as a mathematician. The Ind ictment The charge of
p lagiarism was first and
virulently alleged by Giorgio Vasari, the
artist and architect who is perhaps best known for his collection of biographies
of the principal artists, sculptors, and
architects of the Italian Renaissance [30]. Indeed, Vasari was so outraged by what he saw as Luca's theft of Piero's
work
that
with it:
he begins the Life of Piero
Truly unhappy are those who, after labouring over their studies to give pleasure to others and to leave behind a name for themselves, are not per m itted either by sickness or death to bring to pe1jection the works they have begun. And it often happens that when such a person leaves behind h im works which are not qu ite fin ished or that are at a good stage of develop ment, they a re usurped by the pre smnption of those who seek to cover their own ass's h ide with the noble skin of the lion. And if Time, which is sa id to be thefather ofTruth, sooner or later reveals 1vhat is true, it is none the less possible that for some period of time the man who has done the work can be cheated of the honour due h is labou rs; this is what happened to Piero della Fmncesca from Bargo San Sepolcro. He was regarded as a n uncommon master of the problems of regular bod ies in both arithmetic and geometry, but the blindness which overtook him in old age and finally his death kept him from completing his brilliant ef forts and the many books he wrote 1vhich are still preserved in Bargo, h is native town. The man who should have tried h is best to increase Fiero's glory and reputation (since he learned everything he knew from him), i nstead wickedly and mali ciously sought to remove his teacher Piero 's name and to usurp for himself the honour due to Piero alone by pub lishing u nder h is own name-that is, Fra Luca del Borgo-all the efforts of that good old man who, besides ex celling in the sciences mentioned above, also excelled in pa inting [30, p. 1 63j. And later, he adds:
As I mentioned earlier, Piero was a most diligent student of his art and frequently practised drawing in per spective; he possessed remarkable knowledge of Euclid, to the extent that he comprehended better than any other geometrician all the curves in regular bodies, a nd thus he shed the clearest light yet upon these matters with h is pen. Master Luca del Bargo, the Franciscan monk who wrote about
regular bodies in geometry, was his student. And when Piem reached old age and died after having written ma ny books, the sa id Master Luca usurped them for his own purposes and had them printed as his own work o nce they had fallen into his hands af ter his master's death [30, p. l67]. Vasari is frequently caught telling in teresting anecdotes lacking basis in fact. One such error in his life of Piero concerns Piero's blindness, which Vasari says came on Piero at the age of 60 (circa 1470), though it is known that Piero could see as late as 1482, because he personally made notes to his will, and because his treatises were written between 1480 and 1490 [ 18, p. 634]. Even given the inclusion of errors, however, Vasari's pen could make or break the reputation of an artist. As the translators of his Lives of the Artists note, "Few artists he criticized have been definitively rehabilitated, and al most all the figures he selected for par ticular praise have remained those most popular with collectors, scholars and visitors to the major museums of the world [30, p. x]." But in spite of Vasari's authority, the case against Luca was generally dis missed for centuries, mainly for lack of proof. Piero's books were supposed to have been in the library of Montefeltro in Urbino, but by the end of the eigh teenth century they were no longer to be found. With the passage of time, Vasari's claims grew weaker and weaker, while Luca's reputation grew ever more sterling. The Damning Proof
Finally, 400 years after the accusation, came the first damning discoveries. J. Deunistonn indicated the presence of the De corporibus regu1aribus manu script in the Vatican Library in the 1850s; in 1903 Guglielmo Pittarelli found and identified the manuscript; Pittarelli verified in 1908 that the Vati can manuscript was identical to Luca's Libellus; in 1915 Girolamo Mancini as certained that marginal notes to the manuscript were in Piero's own hand, and finally published the work in Piero's name [20) . Proving that the manuscript in the Vatican Library was
Piero della Francesca's is tantamount to proving that the Libellus published as part of Luca's 1509 edition of De di vina proportione is a copy. Nowhere does Luca give credit to Piero as hav ing authored or even contributed to the work. This effectively constitutes liter ary theft. As if the truth that Luca usurped the De corporibus regularibus of Piero were not enough, in 1989 Enrico Picutti demonstrated that this was not the only plagiary committed by Luca, but one of several (may we call him a "se rial plagiarist"?). First, in his Summa [24), Luca included 54 problems from Piero's Tmttato d'abaco without cred iting Piero. Later he included the other 138 problems in the Libellus [27, p. 76-77]. Finally, according to Picutti, Piero was not the only victim of Luca's ambition. The section titled Geometria in the Summa is a transcription of the manuscript of another mathematician, a maestro Benedetto of Florence [27, p.76). The discovery that Luca copied the work of Benedetto of Florence helps explain one aspect of Luca's work that his readers often decry: his uneven writing style (Bernardino Baldi in the sixteenth century wrote that Luca's style "makes one nauseous"). Bear in mind that in the Renaissance world of independent city-states, differences in dialect and accent were quite marked between cities and regions, as they are to a lesser extent today. It is difficult to believe that in the few years that Luca lived in Florence he would have come to speak a perfect Florentine, but in copying word-for-word the writings of Benedetto, he would naturally be copying the Florentine style as well. Accident or Intent?
By now I think there is no doubt that Luca copied Piero's work as well as the work of the hapless Benedetto of Flo rence. Was he aware that this was wrong? One thing we know is that he took steps to ensure that the same thing didn't happen to him. In the parallels between the cultural context of the late fifteenth and the early twenty-first centuries are lessons to be learned, for the advent of the printing press created at that time a
VOLUME 24, NUMBER 2 , 2002
53
cultural
rea
d
gulf, Fiero in the world
of
divide
imilar to that
by the advent of electronic publishing today . Piero and ide of th
Luca stood
on eith r
the m an us 'lipt, Luca. in the world ofth
pre s, just as scholars today are divid· d
betwe
n tradi tional print publication
and l:he new electronic publication. Pi ro produced his treatise
and figw-e to th
tro of
·,
,
both text
by hand, cons�gni!ng th m
librari
of his patrof\ Monte� 1-
rbino. Luca, on the other hand,
embrac d lh
new medium, establish
ing a firm relationship
with the
Vene
tian publisher Paganino de' Pagan ini who was books
in
to publish all three of his
print, thereby assuring th m
wide disnibution.
B
in mind that when manuscripts we don t
ar
ab ut
t m1 in today's sense of a draft
of
a
'
w
us v
t xt. Man uscripts in the fift
centwy w re finished documen
Figure
1 1 . The
tall< th
ten by hand, omet im
him elf (an
autograph
sometimes by
a
manuscript),
ionaJ ropyist.
The n ew med i u m of pri nt had enormous
f th
page
corporibu.
ript
manu r
the pro m u lgation of knowledge from the late 1 400s forward .
l'h y
writ-
uscripts." Figur
e re sometim
lavi
h1
illus
trated, described as �illuminated man
manusct1pt copy of Piero's
De
11
how
tihe ftrSt
of
Fiero's De
yularibus in the \ atican
Library . F'igure 1 2 , on the other hand,
shows a
pag
from the print edition of
Luca's Smmn o .
The new m clium o J p1int
mous consequ n e
tion
of lmowledg
forward.
had enor
for the promulga
from the late 1 4 00s came much
consequences for
rsion
nth
profe
by the author
iusti Mites
more
Of the wo1·k of mal"hemalics enjoying the grea test
Euclid,
diffusion, the Elemellls of
abo u t 200 mam.lsc,·ipt codices
M·e kiWWII today; even discounl'i llg
the inevilable losses. it ca n be calcu
tha n 300 copi es of at the d isposition of medieval holars, and those 'Were neither easily a e sible nor equally lated that
not
mo1·e
the Elements l'el'e .
reliable. In conh'a l in lite century be
tween
corporibus regularibus In the Vatican Library.
thejir l p1·inled edit ion (Ven ice
ists. Of course, we all know that errors occur in print d documents as well, but errors in print could be easil cor rected in an natum or in future edi tions. For all th n w advantages of print, it raised new probl ms as well. It is at this point that uintellectual property," which we hear so much about today, becomes an issu Today we talk about how easy il is to go on the Internet and "grab" an image and some text and put it up on a web page as one's own. ln Luca's day, ic was easy to take the ext or figure from a manuscript and pub lish them in print under one's own .
name; the fa t that many manuscripts
were one-of-a-kjnd and tucked away in
private
colle tions meant that such
theft would b
hard to
Another probl m
ory, any publi h
he liked. Th
r
bring to light.
was
could
that, in th
print anything
ftrSt mathematics book in
print, Arithmel ica , published in Turtn
in 1478, was in fact an anonymou work, reQuiring no pennission from an author. Was ttl
for
need of a mechanism
the prot ctton of authors and pub lishers recognized , and did a means of protection exist ln the late 1400s? The
answer to both q_uestions is yes; the means of p rot ction was called a prh'
ileg io. Th
rust
ver plivilegio was
granted in Ven:ic in 1 469 to a single publisher, granting him the exclu.sive rights to all publications in the whole territory of the Republic of Venice.
Upon his d ath, others were granted
similar privUegi. After 1480 pri v ilegi were granted for individual works
Figure 12. A page from the print edition of Luca's Summa.
14 '2) a11d the end of lhe sixte
n lh cen
lw'J) abottt 70 edifi.()ns of the Elem nt nen!
p1·inted, for
pt·udenlly about
.
a
total, calculatiug
00 copies per edi tio n . of
0,000 volum es that cin:ulated
thl'ou,qhout Eu mpe [13, p . l 7/. Another
consequence
was
that
printed works introduced a uniformity
that was reviously llllkno wn. Scholars
in disparate areas of Europe,
ample
for
ex
could now possess uniform val page numbers and figure numbers, sur that fellow s •holars could correctly id ntify urn s t hat allowed them ro cite
what
was
tltis is
a
!being discus
feature that i
d.
(lronically,
Ia king in ome
of today's electronic publ icatio
oc um nts produced in hypertext markup language (html) do not in tude page br·eaks; because the in ertion of page .
breaks is detemtined by the configura
tion of each individual's personal com
puter, it is not possible to cite
pages se
Cllrely.) Yet anot11er cons quence of print was reliabilit-y.
the advent
manuscripts, errors
copyist
[ 1 7]. copyright today derives from the pt•i.vilegi of 500 years ago. Either au thor or publi h r· could apply for a Our
cou1d
alter
transmitted unwitting!
In
onunitted by a
the
text; an undiscovered
of
meaning
of a
rror couJd be
by future copy-
privilegio by writing .a letter to the rul
ing
bod of the
tate; in Venice where
Luca PacioLf· works were published,
the ruling body was the Senate of the
Republic of Veni TI1e old
saying goes,
if you want to
protect your house from burglars, ask
a thief what to do. Luca wrote to the
Senate on 19 December 1508 request
ing a p1i vilegio so that, for the next 20 years, no on ould print without Luca's express p rmission either his
Su mma or D
which was to b
ing
year.
1'h
divfna
proportiO'Jle,
published the follow p1•h iiegio was granted,
of Piero's Tra.tta to d. 'abaco and De cor
armouncement of it was on the last page of De d i vi n a proportione published in 1509 as ,.,.·ell as on the last page of the second edi tion of the Summa published in 1523. and
the
printed
v' LUCA P CIO I
Ironically, Luca's claim on the material
meant that not even the heirs of the le· gitimate authors, Piero and Benedetto, could have published the material that was rightfully theirs (27, p. 76]. But Was It Really Plagiary? Surprisingly, the tendency is
not to
condemn Luca for hls usurpation Piero's work, but
of rather to excuse him.
In part this seems to stem from an un
willingness to project modem ideas on the past. Gino L01ia's cormnents are representath·e of the usual arguments:
fLuca 'sf beha cior; i nconC'eivabie to day, i.s new corifirrnation llwt scien tific honesty is a sen timent with mod
ern orig ins; the a n cien ts commJtte(l without scl'uple every so1·t
and,
when
it came
to
ofplagia ry indicating
sou rces draw11 on, they were overcome
with an invLncible
amnesia; i. t is th us no ma rvel tha t such a free-and-easy systEr�n sh ould be adopted !Jy a m a n who w a s not a n original th inker, but a n i ndefatigable compiler [quoted in 6, p. 32/. Loria's
comments appear to ring true
initially.
case of
For
instance.
consider
the
Euclid and Eudoxus. Euclid is said to be, like Luca, a great compiler. And like Luca, Euclid used the work of
earlier mathematicians such as Ett
doxus without giving credit. Yet no.
body tttinks of Euclid as a
plagiarist. yet, Vasari's accusations seem to belie that justification in the Re naissance: if copying without identify ing the original author were indeed And
Figure 1 3. Via Piero della Francesca, Sansepolcro. 56
Tl-lE MATHEt.AATICAJ.. ONTEcLIGENCH\
Figure 14. Via
Luca Paolo�, Sansepolcro.
thought innocuous,
have been
so
why would Vasari vehement? Further, the
maestro BenedeLto, Luca's other vic
por·ib-us regularibus, it
ensUI'ed the therein a wider diffu sion than they wou ld ever have enjoyed in manuscript form. It may even be that Piem's manuscript in the Vatican li bracy would have gone unidentified and Piero's cont1ibution to Renais sance mathematics unrecognized had the material not been so familiar as part of Luca's publication. Whatever the relationship between Piero and Luca in life, certainly they are forever linked in history. A paint ing by Angiolo Tricca, executed in 1927 and on display in the mayor's office in the municipality of Sansepolcro, illus ideas contained
we all
tim, lived at the same time and in the
trates the myth
work of a (·ertain
solids. Titled Piero della Fmncesca dic
same social context as Luca, yet when
he included in his own treatise the maestro Antonio, he
faithfully credited the original author.
If
copying
were so widespread, why
did maestro Benedetto not be have as
Luca clid? I am forced to the conclusion that Luca
did it to arrogate
credit.
In part, the tendency to excuse Luca may arise from a reluctance to attribute mathematical originality to Piero, an artist, rather than to Luca, a mathemat ics teacher. An example of rhis is Fleur Richter
contesting
Vasart's accusation
of a
learned Piero
want to believe,
willingly passing
on
to a reverent Luca the prtnciples of the latf'.s the rules of geometry to Luca Pa
cioli, the painting depicts Piero as an old
man seated in a large chair, cane in hand,
gesturtng
at
Luca
who, dressed in his
Franciscan habit, Vllrites fonnulas on an
ease l, while tlu-ee
fascinated by the
young men the discussion
two great men.
look on,
between
In Sansepolcro there are two paral
lel stTee!S, Via Piero della Francesca
(Fig. 13) and Via Luca Pacioli (Fig. 14).
of Luca: "Piero knew [ Lucaj well, in
The mathematical tourist can walk up
gether, for how
for him- or herself a complicat.ed sto:ry
deed, the two must have worked to
else
could Piero, with
out the instruction of a
teacher, have
handled such complicated material?"
[28, p. 42j. Such a statement shows a fundan1ental. �udgment of Piero's
mathematical skiUs.
Conclusion Luca almost
managed to pull it off. The truth is finally out that he was a schem ing, ambitious man, capable of lying and plagiarism in hopes of going down in history as a great mathematician. There are still those today who don't want to recognize the �hide of an ass covered with the noble skin of a lion' as just that. The important thing is that Piero has fi nally b€en given the credi.t he deserves as one of th e original mathematicians of the Renaissance, in addition to being Oile of the era's great pa:int{'rs. A fmal odd twist: we owe a debt of gratitude to Luca, for if his De d i vina proportione included whole sections
one and down the other, and perhaps
by journey's end will have sorted out begun 500 years ago. Note
I was
working
on this paper when I
learned about the destruction of the
World 'l'rad{' Center and part of the Pen tagon on l l September 2001. I dedicate it to the memozy of those who lost their liv{'s in the horrifying chain of e\•ents. RI!I:&RIINCIS
1 . Baxar1dall, Michael. 1 985. Painting and Ex· perience
in
Rtteenth-Century
Italy:
A
Primer in the Social History of Pictorial Style. Oxford : Oxford University Press.
2. Bouleau . Charles. 1 996. La geometria seg reta dei pi/tori. Milan: Electa.
3. Clark, Kenneth. 1 951 . Piero della cesca . London: Phaidon Press.
Fran
4. Davi s, Margaret Daly. 1 977. The Mathe matical Treatises of Piero della Francesca. Ravenna: Longo Editore.
AUTHOR
quinque corporibus regularibus. Florence:
Giunti Editore. (Anastalic reproduction of
the manuscript found in the Vatican Library with an accompanying modern print tran scription.)
9. Encliclopedia Bompiani. 1 987.
s.v.
"Pacioli
o Paciolo o Paciuolo, Luca". Scienze pure e
applicate, II: 983. Milan: Bompiani.
1 0 . Evans, Robin. 1 995. The Perspective Cast: Architecture and Its Three Geometries. Cambridge
Piero della Fr<�ncesca matematico.
Via Mazzin i, 7
50054 Fuc:ecchio (Florence) I taly Kim Will�ams, trained as an architect. primarily
a
writer and editor on
mathematics in architectural history.
She is the director oflhe biennial
and Mathematics"; tlhe fourth of these conlerene:Gs is to take place 1 5-1 8
JUfle 2002 in Obidos, Portugal. She nal In the
of an electronic jour
same
field (http:/Jwww .
nexuslournal.com). She and her hus
band have
recently renovated a Tus
can famnhouse,
li mes be
found
and she
can
dreaming about her
herb garden. maybe even about
re1irement.
early
matematica del Rinasc;imento. ings
eds. 1 967. L 'opera complete di Piero della
Francesca. Milan: Rizwli Editore.
6. Dell a Francesca.
Piero.
1 970.
Trattato
d'abaco. Gino Arrighi. ed. Pisa: Domus Galilaeana.
7. Della Francesco. Piero. 1 984. De prospec
tiva pingendi. G. Nicco Fasola, ed. Flo rence: Le Lettere.
of tile
conference,
8. Della Francesca. Piero. 1 995 . Libel/us de
Proceed
Sansepolcro,
1 3-16 April 1 994. C i tt a del Castello: Pe
ruzzi Editore.
1 4. Grande Dizionarto Enclfclopedico UTET. 1 989.
s.v.
" Pacioli, Luca·. Vol. 'JW: 2 1 9.
Turin: UTET (Unions Ti pografico-Editrice Torl nese) .
1 5. Jayawardene. SA 1 980. "Pacioli, Luca".
Pp.
269-2 7 1 i n vol 9 of Dictionary of Sci
entific Biography. Scribner's Sons.
New
York:
Charies
1 6. Lessico Universale Italiano. 1 9 75. s.v. "Pa
ciol i . Luca." Vol. XV: 664. Rome: lstituto
della enciclopedia italiana
5. Del Buono. Oreste and Pierluigi De Vecclli.
arti scienze e tecnica.
di
lingua lettera
1 7. Lessico UniVersate Italiano.
1 975.
s. v.
morali, storiche.
e
fllologiche XIV
(1 9 1 5).
21 . Marinoni, A. ·t 994 . Luca Pacloli e it De div
ina proportione. Pp. 1 1 2-1 1 7 in Lengenda di Piero della Francesca, M.G. Rosita. ed. Florence. Edizioni C itta di Vita.
22. Pacioli, Luca. 1 967. De divina proportione.
Maslianico (Como): Editrice Dominion!. An
rections; contains all three parts of the print
23. Pacioli. Luca. 1 982. De divlna proportione.
Mi lan: Silvana Ed itore . (Facsimile repro duction). Contains ooly the divina propor
tione part of the print pubiication, with fine
color plates of Leonardo's Illustrations.
24. Paciol i. Luca. 1 994. Summa di arithmet!ca,
geometria. proportione et proportionafita.
Enrico Giusti , ed. Rome: lstituto Poligrafico
e Zecca dello tion).
Stato.
Maria Garlaschi Peirani. transcription. Mi lan: Ente Raccolta Vlnciana.
26. Peterson . Mark. 1 997. The Geometry of Piero della Francesco. Pp. 33-40 in The
mer 1 997).
27. Picuttl. Ettore. 1 988 . Sui p lag i matematici di frate Luca Pacioli. Pp. 72-79 i n
L e Scienze 246 (February 1 989).
arti scienze e tecnica .
29. Rosita. M .G .• ed.
Piero.
Pp
Ronald.
1 997.
Franceschi.
624-637 in Dizionano bibli
ografico degli ltaliani.
vol. 49. Rome: lsti·
tuto della enciclopedia italiana.
1 9. Ughtbrown.
Ronald.
1 992. Piero della
Francesca . Milan: Leonardo Editore.
20. Mancini . Girolamo. 1 91 5. " L 'opera De cor-
1 9. 3 (Sum
Mathematical lntelfigencer
28. Richter ,
Ligh1brown,
(Facsimile reproduc
25. Pacioli, Luca. 1 997. De viribus quantitatis.
"privilegio." Vol. XVII: 642. Rome : lstrtuto
della encic lopedia italiana di lingua lettera
1 8.
del Lncei. i Memorie della c/asse di Scienze
nascimento. Exhibit Catalogue. Florence:
1 3. Giusti, Enrico. 1 998. Luca Pacioli e Ia
some
Franceschi secondo il Cochce Vaticano"
Pp. 446-580 in Atti della Reale Accademia
publication.
Giunti.
con
Luca Pacioli" and ''II trattato di Pietro
anastatic facsimile reproduction with cor
1 994. Luca Pacioli e Ia matematica del Ri-
ference series "Nexus: Architechxe
is editor-in-chief
Pp.
detto della Francesca. usurpato da Fra
70-77 in Le Scienze 331 (March 1 996).
1 2. Giusti. Enrico and Carlo Maccagni. ed.
e-mail: k.
[email protected]
now
MIT Press.
1 1 . Gamba, Enrico and Vico Montebelli. 1 996.
KIM WILLIAMS
is
MA:
poribus regularibus di Pietro Franceschi
Aeur.
1 995.
Die
Asthetik
Geometrischer K6rper in der Renais
sance. Stuttgart: Verlag Gerd
Hatje.
1 994. legenda di
Ptero della Francesca. Florence: Edizioni
Citta di Vita.
30. Vasari.
Giorgio. 1 99 1 . The
Uves of the
Artists . Julia Conaway Bondanella and Peter Bondanella. trans. Oxford: Oxford University Press.
VOLUME 2A. NUMBER :l:. :l\102
57
STEPHEN R. WASSELL
Red i scoveri ng a Fam 1 y of Means ow to defi ne a mean of ttoo distinct numbers is a question which has been gi.ven diverse answers, depending on the questioner and on the con text. (The concept can be generalized, of course, for we may allow more than two numbers to be gi ven, and we ma.y ass ign 1oeights. These generalizations are left ou t of the present discussion.) The three principal means, the arith metic, the geometric, and the harmonic, were well known to scholars in classical Greece. Calling the two given num bers a and b, and taking 0 < a ::s b, the three means are given by
A = A(a,b)
=
G
G(a,b)
=
H = H(a,b)
=
=
2ab a+b 2 = G /A.
a
;b
-yl(lb a- 1
(1)
( ; r b- 1
1
(2)
Classical Roots
(4) (5)
The Plot
This paper grew out of a gratifying experience: my inde pendent discovery of a family of means, =
Mp(a,b)
= aP+aP1 ++ bbPP + 1
=
(3)
The arithmetic mean, the simplest of the three, was known and used by the Babylonians as early as 7290 BCE [Spi, p. 5, 5 1 ] . The other two presumably came later, but they were known by the beginning of the Hellenic period.
Mp
having the three principal means as members: Mo A, M - 112 = G, and M- 1 = H. The paper begins with a brief history of the study of means, involving Greek mathematics, architecture theory and practice in the Renaissance, and an elegant thread of research woven by such giants as Lagrange, Gauss, and Ra manujan. It then reviews the appearance of families of means in the literature. At the end I recmmt the good luck which led me to the discovery of (6).
(6)
harmonic by Archytas and Hippasus [Eve. p. 94].
The three means were studied by the early Greeks largely within the broader context of the theory of ratio and pro portion, which in tum was motivated at least in part by its application to the theory of music. Pythagoras is generally cited as the first Greek mathematician to investigate how the arithmetic, geometric, and "subcontrary" 1 means relate to musical intervals, though the paucity of contemporane ous literature makes it impossible to know precisely what to attribute to Pythagoras and what to attribute to earlier cultures or to later Pythagoreans. It is generally accepted that Pythagoras himself was reasonably skilled in music theory, knowing, for example, the relationship of the arith metic and harmonic means to the musical fourth and fifth, as illustrated in Figure 1. To put this illustration in context,
This paper is based on research part1ally supported by grant no. 9846 from the Graham Foundation for Advanced Studies 1n the Fine Arts. ' The last was later renamed
58
THE MATHEMATICAL INTELLIGENCER ·i:J 2002 SPRINGER -VERLAG NEW YORK
fifth
fifth
A(
(
6
•
•
8
'
.A
9
• •
• •
'\ •
•
12
'--v----" '--v----" '--v----"
fourth
tone
fourth
Figure 1 . The relationship of the musical fourth and fifth - here shown within an octave from 6 to 12-to the arithmetic and harmonic
means- here 9 and 8, respectively (after Wittkower [Wit, p. 1 1 0]).
if A4 (the A note above middle C on the piano) were 6 units of frequency, then A5 (the A note one octave higher than A.4) would be 12 units of frequency, D5 (the fourth on the scale between A4 and A5) would be 8 units of frequency, and E5 (the fifth on the scale between A4 and A.5) would be 9 units of fre·) quency.� The theory of the three means "was apparently for mulated by Archytas" [Las, p. 59]. This was done in terms of proportions involving the two extremes a,b, together with the mean, which we de note by m . Solving for m in each of the three following pro portions gives m = A, m = G, and m = H, respectively:
ematicians>' In his Timaeus, Plato, who received his math ematical education from Archytas [Kat, p. 86], describes the three means in words, the harmonic, for example, being de scribed as "the mean exceeding one extreme and being ex ceeded by the other by the same fraction of the extremes" (quoted from [Wit, p. 109]). Eudoxus (or perhaps Archytas and Hippasus; cf. [Hea, vol. 1, p. 86]) soon discovered three new means with similar proportional definitions, notewor thy as perhaps the first tin1e a mean was defmed not moti vated directly from applications (cf. [Las, pp. 59-60]). By tak ing other possible combinations of the a, m , and b in (7)-(9), later Pythagoreans would come up with 10 means altogether, though the original three maintained their prominence. Heron used the arithmetic mean to approximate square roots, based on the fact that if n = ab, then v'n is approx imated by (a + b)/2, i.e., G is approximated by A. An itera tion method can then be based on this: If a; is an approx imation to Vn, then
a; + t =
a; + n la; 2
(13)
is a better approximation. (This is called the "Baby lonian method" of extract ing square roots in [Car, p. 496].) By bringing in the harmonic mean as well, the iteration scheme (1 3) can be expressed in the following symmetrical fmm: Given two numbers a and b such that n = ab as above, let a0 = a and b0 = b, and for i = 1, 2, . . . , let
The Renaissance marked a revival of i nterest i n
abstract m athematics .
m - a --b m m-a = b-m m - a = b-m -
---
---
a a a
-
m
a b
-
(7) (8) (9)
These proportions can be simplified into the more common forms:
m-a=b-m b m m a b-m m-a b a
a ; - 1 + bi - 1 2 2a; - tb i - t b ; = H(a; - 1 , b;- 1 ) = a 1 + b i-· i-1
a; = nA (a;- 1 , b ; - 1) =
(10) (11) (12)
Probably because of the elegant sin1plicity of (1 1), the geo metric mean seems to have been the favorite of Greek mafu-
(14) (15)
(cf. [Eve, p. 178], [BoB, pp. 337-338], [BMV, p. 40] , [Kat, p. 151], and [Spi, pp. 43-47]). Then the "compound" (as this process is now called) of the arithmetic and harmonic means converges to the geometric mean, V' ab = \/n. Whether Heron used only the arithmetic mean, or it to gether with the harmonic mean, is uncertain. For our pur poses, at least, this compotmding of means foreshadows the work of Gauss et al., described below. The means were further developed and described by 4 Nicomachus, Pappus, Iamblichus, and Theon of Smyrna. In his highly influential treatise Introduction to Arithmetic, Nicomachus discusses the ten means in chapters xxii through xxviii. In chapter xxix, the last of the treatise, he
2The equitempered scale, which was developed to make all sem1tones equal, gives ratios very closely approximating these: A, is 440 Hz and A5 is 880 Hz, but 3See [Euc. vol. 2,
Os is
p. 292-293] . where the writings of Euclid and later mathematicians Nicomachus, Theon, and lamblichus are compared regarding their uses of the mean.
587.33 Hz (rather than 586 % Hz) and E5 is 659.26 Hz (rather than 660 Hz): see [Ask, p. 78]. terms ratio, proportion, and
4See [Tho, pp. 1 1 1 -1 3 1 ] for relevant excerpts from the works of these mathematicians, both in Greek and in translation. These authors differ slightly in the1r accounts of the ten means, so that there are actually 1 1 means described in all; and Theon erroneously lists a twelfth. Cf. [Hea, vol. 1 , p. 8511.]. especially pp. 87-89 for a com plete comparison of Nicomachus and Pappus. For a comparison of Nicomachus and Theon, see [Nic. p. 28 1 , note 3]. Of course, they all agree on the three principal means (as well as the first three additional means).
VOLUME 24. NUMBER 2. 2002
59
A
B
0
c
Figure 2. Pappus's construction of the three means: the lengths 00, BD, and FD are, respectively, the arithmetic, the geometric, and the har monic means of the lengths AB and BC.
"
describes "the most perlect proportion," which we would write a:H::..4:b; as a numerical example, Nicomachus offers 6:8::9:12, precisely the subject of Figure 1 [Nic, pp. 284-286]. Pappus gives the following geometric construction, which produces all three of the principal means (see Fig. 2):
c
Take B on segment AC, B not being the midpoint 0 of AC. Erect the perpen dicular to AC at B to cut the semicircle on AC i n D, and let F be the foot of the perpendicularfmm B on OD. [Then/ OD, BD, FD represent the a rithmetic mean, the geometric mean, and the harmonic mean of the segments AB and BC. . . . [Eve, pp. 197-198]
f
'
c
We shall see below an illustration based on this construc tion by Andrea Palladio, one of the most important archi tects of the Renaissance. Blossoms in the Great Rebirth
The Renaissance marked a revival of interest in abstract mathematics, which was still considerably intertwined with other disciplines, from both the "arts" and the "sciences," these two areas being divorced only much later, during the Age of Reason. The three means were invoked in the the ories of art, music, and architecture:
[Marsili.oj Ficino in his Commen.tary to the Timaeus had discussed the three means very clearly at considerable length, and possibly through him they became of over whelming importance to Renaissance aesthetics. In the Venetian circle of Palladia's time they were examined by [Francesco} Giorgi as well as by Daniele Barbaro, but it seems probable that Palladia 's source was Alberti toho had treated of them in terms more easily accessible to an ar chitect. [Wit, p. 109] Indeed, Alberti defines the three means in the context of describing how architects can use numbers and propor tions derived from music, geometry, and arithmetic to cre ate buildings which possess a kind of universal harmony
60
THE MATHEMATICAL INTELLIGENCER
Figure 3. Palladio's illustration of the geometric constructions of the
three means (from [Pal, pp. 58-59]). The second illustration, that of the geometric mean, is based on the construction of Pappus quoted above.
[Alb, bk. IX, ch. v-vi]. Alberti states the definition of the arithmetic, geometric, and "musical" means in words, giv ing numerical examples, and then concludes:
By using means like these, whether in the whole buildi ng or within its parts, architects have ach ieved many no table results, too lengthy to mention. And they have em ployed them principally in establishing the vertical di mension. [Alb, p. 309] Palladio expands on Alberti's account in [Pal, bk. II, ch. xxiii] , titled "On the Heights of Rooms." He defines the three means via geometric constructions, both in words and in illustrations (Fig. 3), explaining that they can be used to compute the height of a room given its length and width. Palladio then states:
of the first kind; one of the primary applications of the AGM, then, is in approximating elliptic integrals. Many other applications are described in [BoB], one of which (re lated to the stated application) is the approximation of 1r. AGM-based algorithms provide rapidly convergent approx imations and are among the methods of choice for recent supercomputer techniques to calculate, at this writing, over a billion digits of 1r (see [BoB, pp. 337-342]). The general question of compounding means has also received substantial attention. A key paper here is by D. H. Lehmer [Leh]. He makes an exhaustive study of the compounds of means taken from two major families. The first family, which has been called, in the various refer ences, the fP means, the power means, the Cauchy means, the Holder means, and the Minkowski means, is given by
(.
)
aP + bP llp fJp = Jl.p(a,b) = These heights are related to each other in the following (18) 2 way: thefirst is greater than the second and this is greater than the third; so we should make use of each of these (notation as in [Leh]). Formula (18) and its extensions to higher dimensions are familiar from the study of norms, heights depending on which one will turn out well to en largely developed by Cauchy (with p = 2), Holder, and sure that most of the rooms of different sizes have vaults of an equal height and those vaults will still be in pro Minkowski. (See either [HLP, pp. 12-32] or [BeB, pp. 2-20]. The authors of each of these classics view the arithmetic portion to them, so that they tu m out to be beautiful to the eye a nd practical for the floor or pavement 1.ohich will geometric mean inequality [the fact that the arithmetic mean dominates the geomet go above them because they w ill all end up on the same ric mean] as central to the is Pal lad i a defi nes level. [Pal, p. 59] sues that they consider.) Clearly p.1 = A and p. - 1 = H.
the t h ree means via
While such applications of means did not constitute ad vances in mathematics, they showed the importance of the three means in the minds of those who were looking to an tiquity for inspiration. Mathematical progress in the study of means would take place from the Enlightenment onward.
Moreover, although (18) is not defined for p = 0, a simple ap plication of l'Hopital's rule shows that (18) can be (con tinuously) extended by defining p.0 = G. The second fan1ily of means that Lehmer uses is Mp, al ready defined in (6). Lehmer described the family Mp as "apparently absent from the literature" [Leh, p. 184]. More than 20 years earlier than [Leh], however, it did appear in the literature (see [Bee] and [P61]). Edwin Beckenbach, then an associate editor of the American Mathematical Monthly, introduced this family of means, motivated by a pair of problems posed by P6lya. Thus it seems fair to call Mp the P6lya-Beckenbach means. Lehmer had (as I would do much later) discovered them independently. Thanks to his extensive treatment, his name has become associated with them ([Bob, p. 232 ] , [FP, p. 77]). In addition to a series of results on the compounding of various members of the two families J1.p and Mp (summa rized nicely in [BoB, pp. 263-264]), [Leh] includes a re markable result having a simple proof. Using the homo geneity (defined below in (31)) of the members of both families, writing it
geometric construction .
The Tree of Knowledge Spreads
An important thread of research involving two of the three principal means is the arithmetic-geometric mean (AGM), first discovered by Lagrange and then rediscovered and fur ther developed by Gauss a few years later ([Mie, pp. 29-30], [AB, p. 585]). The AGM is simply the compound of the arith metic and geometric means of a and b by analogy to (14) and ( 15); I shall denote it by M(a,b) after [BoB]. Perhaps the main result of [Gau] concerning the AGM is the fol lowing. Theorem
1. Let lxl < 1 and K(:r)
Then
=
L
rr/2
0
M(1 + x, 1
V1
-
-
.r)
d lJ x2 sin2
=
v·
-. 2
7r
, K(.r)
.
(16)
(17)
(For proofs, see [BoB, pp. 5-7] and [AB, pp. 588-589].) The reader will recognize K(.r) as the complete elliptic integral
/Lp(a,b) = a JLp(1,b/a) = aJ1.p (l, 1
+
t)
(19)
and the analogous equations for Mp, and expanding each of Jl.p and Mp in powers of t, it is shown, by matching co efficients in the first few terms, that the only means (i.e.,
VOLuME 2 4 . NUMBER 2. 2002
61
mean formulae) belonging to both of the families are A, G, and H. Perhaps the ancients would not have been sur prised. A unified approach to the algorithms involving the com pounding of the arithmetic and geometric means is given in [Car], from which the following is taken:
Another example of the breadth of research involving the means is seen in Ramanujan's study of elliptic func tions, where he shows the following (see [Ber, p. 164]): Theorem
3.
For n
F(a,{3) =
Theorem 2.
Let x0 and y0 be positive n umbers. For any fixed choice of the indices i and j from among the num bers 1, 2, 3, 4, we define X, + 1 = f; (Xn,y,. ) Yn + 1 = fJ(x,,y,. ) ,
(22)
·
(
(23) (24)
( X + y ) 112 f4(:x ,y) - 2- y , _
(25)
all square roots taken positive. Then the sequences {x, } and { y , } have a common lim it.
and Re/3 > 0, define
+
__:_ _ _ _ _
_ _ _
11
+
(313/·
2 _ _ ___: ____: :_ _
_ _
n
+
(2 a)
_:_:_ __:_ :_
_ _
n +
F(
(26)
a
-----2 --13
n
Then
r: + y ft (X, y) = -2 r ) = ( ( y) j]_ . ,y x l/2 .r + y 112 j3(x,y) = x� ) .
0, Re a > 0,
(20) (2 1)
n = 0, 1, 2, . . . , ·where
.
>
) a/3 - V-· 2-,
a + 13
=
1
n
2 (F(a,l3)
(4a)2
+ . . .
+
F({3,a)).
Many more mean families other than /1-p and Mp have been developed, as is thoroughly described in [BMV] . For ex ample, the "quasi-arithmetic" means (see [BMV, p. 2 15ff. ]) are obtained by replacing the functions J{.1·) = :rP and f- 1 (x) :c11P in /1-p with an arbitrary pair of inverse func tions, c.p (x) and c.p - 1 (.r) (where c.p(.r) is defined for positive real numbers), as was studied by Kolmogorov and others [BMV, p. 374]. Many such beautiful results involving means =
3
1.5
0.5
-0 . 5
Figure 4. The graph of y
62
0 . 5 -0 . 5
=
f(x), together with those of y
THE MATHEMATICAL INTELLIGENCER
1.5
1
= x and y
=
(27)
a; here a
=
2
1 and b
=
2 . 5
2, so that A
=
3/2, G
3
=
\
2, and H
=
4/3.
3
2 . 5
1.5 - - - - - - - - -
0 . 5
0 . 5
-0 . 5
Figure 5. The graph of y
-0 . 5 =
m(x) = mx(a,b), together with those of y
m(O} = a, m(a) = H, m(G) = G, m(b) = A, and limx-" m(x) = b.
are familiar only to specialists in certain subfields of math ematics. A Newcomer to the Field
My experience with this historically, theoretically, and practically rich subject started with a memory lapse. Dur ing a lecture at the Mathematics and Design 1998 con ference, at a talk concerning proportions used in architec ture, the speaker presented ( 12), after [Wit, p. 109], in defining the harmonic mean. Not certain of the trivial al gebra leading to (4) from ( 12), I wanted to check the pro portion. After the talk, however, I was uncertain as to the final variable of (12), shown below with an .T: m
2 . 5
2
1.5
1
-a a
b-m X
(28)
I tried .r = m (when in fact .r = b is correct) and discovered that (28) then has solution m = \/ab, the geometric mean. Noting that x = a trivially reduces to the defining relation ship of the arithmetic mean, I realized that I had stumbled upon a unification of the three means: (28) is simply an al ternative way to write (7)-(9) in a single equation. Because no one I asked at M&D-98, nor at the next conference on my agenda, Ne.rus '98: Relati.onships Between Architecture and
= x
and y = b; again a
=
1 , b = 2, A
=
3
3/2, G = '\
2,
and H = 4/3. Note
Mathematics, had recalled seeing this tidbit before, I was en
couraged to investigate further to find where it might lead. Does it make sense that different values for x in (28) lead to different mean formulae? The first task is to un derstand what properties must be satisfied for a function to be considered a mean. Although there is no consensus in the literature, [Bob, pp. 230-231] reviews the properties commonly required of means, the first four of which we may paraphrase as follows: 1. A mean is a real-valued function M of two strictly posi tive real variables a and b such that
min(a,b)
:s
M(a,b) :s max(a,b)
(of course, with the assumption that a
M(a,b) :s b). 2. A mean M is strict provided a
:s
M(a,b) = a
or
:s
(29)
b, this becomes
M(a,b) = b
(30)
if and only if a = b. 3. A mean M is homogeneous provided
M(Aa,Ab) = AM(a,b)
for
(31)
A > 0.
VOLUME 24. NUMBER 2 . 2002
63
4. A mean
M is symmetric provided (32)
M(a,b) = M(b,a).
How does my supposed mean measure up? Fixing a and
b, it is given by m = m�.(a,b) =
a(b + x) . a + .r
(33)
with .1: as parameter. If x > 0, it satisfies (29) and (30). When a and b change, the parameter .r has to change in order to
salvage (31). A typical
Moreover, the two values of z which, when plugged into mz yield (35) and (36) are, respectively, z = b\! Wa and z = b 2/a . The common feature leapt into view: we need z of the form bP + 1/aP ; and this leads to the truly special ex pression (6), the P6lya-Beckenbach family of means. One can imagine the thrill of this (re)discovery, even though I was virtually certain that such a splendid result could not be new! The family mp is only the family mx or mz reparame trized by p E ( - :xo,:xo), but the properties of homogeneity (31) and symmetry (32) have been made manifest.
Mathematical prog ress i n
f(x)
graph of = m.r(a,b) is shown in Figure 4. The three points associated with the three means described above are shown with dashed lines: f(a) = A., the arith metic mean; b) = H, the harmonic mean; and the geometric mean is the (attracting) fixed point of the graph, i.e., f( ) (attracting because < 1; see [Dev, p. 39ff. ] for further reading). Further more, f(O) = b, lim_,. __. "' f(x) = a, and every value between a and b is obtained by some choice of x from the interval (0, :xo) . What of property 4 above? Reversing the roles of a and b in (28) or (33) gives a new proposed family of means with parameter z:
the study of means wou l d
A Seemingly Endless Bounty
To underscore the beauty surrounding the means, I will end by describing one more nice result, rediscov ered while investigating the fact that the compound of the arithmetic and harmonic means is the geometric mean. Recall that for a given n > 0 we can approximate \In by compounding A and H as indicated by (14) and (15), e.g., taking a0 = 1 and b0 = n. Of course, I could not resist trying this with, say, n = 2. The resulting double sequence is
take p l ace from the
lf 'CG) I
E n l i g hten ment onward .
f(
G G =
_
mz(a,b) =
b(a + z) . b+z
(34)
This family is in fact the same family reparametrized, for mz(a,b) = rn.,.Ca,b) provided z = ab/x. I might have noted this at once, but the direction I took instead was still more fruitful.
(a;) = ( 1, 3/2, 1 7/12, 577/408, . . . l (bd = [2, 4/3, 24/17, 816/577, . . . lClearly there is a pattern here; in fact, the b; are completely determined from the a; (or vice versa). Letting c; and d; be the numerator and denominator, respectively, of the a;, it is straightforward to deduce that
Ci + l = cT + 2dT d;+l = 2c;d;.
Sensing a nice relationship here, I found that Perseverance Bears Fruit
It seems reasonable, for a given a and b, to view the m.r (x > 0) or the mz (z > 0) as a famly of means. Perhaps, however, continued analysis may lead to a more compelling form for the fantily. The first question that comes to mind is that three values of x give special members of the fam ily (the three principal means), but other possible values for x > 0 may lead to other special mean formulae. One ob serves that the arithmetic mean, A, is the midpoint of the interval [a,b], and that the geometric and the hamwnic means lie in the lower half subinterval [a, A]. Might the val ues in the upper half subinterval [A, b ] that are reflections of and H (about A) be "special"? Direct algebra yields that these two values are, respectively,
G
a\l'a + b\!b a +b-G= v;;i + V b a+b-H=
64
THE MATHEMATICAL INTELUGEt�CER
a Z + b2 a+b "
(35)
(36)
(c; + V2d;)2 = d + 2d7 + 2V2c;d; = Ci + l + V2di + l·
I then generalized this to any n. I had rediscovered a method for approximating Vn with rational numbers: Take 1 + Vn, square this expres-sion, gather the terms as c + v'nd, then square this expression and gather the terms, etc. The upshot is that the sequence of the c/d approaches \ln. Just as with the family of means (6), I am virtually certain that this result has been seen before. I would be most grate ful for reference to a source. Acknowledgments
I would like to thank Karen Parshall for her expert ad vice on historical topics. Thanks as well to Kim Williams, who has worked hard to bring together those interested in the relationships between mathematics and architec ture, and who provides a constant inspiration for all of us.
REFERENCES
[Alb) Leone Battista Alberti. De re aedificatoria . trans. Joseph Ryl<.wert,
Neil Leach, and Robert Tavernor. On the Art of Building in Ten Books, M IT Press. Cambridge, MA,
fl U T H O R
1 988.
[AB] Gert Almkvist and Bruce Berndt, "Gauss, Landen, Ramanujan , the Arithmetic-Geometric Mean, Ellipses,
Math.
[Askl
Monthfy,
v.
11,
and the Ladies Diary," Amer.
95, n. 7 ( 1 988), 585-608.
John Askill, Physics of Musical Sounds, D. Van Nostrand , New
York, 1 979.
(Bee) E . F. Beckenbach, "A Class of Mean Value Functions" Amer. Math. Monthly,
[BeB]
v.
Edwin
57
F.
(1 950). 1 -£. Beckenbach and
Richard
Bellman.
Inequalities,
Springer-Verlag. New York, 1 96 1 .
[Ber) Bruce C . Bemdt. Ramanujan 's Notebooks (Part Ill) , Springer-Ver lag ., New York, 1 985.
(BoB] Jonathan M . Borvve in and Peter B. 801'\vein, Pi and the AGM: A
STEPHEN R. WASSELL
Department of Mathematical Sciences Sweet Briar College
Sweet Bf"iar, VA 24595
Study in Analytic Number Theory and Computational Complexity, Wi ley .
New York,
1 987.
[BMV] P. S. Bullen, D. S . MitrinoviC. and P. M. Vasic. Means and Their lnequalitie:s, D. Reidel, Boston, 1 988.
[Car] B. C. Carlson, "Algorithms Involving Arithmetic and Geo etlic Means: Amer. Math. Monthly, v. 78 (1 971 ), 496-505.
[Devl Robert L. Devaney. A Arst Course in Chaotic Dynamical Systems. Theory and Practice. Addison-Wesley, New York, 1 992.
{Euc] Euclid, The Thineen Books of Euclid's Efement:s, trans. Thomas
L Heath, 3 vols. Cambridge University Press. New York, 1 926. Dover reprint. 1 956.
(Eve] Howard
Eves. An Introduction to the History of Mathematics ,
Saunders College Publishing, New York, 1 990.
[FP] D. M . E. foster and G. M. Phillips, "A Generalization of the
USA
e-mail:
[email protected]
Steve Wassel l received
a
B.S. in architec!ure in 1 984,
a
Ph.D.
in mathematics (mathematical physics) in 1 990, and an M.C.S.
in computer science in 1 999, all from the University of Vll'ginia.
He is currently the chair of the Department of Mathematical Sciences at Sweet Briar College. Steve's research focus is on
the relationships between architecture and mathematics, afld
he is the Malhematics Editor of the online/print journal, Nexus
Network Journal: Architecture and Mathematics. Steve's over
all aim is to explore and extol the mathematics of beauty and
the beauty of mathematics.
Archl medean Double Sequence," J. Math. Anal. Appl .. v. 1 0 1 (1 984) ,
575-581 .
[Gaul Carl Friedrich Gauss. Werke, Georg Olms Verlag . New York , 1 98 1 , v. 3 (357-403) and
[HLP] G.
v.
1 0 (207-23 1 ) .
H . Hardy, J. E. Utllewood. and G. P61ya. Inequalities. Cam
bridge University Press. Cam bridge, 1 952.
[Heal Thomas L. Heath , A History of Greek Mathemallcs. 2 vols. 1 92 1 . Dover reprint. 1 981 .
[Kat] Victor J. Katz, A History of Mathematics , HarperCollins College
Publishers. New York, 1 993.
[las] franyois Lasser re. The Birth of Mathematics in the Age of Plato, American Research Counc i l . Larchmont. NY. 1 964.
[Leh] D. H. Lehmer. ··on the Compounding of Certain Means." J. Math.
Anal.
Appl. ,
v. 36 ( 1 97 1 ), 1 83-200.
(Mie) George Miel. "Of Calculations Past and Present: The Archimedean Algorithm." Amer. Math.
Monthly,
v.
90. n. 1 ( 1 98J). 1 7-35.
(Nic] Nicomachus. An Introduction to Arithmetic. trans. Martin L o·ooge. Macmillan. New York. 1 926.
(Pal] Andrea Palladia. I quattro /ibri delf'archltetture, trans. Robert Tav
enner aJld Richard Schofield. The Four Books on Architecture, MIT Press , Cambridge. MA, 1 997.
[P61] G. P61ya, "On the Harmonic Mean of Two Numbers. " Amer. Math.
Monthly, v. 57 (1 950). 26-28.
[Spi] Maryvonne Spiesser . "Les M8dietes dans Ia Pensee Gr9C{Jue (The Means in Greek. Thought). " Science et Techniques en Perspective.
v. 23 (1 993). 1 -7 1 .
[Tho) lvor Thomas. Greek Mathematics 1 . Harvard University Press. Cambridge. MA. 1 939.
[Wit) Rudolf Wittkower. Architectural Principles in the Age of ism, St. Martin's Press. New York. 1 988.
Human
VOLU� 24. NUMBER 2. roG2
65
Iii§Jh§i.ifj
.Jet Wi m p ,
Ed i t o r
I
A Beginner's Guide to Graph Theory
by W. D. Wallis
YORK: BIRKHAUSER, 200 1 . 240 PP US $34.95: ISBN 081 -7641 76-9 NEW
Feel like writing a review for The
REVIEWED BY BRUCE RICHTER
Mathematical Intelligencer? You are
S
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.
Graph Tlzemy with Applica tions by Bondy and Murty went
ince
out of print many years ago, I have been unable to find a textbook suitable for a first course in graph theory. West's book Introduction to Graph Theory has far more content than I could even think about covering in one
tion to Ramsey Theory. So we have the makings of a very nice text. Here is one surprise. In the chapter "Factorizations," we find Tutte's theo rem about the existence of a 1-factor ( = perfect matching) in a general graph, but do not find Hall's theorem for bipartite graphs. In the introduction, I read, " . . . many colleges and universities provide a first course in graph theory that is intended primarily for mathematics majors but accessible to other students at the se nior level. This text is intended for such a course." It so happened that I was teaching exactly such a course at the time the review copy arrived. Thus, I could compare what I had chosen to do
My ideas of particular topics to choose are d ifferent from Wal l is ' s . But that i s all rig ht .
Column Editor's address: Department of Mathematics. Drexel University, Philadelphia, PA 1 9 1 04 USA.
66
semester, and Diestel's Graph Theory is far too sophisticated for a beginning audience. Don't get me wrong: both of these are excellent books and I refer to them frequently, but they just don't seem right to me for an audience that typically is only taking this one course in graph theory. When I first saw Wallis's A Begin ner's Gu ide to Graph Theory I was fa vorably impressed by its slimness; it does not appear to be overwhelming at first glance. I would say that the pres ent volume has been influenced in style, if not detail, by Bondy and Murty, in the sense that each chapter has a main point (or at most two) and then a small amount of further discussion associated with the topic. A book with this approach is most welcome. We fmd a usual selection of topics basics of paths and cycles, blocks, trees, matchings, colorings, planarity, and directed graphs. There is a nice in troduction to vector spaces associated with graphs, and a readable introduc-
THE MATHEMATICAL INTELUGENCER © 2002 SPRINGER-VERLAG NEW YORK
with what the author chose to include, and I can see that we have similar ideas of what goes into such a course. Hardly surprisingly, my ideas of par ticular topics to choose are different from Wallis's. But that is all right; I would choose a good text with topics different from my favorites rather than have to create my own notes as I go. So the question is, "Is this a good text?" The first thing I noticed was the very poor production of the text. There are far too many typographical errors. This made it hard to read the book at all. The proof of Brooks's theorem (pages 89-90) has so many such errors as to be virtually incomprehensible. If you know the typesetting program TEX, you will likely be able to reconstruct the correct presentation, but phrases such as "Select some color c k (not c; or cj) " will be hard to unscran1ble for the average senior. I almost, but not quite, sent the book back when I got tired of noting typo graphical errors. -
I started thinking more about how it would come across to the average math ematics senior student. Would the book provide a good model of how graph-the oretic ideas can be expressed? Would I be comfortable that a student could read this with comprehension? I list below a munber of points that make me think the answers to these questions is "no." There are unquestionably some good things in the book, but there are too many bad things to lead me to rec ommend this edition. Despite these ob jections, a revised edition of this book might well be suitable for the kind of course it is aimed at. I hope that the au thor is willing to undertake the task of repairing the problems. Here are some that I found.
1 . Page 17: "We define a path P,. to be a graph with n vertices Xt. x2, X11 and v - 1 edges .l'tX:z, X2X3, . . . , x,,_ 1x,," The author wishes v always to mean the number of vertices, but it is hard to break a very long habit of v being a vertex and n being the number of vertices. I think v is too useful as a vertex, whereas n has no other natural use so it is free to de note I VCG) I . 2. Page 131: We have a similar prob lem here. In a directed graph G, for a vertex "v" the sets A( v) and B( v) are defined to be, respectively, the sets {.r : (v,x) is an arc ) and {.r : (.r,v) is an arc ). (It might be helpful to ex plain the notation by saying that A(v) consists of the vertices that come "after" v, while B(v) consists of the vertices that come "before" v.) Then we read, "A vertex v is called a start in the digraph if B(x) is empty and a fin ish if A(x) is empty." The confusion as to what v means continues. And what is wrong with sourse and sink? 3. I found it disconcerting that walks in undirected graphs (alternating se quences of vertices and edges) are defined quite differently from walks in directed graphs (sequences of arcs). It somehow suggests that the two are qualitatively different, which they are not. .
•
•
,
And there are more ftmdamental prob lems.
4.
Pages 59-60: The theorem here is that the set of all unions of edge-dis joint cycles is a vector space (over the integers modulo 2). The main point is to prove that the "sum" (i.e., symmetric difference) of two such unions is again a union of edge-dis joint cycles. A tedious case analysis shows that, in the symmetric differ ence of two cycles, every vertex has degree 0, 2, or 4. Would it not be more straightforward (and convinc ing) to say that if v is a vertex of G, and X and Y are the sets of edges in cident with v in the two unions of edge-disjoint cycles, then the set of edges incident with v in the sum is the symmetric difference X + Y, and I X + Yl = iXI + I Y I - 2 I X n Y l ? Since l X I and I Y I are both even, so is I X + Yj. 5. Pages 61-62: The proof given here of the fact that edge-cuts also make a vector space is so confusing as to be incomprehensible. What we should have is something like the following: LEMMA Let A, B, c be sets. Then A (B + C) = (A n B) + (A n C).
n
For X <:;;; V(G), let B(X) denote the set of edges with exactly one end in X. The lemma implies B(X + Y) = B(X) + B(Y), for if e = uv E E(G), then select A = {u,v), B = X and C = Y. We have that e E B(X + Y) if and only if lA n (B + C) l is odd, which, by the lemma, occurs if and only if exactly one of lA n Bl and lA n Cl is odd. This is equivalent to
e E B(X) + B(Y).
6. There are several places where we are told things like, "the process may be continued" or "because G is finite, the process eventually termi nates." These are just ways of avoid ing doing an induction or a proof by contradiction. Do we really want students to take this as their model? 7. In the section on planar graphs, Theorem 8.5 requires an additional hypothesis to support the statement in the proof that "Each face has at least three edges in its boundary." This is not true of a graph with at most one edge, and I am not sure
what assumptions about connection are being made here. Does the graph with two components, each of which is an edge and its ends, have a face having four edges in its boundary? What makes this error more strik ing, however, is the application to Theorem 8.6, in which G is required to be connected and have at least 3 vertices. The only requirement on G in the proof is that Theorem 8.5 ap ply; so these conditions should be re quired in 8.5 as well. 8. In the introductory material for directed graphs, we see the very natural THEOREM 10.3 Evely acyclic di graph has a start and a fin ish.
On first glance I was impressed that two important lemmas were given first. LEMMA 10.1
If a digraph contains an infinite sequence of vertices (a0, a t, . . . ) such that a; -1a; is an arc for every i, then the digraph con ta ins a cycle. LEMMA 10.2 is the
case
same result in the
a;a; - 1 is an arc for every i.
Unfortunately the proof of Lemma 10. 1 given is wrong; it does not suf fice to pick some a; that is repeated and choose the leastj > i such that a1 = a ;. (The simple correct proof picks the least j such that a1 occurs among ao, a�. . . . , aJ - d But what is perhaps even worse than the incorrect proof of Lemma 10. 1 is the sloppy use of it in the proof of Theorem 10.3. Assunting there is no fmish, we start with an "and so on" (inductive) construction of a se quence a0, a.�. a2, . . . such that, for each i, a;a;+t is an arc. I would then say that Lemma 10. 1 inlmediately im plies there is a contradiction. How ever, what we read is, "Since a di graph has finite vertex-set, the sequence (a0, at, . . . ) satisfies the conditions of Lemma 10. 1, . . . " 9. In the proof of Theorem 1 1. 1 , we see the meaningless "[set] S; = {a : S; <:;;;
VOLUME 24, NUMBER 2. 2002
67
S; -1
U
S; - d," which I cannot deci
pher. Moreover, the proof talks about "transitions" and "transitions [that] can be perfom1ed," without ever explaining what these words mean. As I said above, a revised version of this book would be welcome. I believe there is lots of potential here. Department of Combinatorics and Optimization Faculty of Mathematics U niversity of Waterloo
Waterloo, ON N2L 3G1 Canada
e-mail : brichter@math . uwaterloo .ca
Classical and Modern Methods in Summability
by Joha nn Boos (assisted by Peter Cass)
NEW YORK: OXFORD UNIVERSITY PRESS. 2000, 566 pp
US $ 1 30, ISBN: 0 1 9 -850165-X
REVIEWED BY JEFFREY S. CONNOR
B
oos takes on, and succeeds in, the difficult task of writing an intro ductory graduate-level text on summa bility theory that develops the topic from both a classical and a functional analytic perspective. The book starts at the beginning and, assuming that the reader has had no prior contact with summability theory, provides a broad background in the general theory and introduces recent research topics. The book can also serve as a reference for workers in summability theory or re lated areas. Summability theory has its origins in the study of divergent series. Of course, the convergence of a series is determined by the convergence of the sequence of its partial sums, so it was not much of a shift to move from the study of divergent series to the study of divergent sequences. Hardy attrib utes the first summability results to an 1890 paper of Emesto Cesaro on the multiplications of series. Cesaro used the observation that if a real-valued se quence x = (x, ) converges to a number L then it is also convergent in arith metic mean to L, i.e., 68
THE MATHEMATICAL INTELLIGENCER
lim x, =
"
L
�
lim 11
1
ll
II
) X�; = L.
A":':;;'l
This is now a standard E-N argument that often appears in advanced calcu lus together with the companion exer cise of showing that the sequence ( 1 , 0, 1 , 0, 1, 0, . . . ) is convergent in arith metic mean. The value of this result is that it provides a technique for ex tending the class of convergent se quences. Less trivially, if f is continu ous and 2 1r-periodic on R, then the arithmetic mean of the partial sums of the Fourier series of f converges uni fomlly to f on [ - 1T, 1r]. If we let C = (c, ,�;) be the N X N ar ray defined by cn,k = n - 1 for k :=::: n and 0 elsewhere, then Cesaro's result can be recast as lim X11 = L n
�
lim (Cx)11 = n
L,
where (Cx), = �k c11 .J...X:�;. Observe that we regard C as a matrix and x as a column vector, (Cx), can be regarded as the nth entry in the vector Cx. It is natural to wonder, then, under what conditions an N X N matrix A = (a11 ,k) of scalars has the property that if
lim x11 = "
L
�
A-lim x : = lim(Ax), = n
L
where (Ar), = �k a,,A.xk. An array A with this property is called a regular summability matrix. The answer is found in the celebrated Silverman Toeplitz theorem: Theorem: Let A = (a, , �;) be an N X N array of scalars. Then lim,(Ax)11 ex ists and equals lim, x, for every con vergent sequence (x, ) if and only if
1 . The columns of A
a re null sequences, i.e., for each k, lim, a,,k = 0;
2.
The limit of the row sums is equal to 1, i.e. , lim, �k a,,k = 1; and
3.
The absolute row sums are uni formly bounded, i.e., SUPn 2: �; l a , ,k l < 00 •
It is straightforward to show that the conditions are sufficient and that con ditions (1) and (2) are necessary. It is not so straightforward to show that
condition (3) is necessary; the "classi cal" proof uses a gliding (or sliding) hump argument. Roughly speaking, the strategy is to assume that (1) and (2) hold and that sup, 2:da, ,�; i = x-. Using these assumptions, one constructs two increasing sequences of indices ( v) and (K) in such a way that
) Ki + l -1 I a I K1 Vj,k
"--- h; �
increases without bound while I ) ) .::....k
K, + t I a ,1 ,k I
remain small. Then
can be thought of as a sequence of "humps" gliding off to the right. This construction is then used to define a convergent sequence x such that Ar is divergent. Along with being one of the first re sults of general matrix sUfilffiability the ory, the Silverman-Toeplitz theorem was also one of the first sUfilffiability theo rems to be demonstrated by functional analytic techniques. The proof of the ne cessity of (3) can also be done using the Principle of Uniform Boundedness. As sume that A is reg-ular and define f,(x) = �,a,,k:rk for every convergent sequence x. The convergent sequences form a Banach space (denoted by c), and eachf, is a continuous linear functional on this space. Since Un (x)) is conver gent (and hence bounded) for each con vergent sequence x, the Principle of Uni foml Boundedness (PUB) yields sup, IIJ., II = sup, 2:�;la n,kl < x-. Another arena for the interplay of functional-analytic and classical proof is the realm of consistency theorems, and this theme runs throughout Boos's text. The convergence domain of a ma trix A, denoted c.4, is the set of all se quences x such that Ax is a convergent sequence. Consistency theorems ad dress the question of under what cir cumstances x E c.4 n cs implies that lim, (Ax), = lim" (Bx),. Perhaps the most fanwus of these is the Bounded Consistency theorem:
Let A and B be two regular matrices. If every bounded sequence in c.4. is an elemen t of cs, then, for every bounded sequence x in c.4, lim, THEOREM:
(Ax) , = lim, (B:r),.
One of the first consistency results was established using functional-analytic techniques and is due to Stanislaw Mazur in 1929. He established that the convergence domain of a triangle (a matrix that has a nonzero main diago nal and zero entries above the main di agonal) is a Banach space. If A and B are regular triangles and c.� is con tained in cs, then A-lim and B-lim are continuous linear functionals on cA. Because A and B are regular, A-lim and B-lim agree on c and hence on the clo sure of c in cA. He then went on to char acterize triangles with the property that c is dense in c.�. and also noted that the closure of c in c.4 contains the bounded members of cA, thus obtain ing the Bounded Consistency theorem for triangles. Mazur and Wladyslaw Or licz announced the general Bounded Consistency theorem without proof in 1933. The first published proof of the theorem, however, is due to Brudno in 1945, who established the result by classical means. Mazur and Orlicz eventually published their functional analytic proof in 1955. The convergence domain of a ma trix is not, in general, a Banach space. It turns out, however, that any conver gence domain can be given a unique complete metric topology for which the coordinate functionals are contin uous; this topology is called its FK topology. The basic theory of FK spaces, with applications to summabil ity theory, was developed by Karl Zeller in a series of papers in the early 1950s. A number of properties of ma trices tum out not to be properties of the matrices so much as they are prop erties of their convergence domains or a subspace of their convergence do main. For instance, the following result of Virgil Snyder characterizes "conull" matrices in tem1s of their convergence domains.
Let A be a matrix which maps convergent sequences to cmwer gent sequences. Then thefollowing are equivalent: THEOREM:
conull: �k a,,k - �k lim, a , k = 0. 2. Cf>n ek)n is weakly (i.e., a(c.4, (.D) convergent to 0. 1. A is lim,
,
Observe that the condition (1) is de scribed purely in terms of the matrix A and condition (2) is described purely in tem1s of the convergence domain cA.. Now note that if A is conull and c.4 = cs, then, as there is at most one FK topology for cA = cs, the matrix B must also be conull. The Bounded Consistency theorem can also be extended to the FK-space setting by replacing the convergence domain of cA with the subspace
WE = {(�r= t .rkek>n is
u(E,E')-convergent to xj
of an FK space E and replacing the bounded sequences with any space X satisfYing the "signed pointwise-gliding hump property" (S-PGHP). Several sequence spaces, including the bound ed sequences, p-sunlffiable sequences (0 < p < �), and null sequences, satisfy the S-PGHP property. Theorem: Let E be a n FK-space, X a sequence space with the S-PGHP such that X n E conta ins the finitely non zero sequences, and B be a matrix. If X n WE c Cs, then, for all X E X n WE, lims x = � k(lim, b , k):rk. ,
For the proof-and how to derive the Bounded Consistency theorem from the above-one may (of course!) con sult Boos's book (pages 463 ff.). The book is divided into three parts. The first part presents the classical as pects of the theory and some applica tions of the theory. In addition to pre senting some Silverman-Toeplitz-type results, this section develops some stan dard summability matrices, Tauberian theorems (i.e., when the convergence of Ar implies the convergence of x), and a classical proof of the Bounded Consistency theorem. This section also investigates some special classes of matrices, such as matrices of type M and the relatively recently introduced potent matrices. The last chapter in this section discusses applications such as analytic continuation, Fourier effective methods, and the numerical solution of linear equations. The second part of the book intro duces functional-analytic methods. In its full development, the functional-an alytic approach to summability theory
draws heavily on the general theory of locally convex spaces. A complication is that there are as many as five differ ent natural duals of a sequence space, and as many special subspaces of a se quence space. Chapters 6 and 7 give a quick but thorough introduction to lo cally convex spaces and topological sequence spaces, which includes ver sions of the Banach-Steinhaus theorem and the Closed Graph theorem for bar relled spaces. There is a nice balance between theorems stated with and without proofs. The provided proofs do a good job of demonstrating standard argun1ents that enhance the reader's un derstanding of the material from the viewpoint of sequence spaces. These chapters also provide a good selection of examples. The third chapter of this section focuses on matrix domains as FK and sequence spaces, and gives a functional-analytic view of some of the topics introduced in the earlier chapters. The third part of the book looks at topics that require the use of both clas sical and functional-analytic methods. The first chapter opens by developing a general fom1 of the consistency the orem using gliding hump techniques in conjunction with modem techniques, and closes with considering the ques tion of when a family of regular matri ces is "simultaneously consistent" on the bounded sequences. The second chapter includes consistency theorems using "mixed topologies" on sequence spaces. The third and final chapter dis cusses the links between summability theory and topological sequence spaces. For instance, weak sequential com pleteness with respect to a weak topology is equivalent to an inclusion between sequence spaces, and there is a similar characterization between in clusion and being barrelled with re spect to a Mackey topology. Boos's book is a welcome and timely addition to the summability lit erature. In addition to the high quality of the exposition, the text also fills the need for a contemporary graduate level text in summability theory. The book is well organized and clear, and the proofs are at an appropriate level for such a text. Also to its credit, the book contains a large set of exercises. Even though the book is not compre-
VOLUME 24, NUMBER 2. 2002
69
hensive, Boos often alerts the reader to
a true "life and work" that describes
ter (pp. 37-38) on why he "left astron
further developments of a topic and de
both under all three hats, with empha
omy forever" (namely, it's not rigorous
scribes how to locate further literature
sis toward the end on his ideas as a
enough) would make most mathemati
on the topic. My only complaints are
philosopher of mathematics, especially
cians smile and be informative for non mathematicians. Also, his "just-so sto
more a reflection of my tastes than
ideas on the interaction between math
flaws in the text. The organization of
ematics and society. And yes, it depicts
ries" are fun; e.g., (p. 53): "Quine [a
the book, while conducive to the math
"the education of a mathematician"
famous logician] gave her [his then girl friend, now his wife] an A in the course.
ematical development of the material,
education in
obscures the historical development
schooling, but individual ponderings,
She told me later that she hadn't a clue
and the interplay between the modem
"friendships with people who follow
as to what was . . . going on . . . Quine
and classical approaches to the sub
unique passions" (from the jacket), and
had given me a B . . . and I had thought
As a result, a couple
every form,
not only
chance opportunities. One such was a
I had known what was going on. And
notebook given to him when he was a
that, dear reader, is why ever since I
seemed unmotivated. Sometimes the
child,
notation interfered with the clarity of
record-keeping of horse
the presentation. None of these com
winding up as a place for math j ot
An important part of his "education"
plaints, though, takes away from what
tings-"secret things about numbers, "
has been meeting and knowing inter
ject.
developed
in
the
of the topics
classical
section
intended
by
his
cousin
for
have disliked the philosophy of for
races,
but
malism."
the book accomplishes, and I have no
as h e called them. That brought to
esting people, almost all of them fa
doubt that it will become a standard
mind fond memories of my own child
mous. Roughly a third of the chapter
reference in the years to come.
Department of Mathematics Ohio University
Athens, Ohio 45701
hood "number tricks, " and the fascina
names (at least in the first two-thirds
tion when I took algebra and discov
of the book) are also people names. To
ered that they weren 't "tricks." ("Or were they?" I now ponder in my older
wit: "Ed Block and the Founding of
age. )
Boas, Jr. ," "Modesty Is Not a Virtue:
I n the Prologue (p.
USA
e-mail: [email protected]
The Education of a Mathematician
by Philip J. Davis
NATICK, MA: A. K. PETERS. 2000, 250 US $29.95, ISBN 1 ·56881 · 1 1 6·0
PP.
P
hilip
Davis
mous: Buttonholding Buber," "Power
him the idea of writing about "the slow
Galore: Talking with Leland," "Top o'
steady progression of day-to-day expe
the Morning to You, Professor Synge, "
riences that added up to a set of opin
"Emilie Haynsworth" (about whom I
ions."
reflect, "Being a 'real character' is not
He
then
proceeds
to
write,
usually in short chapters, about his
the same thing as being deep, or inter
childhood, high school and undergrad
esting. Likewise, being tough."), "What
uate years, math-things ("secret things"
I Learned from Mary Lucy Cartwright,"
about other than numbers, such as his
"An Address that Led to Friendship:
encounters with the Theorem of Pap
Reuben
pus and fan10us fommlas he proved in
from Gian-Carlo, " and "Continuing Edu
graduate
school,
Hersh,"
"Taking
Instruction
first
cation: Isaiah Berlin and Giambattista
employment ("at the National Bureau
Vico." Only rarely do I get the feeling that
known.
of Standards in Washington, a traitor
he nanw-drops; however, he leaves me
ous move away from academia and one
with the impression that most of these
chosen to wear three hats: mathemati
that I'm sure raised eyebrows . . . "
friendships were limited, often tempo
cian, philosopher, and writer. From the
p.
"As
well
Norbert Weiner," "Opinions of the Fa
Throughout his lifetime he has
book jacket:
is
Davis de
scribes a motivating incident that gave
dependently),
REVIEWED BY MARION COHEN
ix)
SIAM," "Writing a Thesis under Ralph
a mathematician his
contributions have influenced many
1 12),
and present employment (in
academia). Many
passages
rary, and peripheral. Davis himself says (p.
about
seemingly
44), "It
happens not infrequently:
what begins as a firm and devoted
areas of applications from approxima
small incidents in his development are
As a
amusing and metaphoric of larger is
ated as time goes on and as circum
philosopher he is . . . drawn to a critical
sues and ideas. "Why I Didn't Take Phi
stances change."
tion theory to computer graphics.
analysis of the development of his sci
friendship tapers off and becomes alien
losophy A" (pp. 35-36) goes into detail
In a "light-hearted anecdotal style"
a writer whose work in
about why he chose, and continues to
(from the dust jacket) he writes con
cludes ["Mathematical Encounters of
choose, not to learn philosophy via for
siderably, although sometimes in an
Lore of Large Numbers, Interpolation and Approxi mation, and] the award-winning The Mathematical Experience (with Reuben
mal courses. He states that he retains
in-groupy manner, and with humor typ
"a deep and abiding suspicion of all em
ically dry and not always very funny,
bracing systems of thought, " and that
about the FUN of professional acade
he picked up "general philosophy be
mic life: collaborations, speaking en
hind the garage. " (I do, however, ask,
gagements,
into shining gems that reflect truth and
"What does he mean by 'embracing sys
abroad, and so on. My favorite anec
ence . . . .
As
the Second Kind," The
Hersh), he manages to tum small events
understanding." His new book is an autobiography, 70
THE MATHEMATICAL INTELLIGENCER
tems of thought'? And are
h is
restaurants
and
hotels
systems
dote is "How I Used Blackmail to Be
of thought non-embracing?") His chap-
come a Professional" (p. 336). This is
a play on the double meaning of "pro fessional"-someone who can get complimentary tickets for his friends on occasions for which audiences pay admission to hear him speak versus the kind of professional he'd been all along for many decades. The "blackmail" refers to his sudden pluck when told he could not get the complimentary tick ets: ''I'm not sure what devil got into me . . . but a second soul that I never re alized . . . took charge, and I said . . . in no uncertain terms, 'In that case, I will not speak tomorrow evening' . . . 'Just wait a moment,' Mueller said . . . In five or so minutes Mueller was back. 'Here are your tickets.' " Davis has worn smaller hats as am ateur artist, musician, even a stint in "
ways helpful, because it can over whelm both the beginner and the sea soned researcher with so much infor mation that mental paralysis results from the overdose." A very large focus of the book is his fascination with Thomas Jefferson. Yes, the Thomas Jefferson, third Pres ident of the United States, also amateur scientist and mathematician, who "said that if he had not gotten involved in politics, he would have devoted his life to these things." (p. ix) Perhaps he should have. If I recall correctly, Albert Schweitzer, student and visitor of wild animals, was also originally attracted to mathematics as a career. And-al though I was glad to learn that Jeffer son asked whether the mathematically
way out" or too revealing of their inner selves), and general insecurity and stress-then the sciences and the arts will, like everything else, corrupt rather than purify morals. Davis has little to say about women. For exan1ple, p. 80: "There is a saying an10ng mathematicians that mathe matical talent is often passed down from father to son-in-law, and there are numerous cases to back this up" . . . What about father to daughter-in-law, mother to son-in-law, and so on? But also: What of the daughter through whom it's passed? What was/is her role in this legacy? Davis says that he has learned a lot from his own father-in-law, and I am curious about his wife. It seems she
. he manages to t u rn small events
i nto s h i n i n g gems that reflect truth . "Show Biz" (where the likes of Alexan der Kerensky, Carol Channing, and Imogene Coca's husband Bob Burton showed mild interest in the calculus book he would read on the set). His artist hat has qualified him to make in teresting speculations on computer art, capsulated in the chapter title "A NEW Esthetic?" Also pp. 187-188: "What emotions are stirred up in me? Vener ation, awe, fear, love, hate, surprise? Awe, sometimes. Occasionally amusement." " . . . While I attain a feeling of elation . . . there is little in the images that operates for me in a didactic way, outside of mathematics itself. This may be contrasted to the religious art of the Renaissance. . . . The computer Virgin has not yet appeared to provide peace of mind, comfort, and salvation in a horror-ridden world. . . . " He concludes, "At the technological level, awareness of progress comes fairly easily. With re gard to moral progress, one wonders." Yes, to what extent does computer art depict the human condition? And-a question of my own-to what extent is computer art really art and not craft? (Perhaps to the same extent that craft is art.) Davis has things to say about computers in general, not only com puter art. For example, p. 99: "The amount of computerized database in formation is tremendous. It is not al-
detern1ined motions of the sun and planets would ever, due to these same mathematical laws, come to an end via, perhaps, a crash, and that this made Jefferson a kind of forerunner of "the stability of dynamical systems" (p. 134)-I personally am more fascinated by Schweitzer than by Jefferson. Davis devotes considerable space toward the end of his book to describ ing the mathematics of Jefferson's day and how mathematics has changed since then. He also involves Jefferson in his discussion of the age-old topic, "Have the sciences and the arts puri fied morals?" Basically both he and Jef ferson believe that, despite many ex amples to the contrary, they can. My own feeling is that the answer to that question depends on how the sciences and art are taught (especially to young children), disseminated, and generally treated within society. I've written much on the subject of education, and I repeat here: If individuals (including scientists, artists, and laypersons) are taught and conditioned to associate art or science with negative aspects of life-such as too much discipline (other than self-discipline), hypocrisy, negative self-inmge, fear of diminishing talent, other fears (for mathematicians, of discovering something "trivial," and for artists, of doing something "just too
"
has also written books, and only one sentence is devoted to this. In the Ac knowledgements (p. 353) he con cludes, "My wife, Hadassah F. Davis, has been the absolute without-whom not of this book, and I have often said of her and continue to say, "Thy word is a lan1p unto my feet.' " I'd be inter ested in seeing at least one of her words. Not only women get dismissive jibes. On p. 169 Davis tells us that he once wrote a spoof, "that a certain Bernstein, who was naturally left handed, felt aggrieved because auto mobile traffic in the United States proceeded on the right side of the road. . . . " Now is that fun, or is it mak ing light of minority and disability is sues? Davis himself "detected a slight conservative tinge to it." This tinge did n't seem to bother him, however; rather he simply published the spoof in the National Review. The article was in tended to emphasize the abuses of the class-action suit; I feel that, while there might be some isolated cases of such abuse, class-action suits are important and serious weapons in our society, one of the few that many people have. His article regrettably helped shape public opinion in the wrong direction. (The mathematical "orientations" he mentions do not justify this.)
VOLUME 24. NUMBER 2. 2002
71
Single sentences, "teasers," seem to pepper his book, sentences that I wish were paragraphs. For example, p. 317: "Mathematics often promulgated a spirit and a view of the world that I found unacceptable." p. 291: "I believe that the mathematical spirit both solves problems and creates other problems." p. 280: "The mathematical spirit and mathematical applications are not always benign." And especially in a more personal vein, p. 174: "I al ways considered myself a bad teacher. . . . " In general, I'd like to know more about his vulnerabilities and how he deals with them. Maybe it's a man/ woman thing: if I were to write my own mathematical autobiography, that au-
expository, philosophical, fictional, at about one every three or four years. These books have won me numerous awards and have reached people in many strange places . . . including a high-security prison in Colorado . . . . " There are certainly passages where he goes out of his way to be modest. I have quoted some; here is another, p. 1 10, perhaps more honest than modest: "I naturally inherited a bit of the snob bism described above. I would give my self a 3 [out of 10) in that regard. " "But how could an autobiography possibly avoid sounding arrogant?" asked a mathematician friend of mine when I voiced these thoughts. My an swer, after taking time to reflect, would
he given up anything (as most have) to "get where he is"? A chapter that caught my fancy is "What Are the Dreams?" (p. 206). This is shorthand for "What are the motiva tions, the drives, the questions, per haps the political causes that could in fluence a young person into becoming a mathematician (or philosopher, physicist, or student of some other expression of human intelligence)'?" Davis wrote to experts in various dis ciplines and received answers. For ex ample, from his father-in-law Louis Finkelstein the theologian: "Can we de velop a civilization in which the most pious people are also the most moral?" From Sir Alfred Ayer the philosopher:
Single sentences , "teasers , " seem to pepper h i s book, sentences that I wish were parag raph s . tobiography would be more on the hu man side, and more about the dark side. I both identified with and resisted his passage on p. 109: "I've felt great wonder in the statements of mathe matics, and yet, I have experienced the waning and the disappearance of that wonder as I've succeeded in under standing what lay behind the state ments. Rationalism and wonder are in conflict." Yes, I have felt that polarity, but wonder stays in the lead. First, an swers almost always point to, or al ready are, further questions. Second, a mathematical proof doesn't always convey why a theorem is true and thus doesn't explain away the mystery. Third, a "math-poem" of mine begins, "You can draw pretty pictures WITH OUT Cartesian geometry. I You don't need x-square plus y-square to draw a circle . . . I But," begins the next stanza, "it wouldn't be as pretty. Cu rves look prettier with equations running along side them . . . " The conclusion: "Beauty isn't as pretty without truth." Under standing (and rationalism) can often enhance wonder. In writing about his life, Davis is not always modest. Take p. 156: "It (The Lore of Large Numbers) was my first book and I could hardly have guessed . . . that over the next four decades I would be producing books, technical, 72
THE MATHEMATICAL INTELLIGENCER
be, "By getting in touch with that arro gance, and sharing that, his whole re ality, with the reader." Another question that I asked as I read the book is something I had asked before reading it: Just how attractive is the academic life? For that matter how attractive is the intellectual life? After all, both involve risks and in securities-graduate student blues, tenure-track (or non-tenure-track) woes, reviews or "mere" passing com ments from colleagues; it's sometimes hard not to associate in one's mind the discipline itself with anxieties. Then, too, when academic or intellectual peo ple interact with one another, the un derlying, if vague, assumption often seems to be that everything NOT aca demic or intellectual is somehow laughable. (It is, perhaps, this ethno centrism that provides much of the hu mor in Davis's book.) How does this af fect communication among academic or intellectual people? Are rationalism and human communication in conflict? Davis seems to like his life; still, do I detect a few undertones? It is the omissions that stand out. I've already remarked on the peripheral and tem porary nature of many of his profes sional friendships; and what about non-professional friendships? We are not told how Davis places his work in the context of his life. Has he or hasn't
"(1) To find a secure foundation for our claims to knowledge. (2) To agree upon a criterion for deciding what there is . . . . " And from himself: "(1) The Dream of the Universal Language. . . . (2) The Dream of Indubitability. (3) The Dream of Infinity. (4) The Dream of Description and Oracularity." And, my own favorite, from Josephine Hardin the writer, "the dream of vul nerability conquered." It seemed an interesting and easy ex ercise to list my own "dreams". Some would be more personal, more poetic, and as such deeper than the dreams Davis has in mind. For one thing, "dreams" in Davis's sense of motivation and drive can be psychologically based. I admit that (subconsciously and later consciously) I learned the axiomatic method in order to prove that my par ents, not I, were wrong. And consider p. 178 (in another chapter entirely): "The famous art critic Kenneth Clark conjectured that the impulse for math ematics [at least some mathematics) with its lines and curves came from men's fascination with the female body." (However, I would put Clark's conjecture in more "unisex" terms.) I searched through my "math-poems." According to them, I do math for a va riety of reasons: (1) in order for things to not be simple, nor finished; (2) to pick at things-points, lines, wiry little
x's and y's; yes, math as mannerism; (3) "to love that for which/there is no space"; (4) to have the pleasure of writ ing not in cursive; and (5) "to rescue the insides" (for example, irrationals and transcendental numbers, or re mote theorems and lemmas like some one buried alive on some distant planet). "What are the dreams?" is the title of a book Davis confesses he consid ered but eventually abandoned. Maybe we should revive the project! Speaking of abandoning, Davis ends his book in perhaps the only way one could-by dubbing this ending an abandoning. Perhaps I may do the same with this review. Department of Mathematics and Computer Science Drexel University Philadelphia, PA USA
1 9 1 04
Trigonometry by I.
M.
Gelfand and Mark Saul
BOSTON, BASEL. BERLIN: BIRKHAUSER, 200 1 , x + 229
pp., 1 85 illustrations US $1 9.95 ISBM 0-81 76-39 1 4-4, 3-7643-39 1 4-4 )paperback)
REVIEWED BY EDWARD J. BARBEAU
T
he ancient discipline of trigonometry has many faces. The astronomer's need to make exact calculations gave birth to spherical trigonometry in the work of the Greek mathematician Hipparchus in the second century B.C.E. The subject was systematized by Ptolemy in his Almagest. By the fifth century, Indian mathematicians had produced sine tables, and their contri butions were later consolidated and extended by Arab mathematicians. From the fifteenth century, the subject developed beyond its astronomical ap plications to become an important part of the mathematics of Western Europe. Plane trigonometry was developed and used not only in surveying, cartogra phy, and navigation but also for pure mathematical purposes like solving cu bic equations. All of this still does not account for the subject's prominence in the modem cmriculum. Since the
latter part of the seventeenth century, The book under review is one in the it has been clear how central the Gelfand School Outreach Program se trigonometric functions are in the ries, that includes also Algebra, Func study of analysis and physics. The sub tions and Graphs, and The Method of ject has a richness and elegance that Coordinates, all co-authored by Pro make it well worth the trouble of con fessor Gelfand. Its authors, on the whole, have selected wisely and cre veying to the young. Trigonometry is a bridge between ated a book that gives a gentle and geometry and analysis. Although geom clear introduction to high-school stu etry has much to say about structural dents. One of its goals is to prepare "for relationships, it has few tools designed a course in calculus by directing . . . at to provide quantitative infom1ation. tention away from a particular value of There are criteria for congruence of tri a function to a study of the function as angles, but how can we determine all an object in itself." Written in an easy the measurements of a triangle from a style, it proceeds through generally detem1ining set of data? Larger sides accessible examples and exercises to of triangles rest opposite larger angles, cover the fundamentals of the sub but how are the ratios of the sides re ject-the definition of the trigonomet lated to the measures of the angles? ric ratios, basic identities, the sine and The power of trigonometry rests on cosine laws, Hero's area formula, the observations that each acute angle extending the definition of the ratios corresponds to a class of similar right beyond acute angles, radian measure, triangles and so can be characterized addition and multiple-angle fommlae, by certain ratios. Because each trian sun1-product conversion formulae, gle can be decomposed into right tri expression of the trigonometric func angles, and other figures can be ana tions in tem1s ofthe tangent of the half lyzed through triangles, trigonometric angle, and inverse trigonometric func ratios apply in many geometric set tions. There are some nice excursions tings; trigonometric formulae and ta on the way-a dissection proof of the bles are handy and efficient. The peri Pythagorean theorem, Ptolemy's theo odicity of the trigonometric functions rem on cyclic quadrilaterals and its and their connection with the simple relation to the addition formulae, a differential equation y" + y = 0 en dimensional-analysis appreciation of sure that they intervene in many areas Hero's theorem, the role of trigono of applied mathematics, and their metric identities in approximating 1T close affinity to polynomials and ex and summing trigonometric series, and ponential functions makes still further a preview of the role of lim11 0 relevance. (sin h)lh in finding tangents to and the All of this presents strategic ques area under an arch of the sine curve. tions for the writer of a primer in The teaching experience of the authors trigonometry. How can a treatise of shines through in their anticipation of reasonable size capture the flavor of student confusion or oversights and the subject and prepare the reader for their use of carefully modulated and later developments in analysis? How probing questions. However, their will should the functions be introduced ingness to introduce an idea and then through right triangles or a moving return to it later made me regret the point on a tmit circle? How much em lack of an index. phasis should be placed on surveyor The authors take the reader by the problems, where the context is imme hand, pausing to pose small questions, diate and easily understood? To what providing a commentary on the un extent should complex numbers or po folding mathematics, and occasionally lar coordinates intervene? How exten offering some advice on how to ap sive should be the coverage of fomm proach the subject. On page 46, the lae and technique? How should reader is advised, "From the identities periodicity be treated? How can one we have, we can derive many more. capture the significance of this deep But there is no need for anxiety. We subject? Inevitably choices have to be will not have an identity crisis. If you made, and topics must be left out. forget all these identities, they are eas_,
VOLUME 24, NUMBER 2 . 2002
73
ily available from the fundamental identities. . . . " On page 1 1g, they sell the reader on the advantages of using radians, despite the appearance of the nonrational 7r and its lack of a compact nun1erical representation, and urge, "Don't let this slight inconvenience stop you from using radian measure." This is the sort of book that a student can work through on her own, or that a teacher can take as a text and be sure that the important material is covered. However, I kept getting the feeling that the authors were sometimes too reluctant to push the reader. Without adding much heft to the book, there was more that might have been done in adding content and providing chal lenge in the exercises. Although stu-
completely convincing; a little more argument could have been given to nail it down. One might look at a triangle ilBC with the lengths of CA and CB fixed, CB :s CA, CA horizontal and B a vari able point that moves on a circle of ra dius a and center C. As the angle BCA increases, B moves from a position B1 on CA to a position B2 on .4C produced. Let Bo be a possible position of B. The circle with center A that passes through B0 contains the arc BoB1 in its interior and the arc BoB2 in its exterior. Thus, if LBCA < LB0CA, then B, on the arc BoB 1 is closer to A than B0. Sim ilarly, if LBCA > LB0CA, then BA > BoA . On page 72, it is pointed out that one
the circumference of a unit circle? This would underscore that the trigonomet ric functions, like polynomials and ex ponential functions, are tools of analy sis defined for real variables. However, the reader is quickly made aware of the key fact that sin x - x for small values of x measured in radians. In Chapter 7, the authors wish to ex tend the range of applicability of trigonometric identities, such as sin(a + {3) = sin a cos {3 + cos a sin {3, from positive acute angles to any angle whatsoever. At this point, they have ac cess to such facts as sin
( (} + ; )
=
cos e.
They complain that "checking the for-
The teac h i n g experience of the authors shi nes t h roug h . dents are likely to see only the analyt ical side of trigonometry, it might have been worth providing a few triangula tion problems more substantive than fmding the height of a tree by measur ing its shadow (p. 56). The introductory chapter draws attention to a sinusoidal curve for the hour of sunrise during the course of the year, with a promise that, in a later chapter, the reader will be shown how trigonometry "allows us to describe it mathematically." The ques tion is indeed revisited (p. 203), but the reader is simply told that "if we graph the number of hours of daylight in each day, we get a sinusoidal curve." This is an appropriately challenging question of mathematical modeling, and it might have been worth a few exercises to guide the student to a reasonable set of assumptions that lead to a formula for the number of daylight hours. In some places, there is the question of how much explanation needs to be provided. For example, on page 13, the authors describe how the side of length c opposite the angle C in a triangle ABC gets longer as the angle C increases, given that the lengths, a and b, of the remaining sides are fixed. They justify this by the plausible argument that if the triangle is hinged at C, one can see that the side opposite C gets smaller or larger as one closes down or opens up the hinge. This is insightful, but not
74
THE MATHEMATICAL INTELLIGE�JCER
consequence of the sine law is that the greatest side lies opposite the greatest angle. This is true, but a subtlety is glossed over when one of the angles of the triangle is obtuse, for the sine of the angle decreases when the angle ex ceeds goo. However, once we know that supplementary angles have the same sine, we can note that if a > goo > {3 > y are the angles of a trian gle, then {3 + y is acute and sin a
=
sin({3
+ y) > sin {3 > sin y
by the monotonicity of the sine func tion for acute angles. So, indeed, larger angles have larger sines, and so are op posite longer sides. A tricky issue for anyone teaching trigonometry is how to make the tran sition from degrees to radians and from trigonometric ratios to trigonometric functions. The choice here (on page 103) is to define the radian measure of an angle as the ratio of the arc it cuts off to the radius of any circle whose center is the vertex of the angle. The main reason I can see for this is that the authors want to emphasize that the ra dian measure is dimensionless, but it is not clear what students are to make of this point. On the face of it, measuring angles in radians or degrees is just a matter of choosing a unit. Wouldn't it be better right off the bat to define an gle in terms of the length of arc cut off
mula for sin( a + {3) for general angles becomes very tedious. You can try it for other angles, reducing each sine or cosine to a function of a positive acute angle. But pack a lunch, because such a procedure takes a long time." This, they say, can be avoided by invoking the Principle of Analytic Continua tion, which says that "any identity in volving rational trigonometric func tions that is true for positive acute angles is true for any angle at all." This is a pretty big black box to put in front of high-school students; to me it seems hardly better than the authors simply exerting their authority. Justi fication does not seem to cost more than an induction argument using steps like this:
(
sin a
+ {3 + = =
;)
cos
= a
cos(a + {3) cos {3 - sin a sin {3
;) + sin a cos ( {3 + ; }
(
cos a sin {3 +
On pages 33-35, before the discus sion of angle-sum and double-angle identities, the authors raise the ques tion of what is the maximum of sin a + cos a and sin a cos a. I agree with their decision to have the students explore these and make some conjectures. It is
only much later, however, that they dis
two sinusoidal functions is again sinu
Complex numbers are not men
pose of the question. But the tools are
soidal. It is tempting at this point to
tioned at all in the book Should they
already at hand. Just note that
look at the special case of
2 - (sin a + cos a) 2 = 2(sin2 a + cos2 a) - (sin a = (sin a - cos a)2 2: 0,
+ cos a)2
and (sin a
+
cos a) 2
= 1 + 2 sin
a cos a
sin
k 1.r +
= 2 sin
have been? Many students leave sec
sin k2x
(tul[ + lv2)x) (t cos 2
(k t
- k-2)x
)
ondary school with no acquaintance of these beyond their incidental occur rence in the quadratic formula. The de sire in many quarters to incorporate
and relate it to the concept of "beats"
calculus into the secondary curriculum
when two sounds with very close fre
has preempted time that could have
quencies are sounded together, as might
been used to provide students with
will lead to an expeditious answer. Stu
occur in the tuning of a violin. It can be
a broader experience
dents could be invited to tum to the
seen that the sum
more general question of maximizing
sinusoidal, being now amplitude-modu
a
lated, but it may still be periodic.
sin
x+b
x.
cos
There are a couple of missed op
is no longer strictly
It is pointed out in Chapter
functions,
inequalities,
with algebra, and,
indeed,
trigonometry, so that they can embark on the calculus with a better array of
5 that
technical and conceptual tools. I see fa
portunities that would not have re
the area under an arch of the sine func
miliarity with complex numbers as pati
quired much extra space but would
tion is 2, but the argument is deferred
of this tool box.
have enhanced the book For example,
to a fmal appendix because of the need
Still, it may well be argued that com
8, it
plex numbers are justifiably omitted
might be worth pointing out that the
area under an arch of the graph of y = 2 sin x can be calculated in a much more
from this book There is enough con
elementary way, by exploiting reflec
Moivre's theorem, roots of unity, poly
Chapter
8 treats the graphs of the
trigonometric functions and eventually discusses linear combinations of sinu soidal functions, using the example '\
..!. sin k.r
tive
.::... k
to adumbrate the Fourier series repre sentation. There is a careful explana tion of how to transform the graph of y
= sin x
to
a sin k(.r - {3),
obtain
graphs
to consider a limit. In Chapter
of y
=
and how the sine and
cosine curves are related by shifts. The reader is led to a realization that a lin ear combination of sinusoidal func tions with the same frequency is also sinusoidal with that frequency, before raising the question of combining func tions with different frequencies. In the part of Chapter 8 dealing with linear combinations of sinusoidal func tions, there is a very short section treat ing the question whether the sum of
symmetry
(not
otherwise
tent-absolute value, polar representa tion,
geometrical
applications,
de
dis
nomial equations, use for trigonomet
cussed). Restrict attention to the half
ric results-to warrant a separate com
arch, for which 0 :S x :S 1Tiz. Because 2 2 cos (x - 1Tf2) = sin x, the graph of 2 cos x is the reflection of that of sin2 x
panion volume. The present text has struck a good balance. It is written for a wide range of students, and it ought
about the line x = trf4. Because sin2 x = 1 - cos2 x, the graph of cos2 x is the 2 reflection of that of sin x about the 1 line y = /z. Accordingly, the graph of 2 sin x is carried to itself by the prod
and be purchased by students who
to be negotiable by any who are going on to study mathematics or science at the tertiary level. It should be on the bookshelf in every secondary school,
uct of these two reflections, a rotation
want a succinct introduction to this im
of
portant subject.
equal to the area between the graph
University of Toronto
180° about the point ( 7T/4, 1/z). Thus for 0 :S x :S 7Th the area between the graph of sin2 x and the line y = 0 is and the line y
= 1; both these areas are
equal to 7Th Thus, the area under a
complete arch is
7Tfz.
Department of Mathematics Toronto, Ontario
M5S 3G3 Canada
e-mail: [email protected] .edu
VOLUME 24, NUMBER 2. 2002
75
l$@ii•i§u6hi£11@1§[email protected] i,i,i§
M i c hael K l e b e r a n d Ravi Vaki l , E d i to rs
Cross-Num ber Puzzle Robert Haas
This column is a place for those bits of contagious mathematics that travel from person to person in the community, because they are so
Decimal points are omitted. Decimals are truncated, not rounded. Answers to clues marked ''reversed" are written right-to-left or bottom-to-top.
elegant, suprising, or appealing that one has an urge to pass them on. Contributors are most welcome.
a
b
c
d
k
e
g
h
n
0
u
X
a
Please send all submissions to the
'
h'
Mathematical Entertainments Editor,
Ravi Vakil, Stanford University,
Department of Mathematics. Bldg. 380, Stanford , CA 94305-21 25, USA
Earlier versions of this puzzle appeared in
e-mail: vakil@math. stanford .edu
been received for reproduction.
76
THE MATHEMATICAL INTELLIGENCER © 2002 SPRINGER-VERLAG NEW YORK
The Journal of Irreproducible Results, from which permission has
ACROSS
a. Circle constant i. Mid-April headache in the U.S. j. 116 + 1/12 + 1/24 + 1148 + . . . k. Subtracting it from its reversed dig its gives a multiple of 5; adding it, a multiple of 7. l. a'/1081 - 2£/289 = -857 (see a') 0. 4! - 3! + 2! - 1! + 0! p. Downing prime's number in U. K. [reversed] q. (Sides in Gauss's new polygon)2 r. Plane t. Tchaikovsky overture - 200 u. Liquid butter in India (A = 1, B = 2, . . . ) w. (Total gifts in "12 Days of Christmas") + 1 [reversed] x. (Arab base + computer base) x (Babylonian base - 1)
Vv' 1464 1
y. z.
a'. d'. e'. f'. h'.
(Demuth or Beethoven number) X 10 [reversed] 3a'/1081 - 5£/289 1012 (see f) Adult (Race, or tuning note) X 10 Hastings - 60 Natural constant =
DOWN
a. \/ 10 b. 10 in binary c.
I � �1
n. 385 r. 117 [reversed] s. Product of two triangular non square numbers that's triangular non-square B t. A = H4 (A =F B integers) v. log (6.4339) w. 2(32) y. Futuristic film [reversed] z. Good vision [reversed] b ' . Toll-free [reversed] c ' . Emergency g'. Days around the world [reversed]
-
d . Bad room i n 1984 e. (10 010 001 1 10 0 1 1 )2 f. Fissionable U g. Sexy shape [reversed] h. cos sin- 1 cos sin- 1 (.53881) m. Bond
1 081 Carver Road Cleveland Heights, OH 441 1 2 USA
The solution will appear in Volume 24, Number 3.
MOVING?
We need your new address so that you do not miss any issues of THE MATHEMATICAL INTELLIGENCER. Please send your old address (or label) and new address to: Springer-Verlag New York, Inc., Journal Fulfillment Services
P.O. Box 2485, Secaucus, NJ 07096-2485 U.S.A.
Please give us six weeks notice.
VOLUME 24, NUMBER 2. 2002
77
41£i,I.M9.h.I§i
R o b i n W i l son
Navigational Instruments
here are T
many stamps featuring
navigational instruments, ranging
from
armillary spheres
and astrolabes
to sextants and octants.
A rm illa ry sphaes usually con sisted of metal circles representing the
ST. C H R I STO PHER N£VJS·ANGUI LLA
Armillary sphere
Mariner's astrolabe
main circles of the universe, and were used to measure celestial
coordinates. Other instruments were used by navigators to measure the altitude of a heavenly body, such as the sun or pole star, in order to determine latitude at
seas. The astrolabe reached its matu rity during the Islarnk period and con sists of a brass disc suspended by a ring, fixed or held in the hand. On the front are calculating devices for mea suring the heavenly bodies. The back has a circular scale on the rim and a rotating bar. To measure altitude, the observer views the object along the bar and reads the altitude from the scale. For navigational purposes a more ba sic mariner's ast1'0labe was later de veloped.
Used in Europe from the thilteenth
century, the qu.admnt has the shape of a quwter-cirde (90°); the se.rtant and octant similarly correspond to a sixth (60°) and an eighth (45°) of a circle. To measure altitude, the observer looks along the top edge of the instrument and the position of a movable rod on the circular lim gives the required
Islamic astrolabe
Quadrant
reading.
Around 1300 the mathematician and astronomer Levi ben Gerson invented the Jacob 's staff or cross-staff, for mea suring the angular separation between
two
celestial bodies. Although widely had a m<.ijor disadvantage. To measure the angle between the sun and the horizon one had to look directly at the sun; the back-staff is a clever mod ification in which navigators could keep their backs to the sun. used, it
Sextant
Please send all submissions to
the Stamp Corner Editor,
Robin Wilson, Faculty of Mathematics.
The Open University. Milton Keynes. MK7 6M, England
e-mail: [email protected]
Octant
1n IU c N E 0
Jacob's $t
Back-staff
() 2002 SPRINGER- VERLAG NEW YORK. VOLUME 24. N\JM8ER 2. 2002
79
