Le tters 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.
EN= 5t
Erdos Number Updates*
Marek A. Abramowicz
We give more accurate estimates of
Institute of Theoretical Physics
Erdos numbers (ENs) for three physi
Chalmers University
cists mentioned in the article "Famous
41 2-96 Goteborg
Trails to Paul Erdos" by Rodrigo de
Sweden
Castro and Jerrold W. Grossman (Math
e-mail:
[email protected]
ematical lnteUigencer, vol. 21 (1999), no.
3, 51-63).
Future of Mathematical
3 be
John A. Wheeler has EN=
cause of the paper "On the question of a neutrino analog to electric charge,"
Rev.
Mod. Phys.
which
he
29, p. 516 (1957),
published
with John
R.
Klauder (known to have EN = 2). Roger Penrose has EN
4 at
Literature As an amateur Platonist I am well aware of problems with mathematical and philosophical archives. Much of what you are searching for is simply no longer available. It would be fine
most:
to have it all securely fixed on the
John R. Klauder has published to
Internet for free use by everybody.
gether
with
C.I.
=
Isham
an
article
However,
it
is
getting
increasingly
"Affine fields on operator representa
clear that the problem is not only in the
tions for the nonlinear sigma-model,"
relatively short lifetime of magnetic
JMP 31, p. 699 (1990), and C.I. Isham,
recording, but also in the possibility
3, published
(likely increasing) of vandalization of
with R. Penrose and D.W. Sciama a
electronic media by hackers. Whereas
book, Quantum Gravity, an Oxford
it took a fanatic mob to bum Alexan
Symposium, Clarendon Press, Oxford,
dria Library, it may now take just a few
who therefore has EN
=
1975, in which the Preface is jointly
keyboard
written by the three of them.
electronic Herostratus to erase irre
Stephen W. Hawking, who pub
strokes
by
some
shrewd
placeable records.
lished a book and several papers with
Unfortunately, the back-up problem
Roger Penrose, has therefore EN= 5
is far from trivial. A symbiosis of elec
at most.
tronic and paper storage systems (advo cated by D.L. Roth and R. Michaelson's
Jerzy Lewandowski
EN= 3
letter, Mathematical InteUigencer 22
(2000), no. 1, 5) is unlikely to provide
lnstytut Fizyki Teoretycznej Uniwersytet Warszawski
an adequate long-term solution be
ul. Hoza 69 00-681 Warsaw, Poland
cause of the sheer amount of paper it requires. Instead, more durable read
e-mail:
[email protected]
only techniques are likely to enter the scene. Examples are atomic force mi-
Pawel Nurowski lnstytut Fizyki Teoretycznej
EN = 4
croscopy recording, with a potential capacity of 1012 bits per sq em, or iso
Uniwersytet Warszawski
topic information storage (information
ul. Hoza 69
is coded in the order of stable Si iso
00-681 Warsaw, Poland
topes on a crystal surface).
e-mail:
[email protected]
Standardization and simplification
•This letter must serve as a representative of the hundreds of updates which could be written. The reader is reminded that the Web site http://www.oakland.edu/-grossman/erdoshp.html is perpetually updated. There you may find links joining several chemistry Nobel Prize winners. The updated Erdos numbers of all Fields Medallists are 4 or less, excepting Paul Cohen and Alexander Grothendieck at 5. - Editor's Note. tEN
=
4 after publication of this letter.
© 2000 SPRINGER-VERLAG NEW YORK, VOLUME 22, NUMBER 3, 2000
3
of such technologies (including read ing devices) will allow for inexpensive individual ownership of entire acade mic libraries stored in a briefcase-size box. Potentially, almost every house hold could have one (at a cost, per haps, comparable to a present set of the Britannica); they would be duly updatable. We could then feel there was no longer real danger of our records being permanently lost. Alexander A Berezin Department of Engineering Physics
be translated as, T " he sowerArepo takes pains to hold the wheels." But surely the magical form is more important than any artificially imposed meaning. It is transformed into the Pater Noster in the form of a cross:
s
A
Conceptual Magic Square
I would like to add the standard form of the magic square discussed by A Domenicano and I. Hargittai in The Intelligencer 22, no. 1, 52-53. (I am fol lowing [1], 142.) Usually it reads SAT 0R ARE P 0 T E N ET 0 P ERA R 0 TAS If
.
D-37073 Gottingen
Germany e-mail:
[email protected]
p
A 0 T E R PAT ER NOS T ER 0
Hamilton, Ontario L8S 4L8 e-mail:
[email protected]
Mathematisches lnstitut Bu n senstr 3-5
A
McMaster University Canada
Benno Artmann
T E R
0
Here exactly the same letters are used as in the "magic square." The addi tional copies of letters A and 0, transliterations of alpha and omega, stand as a metaphor for Christ as in the NewTestament,Apocal. Joh. 22,13. It is not clear to me why the authors claim that all the reading forwards, backwards, etc., has anything to do with the regular polyhedra.
Newton Scooped
In my article "Exactly how did Newton deal with his planets" in vol. 18 (1996), no. 2, 6-11, I credited Newton with the following discovery: LetS be a focus of an ellipse and P a point on the ellipse. The lineSP meets the line through the center that is parallel to the tangent at P at a point E.Then the length of PE is independent of P. (Principia, Book II, Proposition XII, Problem VII.) However, Bjorn Thiel of Bielefeld University has pointed out to me that Apollonius had obtained the result some 1900 years earlier. (Conics, Book III, Proposition 50.) Sherman K. Stein Mathematics Department University of California at Davis
1. G. Mazzola, ed.: Katalog: Symmetrie in
one takes the secondary meaning of opera as toil, labor, pains, then it might
Davis, CA 95616-8633
Kunst, Natur und Wissenschaft, Vol. 3.
USA
Darmstadt, Roether, 1 986.
e-mail:
[email protected]
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4
THE MATHEMATICAL INTELLIGENCER
Opinion
Chaotic Chaos Denys A Hill
T
he word "chaos" is used in two dif
with the prevalent literary usage of
ferent contexts.
chaos as confusion!
It is more than two millennia since
Turbulence, understood as absence
"chaos" meaning confusion and empti
of underlying order, is sometimes used
ness was established in use in literary
as an analogy for chaos; so is random
contexts.
ness.
Vergil
and
Ovid
thought
"chaos" (the same word in Latin as in English)
this
Does chaos have a separate identity?
meaning continued in the Middle Ages.
The Opinion column offers
If its methodology comprises (say) dif
Authors such as Shakespeare, Milton,
ference and differential equations, bifur
mathematicians the opportunity to
Pope, Byron, and De la Mare had the
cations, Lyapunov exponents, fractals,
same understanding. A little-known ex
Fast Fourier Transforms, and wavelet
write about any issue of interest to
meant
confusion,
and
This would seem to be giving up
the characterisation by underlying order.
ception is Henry Adams's opinion in
analysis, it does not claim these as its ex clusive reserve. Demoting chaos theory
community. Disagreement and
1907 that chaos brings order. The author of the 1947 Marshall Plan again per
controversy are welcome. The views
ceived
and
theory would be possible, but could lead
and opinions expressed here, however,
present-day newspaper writers do like
to its disappearance as a separate entity.
the international mathematical
are exclusively those of the author, and neither the publisher nor the
chaos
with
foreboding,
wise. Contemporary dictionaries of ma jor European languages support them.
to a subset of another, better delineated
The demand to give it a separate identity is founded on claims for suc
In the mid-twentieth century, "chaos"
cesses in real-life situations: Ruelle
editor-in-chief endorses or accepts
and "chaos theory" began to be used
Takens phase space reconstruction,
responsibility for them. An Opinion
with technical meanings. By now these
mathematical modelling in control of
should be submitted to the editor-in chief, Chandler Davis.
words have appeared in more than
lasers and electronic circuits or car
7,000
diac arrhythmia. Unfortunately some
mathematical
and
scientific
books, dictionaries, and papers. Unfor
exaggerated claims, such as control of
tunately the technical meanings are
El Nifi.o climatic phenomena, under
themselves confused, and there is no
mine its credibility and provide sup
agreement on definitions. The contrast
port for those who regard chaos as
with literature and journalism, which
usurping territory of other theories.
give the word an unambiguous and
The justification of a clean-up effort
embar
is not based only on professional pride,
rassing: mathematics and science, in
on the desire to be at least as succinct
straightforward
meaning,
is
place of their vaunted rigour and pre
and definite as our literary and journal
cision, display ambiguities and even
istic counterparts. It is based on self-in
contradictions. The situation is chaotic.
terest. Techniques of chaos theory are
Confusion over the several technical
allegedly about to be applied effectively
meanings is so rife that some authors
in financial control. We had better re
are on record as abandoning attempts to
solve the present chaotic situation be
define "chaos." One relies on personal
fore we lose out on pecuniary benefits.
know-how to recognise its presence. Other authors depend upon their read ers' knowing the meaning already. Such definitions as have been pub lished have little in common. Is chaos a mathematical phenomenon or condi tion? or is it a range of phenomena? Is
Let me advance this phrasing as workable:
Chaos, the science of non-linear deterministic processes that appear stochastic. It insists on the scientific; it is inclu
chaos something occurring in space
sive enough to expand as the subject
and
it
develops; it may resolve the chaotic
bounded or unbounded? Non-linearity,
chaos dilemma sufficiently to protect
time,
or
irregularity,
just
in
space?
intermittency,
Is
unpredic
tability, transience, changeability, and
the image of mathematics as a rigor ous discipline.
complexity are all invoked in the tech nical literature as attributes of chaos.
1 2, Tulip Tree Avenue
In addition there is the feature of or
Kenilworth
der underlying the appearance of ran
Warwickshire CV8 2BU
domness-a
United Kingdom
usage
which
conflicts
© 2000 SPRINGER-VERLAG NEW YORK, VOLUME 22, NUMBER 3, 2000
5
MARCELO VIANA
What's New on Lorenz Strange Attractors?
esides its philosophical implications on the ideas of determinism and (un)predictability ofphenomena in Nature, E. Lorenz's famous article ''Deterministic nonperiodic flow" [ 1 7], published nearly four decades ago in the Journal of Atmospheric Sciences,
raised a number of mathematical questions that are among the
leitmotifs for the extraordinary development the field of DynamicalSystems has been going through.This paper is about those and related questions, and some remarkable recent answers.The first part is a general overview, mostly in chronological order.The four remaining sections con tain more detailed expositions of some key topics. Modeling the weather
Lorenz, a meteorologist at MIT, was interested in the foun dations of long-range weather forecasting. With the advent of computers, it had become popular to try to predict the weather by numerical analysis of equations governing the atmosphere's evolution. The results were, nevertheless, rather poor.A statistical approach looked promising, but Lorenz was convinced that statistical methods in use at the time, especially prediction by linear regression, were es sentially flawed because the evolution equations are very far from being linear.
To test his ideas, he decided to compare different meth ods applied to a simplified non-linear model for the weather.The size of the model (number and complexity of the equations) was a critical issue because of the limited computing power available in those days. 1After experi menting with several examples, Lorenz learned from B.Saltzmann of recent work of his [35] concerning ther mal fluid convection, itself a crucial element of the weather.A slight simplification
:i; = - ax + ay iJ = rx - y - xz z = xy - bz
(]" =
10
r = 28 b = 8/3
(1)
of a system of equations studied bySaltzmann proved to be an ideal test model. I'll outline later howSaltzmann ar rived at these equations, and why he picked these particu lar values of the parameters O", r, b. The key episode is recalled by Lorenz in [18].At some
1 Lorenz's computer, a Royal McBee LGP·30, had 1 6Kb internal memory and could do 60 multiplications per second. Numerical integration of a system of a dozen differential equations required about a second per integration step.
6
THE MATHEMATICAL INTELLIGENCER © 2000 SPRINGER-VERLAG NEW YORK
.
· .
difference in the final state. 2A similar point was stressed early in the twentieth century by H. Poincare [30], includ ing the very setting of weather prediction:
Why have meteorologists such difficulty in predicti'YI;_{J the weather with any certainty? Why is it that showers and even storms seem to come by chance, so that many people think it is quite natural to pray for them, though they would consider it ridiculous to ask for an eclipse by prayer? [ . . . ] a tenth of a degree more or less at any given point, and the cyclone wiU burst here and not there, and extend its ravages over districts that it would other wise have spared. If they had been aware of this tenth of a degree, they could have known it beforehand, but the observations were neither sufficiently comprehensive nor sufficiently precise, and that is the reason why it aU seems due to the intervention of chance.
Figure 1 . Lorenz strange attractor.
stage during a computation he decided to take a closer look at a particular solution. For this, he restarted the integra tion using some intermediate value printed out by the com puter as a new initial condition. To his surprise, the new calculation diverged gradually from the first one, to yield totally different results in about four "weather days"! Lorenz even considered the possibility of hardware fail ure before he understood what was going on. To speed things up, he had instructed the computer program to print only three decimal digits, although the computations were carried out to six digits. So the new initial condition en tered into the program didn't quite match the value gener ated in the first integration. The small initial difference was augmented at each integration step, causing the two solu tions to look completely different after a while. This phe nomenon, first discovered in a somewhat more compli cated system of equations, was reproduced in (1). The consequences were far-reaching: assuming the weather does behave like these models, then long-range weather prediction is impossible: the unavoidable errors in determining the present state are amplified as time goes by, rendering the values obtained by numerical integration meaningless within a fairly short period of time. Sensitivity and unpredictability
This observation was certainly not new. Almost a century before, J. C. Maxwell [22], one of the founders of the ki netic theory of gases, had warned that the basic postulate of Determinism-the same causes always yield the same effects-should not be confused with a presumption that similar causes yield similar effects, indeed there are cases in Physics where small initial variations may lead to a big
On the other hand, gas environments and, particularly, the Earth's atmosphere, are very complicated systems, in volving various types of interactions between a huge num ber of particles. Somehow, it is not surprising that their evolution should be hard to predict. What was most strik ing about Lorenz's observations was the very simplicity of equations (1), combined with their arising in a natural way from a specific phenomenon like convection. That the so lutions of such a simple set of equations, originating from a concrete important problem, could be sensitive with re spect to the initial conditions, strongly suggested that sen
sitivity is the rule in Nature rather than a particularfea tu1·e of complicated systems. Strange attractors
A few years later, in another audacious paper [33], D. Ruelle and F. Takens were questioning the mathematical inter pretation of turbulent fluid motion. It had been suggested by L. Landau and E. Lifshitz [16], and by E. Hopf before them, that turbulence corresponds to quasi-periodic mo tions inside tori with very large dimension (large number of incommensurable frequencies) contained in the phase space. However, Ruelle and Takens showed that such quasi-periodic tori are rare (non-generic) in energy dissi pative systems, like viscous liquids. Instead, they main tained, turbulence should be interpreted as the presence of some strange attractor. An attractor is a bounded region in phase-space, in variant under time evolution, such that the forward tra jectories of most (positive probability), or even all, nearby points converge to it. Ruelle and Takens did not really try to define what makes an attractor strange. Eventually, the notion came to mean that trajectories converging to the
attractor are sensitive with respect to the initial condi tions. Lorenz's system (1) provides a striking example of a strange attractor, and several others have been found in
2Even earlier, the same idea appeared in E. A Poe's The mystery of Marie Roget, in the context of crime investigations ... So much for priorities on this matter.
VOLUME 22, NUMBER 3, 2000
7
various models for experimental phenomena as well as in theoretical studies. However, not many examples were available at the time [33] was written.Still unaware of the work of Lorenz, which came to the attention of the math ematical community only slowly,R uelle andTakens could only mentionS.Smale's hyperbolic solenoids [38], which, although very important from a conceptual point of view, had no direct physical motivation. Hyperbolic systems
Throughout the sixties,Smale was much interested in the concept of structurally stable dynamical system, intro duced byA.Andronov and L. Pontryagin in [2].The reader should be warned that the word "stability" is used in DynamicalSystems in two very different senses.One refers to trajectories of a system: a trajectory is stable (or at tracting) if nearby ones get closer and closer to it as time increases.Another applies to systems as a whole: it means that the global dynamical behavior is not much affected if the laws of evolution are slightly modified.3 Structural sta bility belongs to the second kind: basically, a system is structurally stable if small modifications of it leave the whole orbit structure unchanged, up to a continuous global change of coordinates. In an insightful attempt to identify what the known sta ble systems had in common,Smale introduced the geo metric notion of hyperbolic dynamical system. I will give a precise definition of hyperbolicity later; for now let me just refer to Figure 2, which describes its basic flavor: ex istence, at relevant points x in phase-space, of a pair of sub-manifolds that intersect transversely along the trajec tory of x, such that points in one of them (the horizontal "plane") are forward asymptotic to x, whereas points in the other submanifold (the vertical "plane") are backward asymptotic to x. Most remarkably, hyperbolicity proved to be the crucial ingredient for stability: the hyperbolic sys tems are, essentially, the structurally stable ones. More over, a beautiful and rather complete theory of these sys tems was developed in the sixties and the seventies: hyperbolic systems and their attractors are nowadays well understood, both from the geometric and the ergodic point of view.The reader may find precise statements and ref erences to a number of authors, e.g., in the books [6, 28, 36]. Yet, not every system can be approximated by a hy perbolic one. . . The flow described by equations (1) is not hyperbolic, nor structurally stable, so it doesn't fit into this theory.On the other hand, its dynamical behavior seems very robust. For instance: Figure 1, which represents a solution of (1) integrated over a long period of time,4 would have looked pretty much the same if one had taken slightly different values for the parameters u, r, b. How can this be, if these systems are unstable (and sensitive with respect to initial data!)?
Figure 2. Hyperbolicity near a regular trajectory.
It was probably a fortunate thing thatSmale and his stu dents and colleagues did not know about this phenomenon at the time they were laying the foundations of the theory of hyperbolic systems: it might have convinced them that they were off in a wrong direction. In fact, a satisfactory theory of robust strange attractors of flows would come to existence only recently, building on several important ad vances obtained in the meantime. But I'm getting ahead of myself! Lorenz-like flows
For now, let us go back to the mid-seventies, when the work of Lorenz was finally becoming widely known to dy namicists, and, in fact, attracting a lot of attention.So much so, that by the end of that decade C.Sparrow could write a whole book [39] about the dynamics of equations (1) over different parameter ranges. Understanding and proving the observations of Lorenz in a rigorous fashion turned out to be no easy task, though. A very fruitful approach was undertaken, independently, by V. Afraimovich, V. Bykov, L.Shil'nikov [1], and by J. Guckenheimer andR . Williams [8, 44]. Based on the be havior observed in (1), they exhibited a list of geometric properties such that any flow satisfying these properties
must contain a strange attractor, with orbits converging to it being sensitive with respect to initial conditions. Most important for the general theory, they proved that such flows do exist in any manifold with dimension 3. These examples came to be known as geometric Lorenz
models. The strange attractor has a complicated geometric struc ture like the "butterfly" in Figure 1.5Sensitivity corre sponds to the fact that trajectories starting at two nearby states typically end up going around different "wings" of the butterfly.There are orbits inside the strange attractor that are dense in it.This means that the attractor is dy namically indecomposable (or transitive): it can not be split into smaller pieces closed and invariant under the flow.Another very important feature of these models is that the attractor contains an equilibrium point.
3For instance, by altering the values of parameters appearing in the evolution equations. 4An initial stretch of the solution is discarded, so that the part that is plotted is already close to the strange attractor. 5The figure was produced by numerical integration of the original equations (1).
8
THE MATHEMATICAL INTELLIGENCER
Now, one might expect that small modifications of the flow could cause such a complicated behavior to collapse. For instance, the attractor might break down into pieces displaying various kinds of behavior, or the different types of trajectories (regular ones and equilibria) might be set apart, if one changes the system only slightly. Surprisingly, this is not so: any flow close enough to one of these also has an attractor containing an equilibrium point and ex hibiting all the properties I described, including sensitivity and dynamical indecomposability. A theory of robust strange attractors
Vz
VI
�""""'"-
�
L
I
!.
IP(VJ) J
-
-
P(\-S)
r Figure 3. Suspended horseshoe.
As 111
explain later, the presence of equilibria accumulated by regular orbits of the flow implies that these systems can not be hyperbolic. On the other hand, they can not be dis regarded as a pathology because, as we have just seen, this kind of behavior is robust. Indeed, these and other situa tions, often motivated by problems in the Natural Sciences, emphasized the need to enlarge the scope of hyperbolicity into a global theory of Dynamical Systems. Profiting from the success in the study of specific classes of systems like the geometric Lorenz flows or the Henon maps (M. Benedicks and L. Carleson [3], after pioneer work of M. Jakobson [14]), as well as from fundamental advances like the theory of bifurcations and Pesin's non-uniform hy perbolicity [29], a new point of view has been emerging on how such a global theory could be developed. In this direc tion, a comprehensive program was proposed a few years ago by J. Palis, built on the following core corijecture: every
smooth dynamical system (diffeomorphism or flow) on a compact manifold can be approximated by another that has only finitely many attractors, either periodic or strange. I refer the reader to [27] for a detailed exposition. In the context of flows, decisive progress has been ob tained recently by C. Morales, M. J. Pacifico, and E. Pujals [25, 26], whose results provide a unified framework for ro bust strange attractors in dimension 3. While robust at tractors without equilibria must be hyperbolic [11], they prove that a ny robust attractor that contains some equi
librium poi nt is Lorenz-like: it shares aU the fundamen tal properties of the geometric Lorenz models. A key in gredient is a weaker form of hyperbolicity, that Morales, Pacifico, and Pujals call singular hyperbolicity. They prove that any robust attractor containing an equilibrium point is singular hyperbolic [26]. 6 This is a crucial step lead ing to a rather complete geometric and ergodic theory, ap plying to arbitrary robust attractors of 3-dimensional flows. More on this will come later. Back to the original equations
While they were catalysing such fundamental develop ments in Dynamical Systems, equations (1) themselves kept resisting all attempts at proving that they do exhibit a sensitive attractor.
On the one hand, no mathematical tools could be de vised to solve such a global problem for specific equations like (1). For instance, M. Rychlik [34] and C. Robinson [32] considered systems exhibiting certain special configura tions (codimension-2 bifurcations) and, using perturbation arguments, proved that nearby flows have strange attrac tors like the geometric Lorenz models. This enabled them to exhibit the first explicit examples (explicit equations) of systems with strange attractors of Lorenz type: those special configurations occur in some families of polyno mial vector fields, with degree three, for appropriate choices of the parameters. However, it has not been pos sible to find parameter values u, b, and r, for which (1) sat isfy the assumptions of their theorems. Another approach was through rigorous numerical cal culations. Here a major difficulty arises from the presence of the equilibrium: solutions slow down as they pass near it, which means that a large number of integration steps are required, resulting in an increased accumulation of in tegration errors. This could be avoided in [9, 10, 23, 24], where all the relevant solutions remain far from the equi librium point (error control remains delicate, neverthe less). In these works, the authors gave computer-assisted proofs that the Lorenz equations have rich dynamical be havior, for certain parameters. More precisely, they used numerical integration with rigorous bounds on the inte gration errors, to identify regions V = V1 U V2 inside some cross-section of the flow, such that the image of V under the first-return map consists of two pieces that cross V as in Figure 3. By a classical result of Smale, see [38], this con figuration ("suspended horseshoe") implies that there is an infinite set of periodic trajectories. Still, the original question remained: do equations (1) re ally have a strange attractor? These equations have no par ticular mathematical relevance, nor do the parameter val ues (10, 28, 8/3): their great significance is to have pointed out the possibility of a new and surprising kind of dynam ical behavior that we now know to occur in many situa tions. Nevertheless, many of us felt that answering this question, for parameters near the original ones, was a great challenge and a matter of honor for mathematicians. 7
6See [4, 7] for related results about discrete-time systems, in any dimension. These and other important recent developments are surveyed in my article [43]. 7See Smale's list of outstanding open problems for the next century in vol. 20, no. 2 of The Mathematical lntelligencer, and his contribution to the book Mathematics: Frontiers and Perspectives, tube published under the aegis of the International Mathematical Union as part of the celebrations of the World Mathematical Year 2000.
VOLUME 22, NUMBER 3, 2000
9
The Lorenz attractor exists!
Remarkably, a positive solution was announced about a year ago, by W. Tucker, then a graduate student at the University of Uppsala, Sweden, working under L. Carleson's supervi sion. Theorem 1 (Tucker [41, 42]) For the classical parame
ters, the Lorenz equations (1) support a robust strange attractor. Tucker's approach is a combination of two main ingre dients. On the one hand, he uses rigorous numerics to find a cross-section I and a region N in I such that orbits start ing in N always return to it in the future. After choosing reasonable candidates, Tucker covers N with small rec tangles, as in Figure 4, and estimates the forward trajec tories of these rectangles numerically, until they return to I. His computer program also provides rigorous bounds for the integration errors, good enough so that he can safely conclude that all of these rectangles return inside N. This proves that the equations do have some sort of attractor. A similar strategy is used to prove that the attractor is sin gular hyperbolic in the sense of [26]. The other main ingredient, normal form theory, comes in to avoid the accumulation of integration errors when trajectories are close to the equilibrium sitting at the origin. Tucker finds coordinate systems near the equi librium such that the expression of the vector field in these coordinates is approximately linear. Thus, solu tions of the linear flow (which are easily written down in analytical form) can be taken as approximations of the true trajectories, with efficient error estimates. Accord ingly, Tucker instructs the computer program to switch the integration strategy when solutions hit some small neighborhood of the equilibrium: instead of step-by-step integration, it uses approximation by the linear flow to estimate the point where the solution will exit that neigh borhood. Verifying that a computer-assisted proof is correct in volves both checking the algorithms for logical coherence, and making sure that the computer is indeed doing what it is supposed to do. The second aspect is, of course, less familiar to most mathematicians than the first. In fact, as
Figure 4. Covering a possibly invariant region with small rectangles.
10
THE MATHEMATICAL INTELLIGENCER
computer-assisted proofs are a rather new tool, there hasn't been much time to establish verification standards for computer programs. A basic procedure is to have the codes recompiled and rerun on different machine archi tectures. Preferably, beforehand the algorithm should be reprogrammed by different people. As far as I know, such a detailed independent verification of Tucker's computer programs has not yet been carried out. The first version did contain a couple of "bugs," which Tucker has fixed in the meantime [42], and which I will mention briefly near the end. He has also made the text of his thesis and the computer codes, as well as the initial data used by his pro grams, available on his web page [41]. An outline of Tucker's arguments is given in the last sec tion of this work. Right now, let us go back to where it all started, for a closer look. From Thermal Convection to the Equations of Lorenz
Most of the motion in the Earth's atmosphere takes the form of convection, caused by warming of the planet by the Sun: heat absorbed by the surface of the Earth is trans mitted to the lower layers of the atmosphere; warmer air being lighter, it rises, leaving room for downward currents of cold air. A mathematical model for thermal convection was pro posed early in the twentieth century by the British physi cist R. J. Stutt, better known as Lord Rayleigh. This model [31] describes thermal convection inside a fluid layer con tained between two infinite horizontal plates that are kept at constant temperatures Ttop and Tbot · It is assumed that the bottom plate is hotter than the top one, in other words, Tbot > Ttop · If the temperature difference 11T = Tbot - Ttop is small, there is no fluid motion: heat is transmitted up wards by conduction only. In this case, the fluid tempera ture Tsteady varies linearly with the vertical coordinate YJ. As 11T increases, this steady-state solution eventually be comes unstable, and the system evolves into convective motion. Convection cells are formed, where hot fluid is cooled down as it rises, and then comes down to get heated again. See Figure 5. B. Saltzmann [35] analyzed a simplified version of Rayleigh's model. Firstly, he assumed that the system is in variant under translations along some favored direction, like the direction of convection rolls in Figure 5, so that the corresponding dimension in space may be disregarded.
Figure 5. Convective motion.
This brings the problem down to two spatial dimensions;
metical integration showed that, for convenient choices of
the evolution equations reduce to
the parameters, all but three of the dependent variables are
a( 'I',V2'1')
aV2'1' at
-- =
a(g, TJ) a('¥,8)
a8 at where
g
TJ
and
(2)
D.T a'l' 2 -+KV8 H ag
( 3)
+
are the spatial coordinates,
'I'
and
8
t
is time, and
are interpreted as fol
lows: •
although the phase-space has dimension
tor" contained in a three-dimensional subset of the phase space. On the other hand, these three special non-transient modes seemed to have a rather complicated (nonperiodic) evolution with time. Lorenz took the system of equations obtained in this manner, by truncating the original infmite system right
'l'(g, TJ, t)
is a stream function: the motion takes place
along the level curves of
( •
other words,
seven, many solutions seem to converge to some "attr3;c
-
a(g,TJ)
the dependent variables
transient: they go to zero as time increases to infmity. In
a8 +vV4'I'+g aag
8(g, TJ, t) = T(g, TJ, t)
'I',
a'lt a'¥
-
aTJ' ag -
from the start to only these three variables. This corre sponded to looking for solutions of
with velocity field
).
'l'(g, TJ, t)
Tsteady(g, TJ, t) is
the temperature
ecg, TJ, t)
departure from the steady-state solution mentioned above. The other letters represent physical parameters: height of the fluid layer,
g
H
is the
is the constant of gravity,
a
is
the coefficient of thermal expansion, v is the viscosity, and
K
is the thermal conductivity. If the Rayleigh
However, as observed by Rayleigh, as
Ra
where
A0, Eo, C0,
where X0 and
are created, of the form
t) = Yo cos
( ;; g) (; TJ) ( ;; g) (; TJ}
( 4)
sin
(5)
constants. They describe the motion
5, the parameter a
being related to the eccentricity of the cylinders. They are
stationary solutions,
as time
on the right hand side of
( 4)
t
does not appear explicitly
and
g, a,
v,
K,
and
D.T.
a and H.
The parameters
cr,
r, bin (1)
v
b
cr =
K
=
a
To obtain
and
(1)
D o depending
are given by
4
. 1 +a2
Some simple facts about equations ( 1) are easy to check
sin
in cylindrical convection cells in Figure
Z = C oZ,
one also rescales time, by another constant on
t) = Xo sin
Yo are
one obtains three
are constants depending only on
the physical quantities H,
Ra r =Rc
TJ,
(2), ( 3),
Y = B oY
X = A oX
crosses a threshold
8 (g,
sin
rescaling:
is small, the system remains in the steady-state equilibrium
TJ,
sin
X(t), Y(t), Z(t) which are equivalent to equations ( 1). Actually, X, Y, Z are not exactly the same as x, y, z in (1), but the
VK
'l'(g,
sin
two sets of variables are related to each other, simply, by
number
(2) -( 3)
sin
ordinary differential equations in the coefficients
D.T
new solutions of
X(t)
Inserting these expressions in
R a = g a/{J 'I' = 0, 8 = 0.
of the form
( ;; g) (; TJ) (7: g) (; TJ) +Z(t) (� TJ) ·
=
Y(t) cos
=
(2), ( 3),
(5).
For instance, of
(0,0,0) is an equilibrium point,
for every value
r. This equilibrium is stable (attracting) when r < 1, cor
responding to the stability of the steady-state solution. As
r crosses the value 1, the origin becomes unstable,
and two
new stable equilibrium points P 1 and P 2 arise. They corre spond to the stationary solutions given by one further increases
r,
( 4)
and
(5). If
these two solutions become un
stable. This suggests that, for large Rayleigh number, the convection motion described by
( 4) and (5), is replaced by
some different form of dynamics. Let me also comment on the particular choice of para
Nonperiodic behavior
Aiming to understand what happens when
D.T is further in
creased, Saltzmann looked for more general solutions space-periodic in both dimensions. For this, he expanded
'I'
and
8
as formal Fourier series in the variables
g and TJ,
with time-dependent coefficients. Inserting this formal ex pansion into
(2), ( 3),
(1). Saltzmann took a = 1/ v2, which is the a for which R c is smallest. This gives b = 8/3. The Prandtl number cr = 10 is typical of liquids (cr = 4. 8 for water); for air cr 1 . Finally, the relative Rayleigh num ber r = 2 8 is just slightly larger than the transition value r = 2 4. 7368 at which the two equilibria P 1, P 2 become meter values in
one fmds an infmite system of ordi
value of
=
unstable.
nary differential equations, with the Fourier coefficients as unknowns. Saltzmann truncated the system, keeping only
Geometric Lorenz Models
a finite number of these equations.
Here is an outline of the main facts about the geometric
He tested several possibilities, but one particular case, involving seven equations, was especially interesting. Nu-
Lorenz models in
[ 1 , 8, 44],
that are also relevant for the
next two sections.
VOLUME 22, NUMBER 3, 2000
11
r
:::J
)P(I\f)
c: I
Figure 6. A geometric Lorenz flow.
Figure 8. The image of the first-return map P.
Equilibrium point
A first condition in the definition of the geometric models
is that the flow should have an equilibrium point 0. If X de notes the vector field associated to the flow, the derivative
DX(O) should have one positive eigenvalue A1 > 0 and two negative eigenvalues -A2 < -A3 <
0.
As a consequence,
some future time. This means that
I
acts as a
trap:
tions that hit it cannot escape from returning to
I
solu
indefi
nitely (unless they happen to hit r, in which case they sim ply converge to 0 and never come back to
I).
The behavior of these trajectories can be understood by
there are two trajectories 'Y+ and 'Y- moving away from 0
looking at the points where they successively meet
in opposite directions as time increases, as shown in Figure
other words, under this assumption the evolution of the
6. The union of 'Y- and 'Y+ with the equilibrium point 0 is called the
unstable manifold
of 0, and denoted
wu(O).
� P(z),
that assigns to each z in I\f the pointP(z) where
I.
Moreover, there is a two-dimensional surface containing the
the trajectory of z first intersects
equilibrium point and formed by solutions that converge to
schematic representation of the image of
0 as time goes to 0, and denoted
+ oo. This
WS(O).
is called the
stable manifold of
Check also Figure 7.
weakest contracting one. In other words, one also asks that
(6) It is a simple exercise to check that the equilibrium point 0
=
I\f
under P.
takes
I
(1). For instance, Sparrow
contained in the horizontal plane {z
=
r-
[39]
1}. His
computations of the first-return map, e.g., Figure 3.4(a) in [39, page 34], do suggest that P is well-defined, and its im age has two connected components. The shape of these components is less clear from those pictures: they look
(0, 0, 0) of (1) satisfies these eigenvalue conditions for
more like curve arcs; compare Figure 4. The reason is that
b close to the ones con
the return map is strongly area-dissipative, because the di
all values of the parameters
r,
u,
sidered by Lorenz.
vergence of the flow
Next, one assumes that there exists some two-dimensional domain
I
+ ay, rx- y -xz, xy
div(-ax
Cross-section
that is cut transversely by the flow trajectories,
and also intersects the stable manifold W8(0) along some curve f. In addition, both trajectories 'Y- and 'Y+ intersect
I.
8 contains a
Figure
It is not difficult to find reasonable candidates for such a cross-section in equations
In addition, the expanding eigenvalue should dominate the
In
first-return map
original flow can be reduced to that of the z
I.
- bz)
= -u -
1-
b
is negative, for the parameter values we are interested in. Consequently, the image of P has very small area, which means that the cuspidal triangles represented in Figure
8
must be very thin.
Actually, all the trajectories starting in any of the two
connected components of
I\f
should also intersect
I
at
Invariant contracting foliation
I move on to a more subtle condition in the definition of geo metric Lorenz models, and one that is definitely harder to check in specific situations. One assumes that there exists a
foliation,
that
is, a decomposition of the cross-section I
into roughly parallel curve segments, the ation, which
leaves of the foli
is invariant under the first-return map P: if two
points z1 and z2 are in the same leaf, then so are P(z1) and P(z2). Think of the leaves as vertical lines in Figure
8.
Moreover, the foliation should be contracting: the distance from P"(z1) to P"(z2) goes to zero exponentially fast as goes to
n
+ oo, for any pair of points Z1 and z2 in the same leaf.
The reason this property is very useful is that it allows Figure 7. Stable and unstable manifolds at an equilibrium point.
us to reduce the dimension of the problem even further.8
BPreviously, existence of a cross-section permitted us to go from the 3-dimensional flow to the 2-dimensional map P.
12
THE MATHEMATICAL INTELLIGENCER
Roughly speaking, this goes as follows: points in the same leaf of the foliation have essentially the same behavior in the future, because their trajectories get closer and closer; so, for understanding the dynamics of P it is enough to look at the trajectory of only one point in each leaf, for instance, the point where the leaf intersects a given horizontal seg ment. Expanding map of the interval
Let me make the idea a bit more precise, with the aid of Figure 9. Let I be a horizontal segment in I, e.g., the bot tom side of I. It is convenient to think of I as an interval in the real line, for instance I = [0, 1]. Given any point x in I, let y1 be the leaf containing it. By the invariance prop erty, P( y1) is contained in some leaf y2. Let.ftx) be the point where y2 crosses I. This is how the map f is defined. The graph offis represented on the left-hand side of Figure 10. Note that the map has a discontinuity at the point c of I, corresponding to the special leaf r. As a final condition, the mapfmust be expanding: there exists some constant r > 1 such that dist(ftx), .fty))
:.::::
T
dist(x, y)
(7)
for any two points x, y located on the same side of the dis continuity point c. It is interesting to point out that Lorenz computed such a map numerically for (1) in [17). The cross-section he was considering (implicitly) is not quite the same as our I, so that he got a seemingly different picture, shown on the right-hand half of Figure 10. Nevertheless, the information provided by either of the two maps is equivalent. A sensitive attractor
Under these assumptions, [1] and [8, 44) prove that the flow exhibits a strange attractor A, that contains the equilibrium point 0. The attractor is the closure of the set of trajecto ries that intersect the cross-section I infinitely many times in the past (as well as in the future). Any forward trajec tory that cuts I accumulates in A, and these form a whole neighborhood of the attractor.
r
1'2
'}'1
P( 'YI) I
I
X
c
Figure 9. Definition of the interval map f.
f(x)
c
c Figure 10. Interval maps related to Lorenz flows.
Moreover, these trajectories are sensitive with respect to initial conditions. Indeed, suppose you are given two nearby points z and w, whose forward flow trajectories in tersect I. Typically, the intersection points will be in two different leaves Yz and Yw of the invariant foliation (corre sponding to nearby points x =I= y in /). The next time the two flow trajectories come back to I, the new intersection points will be in leaves y� and y;_,, corresponding to the points.ftx) and .fty) in I. Now, because of the expansive ness property (7), the distance dist( y�, y;_,) is larger than dist( Yz, Yw). Thus, the distance between the two flow tra jectories at successive intersections with I keeps increas ing; one can check that it eventually exceeds some uniform lower bound that does not depend on how close the initial points z and w are.9 Another important conclusion is that the attractor con tains dense orbits (dynamical indecomposability): there ex ists some z0 E A whose forward trajectory visits any neigh borhood of any point of A. (For this one requires the constant r in (7) to be larger than V2.) In particular, the trajectory of z0 accumulates on the equilibrium point 0. Lorenz models without invariant foliations
Numerical investigations of the Lorenz equations carried out by M. Henan andY. Pomeau [12, 13], showed that when the relative Rayleigh number increases beyond r 30 the image of the first-return map develops a "hooked" shape, described in Figure 11. This indicates that for such para meter values there is no longer an invariant foliation, as as sumed in the geometric Lorenz models. As a simple, easy to-iterate model of the behavior of the first-return map near the "bends," they introduced the family of maps of the plane =
(x, y) � (1 - a:i2
-
y,
bx),
(8)
that is now named after Henon. Based on their computations, and especially on Chapter 5 of Sparrow's book [39], S. Luzzatto and I proposed an ex tended geometric model for Lorenz equations, including the creation of the hooks. The main result [21, 20) is that
the attractor survives the destruction of the invariant fo liation, at the price of losing its robustness: after the hooks are formed, a strange attractor exists for a positive-
90f course, the growth must stop at some point: these distances cannot exceed the order of magnitude of the attractor's diameter.
VOLUME 22. NUMBER 3, 2000
13
r
A=
{x :
'Pt(x) E U for all t E IR}.
Secondly, given any vector field Y close to the original one X, the set
Ay : =
{x : 'f' Kx)
E U for all t E IR}
of 'P �-trajectories that never leave U is dynamically inde composable for the flow 'P � associated to Y. Here close ness means that the two vector fields X and Y, and their first derivatives, are uniformly close over M. Finally, A is a robust attractor if the neighborhood U can be chosen to be trapping (or forward invariant): 'Pt(U) c U for all t > 0. Hyperbolicity
Figure 1 1 . Hooked return maps in Lorenz equations.
Lebesgue-measure set of parameter values. This set is nowhere dense, which is related to the lack of robustness: near the parameter values for which the strange attractor exists there are others corresponding only to attracting pe riodic orbits. Moreover, the conclusions of [2 1], which treats a version of the problem for interval maps, have been further extended by Luzzatto and Tucker in [19].
Now I define when an invariant set A is hyperbolic. At each point x of A it should be possible to decompose the tan gent space into three complementary directions (sub spaces)
depending continuously on x, such that •
•
Robust Strange AHractors
In order to state the results of Morales, Pacifico, Pujals, mentioned before, I need to introduce the precise defini tion of a robust attractor. It makes sense also for discrete time systems (diffeomorphisms, or just smooth transfor mations), but here I restrict myself to flows.
1. qP(x) = x for every x E M, and 2. 'fit qf(x) = 'Pt+s(x) for all x E M and o
s, t E IR.
Denote by X the associated vector field on M, which is de fined by
D'Pt E/1 = E�'Cx)
is dense in A. This property is often called transitivity in the specialized literature. An invariant set A is called robust if it admits a neigh borhood U such that the following two conditions are sat isfied. Firstly, A consists of the points whose trajectories under 'fit never leave U:
14
THE MATHEMATICAL INTELUGENCER
and
D'Pt E!f = Eif'Cx)
and there are constants C > 0 and A > 0 such that
IID'Pt
I E/1 11 ::s ce - At
and
IICD'Pt
I EJ;')- 111 ::s ce - At
for all t > 0 and all x E A. I already mentioned the following important conse quence, described in Figure 2. If an invariant set A is hy perbolic, in the sense of the previous paragraph, then every point x in it is contained in a pair of local sub-manifolds of M, the stable manifold WS(x) and the unstable manifold WU(x) such that •
We say that a subset A of M is invariant if trajectories starting in A remain there for all times: if x E A then 'Pt(x) E A for all t E IR. In what follows I always consider compact invariant sets. An invariant set A is dynamically indecomposable if there exists some point x E A whose forward orbit
tion of the vector field X at x; the linearized flow ll'f't preserves the subbundles E8 and Eu (this is clear for Ffl); moreover, it contracts E8 and expands Eu exponentially fast.
This last condition means that
Robust invariant sets
Let q/ : M � M, t E IR, be a flow on a manifold M, that is, a one-parameter group of diffeomorphisms satisfying
EJ is the one-dimensional subspace given by the direc
•
if y is in W8(x), then d('f!t(x), 'Pt(y)) :::; ce - At, for every t > 0; if z is in wu(x), then d('f!-t(x), 'P-t(z)) :::; ce- At, for every t > 0.
The existence of these manifolds also determines the local behavior of solutions close to the one passing through x. So far, I have implicitly assumed that x is a regular point of the flow, that is, X(x) is not zero. When, on the contrary, x is an equilibrium point, then the definition has to be re formulated: in this case, the directions Ef:; and E!f; alone must span the tangent space:
Figure 7 describes stable and unstable manifolds through an equilibrium point.
Observe that if x is an equilibrium then dim EJ
+ dim EJ:: =
dim M,
whereas in the regular case dim Ei: + dim EJ:: = dim M
-
1.
This has a simple, yet important, consequence: an invariant indecomposable set containing equilibria is never hyperbolic (except in the trivial case when it consists of a unique point). This is because the dimension of either Ei: or Ei'would have to have a jump at the equilibrium, contradicting the require ment of continuity. In particular, the geometric Lorenz at tractors that I described above can not be hyperbolic. Singular hyperbolicity
Morales, Pacifico, and Ptijals propose a notion of singular hyperbolic set, which plays a central role in their results. Let A be an invariant set for a flow q/. We say that A is singular hyperbolic if at every point x E A there exists a decomposition TxM Ex EB Fx of the tangent space into two subpsaces Ex and Fx such that the linear flow con tracts Ex exponentially, and is exponentially volume ex panding restricted to Fx: =
-
det (D
-
At
for all t > 0. The decomposition must depend continuously on the point x. Note that the linear flow is allowed to ei ther expand or contract Fx, or both. However, we also re quire that whatever contraction there is in this direction, it should be dominated by the one in the Ex direction:
IID'Pt(x)e[[ IID'Pt(x)f[[
-
<
Ce - At
for all norm-1 vectors e E Ex and f E Fx. Finally, if there are equilibrium points in A, they should all be hyperbolic (no eigenvalues with zero real part). The strange attractors in the geometric Lorenz flows I mentioned before are singular hyperbolic. In fact,
vice-versa. So, from now on I'll speak only of Case 1. In this case, the eigenvalues satisfy the relation (6), just as for geometric Lorenz systems. Furthermore, the invariant set A must be an attractor for the flow. And this attractor is singular hyperbolic! These results have several important implications. For one thing, every non-trivial robust attractor of a three
dimensional flow that contains an equilibrium point is sensitive with respect to initial conditions. Moreover, the flow admits a contracting invariant foliation in a neighbor hood of the attractor. Finally, there exists a cross-section I, with a finite number of connected components, and the first return map induces an expanding one-dimensional map in the space of leaves restricted to I. The statements in the last sentence are being proved by a graduate student at the Federal University of Rio de Janeiro, as part of showing that these attractors are stochasticaUy stable: time averages don't change much when small random noise is added to the system. IO In fact, having reached this point, the main geo metric and ergodic properties of the classical Lorenz mod els extend to this whole class of robust attractors. Higher dimensions
Not much is known about attractors of flows on manifolds of dimension larger than 3, apart from the hyperbolic case. Robust nonhyperbolic examples can be constructed in a sort of trivial way: the attractor lies inside a 3-dimensional submanifold that is invariant under the flow, and attracts all nearby trajectories. On the other hand, until recently one didn't know whether there are truly high-dimensional attractors of Lorenz type, that is, containing equilibria and yet robust. This was answered, affirmatively, by C. Bonatti, A Pumarifio, and myself in [5]: for any k 2: 1 there exist
smooth flows exhibiting robust strange attractors that contain equilibria with k expanding directions; in par
ticular, the topological dimension of the attractor is at least Needless to say, pictures of these multidimensional at tractors are not easy to make . . .
k.
Theorem 2 (Morales, Pacifico, Pujals [25, 26]) Any ro
bust attractor of a three-dimensional flow that contains an equilibrium point (and is not reduced to it) is sin gular hyperbolic. Actually, they prove an even stronger statement. Let A be any robust invariant set. If A contains points of equi librium, then they must all have the same stable and un stable dimensions. More precisely, the derivative DX has (1) either two negative eigenvalues and one positive eigenvalue, at every equilibrium point; (2) or two positive eigenvalues and one negative eigenvalue, at every equilibrium point. These cases can be interchanged: replacing the vector field -X (in other words, reversing the ori entation of trajectories) transforms Case 1 into Case 2, and
X by its symmetric
The Lorenz Attractor In order to prove Theorem 1, Tucker begins by rewriting the equations in more convenient coordinates (xi, x 2, X3), related to the original ones (x, y, z) by a linear transfor mation, such that the linear part of the vector field at the origin now takes a diagonal form:
DX(O)
=
For equations (1), with u gives XI X2
x3
= = =
l l .Sxi
-
(
AI 0 0
=
10,
0 0 -A 2 0 0 - A3
b
=
8/3, r
)
=
28, this roughly
0.29(xt + x )2X3 -22.8x 2 + 0.29(XI + X 2)X3 - 2.67x3 + (xi + x )2 (2.2xi - 1. 3 x )2
·
(9)
' D"fhe study of systems with small random noise goes back to A. Andronov, L. Pontryagin, A. Kolmogorov, and especially Ya. Sinai [37]. It was much developed by Yu. Kifer who showed, in particular, that hyperbolic attractors as well as the geometric Lorenz attractors are stochastically stable [15].
VOLUME 22, NUMBER 3, 2000
15
The next step is to choose a cross-section � for the flow: Tucker takes � c
{x3
=
r-
arbitrary. Now Theorem
1 = 27), although this is fairly 1 can be restated in terms of the
first-return map P of the flow to this cross-section. There are three essential facts to prove:
(A) There exists a region N c � that is forward invariant under the first-return map, meaning that P(N\f) is con
:L '
tained in the interior of N.
(B) The return map
P
admits a forward invariant cone
field. ln other words, there exists a cone
't5(z) inside the tangent space of � at each point z of N\f, such that D P(z)'t5(z) is strictly contained in 't5(P(z)), for every z in N\f.
c
(C) Vectors inside this invariant cone field are uniformly expanded by the derivative DP of the return map: there
c > 0 and T > 1 such that
exist constants
for every
v
Poincare map from � to � · must be contained in a rectangle
E 't5(z) and n 2:: 1.
Indeed, statements
(A), (B), (C), together with some extra
information on the value of the expansion constant
T,
im
ply that the flow has a strange attractor. Since these three statements are robust, that is, if they hold for a given flow then they hold also for any nearby one (possibly with a slightly smaller
r), so is the attractor.
Let me begin by explaining how property
Figure 12. Step by step integration (x2 direction not represented, for clarity).
Ri of size not larger than E1 + E2 around ci.
See Figure 13.
The inductive step
Now the idea is to proceed for Ri in the same way as for Ri. There are, however, several points one has to take into account. To start with, Ri may be much larger than Ri. So, by repeating this step in a naive way, one is likely to end
(A) is ob
up with rectangles that are too big for the Taylor expan sion estimates to be of any help. To solve this, Tucker sub
tained; I'll talk about the other two later on.
divides each Ri into sub-rectangles Ri,j of size at most B
max,
Existence of a forward invariant region
and then treats each one of them individually. That is, the
A first non-rigorous computation of the return map is used to guess what the region N could be. As I mentioned be
program goes on to integrate the trajectories of the central
fore, trajectories returning to � cut it along two "arcs."
plane ���, located at distance
Tucker covers the approximate locations of these "arcs"
points of each
Ri,j
up to another intermediate horizontal
h from � · , and so on.
Another problem is that, as one moves down, trajecto
by small rectangles Ri, of size Bm= = 0.03. Then he takes N to be the union of these Ri· Recall Figure 4.
ries tend to approach the horizontal direction. See Figure
At this initial stage, Tucker has to deal with 700 rectan
mating these Poincare maps between horizontal planes in
gles, although he takes advantage of the system's symmetry
14. As a consequence, integration errors involved in esti crease dramatically. The way to avoid this is by switching
with respect to the X3 axis to cut the number down to half,
from horizontal cross-sections to vertical ones, whenever
that
trajectories are far from the vertical direction. More pre
is, 350 rectangles. ln any case, this number is soon go
ing to increase, as we'll see. For simplicity, he always works with rectangles with sides parallel to coordinate axes. The
points in every one of the Ri wiU return to the cross-section � inside N. ultimate goal is to prove that the
cisely, at each step the program checks whether the verti cal component
x3 is larger than the horizontal one x1. In
the affirmative case, the integration goes on as I described.
Otherwise, a Poincare map is computed, from the hori-
For this, he estimates the future trajectories of points in each of these rectangles in the way I now describe. First, the trajectory of the central point the Euler method) from time
t=
ci of Ri is integrated (by
0 up to the moment where
R!,!
it intersects an intermediate horizontal plane � ' placed at some small distance
h=
w- 3 underneath �- Let 12.
ci be the
intersection point. See Figure
The place where the trajectories of the other points of Ri intersect � · cause Ri
is estimated from ci using Taylor expansion: be
is small, the distance from intersection points to ci E1, that depends on 8,aa;
cannot exceed some small number and
h.
One also has an upper bound E2 for the error com
mitted in the integration of the trajectories of the central points
16
ci.
This means that the whole image of
THE MATHEMATICAL INTELLIGENCER
Ri
by the
[] I
Ri,2
Ri Ci.' Figure 13. Accommodating all possible errors, then subdividing.
goals one has in mind, and also upon specific properties of the system. In the present situation Tucker uses 'V =
Figure 14. Switching from horizontal to vertical cross-sections.
zontal cross-section that is currently under consideration, to a nearby vertical plane I . Bounds for the errors involved in this step are also computed. One ends up with another rectangle Ri inside I, that contains the image of the previ ous one under this Poincare map. Now, the algorithm pro ceeds just as before (apart from the fact that we are now dealing with vertical objects): Ri is subdivided into sub rectangles with size less than Dmax, the trajectories through the centers of these sub-rectangles are integrated up to in tersecting another vertical plane I ' located at distance h to the side of I, and so on. The question concerning the relative strength of the hor izontal and the vertical components of the vector field is asked at each step. If the vertical component later becomes the stronger one once more, the program goes back to con sidering horizontal cross-sections. Switching back and forth from horizontal to vertical cross-sections may happen sev eral times before one gets back to the original plane I. The whole process stops when the trajectories hit I again. It is clear that, due to subdivision, the program has to deal with an increasing number of rectangles: the total number reaches some tens of thousands, from the initial 350. The point is that, iffor every one of the rectangles that are cre
ated along the way, the algorithm finishes in finite time, and the return to I occurs inside N, then one is certain that N is indeed a forward invariant region for the flow. Passing close to the equilibrium
So far, I deliberately avoided talking about trajectories that go close to the origin. It is time to explain how this is dealt with. For the computer program, "close to the origin" means "inside a cube C of size 115 around the origin." If in the course of the integration, the trajectories do not hit the cube C, then the algorithm is precisely as I described. For trajectories that enter C at some step, the computer pro gram calculates the exit point Pexit directly, as follows. A key point is that the eigenvalues AI> -A2, -A3 of the vector field at the origin are sufficiently far from being res onant. What I mean by this is that linear combinations (10) are not zero, nor too close to zero, for many positive inte ger values of n 1, n2, n3. The precise set 'V of values of n1, n2, n3 for which this must be verified depends upon the
{(n1, n2, n3) E N 3 : n1
+ n2 + ns 2: 2 and either n1 < 10 or n2
+ n3 < 101.
This set is actually infinite, so it may seem hopeless to check these conditions by means of actual computations. However, it is clear that ( 1 0) is far away from zero both when n1 is much larger than n2 + ns (in which case it is positive) and when n2 + n3 is much larger than n1 (in which case it is neg ative). This observation leaves only a finite number of triples (nt. �. n3) for which non-resonance must be checked (al most 20,000 triples), which is readily carried out by an aux iliary computer program. The eigenvalues A1, A2, A3 can be computed with as much precision as required, of course. Having checked this, a classical theory developed by H. Poincare, S. Sternberg [40], and a number of other math ematicians, can be applied to conclude that there exist co ordinates y = (Yt, Y 2, Ys) in C such that the expression of the vector field X in these coordinates is very close to be ing linear:
DX(O)y + 1/Jl(y) 1 I PA(y)l ::::; const 1Y t l 10 CIY 2 I + IY31) 0. X(y)
=
(11)
Why is this useful? The trajectories of points under the lin ear flow L(y) = DX(O)y can be expressed in a simple ana lytical form. Then, in view of (l l), the trajectories of points in C for the actual flow X can be estimated with a good de gree of accuracy. Note that we are talking of computations in the new y coordinates. One still has to go back to the x coordinates. This involves estimates for how far the trans formation x = 'l'(y) is from being the identity, which Tucker derives from explicit lower bounds for the norms of (10). According to his estimates, the errors involved in this step 2 are bounded by w- . So, he replaces the exit rectangle ob tained for the linear flow L by another with size increased 2 by w- on each side. The latter rectangle is then treated in the same way as described above: subdivision into smaller rectangles, integration of small pieces of trajectories start ing at the corresponding central points, and so on. As explained before, this program should stop in finite time, with all the rectangles that were introduced in the course of the algorithm returning inside N. Tucker imple mented this algorithm in C, with double-precision floating point arithmetic (relative accuracy w- 1 5). The program ran for about 30 hours on a Spare Server station using both UltraSparc II 296 Mhz processors. And it stopped! Invariant expanding cones
The steps I described so far show that equations (1) admit a trapping region, namely, the union of all the trajectories through the points of N. The set A of points that never leave the closure of this region, neither in the future nor in the past, is our candidate for being the attractor. However, at this point we still don't know much about A. For instance, it could be just an attracting periodic orbit. To rule out this possibility, and conclude that A is indeed a strange attractor, we still
VOLUME 22, NUMBER 3, 2000
17
need properties (B) and (C). In fact, Tucker's program deals with all three statements simultaneously, as follows. Initially, besides feeding the program with a covering of N by small rectangles Ri, as explained before, Tucker also assigns to each Ri a cone Ci inside the tangent plane of Ri, represented by the slopes cii and a{ of its sides with re spect to the x1-direction. All these initial cones are chosen with total angle 10 degrees, but their inclination varies from one rectangle to another: their axes are more or less along the tangent directions to the two "arcs" that form the at tractor; recall Figure 4, obtained from a preliminary (non rigorous) computation. 1 1 Then, in parallel to finding a rigorous upper bound Ri for the image of the rectangle R i under the Poincare map 7T : I � I', the program also looks for a corresponding up per bound for the image of ci under the derivative of 7T. This derivative is computed through the formulae a1rk -
dXj
(x)
=
a
where 7Tk,
=
(DX
a
·
D
with D
=
Id
(12)
of (l). Note that D
yield a cone Ci inside the tangent plane of Ri that contains D7T(x)Ci for every x E Ri · See Figure 15. As before, this cone Ci is described by the slopes f3i and f3t of its sides. Finally, the program also computes a rigorous lower bound for the expansion inside ci, that is, a positive number ei such that
for all x E Ri and v E Ci. Now Ri is subdivided into rectangles Ri1, as explained before, and the program carries out the integration for each (Ri1,{3:;,{3t) in place of the initial (Ri,a:;,at). There is no subdivision of cones. Every integration step uses the same procedure as I just described for the first one, except that (i) the roles of x1 and x3 are interchanged when dealing with vertical cross-sections, and (ii) linear approximation is used, instead, when solutions pass close to the origin. The program keeps track of successive expansion lower bounds, so that property (C) can be readily checked at the time of return to I: the product should be larger than some
s+ I
�. R1,]
�
I'
.... �
? I
Figure 1 5. Invariant cones.
constant r > 1. In fact this is not quite so: somewhat sur prisingly, vectors in the invariant cone field may even be contracted by the first-return map, if the starting point is close to the tips of the attractor. 12 However, this is com pensated by expansion in subsequent returns: according to the computer output, the accumulated expansion exceeds 2 before the trajectory can cross r. This sort of estimate is sufficient for most purposes, including the proof that the attractor is indecomposable. Moreover, the invariance property (B) translates into the following statement:
[ YT >Y/l is contained in [ ai , a{] for every one of the initial (Ri,ai ,a{) and every returning (SJ, rT , y/) such that S1 intersects Ri. Successful verification of all these inequalities shows that A is a robust strange attractor for (1). Thus A is sin gular hyperbolic, and so it fits directly into the theory I de scribed in the previous section. Its geometry and its ergodic properties can be well understood, and they do correspond to those of the classical geometric Lorenz models. A happy conclusion to a beautiful story! Acknowledgments
I'm most grateful to Maria Jose Pacifico and Warwick Tucker for explaining their work to me, and providing some of the bibliographical references. Many thanks also to Mattias Lindkvist, for caring to produce the 3-D pictures. This work was partially supported by Faperj and Pronex-Dynamical Systems, Brazil. REFERENCES
[1 ] V. S. Afraimovich, V. V. Bykov, and L. P. Shil'nikov. On the ap pearance and structure of the Lorenz attractor. Dokl. Acad. Sci. USSR, 234:336-339, 1 977.
[2] A. Andronov and L. Pontryagin. Systemes grossiers. Oak/. Akad. Nauk. USSR, 1 4:247-251 , 1 937.
[3] M. Benedicks and L. Carleson. The dynamics of the H€mon map. Annals of Math . , 1 33:73-169, 1 991 .
[4] C. Bonatti, L. J. Dfaz, and E. Pujals. A C 1 -generic dichotomy for diffeomorphisms: weak forms of hyperbolicity or infinitely many sinks or sources. Preprint, 1 999.
1 1 First, Tucker tried to use a constant cone field: axis everywhere horizontal, total angle 20 degrees. This turned out not to be invariant under the first-return map, but the computer program failed to realize it, because cones were represented in terms of slopes only, so that the program was unable to tell a cone from its comple ment. . . Tucker modified the program to include orientation in the internal representation of cones, and he also added a safety device that would fire if cones were ever to get larger than 20 degrees in the course of the iteration. This device remained silent when the program was run for the new initial cone field. 1 2This is the reason why we need a constant c in the statement of condition (C). This fact was also not apparent when the program was first run, because the ex pansion was not correctly estimated.
18
THE MATHEMATICAL INTELLIGENCER
(18] E. N . Lorenz. On the prevalence of aperiodicity in simple systems. Lect. Notes in Math. , 755:53-75, 1 979.
(19] S. Luzzatto and W. Tucker. Non-uniformly expanding dynamics in maps with singularities and criticalities. Pub/. Math. IHES. To appear. [20] S. Luzzatto and M . Viana. Lorenz-like attractors without inva!i ant foliations. In preparation. [21 ] S. Luzzatto and M. Viana. Positive Lyapunov exponents for Lorenz like maps with criticalities. Asterisque, 1 999. [22] J. C. Maxwell. Matter and motion. Dover Publ . , 1 952. First edition in 1 876. [23] K. Mischaikow and M. Mrozek. Chaos in the Lorenz equations: a com puter assisted proof (1). Bull. Amer. Math. Soc., (2) 32:66-72, 1 995.
MARCELO VIANA
[24] K. Mischaikow and M. Mrozek. Chaos in the Lorenz equations: a
IMPA Est.
D.
Castorina
22460-320
computer assisted proof (II). Math. Camp . , 67: 1 023-1 046, 1 998.
1 10
[25] C. Morales, M. J . Pacifico, and E. Pujals. Partial hyperbolicity and
Rio de Janeiro
persistence of singular attractors. Preprint 1 999.
Brazil
e-mail:
[26] C. Morales, M. J. Pacifico, and E. Pujals. On C 1 robust singular
[email protected]
transitive sets for three-dimensional flows. C. R. Acad. Sci. Paris .
web: www .impa.br/-viana/
326, Serie 1:81-86, 1 998. [27] J. Palis. A global view of Dynamics and a conjecture on the dense
Marcelo Viana got his doctorate at IMPA in 1 990, under the
ness of finitude of attractors. Asterisque, 261 :339-351 , 1 999.
direction of Jacob Palis, and is now a Professor there. He is
[28] J. Palis and F. Takens. Hyperbolicity and sensitive-chaotic dynam
well known for his work in dynamical systems and ergodic
ics at homoclinic bifurcations. Cambridge University Press, 1 993.
theory, especially strange attractors and their statistical prop
[29] Ya. Pesin. Families of invariant manifolds corresponding to non-zero
erties. His musical avocation extends to many styles; his read
characteristic exponents. Math. USSR. lzv. , 1 0: 1 261-1 302, 1 976.
ing extends to fiction, history, and comics.
[30] H. Poincare. Science and method. Dover Publ. , 1 952. First edi [5] C. Bonatti, A Pumari no , and M. Viana. Lorenz attractors with ar
tion i n 1 909. [31 ] Lord Rayleigh. On convective currents in a horizontal layer of fluid
bitrary expanding dimension. C. R. Acad. Sci. Paris , 325, Serie I ,
when the h i gher temperature is on the under side. Phil. Mag.
Mathematique: 883-888, 1 997.
32:529-546, 1 91 6.
[6] R. Bowen. Equilibrium states and the ergodic theory of Anosov diffeomorphisms, volume 470 of Lect. Notes in Math. Springer
Verlag, 1 975. [7] L. J. Diaz, E. Pujals, and R. Ures. Partial hyperbolicity and robust transitivity. Acta Math. , 1 999. To appear.
.
[32] C. Robinson. Homoclinic bifurcation to a transitive attractor of Lorenz type. Nonlinearity, 2:495-5 1 8, 1 989. [33] D. Ruelle and F. Takens. On the nature of turbulence. Comm. Math. Phys. , 20:1 67-1 92, 1 971 .
[34] M. Rychlik. Lorenz attractors through Shil'nikov-type bifurcation.
[8] J. Guckenheimer and R. F. Williams. Structural stability of Lorenz attractors. Pub/. Math. IHES, 50:59-72, 1 979.
Part 1 . Erg. Th. & Dynam. Syst. , 1 0:793-821 , 1 989. [35] B. Saltzmann. Finite amplitude free convection as an i n itial value
[9] B. Hassard, S. Hastings, W. Troy, and J. Zhang . A computer proof
problem. J. Atmos. Sci. . 1 9:329-341 , 1 962.
that the Lorenz equations have "chaotic" solutions. Appl. Math.
[36] M. Shub. G/obal stabi/ity ofdynamicalsystems. Springer Verlag , 1 987.
Lett., 7:79-83, 1 994.
[37] Ya. Sinai. Gibbs measure in ergodic theory. Russian Math.
[1 0] S. Hastings and W. Troy. A shooting approach to the Lorenz equa
tions. Bull. Amer. Math. Soc (2) 27:298-303, 1 992. (1 1 ] S. Hayashi. Connecting invariant manifolds and the solution of the C 1 stabil i ty and .{} -stability conjectures for flows. Annals of Math. ,
1 45:81-1 37, 1 997.
73:747-81 7, 1 967. [39] C. Sparrow. The Lorenz equations: bifurcations, chaos and strange attractors. volume 41 of Applied Mathematical Sciences. Springer
(1 2] M. Henon. A two dimensional mapping with a strange attractor. Comm. Math. Phys. 50:69-77, 1 976.
Verlag, 1 982. [40] S. Sternberg . On the structure of local homeomorphisms of eu
[1 3] M. Henon and Y. Pomeau. Two strange attractors with a simple structure. In Turbulence and Navier-Stokes equations, volume 565, pages 29-68. Springer Verlag, 1 976.
clidean n-space- 1 1 . Amer. J. Math. , 80:623-631 , 1 958. [41 ] W. Tucker. The Lorenz attractor exists. PhD thesis, Univ. Uppsala, 1 998.
[14] M . Jakobsen. Absolutely continuous invariant measures for one parameter families of one-dimensional maps. Comm. Math. Phys .
Surveys, 27:21-69, 1 972.
[38] S. Smale. Differentiable dynamical systems. Bull. Am. Math. Soc. ,
. •
Text
and
program
codes
available
at
www.
math.uu.se/-warwick/.
.
[42] W. Tucker. The Lorenz attractor exists. C. R. Acad. Sci. Paris,
(1 5] Yu. Kifer. Random perturbations of dynamical systems. Birkhii.user,
[43] M. Viana. Dynamics: a probabilistic and geometric perspective. In
81 :39-88, 1 981 . 1 988. ( 1 6]
L.
D. Landau and E. M. Lifshitz. Fluid mechanics. Pergamon, 1 959.
[1 7] E. N . Lorenz. Deterministic nonperiodic flow. J. Atmosph. Sci. , 20:1 30-1 4 1 , 1 963.
328, Serie I, Mathematique: 1 1 97-1202, 1 999. Procs. International Congress of Mathematicians ICM98-Berlin,
Documenta Mathematica, vol l , pages 557-578. DMV, 1 998. [44] R . F . Williams. The structure of the Lorenz attractor. Pub/. Math. IHES, 50:73-99, 1 979.
VOLUME 22, NUMBER 3, 2000
19
M?•ffli•i§i•bhl£11@%§4£"1'!•1§1'¥1
This column is devoted to mathematics for fun. What better purpose is there for mathematics? To appear here, a theorem or problem or remark does not need to be profound (but it is allowed to be); it may not be directed only at specialists; it must attract and fascinate. We welcome, encourage, and frequently publish contributions from readers-either new notes, or replies to past columns.
Please send all submissions to the Mathematical Entertainments Editor, Alexander Shen,
Institute for Problems of
Information Transmission, Ermolovoi 1 9, K-51 Moscow GSP-4, 1 01 447 Russia; e-mail:[email protected]
20
Alexan der Shen ,
Editor
Lights Out T
he board for this game is an
mX
n rectangular array of lamps. Each
lamp may be on or off. Each lamp works as a button changing the state (on/off) of the lamp and all its neigh bors. Thus the maximal number of lamps affected by one button is five, the minimal number is three (for the comer button). Initially all lamps are on; the goal is to switch all the lamps off by a succession of button-pushes. I heard about this game about ten years ago from Michael Sipser (MIT), who told me that it is always solvable and there is a very nice proof of this using linear algebra. Recently Prof. Oscar Martfn-Sanches and Cristobal Pareja-Flores wrote an article about this puzzle (to appear; see also their site http://dalila.sip.ucm.es/miembros/ cparejallo), where they provide a de tailed prooffor the 5 X 5-game. (By the way, they have found this puzzle in toy stores!) Here is the solution using linear al gebra. First of all, we may forget about the rectangle; let V be the set of ver tices of an arbitrary undirected graph. Each vertex has a lamp and a button that changes the state of this lamp and all its neighbors. The set of all config urations of lamps forms a linear space over 7L/27L. Each vector is a function of type V ---? {0, 1 }. Here 1/0 means on/off, and vector addition is performed mod ulo 2. The dimension of this space is the number of lamps, i.e., ll'j. For each vertex v we consider a functionfv that equals 1 in the neighborhood of v and 0 elsewhere. We need to prove that the
THE MATHEMATICAL INTELLIGENCER © 2000 SPRINGER-VERLAG NEW YORK
I
function u that is equal to 1 every where can be represented as a linear combination of functions fv . It is enough to show that any linear functional a that maps all fv to zero equals zero on u. Any linear functional a:{O, l }v ---? {0, 1 } can be represented as a(f) = I{.ftv)lv E A ) for some A � V (the sum is computed modulo 2). Therefore the statement can be refor mulated as follows: if A has even-sized intersection with the neighborhood of any vertex v, then IAI is even. To see that this inference holds, consider the restriction of our graph to A. Each ver tex a E A has odd degree in the re stricted graph, but the sum of the de grees of graph A is of course even; therefore the number of vertices of the restricted graph, IA I, is even. We get also the criterion saying whether the state c E { 0, 1 } vis solvable. Here it is: I{c(v)lv E A) 0 for any subset A � V having even-size inter section with the neighborhood of any vertex. (Such a set A can be called "neutral": if we press all buttons in A, all lamps return to their initial state.) One may ask for an "elementary" solu tion; indeed Sipser reports, =
. . . An epilog to the lamp problem. A generalization (which may make the problem easier) appeared in the problem section of American Mathematical Monthly [see problem 10197, vol. 99, no. 2, February 1992, p. 162 and vol. 100, no. 8, Oct. 93, pp. 806--807]. A very sharp new student here (named Marcos Kiwi) found a nice solution to it. Say that the lamps are nodes of a given undirected graph. The button associated with a lamp switches both its state and the state
of all its neighbors. Then we prove that there is a way to switch all states as follows. Use induction on n (the number of nodes of the graph). First say n is even. For each lamp, remove it, and take the inductively given solution on the smaller graph. Replace the lamp and see whether the solution switches it. If yes, then we are done. If no for every lamp, then take the linear sum of all the above solutions given for all the lamps. Every lamp is switched an odd number of times (n - 1), so we are done. If n is odd, then there must be a node of even degree. Do the above procedure for only the nodes in the neighborhood of this node, including itself. In addition press the button of this lamp. This also switches all lamps an odd number of times. Last year this problem appeared on the All-Russia Math Olympiad. One of the participants, Ilia Meszirov, redis covered Kiwi's argument. He also gave an elementary proof (not using linear al gebra) for the statement mentioned above (a state having even-sized inter section with any neutral set is solvable).
W I N N E R OF T H E 1 998 ASSOC IATION OF A M E H I CA N P U B LI S H ERS I)EST N E W TITLE I N MATH EM;\T I C S !
JAMES. KEENER, University of Utah and JAMES SNEYD, University of Michigan
I IIIUitCIU I U U UrU U U I .fiUUt iU
Mathematical Physiology
Mathematical Physiology
James Keener james Sneyd
This book provides an overview of mathematical physiology containing a variety of physio logical models. Mathematical Physiology is divided into rwo parts: the Arst pan is a pedagogical presen tation of some of the basic theory: the second part is devoted to an exten sive discussion of particular physiological sys tems. Mathematical Physiology will be of interest to researchers and graduate students in applied mathematics interested in physiological prob lems. and to quantitative physiologists wishing to know about current and new mathematical techniques. 1 998/785 PP. • 360 ILLUS. HARDCOVERI$69.95 ISBN 0-387-9838 1 -3 I TERDISCIPU ARY APPUEO MATHEMATICS. VOLUME 8
•
I ,I\
Order
Springer
I odayr
1 11l I 1 l \H 1 �)()')
•
( ,1 1\ I ·HIHI·SI'I\1\(,1 1\ \ 1 ...,1 1 tlltp \\\\\\ '-'PilrH�t r
PromoOOn IH274
•
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Revisit the Birth of Mathematics . . . EU C L I D
VOLKER R. REMMERT
M ath emati cal Pu b l i s h i ng i n the Th i rd Re i c h : S pri ng e r-Verl ag an d the Deutsc h e M athe m ati ke r-Vere i n i g u ng
� •
n October 193 7 the Freiburg mathematician Wilhelm Suss (1895-1958) was chosen president of the Deutsche Mathematiker-Vereinigung (DMV) . When the members of the DMV board, Helmut Hasse, Conrad Muller, and Emanuel Sperner, were looking for a suitable candidate in August 1937, they found strong arguments in favour of
Suss. He was known to take interest in DMV affairs and they believed his views coincided with those current at the DMV board, in other words, with their own. Suss had been a pupil of Ludwig Bieberbach (1886-1982), who in the Third Reich propagated an anti-Semitic, racial theory of Deutsche Mathematik and led a group of National Socialist mathe maticians strongly opposed to the DMV. The DMV board hoped that Suss might be able to reconcile his former teacher with the DMV, or at least safeguard it and its pol itics against the threat of political attack from Bieberbach's faction. In addition, Stiss had recently become a member of. the Nazi Party (Nationalsozialistische Deutsche Arbeiterpartei) and was thought to have good relations to the Ministry of Education and Research, which, accord ingly, approved Suss's election in October 1937. Stiss grew to be one of the most influential representa tives of the German mathematical community in the Third Reich, and collaborated closely with Nazi authorities partic ularly in the Ministry of Education and Research. In the course of this collaboration the DMV's professional policy became closely entangled with issues that stood at the very core of the Nazi state, notably its anti-Semitism and its anti internationalism. The Ministry of Education and Research was dedicated to transmitting these values to the sphere of
22
THE MATHEMATICAL INTELLIGENCER © 2000 SPRINGER-VERLAG NEW YORK
the sciences. The collaboration of the DMV board and espe cially of Stiss in this program, which in its consequences lay beyond their control, was a precondition of their influence and successful professional policies during the war [10]. During his DMV presidency, which lasted until 1945, Suss was repeatedly at odds with Springer-Verlag. Springer was amongst the leading scientific publishers in Germany, and their mathematics branch enjoyed a worldwide repu tation. After World War I Springer had enrolled outstand ing mathematicians to launch a first-class publication pro gram in mathematics. Felix Klein, David Hilbert, and Richard Courant stood for the Gottingen influence in this enterprise. Springer's yellow books were soon known in the most remote places and often alluded to as the yellow plague, which no mathematician could escape. The many instances of conflict between Suss, as the representative of the DMV, and Ferdinand Springer and his main mathe matical adviser Friedrich Karl Schmidt (1901-1977), can serve well to illustrate how the independence of scientific publishing was shaken under National Socialism. Siiss and the Mathematische Zeitschrift
On March 3, 1938, Suss, as president of the DMV, went to Berlin to meet Dr. Dames, who was responsible for math-
matical community. Suss reports that when he called Dames's attention to the role of Jewish mathematicians in mathematical publishing, he gave a "personal opinion" about the
situation
in
the
editorial
boards
of the
Mathematische Annalen and the Mathematische Zeit schrift (MZ), both published by Springer-Verlag. He stated that German journals should not be represented by Jews any more, although Jews should still be allowed to publish, and he hoped that his personal stance on this matter was clear to Dames. Dames, for his part, promised to put pres sure on Springer, so that authors would no longer be com pelled to negotiate with Jewish editors. Although this was what he said in the circular to Hasse, Mi.iller, and Sperner, Stiss had evidently been more explicit in his conversation with Dames. When he wrote to Dames a few days later, he came back to problems, as he saw them, of the editorial organisation in Springer-Verlag. He de scribed the arrangements for Richard Courant's "yellow se ries" Grundlehren der mathematischen Wissenschaften in EinzeldarsteUungen, which had F. K. Schmidt as general editor and the Jewish emigre Courant as editor for the "Anglo-Saxon domain. " He saw similar arrangements being developed for the
Mathematische Annalen,
whose manag
ing editor, Otto Blumenthal, was Jewish. According to Suss, Springer was looking for a co-editor in England. Suss's opin ion was that in the interest of the "German reputation," all means should be used to prevent foreigners having influ Wilhelm Siiss in Oberwolfach.
ence on "this leading j ournal founded by our champion Felix
ematics in the Ministry of Education and Research. Suss
pointed to the case of the Zentralblattfiir Mathematik und
reported on this meeting in a circular to the members of
ihre Grenzgebiete,
the DMV board, Hasse, Mi.iller, and Sperner. 1 At first Suss
Copenhagen by the emigre Otto Neugebauer.
Klein."2
As an example
of what was already happening, he
which was now being managed from
and Dames discussed the organization of the annual con
Concerning the MZ, according to Suss, things were bet
ference of the DMV to be held in September 1938 in Baden
ter. But he reminded Dames that he had asked him and the
Baden. Then they talked about the more politically rele
ministry, "to oblige the publisher, to remove Professor Issai
vant question of how to handle Jewish and emigre DMV
Schur from the
members (the "Judenfrage") and how to restrict the influ
he himself would raise the matter with MZ's managing ed
ence of Jewish mathematicians in mathematical publish
itor, Konrad Knopp.
MZ board. "3 He
continued by adding that
ing. The Ministry of Education and Research did not di
On March 1, 1938, two days before his meeting with
rectly dictate the agenda or any rules of conduct for dealing
Dames, Knopp had written to Stiss, inviting him to join the
with these issues. Rather Stiss, supported by the DMV
MZ
board, took the lead in shaping DMV policies on the
throws a somewhat ambivalent light on Stiss's strivings to
"Judenfrage." Amongst his motives was the fear that the
Aryanise Springer's editorial boards.4 There had been an
advisory board (wissenschajtlicher
Beirat),
which
position of the DMV as an instrument of professional lob
earlier offer from Knopp, but Suss had declined because
bying would lose ground against Ludwig Bieberbach, par
there were two Jewish members on the MZ board, Edmund
ticularly as Bieberbach and his a
National
Socialist
faction had planned
Mathematicians
League
(NS
Landau and the aforementioned Issai Schur (1875-1941), who was a co-founder of the MZ in 1918, along with Knopp,
Mathematikerbund) and had explicitly pointed to the prob
Leon Lichtenstein, and Erhard Schmidt. Edmund Landau
lem of Jewish and emigre DMV members to legitimise their
had died in February 1938, which changed the situation,
plan, threatening in effect a schism in the German mathe-
and, as Knopp explained to Stiss, this induced him to renew
1March 9, 1 938, University Archives Freiburg (UAF), E4/46. For the following ct. [1 0] ; on Bieberbach [6]. I gratefully acknowledge the kind support of the Volkswagen Stiftung and the friendly cooperation of the Springer Archives at Heidelberg and the University Archives Freiburg. Thanks for their help and comments go to Stefan Hanke, Otto H. Kegel, Ben Kern, and Reinhold Remmert. 2Actually the Annalen had been founded by Alfred Clebsch and Carl Neumann in 1 868; Klein only joined the editorial board in 1 873.
3Suss to Dames, March 15, 1 938, UAF, E 4/75: Fur Sie wiederhole ich hier nur meine Bitte, den Verlag zu veranlassen, aus der Redaktion Prof. lssai Schur zu ent femen. 4For the following ct. the correspondence Suss-Knapp in March and April 1 938, UAF, C 89/318.
VOLUME 22, NUMBER 3, 2000
23
the invitation. It is open to speculation whether Suss had already read Knopp's second invitation by March 3 when he spoke to Dames demanding that Schur be removed from the MZ board, but he did not mention the question of Schur in his report to Hasse, Milller, and Sperner at the DMV, nor in his subsequent reply to Knopp. In this reply, he gives an account of his conversation with Dames; but it is a differ ent version from that given in his report to Hasse, Milller, and Sperner, and from that in his earlier letter to Dames. To Knopp he implies that it was the Ministry of Education and Research that had taken the lead concerning the role of Jewish mathematicians in mathematical publishing, and he says that he understood from the Ministry that they would see to Schur's expulsion from the MZ board. He did not, however, mention that it was he himself who had ex plicitly demanded this course of action. His main concern, as portrayed in the letter to Knopp, was the possible re striction of his freedom to implement DMV policies con cerning Springer if he joined the MZ board. Knopp reas sured him that this would not be the case, and regarding Schur, he would welcome the ministry taking a hand in the matter, as he too did not consider it practicable any more for Jews to participate officially in mathematical journals.5 In April the Reichsschrifttumskammer, a division of Goebbels's Ministry of Propaganda, whose function was to control writers and publishers in Germany, demanded to know from Springer why there were still Jewish editors on the boards of the Mathematische Annalen and the MZ. The Reichsschrifttumskammer referred specifically to an in quiry made by the Ministry of Education and Research. By the end of April, it was clear that Schur would have to leave the MZ board. When Knopp reported this to Suss, the in vitation to join the MZ board was finally accepted. Schur's name did not appear on the title-page of the MZ in 1939, and he emigrated in the same year. But this was not the only pressure put on Schur. In late March 1938 Bieberbach in the Prussian Academy of Sciences had found "it surprising that Jews are still mem bers of academic commissions," meaning Schur. The math ematician Theodor Vahlen (1869-1945), a long-standing Nazi and close ally of Bieberbach, had asked for a change, and Max Planck had promised to take care of the matter. Within a week Schur had resigned from the commissions [12, p. 122]. Zentralblatt and the Schmidt affair
Stiss's report on the relations between Jewish mathemati cians and Springer-Verlag, and his denunciation of Issai Schur in March 1938, were not his only attempts to inter-
vene in mathematical publishing and to attack Springer in particular. Springer had entered the scene of mathematical re viewing in Germany in 1931 by publishing the Zentralblatt under the auspices of Otto Neugebauer and Richard Courant. From the beginning the Zentralblatt stood in di rect competition with the traditional Jahrbuch iiber die Fortschritte der Mathematik published by the Prussian Academy of Sciences in Berlin. The Jahrbuch was notori ous for its constant delays in reviewing, whereas the Zentralblatt was soon known to be more efficient. By 1939 the Jahrbuch was having to cope with the constant ideo logical interventions of Ludwig Bieberbach, who had in stalled himself as spokesman of the Academy's Jahrbuch commission. The Zentralblatt also had problems with Nazi racial and nationalistic policies. It had non-Aryan members on the editorial board: the Italian mathematician Tullio Levi-Civita for example, who had to be expelled in October 1938. Its managing editor, the emigre Otto Neugebauer, re signed in November 1938 as a result of the Levi-Civita af fair. In addition, the journal was explicitly international in character; and Schmidt had reported to Springer in December 1937 that collaboration with the Zentralblatt had been held against a mathematician in negotiations about a professorship because of this.6 Given these cir cumstances, it might have seemed reasonable for the two journals to have formed some kind of alliance, if it had not been for the economic competition between the respective publishing houses, de Gruyter and Springer, and the ideo logical incompatibility between Bieberbach at the Jahrbuch commission and Springer. By the end of the thir ties, however, a fusion or at least a co-operation of the Zentralblatt and the Jahrbuch was under discussion. In late 1938 the news spread in the German mathemat ical community that a new journal, the Mathematical Reviews, was about to be founded by the American Mathematical Society in the United States. Naturally this caused a stir among German mathematicians and publish ers, Nazi or not. In January 1939 Bieberbach urged de Gruyter and Springer to consider a fusion and even made detailed suggestions as to the procedure. Also in early 1939, the DMV, that is to say its president Suss, tried to put di rect pressure on de Gruyter and Springer in order to in duce them to fuse [12, pp. 167ff], [9, pp. 327-333]. It has been argued that Springer considered the idea of a fusion in January 1939 and still as late as May 1939 [12, p. 168-170]. But Springer definitely had plans of his own, which were to discuss the situation with the Americans first and, if possible, to bring about a co-operation between
5Knopp to Suss, March 1 6, 1 938, UAF, C 89/318: Nur fUr Sie und ganz vertraulich lUge ich hinzu, daB ich im Grunde froh ware, wenn es endlich zu der vom Ministerium angestrebten Regelung kommen wurde. Denn auch ich halte es kaum mehr fUr angangig, daB bei mathematischen Zeitschriften - und allein bei diesen, soviel ich weiB-in der jetzigen Zeit noch Juden offiziell mitwirken. lch weiB auch gar nicht, warum das Ministerium hier eine Angstlichkeit zeigt, die doch sonst nicht Sache der heutigen Regierung ist. Aber ich selbst kann und will Schur gegenuber nicht die Initiative ergreifen. 6Schmidt to Springer, December 26, 1 937, Springer Archives, C1 1 64, Wintner: Zum SchluB noch eine streng vertrauliche Mitteilung. AnlaBiich der Wiederbesetzung von Lehrstlihlen ist zwei aussichtsreichen Kandidaten die Mitarbeit am Zentralblatt f. Mechanik bez. am Zentralblatt f. Mathematik zum Vorwurf gemacht worden.
Beim Zentralblatt f. Mechanik wurde der Vorwurf auf die Beteiligung von Karman gegrundet, beim Zentralblatt f. Mathematik auf dessen angeblichen internationalen Charakter. Der erste der beiden Herren hat den fraglichen Lehrstuhl nicht erhalten; im zweiten Fall steht die Entscheidung noch aus. Die geruchteweise Verbreitung dieser Vorwurfe, die tatsachlich gefolgt zu sein scheinen, droht natlirlich die jungen Mathematiker von der Mitarbeit abzuschrecken.
24
THE MATHEMATICAL INTELLIGENCER
the Zentralblatt and the Mathematical Reviews. Ferdinand Springer discussed the founding of the Mathematical Reviews with Oswald Veblen in December 1938 and pro posed to send his main mathematical adviser, Schmidt, to the United States as a spokesman for his interests in math ematical reviewing [9, p. 331]. Suss learned about this in March and immediately pressed Dames in the Ministry of Education and Research to refuse Schmidt permission to travel, so long as Springer left Suss in the dark about his motives. 7 Suss also suggested to Dames that both his su perior in the ministry and the Reichsschrifttumskammer be informed of Springer's plans. When Suss found out in April that Springer had procured the Ministry's permission for Schmidt to go to the United States, he wrote to Dames's superior, Ministerialrat Kummer, and again strongly op posed Schmidt's journey. He pointed to the competition be tween the Jahrbuch and the Zentralblatt and the DMV's in terest in their fusion. Stiss expressed his fear that Springer's contacts with the Mathematical Reviews would run contrary to this idea, as they actually did. To strengthen his argument, he characterized the Zentralblatt as an or ganization of "a group of Jewish mathematicians and their friends" and emphasized that Springer still had close ties to Jewish emigrants, especially to Richard Courant. Therefore it was to be doubted that "the German cause would be represented at all in America by a spokesman of Springer." Finally he suggested that Schmidt's travel per mission should be revoked immediately as he intended to leave for the States the following week.8 Two days later, on April 29, Suss phoned Ministerialrat Kummer in Berlin to inquire how things stood. When Kummer informed him that Schmidt had already left, Suss told him that to his knowledge Schmidt was only on his way to Bremen to board the ship, which was due from America on May 1 or 2, implying that Schmidt could still be stopped. Kummer did not take this up, but explained that his superior in the Ministry of Education and Research had definitely decided to let Schmidt go, as he was not only to discuss Zentralblatt matters, but also to evaluate the at mosphere among American mathematicians and if possi ble to soften their minds. At this point Suss lost his tem per and berated Kummer that this was an unsuitable job for Schmidt and that the ministry would have done better to consult an authoritative source before allowing Schmidt to go on this mission, because to send this particular man was asking for trouble.9 After his return from the United States Schmidt con vinced the Ministry of Education and Research that the col laboration of at least some of the American mathematicians could only be secured if the fusion of the Zentralblatt and
the Jahrbuch were prevented. He argued that the fusion would nourish anti-German propaganda and promote the foundation of the Mathematical Reviews. Apparently the ministry was impressed by Schmidt's report, as it wrote to Springer in late July 1939 that the fusion of the Zentralblatt and the Jahrbuch was not considered "advisable," taking into account that the situation of German scientific litera ture outside Germany should not be made more difficult than it already was. 10 Whatever Schmidt's activities in the United States, the committee of the American Mathematical Society decided in May 1939 to commission Veblen to launch and supervise the Mathematical Reviews [9, 332f]. Word of this soon reached the German mathematical community and put Schmidt, who had reported that the foundation of the Mathematical Reviews could still be prevented, in an awk ward position. In a letter of September 22-three weeks after the declarations of war-Suss, who had just learned about the ministry's decision of July which ran contrary to his own plans, accused Schmidt of having given false in formation on his return from the United States. He ex plained that the foundation of the Mathematical Reviews and the outbreak of the war had considerably changed the situation. As a consequence he no longer saw the neces sity to consider the American and most of the foreign coun tries' positions with regard to the fusion of the Zentralblatt and the Jahrbuch. 1 1 One day later, o n September 23, Stiss gave Springer and de Gruyter an ultimatum to consider the fusion [12, p. 223]. Stiss's course of action was fully approved of by Bieberbach and Vahlen. Bieberbach wrote to Suss that the Jahrbuch commission of the Prussian Academy of Sciences had de cided to publish a preprint journal which was to include the most recent reviews in order to find a remedy against the Jahrbuch's slowness in reviewing. De Gruyter had con sented to this plan, but the war and the ensuing shortage of paper had stalled it. Springer, on the other hand, had not given his consent. His aversion against a fusion had been backed by the Ministry's letter from July, but, according to Bieberbach, had been known long before. The war having changed the circumstances, Bieberbach now proposed to pursue further the idea of a fusion with the committed Nazi Harald Geppert as managing editor, and to add the name of the Zentralblatt to the title of the preprint journal. 12 Springer for his part replied to Suss's ultimatum, which had set October 3 as a deadline, on October 4. He explained that he could not respond to inquiries which were com bined with an ultimatum; and that Suss's letter indicated that he had insufficient knowledge of the publishing busi ness. He declined to discuss the fusion as long as he did
7Excerpt from a letter of Suss to Dames, March 24, 1 939, UAF, C89/36. 8Excerpt from a letter of Suss to Kummer, April 27, 1 939, UAF, C89/36. 9Suss to Hasse, May 1 , 1 939, UAF, C89/36. 10Ministry of Education and Research to Springer, July 27, 1 939, Springer Archives, ZS 29; this interpretation of the Ministry's decision disagrees with [ 1 2 , p. 1 70fj. 1 1 Suss to Schmidt, September 22, 1 939, UAF, E4/78. 12Bieberbach to Suss, September 25, 1 939, UAF, E4/78.
VOLUME 22, NUMBER 3, 2000
25
not know whether the Ministry's position had changed alluding to the July letter. 13 When Suss wanted to talk about the Zentralblatt and the Jahrbuch in the Ministry of Education and Research in Berlin on November 3, he was informed that the Foreign Ministry had forbidden all fusions of scientific journals dur ing the war as a propagandistic means to keep up the num ber of scientific publications. 1 4 Thus the actual fusion of Zentralblatt and Jahrbuch was out of the question, in spite of the new situation after the outbreak of the war and the launching of the Mathematical Reviews. However, it was decided to bring about a collaboration of Zentralblatt and Jahrbuch along the lines Bieberbach had proposed, and which Suss had also discussed with Geppert. The review ing principle was to be speed with the Zentralblatt and completeness with the Jahrbuch. On November 15, Bieberbach, Geppert, Schmidt, Springer, and representa tives of de Gruyter met in Berlin to agree on the reorgani zation of the Zentralblatt and the Jahrbuch under a joint editorial office (Generalredaktion) in Berlin with Geppert as managing editor [12, pp. 224-226], which came near to the fusion that the DMV, Suss, and Bieberbach had wanted. Reorganization of Mathematical Journals
Springer's independence as a publisher had not only been threatened by the Zentralblatt and the Schmidt affairs, but also by Suss's intentions to reorganise the system of math ematical journals as a whole. After the Nazis' rise to power, there had been discussions about reducing the number of scientific journals in order to put an end to fragmentation as the Nazis saw it, but nothing specific had been done [3, p. 418]. In particular, the physi cist and Nobel prize winner (1919) Johannes Stark, who had joined the Nazi party in 1930 and stood for the German Physics movement along with Philipp Lenard, had in vain called for a reorganization of the scientific literature in physics under a joint editorial office (Neuordnung des physikalischen Schrijttums) in autumn 1933 [11, 329-331]. Probably in spring 1939, Bieberbach drafted a detailed proposal, highly reminiscent of Stark's 1933 plans, of how to reorganize the systems of scientific journals, a copy of which he sent to Suss. 15 He chose mathematical journals as an example to illustrate his ideas. He deplored, what had earlier been conceived of as fragmentation, namely that ar ticles belonging to specific fields of mathematics were scat-
tered in more than half a dozen journals, which was nei ther effective for the scientists interested in the field nor for the editorial offices. In addition this had the negative economic effect that personal subscriptions were rare. Bieberbach's proposal of reorganization, which by the way included Zentralblatt and Jahrbuch, culminated in the idea to centrally supervise the journals, with the DMV as a su pervisor. Siiss was in favour of Bieberbach's initiative. When he met Ministerialrat Kummer in the Ministry of Education and Research in November 1939 to discuss Zentralblatt and Jahrbuch, he also talked about the possible restructuring of the production of mathematical journals. 16 N otwith standing the Foreign Ministry's order against fusions of sci entific journals, Kummer and Suss discussed a new or ganising principle, that mathematical journals should be specialised. This would have put an end to the traditional journals of broad mathematical variety, for example the
Journal filr die reine und angewandte Mathematik (Grelle's Journal), the Mathematische Annalen and the Mathematische Zeitschrift. Bieberbach had suggested that Grelle's Journal specialize in algebra and number theory, Mathematische Annalen in analysis, and Mathema-tische Zeitschrift in geometry. Suss had immediately pursued this
idea and decided to negotiate with the managing editors. 1 7 Hasse, the managing editor of Grelle's Journal, felt sym pathetic towards Bieberbach's idea, and thought it "good and healthy," though "a bit stormy, as always with Bieberbach." But, apparently stalling for time, he pointed out that the plans would be "difficult to realise practi cally." 18 Suss also discussed the fate of the Mathematische Annalen with its managing editor Heinrich Behnke. Behnke, however, was not very enthusiastic about Bieberbach's and Suss's plans, although he professed readi ness to "theoretically discuss a desirable reorganization of the German mathematical journals." He reminded Suss that Erich Heeke and B. L. van der Waerden were his seniors on the Annalen board and that they would never consent to Suss's plans. DMV board member Emanuel Spemer per sonally explained the plans to Springer, who flatly refused to discuss the envisaged reorganization and any interven tion with his independence as a publisher. 19 However, whatever these aspirations, eventually they were to be thwarted during the course of the war.
13Ferdinand Springer to Si.iss, October 4 , 1 939, UAF, E4/68: 1 .) lch bin grundsatzlich nicht in der Lage, auf Anfragen naher einzugehen, die mit einer ultimativen Fristsetzung verkni.ipft sind. 2.) lch bin solange nicht in der Lage, zu der Frage der Vereinigung der beiden Organe Stellung zu nehmen, als ich nicht weiss, ob das Ministerium seinen bisherigen Standpunkt aufrechterhalt oder andert. lch darf Ihnen anheimgeben, hieri.iber zunachst Klarheit zu schaffen. 3.) Ein weiterer Hinderungsgrund fi.ir mich, auf lhr Schreiben naher einzugehen, ist die Tatsache, daB es nach Form und lnhalt auf mangelnde Kenntnis der Stellung des wissenschaftlichen Verlegers im allgemeinen, der Aufgaben und Pfiichten im speziellen schlieBen laBt, die im dritten Reich dem deutschen Verleger auferlegt sind. 1 4Si.iss's report to Hasse, Muller, and Sperner, November 1 1 , 1 939, UAF, E4/68. 1s"Vorschlage zu einer Planung auf dem Gebiete der wissenschaftlichen Zeitschriften," UAF, E4/78, probably April 1 939. 1 6Cf. Si.iss to Kummer, May 28, 1 940, UAF, E4/45. 1 7Si.iss to Sperner, December 1 4, 1 939, UAF, E4/76. 1 BHasse to Si.iss, December 8, 1 939, UAF, E4/76: lm Grossen und Ganzen Iinde ich die Bieberbachschen Vorschlage gut und gesund. Sie scheinen mir nur, wie alles was von B. ausgeht, etwas sti.irmisch, und es di.irfte schwierig sein, sie in die Praxis umzusetzen. 19Behnke to Si.iss, February 1 7, 1 940; Si.iss to Behnke, February 27, 1 940, UAF, C89/42.
26
THE MATHEMATICAL INTELLIGENCER
Much more could be said about Suss's and the DMV's ideas on mathematical publishing and mathematical re viewing, and the ideological background, but it is clear that they were not content to hold these ideas merely as points of view, but decided to pursue them actively in order to gain control over mathematical reviewing and publishing. Against this, Springer Verlag was in an awkward position because of its close ties to Jewish mathematicians and the international mathematical community, and the fact that Ferdinand Springer himself had Jewish ancestors [ 1 1 ] . Therefore the DMV and Stiss could not only openly oppose Springer's policies and his representative Schmidt, but in doing so could take recourse to what the regime offered, as shown for example by Suss's behavior concerning Springer's editorial organization, his denunciation of Schur in 1938, and his attempt (in this case unsuccessful) to stop Schmidt on his way to America after he had already left for Bremen to board his ship. The DMV's professional policies had in fact become closely entangled with issues that stood at the very core of the Nazi state: its anti-Semitism, its anti-internationalism, and its striving for autarky. The objective of the Ministry of Education and Research was to transmit these issues to the whole sphere of the sciences. Although its ultimate con sequences lay beyond their control, the collaboration of the DMV board and especially of Suss in this program was re warded by their influence and the success of their policies during the war [10]. Conflicts in World War II
In late 1941 the physicist Dr. Johannes Rasch sent two memoranda to the Reich Research Council (Reichs forschungsrat), a government office charged with the or ganization of scientific research in Germany. Rasch, who worked as an engineer with the Siemens & Halske com pany, complained about the lack of mathematical refer ence-works for the use of physicists and engineers in in dustry. He pointed out the better situation in other countries, especially in the United States [7, pp. 1 15f] . Rasch's memoranda were reacted to quickly, and in early 1942 the Reich Research Council initiated a program to pro cure important mathematical reference-works and litera ture for the interested parties. Most of these works were to be obtained by specially commissioning mathematicians to produce them, and the publication program was en trusted to Suss. In the preceding years he had repeatedly, but always in vain, tried to get the Reich Research Council to be more interested in mathematics, and in particular to found a special department for mathematics, which was then only represented in the Council via the department for physics. Rasch's initiative had provided a welcome op portunity to bring about a "practical liaison of the Reich Research Council and the DMV," and naturally Suss sought to profit from this sudden chance "in the interest of the sta tus of mathematics. "20
Gustav Doetsch (middle, in uniform).
But although Suss had an official task from the Reich Research Council, he had not yet obtained sufficient fund ing to carry it out. He tried to interest Hermann Goring's powerful Aviation Ministry and its resources in the pro gram, but problems arose. In the Forschungsfiihrung of the Aviation Ministry, questions pertaining to mathematics were under the charge of the Freiburg mathematician Gustav Doetsch (1892-1977). Doetsch, who worked closely to the demands of engineering, especially as formulated by the aviation industry, had already started a similar publi cation program, albeit on a smaller scale, and he had him self worked on a book on Laplace Transforms [2]. Doetsch and Suss had seriously quarrelled in the early years of the war, and it was extremely difficult for them to cooperate, even on matters of obvious importance [ 10]. Notwithstand ing, they met to discuss their respective ideas in September 1942. In this meeting Suss announced that he had now raised his funds from the recently reorganized Reich Research Council and he could therefore follow his own plans. Nonetheless, they agreed at least to coordinate what they were doing, and in that sense their activities coexisted during the remaining years of the war. Suss's program was clearly the more ambitious one and, in terms of the num ber of projects and monographs printed or ready to print by the end of the war, also the more successful [7, p. 1 15]. The rivalry of Doetsch and Suss in mathematical pub lishing was reflected by their choice of publishers. Doetsch intended to collaborate with his own publisher Springer, whereas Suss, following a suggestion of Behnke, began
20Suss to Behnke, Feigl, Hamel, and Sperner, February 25, 1 942, UAF, C89/1 9: Zum ersten Mal kommt jetzt eine praktische Verbindung des Reichsforschungsrates mit der DMV anscheinend zustande, die ich natUrlich gern im Interesse der Stellung der Mathematik ausnutzen miichte.
VOLUME 22, NUMBER 3 , 2000
27
working with the Akademische Verlagsgesellschaft in
formed the impression that Suss's projects would be of lit
Leipzig, although Georg Feigl
tle relevance after the war. Therefore Schmidt again pro
(1890-1945) had begged him
to negotiate also with Springer. Doetsch on his part, vis
posed that Springer should concentrate on plans for the
ited Springer in October
post-war years in order to safeguard its position in the long
1942,
and discovered that some of
Springer's projects would fit perfectly into his own pro gram: a formulary by Wilhelm Magnus, books on elliptic
run.24 Suss's various attempts to interfere with the system of
functions by Wilhelm Magnus, on conformal mappings by
mathematical publishing since
Albert Betz, on developments by real functions by Georg
trying to bring about a planned economy. The DMV was to
1938
were somehow like
Feigl and Erhard Schmidt, and a table of integrals by
become the absolute center of all professional influence in
Walther Meyer zur Capellen. Magnus's formulary appeared
mathematics. He made this perfectly clear in a letter to
in
1943 [4], and Springer eventually published the books by
Georg Feigl in April
1941:
"I have the imperialistic goal to
Betz, Magnus, and Meyer zur Capellen some years after the
gain exclusively for the DMV all rights and responsibilities
In the meantime, Suss included the mono
for mathematics."25 Naturally this ambitious goal was not
war
[ 1 ] , [5], [8] .
graphs of Magnus
[5] and of Feigl/Schmidt in a list of works commissioned in April 1944. The latter was never pub
compatible with
he
Aviation Ministry. But the course the publication program
lished.
Doetsch's
influential position in the
took was an unmistakable sign of Doetsch's deteriorating
Springer's mathematical adviser, Schmidt, was well aware of the competition between Doetsch and Suss, and
power base February
and
1944
Siiss's seemingly ever rising star. In
Suss even became official "censor for math
was partisan in his views of them. Suss kept the tactics of
ematical publications" for Goebbels's Ministry of Propaganda
his far-reaching plans for mathematical publishing strictly
and Speer's Ministry of Armaments.26 This meant that all
to himself, which made Schmidt anxious, especially with
applications to put mathematical works in print required
regard to the independence of Springer's publishing poli
his approval, which increased his influence in mathemati
cies. Aside from these fears, Schmidt considered Doetsch
cal publishing even more.
more factual and businesslike than Suss, so in any case, if there was a choice, he believed him to be a better partner
Springer and Siiss After the War
for Springer. 21 But Schmidt clearly recognised that Suss
In June
was in a strong position and that Doetsch would need
to Schmidt that Suss had inquired about Feigl's book In
Springer's support against him.22
particular he was eager to know whether she had a con
In early
1943
Doetsch's influence in the
Forschungs
1946
Maria Feigl, the widow of Georg Feigl, wrote
tract with Springer. Suss told her that he himself was about
fiihrung drastically diminished, and Suss obtained a prac
to publish a series of monographs, in which Feigl's book
tical monopoly on the commissioning of mathematical
could well be included. This series materialized as Studia
monographs
Mathematica,
[ 10].
Springer saw the danger that their pre
dominance in the field of mathematics might collapse, es
published by Vandenhoeck
&
Ruprecht in
Gottingen.
pecially because simultaneous work by mathematicians on
Schmidt enclosed a copy of Maria Feigl's letter when he
the particular topics was virtually impossible during the
wrote to Springer in July, asking him to send a letter of rec
war, which ruled out the possibility of direct competition.
ommendation to the University of Miinster where Schmidt
Therefore Schmidt was to negotiate with potential post
was being considered for a professorship. In the new post
war authors regardless of Siiss's activities.23 By September,
war political situation, it was obviously important that his
Schmidt became more optimistic about Springer's future.
attitude during the Nazi period be made clear. He suggested
The Aviation Ministry had accepted three projected mono
that Springer mention that he had been known to cooper
graphs of Walter Brode!, Gerhard Damkohler, and Eberhard
ate with Jewish mathematicians as late as the end of
Hopf as being important to the war effort, which allowed
and that his journey to the United States in May
the authors to start working on them. When Suss had re
been heavily opposed. 27 In his letter of recommendation to
ported on his publication program at the DMV meeting in
the University of Munster, Ferdinand Springer, who had
1938, 1939 had
Wiirzburg, it seemed clear that Suss was to work with the
been exasperated with Suss's attempt to lure Feigl's book
Akademische
away from Springer-Verlag, followed Schmidt's outline, but
Verlagsgesellschaft
exclusively.
Schmidt
2 1 Schmidt to Springer, October 30, 1 942, Springer Archives. C1 52, Doetsch: Die Plane von Suss, die er streng geheim hii.lt, scheinen sehr weit zu gehen. Ob sich mit seinen Absichten das Bestehen selbstii.ndiger Sammlungen noch vertragt, vermag ich nicht zu sagen. Doetsch scheint mir der sachlichere. 22Schmidt to Springer, November 3, 1 942, Springer Archives, C1 52, Doetsch. 23Memorandum of a conference at Springer Verlag on February 9, 1 942, Springer Archives, C408, Hasse: Professor Schmidt berichtet uber die Schwierigkeiten, die durch die Aktion von Suss entstanden sind. Dadurch, dass das Gesamtgebiet der Mathematik im Hinblick auf die Kriegswichtigkeit aufgeteilt und zur Bearbeitung an verschiedene Autoren abgegeben wurde, besteht die Gefahr, dass ein Einbruch in die Vorherrschaft Springer's auf dem Gebiete der Mathematik entsteht. [ . . . ] Es wird empfohlen, dass Professor Schmidt, ohne Rucksicht auf Suss, diejenigen Autoren mit Ablieferungstermin nach dem Kriege verpflichtet, die ihm am besten zu sein scheinen. 24Schmidt to Rosbaud, September 1 7, 1 943, Springer Archives, C707, Magnus. The books of Brbdel, Damkbhler and Hopf never materialized in print. 25Suss to Feigl, April 3, 1 941 , UAF, C89/51 : Fur die DMV habe ich das imperialistische Ziel, ihr allein aile Rechte und Pflichten fOr die Mathematik zu verschaffen. 26Suss to Heisig of the Teubner publishing house, February 3, 1 944, UAF, C89/21 . 27Schmidt to Springer, July 22, 1 946, Springer Archives, C931 , Schmidt; Maria Feigl to Schmidt, June 20, 1 946, Springer Archives, C230, Feigl.
28
THE MATHEMATICAL INTELLIGENCER
whereas Schmidt did not mention Si.iss as an interested
A U T H O R
party, Springer did.28 In September the story reached the rector of Freiburg University,
who
immediately
demanded
details
from
Springer,29 as Si.iss was held in high esteem in Freiburg af ter the war and had only been suspended from his profes sorship for two months in summer
1945.
In spite of the
many "good deeds" credited to Si.iss during the denazifica tion, it was particularly important to him, naturally enough,
r
that the scope of his collaboration with the Nazis not be generally known, either in Freiburg or within the German mathematical community, even if most of those who may
VOLKER R. REMMERT
have known about these things, had no interest in dragging them to light
AG Geschichte der Naturwissenschaften
[ 10].
Springer replied to Arthur Allgeier, the rector of Freiburg i University, in October, accusing S .iss of having tried to have Schmidt arrested in
1939
FB 1 7 - Mathemati k Johannes Gutenberg-Universitat Mainz
on his way to the United States
D-55099 Mainz
and of having denounced Springer in the Ministry of
Germany
Education and Research for his contacts with Jewish emi
e-mail: [email protected]
grants, especially with Richard Courant. Schmidt also wrote to Allgeier, saying that when Springer's partner, Tonjes
Volker Remmert was trained as a mathematician and as a his
Lange, tried to secure his travel permission from the Ministry
torian. In 1 997 he received a Ph.D. in early modem history
of Education and Research, Ministerialrat Kummer told him
from Freiburg University for worlk on Galilee and the mathe
that Si.iss strongly opposed his journey because he still held
matical sciences in the seventeenth century. Apart from the
close ties to Jewish emigrants, and that Si.iss had suggested
history of mathematics in the first half of the twentieth cen
that he himself should go to the United States instead.30 i S .iss denied all of the charges Springer and Schmidt
tury, his main research interests are in the history of early mod ern European science and culture.
brought against him. He was questioned by the Freiburg Committee of Denazification
(Selbstreinigungsausschuj5)
and completely exonerated.31 He testified that he had not
a suitable publisher, his friends pressed him to negotiate
mentioned Springer in the Ministry of Education and
with Springer. Si.iss evidently preferred not to collaborate
Research at all, prior to the discussion of the fusion of the
with Springer. The new journal eventually came out in
Jahrbuch
under the title Archiv
and the
Zentralblatt
in late
1938.
And further,
Si.iss characterized Kummer as an ally of Springer and ac
Verlag Braun in
cordingly as an unreliable witness, without giving any
Birkhauser in Basel in
der Mathematik,
Karlsruhe,
1948
published by the
and was taken
over by
1952.
proof. Naturally he did not volunteer information about his meeting with Dames or his denunciation of Schur in March
Concluding Remarks
1938.
The history of mathematics and mathematicians in Nazi
Springer was notified by letter in late January that Si.iss
Germany is often highlighted as a history of extremes, ex
had been e�onerated. Enclosed with the letter was a copy
emplified by Bieberbach and Deutsche
Mathematik or the
of the report of the Committee of Denazification, which
abolition of the Gottingen mathematical tradition. But
suggested that Springer himself had distorted the facts,
these phenomena, however depressing, were only those of
which Springer firmly repudiated. But nothing came of his
high visibility. The threats to Springer's independence in
protest.32
scientific publishing on the other hand, could not generally
In the post-war years, relations between Springer and
be seen by the public, nor did they result from official di
1946 Si.iss
rectives. Rather they stemmed from everyday collaboration
pursued the plan to publish a new mathematical journal.
with Nazi party and government officials. This collabora
Si.iss showed no sign of improvement. In summer
Mathematisches
tion, however motivated, was essential to the purposes and
Forschungsinstitut Oberwolfach, which Si.iss had founded
functioning of the Nazi bureaucracy. From this point of
The journal was to be edited by the
in late 1944. Schmidt and Springer considered the new jour
view the story of the conflicts between the Springer-Verlag
MZ. When Si.iss looked for
and Si.iss and his colleagues at the DMV is not just a curi-
nal an overt competitor for the
28Springer to Schmidt, July 30, 1 946, Springer to the Dean of the Faculty of Mathematics and Natural Sciences of MOnster University, July 30, 1 946, Springer Archives, C931 , Schmidt. 29Rector Allgeier to Springer, September 30, 1 946, Springer Archives, C1 039, Suss; most of the correspondence is also to be found in UAF, 834/74. 30Springer to Allgeier, October 2 1 , 1 946; Schmidt to Allgeier, December 6, 1 946, Springer Archives, C1 039, Suss. 31 Report of the committee to von Dietze, December 1 1 , 1 946, Springer Archives, C1 039, SOss. 32Correspondence von Dietze-Springer, January to May 1 947, Springer Archives, C1 039, Suss; cf. UAF, 824/391 9.
VOLUME 22, NUMBER 3, 2000
29
ous collection of passing historical details, but illustrates how mathematicians' professional policies and Nazi poli cies actually interacted.
[7] Mehrtens, H . : Mathematics and War: Germany 1900-1945, in:
Sanchez-Ron, J . M./Forman, P. (eds.): National Military Establish
ments and the Advancement of Science and Technology: Studies in Twentieth Century History, Dordrecht/Boston/London: Kluwer
BIBLIOGRAPHY [1 ] Betz,
A.:
Academic Publishers, 1 996, 87-1 34. Konforme Abbildung,
Berlin/G6ttingen/Heidelberg:
[8] Meyer zur Capellen, W.: lntegraltafeln. Sammlung unbestimmter lntegrale elementarer Funktionen, Berlin/G6ttingen/Heidelberg:
Springer Verlag, 1 948. [2] Doetsch, G.: Tabellen zur Laplace-Transformation und Anleitung
Springer Verlag, 1 950. [9] Reingold, N . : Refugee Mathematicians in the United States of
zum Gebrauch, Berlin/G6ttingen: Springer Verlag, 1 947. [3] Knoche, M . : Scientific Journals under National Socialis m, in: Libraries & Culture, 26(1 991 ) , 41 5-426.
America, 1933- 1 94 1 : Reception and Reactio n, in: Annals of Science, 38(1 981 ), 31 3-338.
[4] Magnus, W./Oberhettinger, F.: Forrneln und Satze fUr die speziellen
[1 0] Remmert, V. R.: Mathematicians at War. Power Struggles in Nazi
Funktionen der mathematischen Physik, Berlin: Springer Verlag,
Germany's Mathematical Community: Gustav Doetsch and Wilhelm Suss, in: Revue d'histoire des mathematiques 5 (1 999),
1 943. [5] Magnus, W./Oberhettinger, F.: Anwendung der elliptischen Funk tionen
in
Physik und
Technik,
Berlin/G6ttingen/Heidelberg:
Springer Verlag, 1 949. [6] Mehrtens, H . : Ludwig Bieberbach and "Deutsche Mathematik ," in:
Phillips, E. R. (ed.): Studies in the History of Mathematics,
7-59. [1 1 ] Sarkowski, H . : Der Springer Verlag. Stationen seiner Geschichte. Tei/ 1: 1842- 1945, Berlin, et a/. : Springer Verlag, 1 992. [1 2] Siegmund-Schultze,
R . : Mathematische Berichterstattung in
Hit!erdeutschland. Der Niedergang des "Jahrbuchs Qber die
Washington: Mathematical Association of America, 1 987, 1 95-
Fortschritte der Mathematik", G6ttingen: Vandenhoeck & Ruprecht,
241 .
1 993.
Ma them a tica l Olym piad Cha l lenges Titu Andreescu, American Mathematics Competitions, University ofNebraska, Lincoln, NE Razvan Gelca, University ofMichigan, Ann Arbor, MI This is a comprehensive collection of problems written by two experienced and well-known mathematics educators and coaches of the U.S. International Mathematical Olympiad Team. Hundreds of beautiful, challenging, and instructive problems from decades of national and international competitions are presented, encouraging readers to move away from routine exer cises and memorized algorithms toward creative solutions and non-standard problem-solving techniques. The work is divided into problems clustered in self-contained sections with solutions provid ed separately. Along with background material, each section includes representative examples, beautiful diagrams, and lists of unconventional problems. Additionally, historical insights and asides are presented to stimulate further inquiry. The emphasis throughout is on stimulating readers to find ingenious and elegant solutions to problems with multiple approaches. Aimed at motivated high school and beginning college students and instructors, this work can be used as a text for advanced problem-solving courses, for self-study, or as a resource for teach ers and students training for mathematical competitions and for teacher professional develop ment, seminars, and workshops. From the foreword by Mark Saul: "The book weaves together Olympiad problems with a com
mon theme, so that insights become techniques, tricks become methods, and methods build to mastery. . . Much is demanded ofthe reader by way ofeffort and patience, but the investment is
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Do Mathematical Equations Display Social Attributest Loren R. Graham
This column is a forum for discussion of mathematical communities throughout the world, and through all time. Our definition of "mathematical community" is the broadest. We include "schools" of mathematics, circles of correspondence, 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.
M arj orie S e n echal , Ed itor
In the aftermath of the publication in 1996 of the famous spoof article by the New York University physicist Alan Sokal1 many scientists and mathe maticians have considered the thesis of social constructivists-those spe cialists in science studies who main tain that science is shaped by social and cultural factors-as discredited. Sakal's article was, indeed, very clever, and he was correct in ridiculing the views of the most extreme social con structivists.2 The basic issue of the controversy remains, however, unre solved. To what extent are science and mathematics affected by the society in which they are developed? In this article I will maintain that even the mathematical equations used by out standing physicists sometimes display attributes of their social environment. In a book I published in 1998 I ar gued that Russia was a particularly ap propriate environment in which to test the social constructivist hypothesis. 3 In the twentieth century Russia has had a large scientific establishment in a social, economic, political, and philo sophical environment quite different from that of leading Western countries. If environmental factors affect science, the effects of those factors should show up in Russia. I concluded that these effects do indeed manifest them selves in Russian science, but that, per haps surprisingly, Russian science re veals both the strengths and the
I
weaknesses of the social constructivist hypothesis. An example of the weak ness of the social constructivist thesis can be found in the overthrow of Lysenkoist biology in 1966, when Soviet biologists embraced the same biological principles as Western biolo gists after decades of rejecting these principles. This tardy but eventually successful worldwide agreement illus trated the universalism of much of sci ence when open debate is permitted, and the existence of an objective bio logical world studied by biologists everywhere. The rejection of Lysen koism showed that reality in nature does matter. However, Russian science contin ued to be different from Western sci ence in a number of ways, and some times these differences were creative and enriching to science. The strengths of some of the great schools of Soviet science-such as the Vygotsky-Luria Leontiev school in psychology, or the Vladimir Fock school in relativistic physics, or the A D. Aleksandrov school in geometry, or the Luzin school of mathematical functions, or the Vernadsky school in geochemistry can not be adequately explained with out reference to social and political factors. Marxism was one important factor influencing Soviet science, and sometimes in positive ways. Anti Marxism, hidden opposition to Soviet policies, and religion also sometimes
1Aian D. Sakal, "Transgressing the Boundaries-Toward a Transformative Hermeneutics of Quantum Gravity," Social Text (Spring/Summer 1 996): 2 1 9-52. Sakal revealed that this article was a hoax designed to parody
science studies in his "A Physicist Experiments with Cultural Studies," Lingua Franca (May/June, 1 996): 62--£4. 2As an example of extreme views, I would cite Bruno Latour's confused critique of relativity in which he fails to understand what frame of reference means in physics: I also disagree with his statement in another place: "Since the settlement of a controversy is the cause of Nature's representation, not the consequence, we can never (my emphasis- LRG) use the outcome- Nature-to explain how and why a controversy has been set
tled." While I differ with Latour's views, I believe that a more intelligent application of social constructivism can be fruitful, and I attempt to achieve such an application in this article, showing how even equations can be af
Please send all submissions to the
fected by social context. See Bruno Latour, "A Relativistic Account of Einstein's Relativity, " in Social Studies
Mathematical Communities Editor,
of Science 1 8 (1 989), pp. 3-44; his Science in Action: How to Follow Scientists and Engineers Through Society,
Marjorie Senechal,
Department
Harvard University Press, Cambridge, p. 99; also, see Noretta Koertge (ed.), A House Built on Sand: Exposing Pas/modernist Myths about Science, Oxford University Press, New York-Oxford, 1 998, especially the essays
of Mathematics, Smith College,
by Alan Sakal and Philip Kitcher.
Northampton, MA 01 063, USA;
3Loren R. Graham, What Have We Learned About Science and Technology from the Russian Experience?, Stanford University Press, Stanford , 1 998.
e-mail: [email protected]
© 2000 SPRINGER-VERLAG NEW YORK, VOLUME 22, NUMBER 3, 2000
31
played roles, as I have discussed in ear lier publications. 4
A thorough explo
ration of the social
dimensions of
Russian science must include
all social
factors, of which Marxism and religion are important examples. In response to my book Alan Sokal
simplicity or ad-hocness, etc., as a de scription ofphenomena it purports to describe; so ANYTHING unrelated to those phenomena (be it social, politi cal orfrom some unrelated area ofsci ence) is irrelevant to the question of justification.
one which I will defend, is based on the assumption that it is a mistake to try to draw a circle around specific lo cations within science-such as core theories or mathematical equations and hold them immune from social in fluences. Rather, the impact of social influences on science should be seen
sent me two long e-mail messages. earlier message
Moving on to some of the specific
(November 17, 1998) first, and his sec
examples I gave in my book, Sokal
entific explanations, just as "color" can
ond message (January 15, 1 999) near
maintained that the impact of social in
be an attribute of physical objects or
I will
discuss his
as the imparting of "attributes" to sci
the end of this article. In his first mes
fluences, such as Marxism, on the
"flavor" can be an attribute of an herb.
sage, Sokal wished that I had made a
work of Soviet scientists related only
Reducing a flower to its color or an
greater distinction between "the con
to the "soft parts" of their theories,
herb to its flavor would obviously be
text of discovery" and the "context of
such as philosophical interpretations,
absurd, just as would be reducing a sci
justification," as well as a distinction
not the core theories themselves, and
entific theory to the attributes which
between descriptive
certainly not to mathematical equa
may have been given to it by social in
tions that lay at the heart of physical
fluences. But to ignore those attributes
and normative
claims. 5 He then continued:
theories.
Regarding the context of discovery, everyone agrees (I think) that social, political, religious and philosophical ideas can of course affect what topics get studied and what theories get con ceived [descriptive claim} and that there is nothing wrong in such influ ence [normative claim}. (E.g., the case of Malthus' influence on Darwin's thinking.) People get their inspira tionjrom all sorts of things, and that's all to the good. Historians of science can then jill in the empirical details about how often, and in what way, such influences affect scientific work in particular fields, and under what circumstances such influences are more or less likely to lead to fruitful orfruitless, good or bad, work [judged normatively by the criteria ofjustifi cation of scientific theories}. Regarding the context ofjustifica tion, everyone agrees (I think) that social, political, religious and philo sophical ideas SOMETIMES affect scientists' evaluation of the evidence for and against particular theories [descriptive claim}; however, I and most philosophers of science would agree that such influences are delete rious [normative claim}. Indeed, we would argue that the ONLY criteria relevant to judging a scientific theory are those that bear on its truth orfal sity (or approximate truth orfalsity),
for such attributes is a significant and
proach the history of science and the
rewarding endeavor. Furthermore, to
problem of social constructivism? I
see
share with Sokal a skeptical view to
"harmful" would be to impoverish sci
attributes
as
universally
entific thought, to rob it of its color and
constructivism. I agree with him that
flavor. And if these attributes are in
science is the most reliable form of
herent, as I think they are, they cannot
knowledge that we possess. I also be
be separated from the theories them
lieve in the existence of objective re
selves without doing violence to our
ality. But I differ with the view Sokal
understanding of science.
presented on the relative roles of nat
A critic of my distinction between
ural reality and social context in the
the locational and the attributive ap
formation of science. His view was
proaches might say: "But you are be
what I call a "locational" approach,
ing just as locational as Sokal. He
while I follow an "attributive" ap
pointed to some locations in science
proach. Let me try to explain the dif
such as mathematical equations as
ference.
being
Scholars who follow the
touched
by
social
not
construc
"loca
tivism; you point to specific locations
tional" approach believe that in the en
in science which are touched by social
tire corpus of scientific knowledge
constructivism. You are equally loca
there are core theories, certainly math
tional."6 My reply would be that I do
cate
ematical equations, which are unaf
not accept the view that certain
fected by the social context in which
gories
they were developed. The philosophi
ments, be they mathematical equations
or
types
of scientific state
cal, economic, religious, and political
or core theories, are immune to social
ideas and beliefs of a given society can
influence. In my view,
affect the motivations of a scientist;
ence may be so influenced. It is in
any part of sci
they can explain, at least in part, his or
evitable that when I identify such a
her verbal presentations of theories,
specific influence, it will have a loca
especially the "soft parts" of them, but
tion, but it could show up anywhere.
these social factors do not affect the
Therefore, social influence on science
core theories, and certainly not the
is best described as an "attribute." If
mathematical equations which lie at
we accept the view that social influ
the heart of modem physics. Another approach, the "attributive"
5E-mail message from Alan Sakal to Loren Graham, November 1 7, 1 998. 61 am grateful to Michael Gordin for this point.
THE MATHEMATICAL INTELLIGENCER
such
ward the most extreme forms of social
4/bid., pp. 28-31 .
32
would be mistaken; indeed, the search
Is this a valid and useful way to ap
ence is an attribute of scientific expla nations, the distinction between "hard"
and "soft" science, and the view that
the old absolutes of "space" and "time"
opposition to completely relativistic
society influences only the latter, loses
were relative to frames of reference
concepts of truth. 13 And Fock there
and therefore could no longer serve as
fore tried to develop a mathematical
reliable
standards.
framework for general relativity that
of the attributive point of view the
However, the new theory that Einstein
would reflect these commitments. This
most important test cases, no doubt,
presented was
also, thought Fock,
new framework had to be compatible
will
based on absolute standards, just as
with that of Einstein, but carry differ ent philosophical implications.
its validity. In the exploration of the adequacy
involve
mathematics.
Can
we
and
unchanging
show that the mathematical equations
Newton's had been. Echoing the views
at the core of certain physical theories
of Max Planck9, Fock remarked, "The
Fock agreed with Einstein's equiva
may display social attributes? I would
theory of relativity, after showing that
lence of acceleration and gravitation on
like to propose a case, taken from my
a whole series of concepts earlier con
the local, infinitesimal, level, but he
research,
where I believe that the
sidered absolute were actually relative,
added that if one wishes to consider the
mathematical core of a physical theory
at the same time introduced new ab
universe as a whole, then one must set
solute concepts. The majority of crit
initial or boundary conditions that re
displays such social attributes. The leading Soviet physicist V. A
ics of the theory of relativity forget
flect the
Fock was a prominent defender of rel
this."10 He pointed out that Einstein
whole.14 By doing so he believed that
properties
of space as
a
ativity theory. To question the validity
found new absolutes in the constant
he could show that even in general rel
of relativity theory, he once wrote, was
velocity of light and in the concept of
ativity there are preferred or privileged
on an intellectual level with question
space-time--the conjoining of Newton's
systems of coordinates. His position
ing the roundness of the earth. 7 But
terms in a four-dimensional manifold.
here was connected with his desire to
Fock was a convinced Marxist, 8 and in
The "theory of general relativity" could
show that relativity theory was not en
order to make relativity theory more
thus be just as easily called the "theory
tirely relativistic and that it supported materialism, that even general relativity
compatible with his philosophical be
of absolute space-time" or the "theory
liefs he was willing not only to suggest
of gravitation" (Fock's preferred term).
contained reference standards deter
changes in Einstein's terminology but
Fock thought a strong argument could
mined, ultimately, by matter. In each of
also to modify the mathematical elab
be made that the new title would be
the
oration of relativity theory, always
more accurate than was Einstein's.
(As
Fock-that
types
of
space
envisioned
by
is, Galilean space, space
making certain that his new version of
Gerald
out,
uniform at infinity, and Friedmann
the mathematics did not contradict
Einstein was aware of the problem, but
Lobachevsky space-he said that there
that of Einstein, even if it carried dif
thought it was too late to change the
"probably" exists a preferred system of
ferent implications. Fock was particu
title of his theory. 1 1) The one thing that
coordinates. 15 The word "probably" in
Holton
has
pointed
that
the theory of general relativity was not,
dicates Fock's continued hesitation in
Einstein's formulation of general rela
maintained Fock, was a purely rela
the case of Friedmann-Lobachevsky
tivity does not use absolute coordi
tivistic theory.12 He believed that a
space; in the case of Galilean space and
nates; Fock believed he could clarify
purely relativistic theory would be dif
space uniform at infinity, he was confi
the significance of this approach and
ficult to accept by dialectical material
dent of the existence of preferred sys
point to another possibility. Fock agreed
ists like himself, with their commit
tems of coordinates. The existence of
completely that Einstein showed that
ment to objective reality and their
such preferred systems of coordinates
larly
interested
in
the
fact
7V. A. Fock, "Protiv nevezhestvennoi kritiki sovremennykh fizicheskikh teorii," Voprosy filosofii (No. 1 , 1 953). 81n dozens of conversations, both with Fock himself and with others, I have been told by many people-both Marxists and non-Marxists-that Fock's commitment to dialectical materialism was sincere. Some of these conversations with Russians who knew Fock occurred after the fall of the Soviet Union, when there was no longer any political reason to affirm Fock's Marxism. Two years after the end of the USSR, in 1 993, the historian of Russian physics Gennady Gorelik wrote, "There is no doubt that Fock in the nineteen thirties was already sincerely devoted to dialectical materialism." G. E. Gorelik, "V. A. Fok: filosofiia tiagoteniia i tiazhest' filosofii," Priroda (No. 1 0, 1 993), p. 92. E. L. Feinberg went in 1 993 even further, stating Fock's " 'love' for diamat was, without question, sincere. I will say even more- maybe people will disdain me, but I also am in agreement with diamat. In itself, it makes sense." E. L. Feinberg, "Fok govoril to, chto dumal," ibid., p. 94. For a strong defense by Fock of dialectical materialism as a way of seeing physics, see his reply to my article, "Quantum Mechanics and Dialectical Materialism," and "Reply" and Paul K. Feyerabend, "Dialectical Materialism and Quantum Theory," Slavic Review 25 (September, 1 966), pp. 381-41 7 . 9Pianck commented, "The concept of relativity i s based o n a more fundamental absolute than the erroneously assumed absolute which i t has supplanted." M. Planck, The New Science, Meridian Books, Greenwich, Connecticut, 1 959, p. 1 46. Planck expressed the same idea in early publications, for example, Oas Weltbild der Neuen Physik, Leipzig, 1 929, p. 1 8.
1 °Fock, "Protiv nevezhestvennoi kritiki sovremennykh fizicheskikh teorii," p. 1 72. 1 1 See Gerald Holton, "Introduction: Einstein and the Shaping of Our Imagination," in Gerald Holton and Yehuda Elkana (eds.), Albert Einstein: Historical and Cultural Perspectives, Princeton University Press, 1 982, Princeton, p. xv. 12Fock liked to put his viewpoint in French: "(1 ) La relativite physique n'est pas general; (2) Ia relativite generale n'est pas physique." V. A. Fock, "Les principes me caniques de Galilee et Ia theorie d'Einstein," in Atti de convegno sui/a relativita generale: Problemi dell'energia e onde gravitaziona/i, Florence, 1 952, p. 1 2. 1 3"Es ist nicht uberflussig zu unterstreichen, dass das Verhaltnis von Kbrpern oder Prozessen zum Bezugssystem ebenso objek1iv ist (d.h. unabhangig von unserem Bewusstsein) wie uberhaupt aile physikalischen und anderen Eigenschaften der Kbrpern." V. A. Fock, "Uber philosophische Fragen der modernen Physik," Deutsche Zeitschrift fur Philosophie (No. 6, 1 955), p. 742.
14V. A. Fock, The Theory of Space, Time and Gravitation, trans. N. Kemmer, New York, Pergamon Press, 1 959, p. xv. 1 5/bid., p. xvi.
VOLUME 22, NUMBER 3, 2000
33
monic coordinates. Fock wanted to set up a comparison of different equations for the curvature tensor, Einstein's and his own, and demonstrate that his equation had advantages. He began with Einstein's gravitational equation
RILV - ig�LVR
Vladimir Fock
in each case would be, of course, con trary to Einstein's concept of the com plete relativization of motion. Just as the special theory of relativity is asso ciated with the relativization of inertial motion (and therefore the equivalence of inertial reference frames), so the gen eral theory of relativity is associated with the relativization of accelerated motion (and therefore the equivalence of accelerated reference frames). But now Fock questioned whether the gen eral theory actually was a generaliza tion of the special theory in this sense. Fock was a formidable mathemati cian and he devoted much of his re search over many years to the task of proving that in space uniform at infin ity there is also a preferred system of coordinates, that of harmonic coordi nates; such a system of coordinates, he maintained, reflected "certain intrinsic properties of space-time."16 Thus, Fock's commitment to Marxism influ enced his mathematics, the very equa tions which he used to elaborate har-
=
- xT!LV
( I)
From the form of the full curvature ten sor, Rf:$, Einstein and others could eas ily derive an expression for R 11-v which involved a long series of different com binations of second derivatives of the metric tensor, g11-"" This expression, with all of its combinations of deriva tives, seemed to Fock to be too much needless combination: Einstein and others kept all of these terms because they were necessary to maintain the full generality of all the possible coor dinate systems a physicist might choose to use. But they struck Fock as irrelevant, given his strong, principled preference for a particular coordinate system, his beloved harmonic coordi nates. By way of some intricate math ematical calculations, including the in troduction of his own simplifying notation, Fock therefore transformed the curvature tensor to a form that would make it especially convenient for expression in harmonic coordi nates:
f�"•"'/3f�f3 (2) The convenience of equation (2) be comes clear when one sees that in har monic coordinates f�"v, a term Fock in vented as a convenient shorthand, becomes 0, so that term drops out, and equation (2) becomesP
R�LV = -tg "'/3
a2g"' v axa ax /3
+ [11-•"'13[�{3 (3)
Fock's preference for (3) over (1) is an example of sociophilosophical con text affecting equations. Fock pointed out that his gravitational equation (3) was "compatible with Einstein's" and did not "impose any essential limita tion on the solution of the latter, serv ing only to narrow down the class of permissible coordinates." And the rea son Fock preferred harmonic coordi nates was because he believed them to reflect objective properties linked to "the distribution and motion of pon derable matter." Therefore he believed he had developed a theory of gravita tion that permitted unique solutions, was compatible with Einstein's theory, and also was in accord with philo sophical materialism. This philosophical and mathemati cal difference with Einstein affected Fock's entire approach to general rel ativity. As he wrote in his most famous book, "All concrete problems in gravi tational theory discussed in this book are solved in harmonic coordinates. This ensures that the solutions are unique." 18a His preference for har monic coordinates was an effort to avoid complete physical relativism even within general relativity, an avoidance that he connected to Marxism's effort to preserve a coher ent picture of the world that possessed a preferred system of coordinates. His original work here cannot be dis missed as frivolous or insignificant. At international conferences such leaders in the field as John Wheeler and Stanley Deser of the United States and Andre Lichnerowicz of France praised the work of Fock for its originality and insight. In discussion with me, Wheeler agreed that the reasons for using har monic coordinates are not all philo sophical; some Western physicists
1 6/bid., p. 351 . See also V. A. Fock, "Poniatiia odnorodnosti, kovariantnosti i otnositel'nosti," Voprosy filosofii (No. 4, 1 955), p. 1 33. 1 7Furthermore, the form of equation (3) impressed Fock because of its very close similarity with the ordinary wave equation for the propagation of light, being essen tially the relativistic d'Aiembertian ga,J3 d/dxa d/dxi3, with the cross-terms vanishing. And the propagation of light was particularly significant to Fock because he saw it as confirmation that space and time were indeed linked objectively in nature, adding extra physical-philosophical relevance to equation (3). See Fock, The Theory of Space, Time and Gravitation, 2nd ed., N.Y., 1 964, p. 1 58, in the section entitled "Properties of Space-Time and Choice of Coordinates." Here Fock uses the propa gation of light to show that "the properties of space-time are objective, they are determined by Nature and do not depend on our choice . . . . The equation for the propagation of a wave front in free space characterizes not only the properties of the kind of matter being propagated (e.g., of the electromagnetic field) but also the properties of space-time itself. . . . Consequently, the concepts of geometry and the notion of time are very closely connected with the law of wave-front propagation in free space." 1 88V. A. Fock, Teoriia prostranstva, vremeni i tiagoteniia, Gosudarstvennoe izdatel'stvo tekhno-teoreticheskoi literatury, Moscow, 1 955; translated into English by N. Kemmer as The Theory of Space, Time and Gravitation (2nd revised edition), Pergamon Press, New York, 1 964; see pp. 4, 1 92, 263-267, 365-375, 425.
34
THE MATHEMATICAL INTELLIGENCER
have shown interest in them for phys
ical reasons. 1 8b
But this is a place where philosophy,
a study of Fock with the intention of
and the "context of justification," and
showing how his physics had been
the belief that social influences are ger
"distorted" by Marxist philosophy, he
mane only to the first, not to the sec
and physics come to
was so impressed by what he found
ond. The importance of this topic is un
gether. When Westerners chided Fock
that he ended up defending Fock.21
derlined by an interesting shift that
for allowing his philosophical views to
Recently, on January
occurred in Alan Sokal's correspon
"distort"
he
ing Russian science newspaper pub
dence with me. In Sokal's two long
replied that to leave general relativity
lished an article on Fock entitled "On
messages to me he emphasized the dif
mathematics,
his
physical
theories,
29, 1999,
a lead
entirely as Einstein had bequeathed it,
an Equal Footing With Einstein," in
ference between discovery and justifi
without any system of preferred coor
which his interpretation of relativity
cation, but in the first message he also
dinates and with an assumption of com
was
cast doubt on the idea that social in
plete physical relativity, was also to
Russian defense of Fock, which re
fluences could affect the mathematics
make a philosophical statement. The
ferred to his dialectical materialism,
of physics, and he made a distinction
praised. 22
This
contemporary
choice, he said, was not whether to
cannot be described as politically re
between the "soft" and the "hard" parts
make a philosophical statement, but
quired, since Marxism no longer has of
of science. After seeing the first
which one to make, since each choice
ficial status in Russia.
of this paper, he revised that view. In
inevitably carried with it such a state ment,
implicit
or explicit.
And
the
choice, he continued, has implications
But the point here is not just about
draft
his latter communication he said "I do
Russia, but about the influence of so
not
cial
tions are necessarily immune to social
environments
on
physics
and
believe that mathematical equa
for the mathematics that the physicist
mathematics everywhere.
I look
influence (within the context of dis
finds most useful. 19 If Fock's view here
back over the entire history of physics
covery). "24 He now bases his entire
As
has validity, we can hardly continue to
I see quite a few examples of such in
logical
defend a "locational" critique of social
teraction between mathematics and
tivism on the distinction between dis
critique
of social
construc
constructivism in which the mathemat
philosophy.
ical cores of theories are held immune
their belief in the relationship of math
maintain that only Alan Sokal has shifted views in our very congenial and
The
Pythagoreans
musical
harmonies,
and the
covery and justification.
(I do not
from social influences. Indeed, instead
ematics to
of the mathematics of general relativity
Ptolemaic astronomers and their at
interesting interchange; he has had an
being immune from social influences, it
tachment to the spherical description
impact on me, also, causing me to fo
seems to have been particularly prone
of the heavenly bodies, Kepler and his
cus much greater attention on the dis
to them.
tinctions which are so important to
As David Kaiser wrote in a re
geometric architecture of the solar sys
cent article in which he surveyed pre
tem, Maxwell and his explanation of
him. I am grateful to him for his con
sentations of general relativity in the
electromagnetic fields on the basis of
tributions to this discussion.) We now need to analyze this "discov
post-World War II period in Europe and
mechanical and hydrodynamic analo
America, " 'General relativity' became a
gies23, and a host of other examples
ery"-"justification"
playing field upon which many different
of the
more carefully. I start out by recogniz
interaction
of mathematics
distinction
much
physicists, speaking different kinds of
and philosophical or social influences
ing that the distinction has some legit
mathematical languages, could renego
could easily be found. To say that all
imacy. I have used it in my previous re
tiate what it meant to do gravitational
physics. "20
Fock has been dead twenty-five years, but the issue of his mathemati
such influences are deleterious is to
search, and found it useful.25 However,
miss the point that science develops
the understanding of historians and
within a social context and can never
philosophers of science of this original
escape totally from it.
Reichenbachian distinction has greatly
cal innovations in interpreting general
But we still have not dealt ade
deepened in recent years, and many of
relativity is still alive. When the West
quately with one other objection to the
us now realize that the distinction is
German historian and philosopher of
social history of science, the division
only relative, not absolute. Karl Popper
science Siegfried Muller-Markus began
between the "context of discovery"
told us that there is no such thing as
1 8bJohn Wheeler's response to a paper by Loren Graham, The Einstein Centennial Symposium in Jerusalem, March 1 4-23, 1 979. See Gerald Holton and Yehuda Elkana (eds.), Albert Einstein: Historical and Cultural Perspectives, Princeton University Press, Princeton, 1 982, p. 1 35. 19Discussions of Loren Graham and Vladimir Fock, Leningrad, spring 1 96 1 . 20David Kaiser, "A 1/J is just a 1/J? Pedagogy, Practice, and the Reconstitution of General Relativity, 1 942-1975," Studies in the History and Philosophy of Modern Physics, Vol. 29 (No. 3, 1 998), p. 336. I am particularly grateful to David Kaiser for comments and suggestions for my interpretation in this article. 21 Siegfried Muller-Markus, Einstein und die Sowjetphilosophie, Vol. II, Dordrecht-Holland, 1 966. 22"Naravne s Einshteinom," Poisk (January 29, 1 999), p. 8. 23Discussed in many sources, and recently in Jed Z. Buchwald, From Maxwell to Microphysics: Aspects of Electromagnetic Theory in the Last Quarter of the Nineteenth Century, University of Chicago Press, Chicago, 1 985. 24Aian Sokal, e-mail to Loren Graham, January 1 5, 1 999. 251n my research on Boris Hessen the distinction helped me in seeing that scholar in a different way than most scholars do; see Loren Graham, "The Socio-Political Roots of Boris Hessen: Soviet Marxism and the History of Science," Social Studies of Science 1 5 (1 985), 705-722.
VOLUME 22, NUMBER 3 , 2000
35
evidence outside of the circumstances
matical equations and holding them sep
in which it was created. Ludwig Fleck
arate from their social context. Nor can
deepened our understanding of the
it be done by drawing a sharp line be
ways in which "facts" are often imbed
tween the context of discovery and the
ded in those circumstances. And Peter
context of justification and holding the
Galison tells us:26
latter separate from social context.
As
Between first suspicion and final ar gument there is a many-layered process through which belief is pro gressively reinforced. These interme diate stages in the construction of a demonstration belie the radical di chotomy between a "psychologistic logic of discovery"-where arguments are no more than the arbitrary, com pletely idiosyncratic prejudices of the discoverer-and the formal, fully per suasive "logic of justification" that finds its way to print.
A U T H O R
I look over the works of recent
historians and philosophers of science, including those of Peter Galison, Mario Bagioli, Ian Hacking, Ann Harrington, Norton Wise, and Simon Schaffer, I think that we are gradually finding a path toward recognizing simultane ously that science is a part of culture
LOREN GRAHAM
and displays the pluralities of culture,
Program
but also seeing that science gives us a
on
Science.
Technology, and Society
way of winnowing the unreliable forms
MIT
of knowledge from the reliable ones.
Cambridge, MA 02 1 39
We are learning to reject the extreme views of some social constructivists
USA
who would deny the potency of the sci
e-mail: [email protected]
Does this open the door to complete
entific way of knowing while at the
relativism? In my opinion, not at all. The
same time we recognize that all of sci
scientific way of knowing, with all the
ence-not just some parts of it-is sub
History of Science at MIT. He is the
disunity and intercalation of different
ject to social influence. Describing the
author of numerous books and arti
beliefs that it involves, is by far the best
relationship between science and the
cles,
social context in "attributive" rather
Russian
(1 993) and What Have We Learned
approach to natural reality that we pos
Loren Graham is Professor of the
including A Short History of Science
and
Technology
sess, relative to the operational needs
than in "locational" terms may be one
and goals that societies put to that nat
way to do this. All of science displays
ural reality. In the long
social attributes-yes, even the "hard
the Russian Experience? (1 998). He
Lysenko Affair and many other episodes
est" and most mathematical parts of sci
and his family restored a remote ru
in the history of science illustrate, the
ence-but to reduce science to those at
ined lighthouse on an island in Lake
scientific way of knowing is our most re
tributes would be silly. To deny the
Superior; this article was written there,
liable guide. But the defense of this ap
existence and significance of these so
on
proach to knowledge cannot be based
cial attributes in the body of science
power.
on fencing off core theories and mathe-
and mathematics would be misleading.
run,
as the
about Science and Technology from
a
computer
running
on
26Peter Galison, How Experiments End, University of Chicago Press, Chicago and London, 1 987, p. 3.
"Running dry" is normal in all research. It can of course be painful if it goes on too long. But to some extent I like running dry. After all, when things work out all by themselves you tend not to make enough effort. But as soon as you run dry, you get desperate to overcome the difficulty, you throw all your resources into it, and you end up finding something. Not necessarily what you were looking for, true, but surely something interesting. I am happy whenever I run dry, but unhappy when it becomes long and painful. After the enthusiasm of sudden discovery, you are bound to run
·
into some dead ends. from Laurent Schwartz:
1997
36
THE MATHEMAnCAL INTELLIGENCER
Un mathematicien aux prises avec le siecle: Odile Jacob,
solar
M a t h e n� a tic a l l y B e n t
The proof is i n the pudding.
Col i n Ad a m s , Ed itor
back home. The calling plan reverts to
The Pepsi Putnam Challenge ''
the standard one after today. Internet access has been provided by Amer ica Online available to the students through the SGI workstations provided by Silicon Graphics." 'Thanks a lot, Cranston, for filling us
in on which corporations have made this competition possible. Now, as you all know, the Pepsi Putnam Challenge pairs
L
adies and gentlemen. Welcome
talented math students with celebrity
to the Nabisco Amphitheater
teammates as they take the Putnam
Opening a copy of The Mathematical
and the ceremonies to honor the top
Exam, which consists of a set of twelve
lntelligencer you may ask yourself
contenders in the Pepsi Putnam Chal
formidable math problems. Let me in
lenge. We are pleased to have Karen
troduce some of the contestants.
uneasily, ''What is this anyway-a mathematical journal, or what?" Or you may ask, "Where am !?" Or even "Who am !?" This sense of disorienta tion is at its most acute when you open to Colin Adam's column. Relax. Breathe regularly. It 's mathematical, it 's a humor column, and it may even be harmless.
Parton, Nike's director of marketing, who
will
"Here is the Pennzoil Team, Anant Lerosky from Harvard University and
emcee today's festivities."
"Thanks, Bill. What a thrill it is to be here today with these future profes
his teammate John Tesh. Anant, did you fmd John helpful?"
sional matheletes. I am so glad that
"Actually, he kind of made me ner
Nike can contribute in its small way to
vous, sitting there looking over my
these intellectual giants. Knowing that
shoulder."
their Nike footwear was gripping the floor, that their Nike warm-up jackets prevented them from getting a chill
"And John, how did you find it?" "Oh,
it
was
very
exciting.
We
worked together. Anant figured out the
and that their Nike superabsorbent
problems and wrote up the solutions.
athletic wear were soaked with their
Then I rewrote them to look nicer and
sweat as they pored over these diffi
added some doodles. When it looked
cult problems, well it just makes me
like Anant's energy was flagging, I sang a few tunes from my upcoming CD,
proud." "I am now pleased to present to you
Songs for SeaguUs."
of
"Yes w e are all looking forward to
Mathematics from Harvard University,
that. And here we have the Kellogg's
the
Nestle's
Crunch
Professor
Cranston Lamont, who will say a few
Pop-Tart Team, Kathy Panaur of MIT
words."
competing
"Thank you, Karen. And thanks to the innumerable corporations,
who
through their sponsorship, make this
with
Richard
Simmons.
Kathy, how much help was Richard on the math portion of the exam?" "He didn't show up."
event possible. Special mention goes
"Ha ha, Richard didn't show up.
to American Airlines for flying these
What a cut-up. How did you feel while
math students out here for the cere
you were taking the exam?"
mony and giving them plenty of Pepsi
"I guess I wish that the camera guy
along the way to keep up their caffeine
could have stopped shoving that cam
levels. Express
And for
thanks
to
providing
American them
with
AMEX cards to pay for the necessary
era in my face. It made it hard to con centrate." "Yes, well, Kathy, without that cam
this would just be
incidentals like erasers, pencils, and
eraman,
Department of Mathematics, Bronfman Science Center, Williams College,
pads. Let American Express take care
people taking a test, now wouldn't it?"
of you. MCI long distance is proud to
"And here is Jeff Terwilliger from
Williamstown, MA 01 267 USA
have contributed cellular phones so
the University of Waterloo competing
e-mail: [email protected]
students could contact their coaches
with Ed McMahon on the Kitchenaid-
Column editor's address: Colin
Adams,
a bunch of
© 2000 SPRINGER-VERLAG NEW YORK, VOLUME 22, NUMBER 3, 2000
37
I'm
the results tabulated by the Mathemat
sure your audience members remem
ics Department of the University of
Appliances-for-the-Home
Team.
ber the slow mo of Jeff's pencil point
California at Berkeley. They have an
breaking on Problem
800
7.
How did you
feel when that happened, Jeff?" "What do you mean? I just picked up another pencil." "Yes, we saw the replay. Besides
number if you are interested in
graduate school." "I'll just open the envelope provided
I am thrilled to announce that
that near disaster, how do you think
this year's winner is the Kellogg's-Pop
you did?"
Tart
"Well, I would have done okay, but
Team
Richard
of
"Ha, ha, very funny. Now come on Kathy,
tell them what station you
watch as you study, with call letters the
by Office Max. Well, what do you know?
"What do you mean? I can't study with the TV on."
Kathy
Simmons.
Panaur
and
Congratulations
first three letters of the alphabet." "Oh, yes, I watch ABC television, where I'm part of their family." "That's right. And what do you eat while you are studying?" "I like to eat Kellogg's
Frosted
Ed kept bumping me with that giant
Kathy. This means a full fellowship to
Sparkly Pop Tarts, the ones that glow
cardboard check Why does he always
study mathematics at Harvard, a life
in the dark "
carry that thing around?"
time supply of Rayovac batteries, and
"Nobody knows, Jeff. How did you like the half-time show?"
"Thanks,
Kathy, for those precious
a lucrative endorsement contract with
words of wisdom. Well that wraps it up
Fischer Toys. What would you like to
for us today. Stay tuned to ESPN for the
"I would have preferred spending
say to the young matheletes in our au
Frito-Lay Nobel Prize chicken fights.
the time boning up on my integration
dience who would aspire to be as suc
We'll have the finals between Economics
tricks, but that's hard when they're
cessful as you?"
shooting live animals out of cannons."
"I liked that part, too. But now, it's
the moment we have all been waiting
"Well, I guess
and the Peace Prize Winners. You won't
I would just say that
it is important to work lots of prob lems, and study the past exams. "
for. It's time to fmd out who has won
"And when you are studying, what
the Pepsi Putnam Challenge. Here are
station do you have your TV tuned to?"
want to miss it. We
will see you all next
year, when mathematics and corporate sponsorship again join hands to present to you the Pepsi Putnam Challenge." (Cut to Commercial.)
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CURTIS D. BENNETT
A Paradoxi ca View of Escher' s Ange s and Devi s
"A circular regular division of the plane, logically bordered on all sides by the infinitesimal, is something truly beauti ful . . . " -M.C. Escher
This paper is centered around figures 1 and 2: both the mathematics behind them and their creation. The backdrop for these figures is the Circle Limit IV woodcut of M.C. Escher (1898--1971). One of Escher's goals was to repre sent infmity artistically. Planar tessellations did so but they could only represent a part of infinity since they could "be expanded as far as one wishes" [B]. When H.S.M. Coxeter sent Escher the article Crystal symmetry and its gener alizations ([C1], [C3]), Escher realized how to avoid the limitations of planar tessellations using the Poincare model of hyperbolic space. Using the Poincare disc as a template, Escher created his four Circle Limit woodcuts, the last of which, Circle Limit IV, is the "Angels and Devils" print that lies behind our pictures. (For a further discussion of the history and mathematics behind Escher's prints, see [C2], [C3], [C4], and [S].) The backdrop of the figures is the Poincare disk rep resentation of hyperbolic space. The points in the Poincare disk model are the interior points of the unit disk. The lines are arcs of circles which are perpendicu lar to the boundary of the space (see figures 3a and 3b).
Note in the Poincare disk model, figures close to the boundary of the disk appear smaller than they actually are. In fact, all of the angels in figure 1 (and figure 2) are the same size in hyperbolic space. One way to visualize this is to think of the hyperboloid model for hyperbolic space: it has as point set the points from a single sheet of a hyperboloid of two sheets (z2 - x2 - y2 = 1), and as lines the intersection of planes through the origin with the hyperboloid [R]. From the hyperboloid model, the Beltrami-Klein model is obtained via central projection to a disk in the plane z = 1. (The Beltrami-Klein model has as points the interior of a disk and as lines straight lines on the disk.) The Poincare model arises from the Beltrami-Klein model as a composition of two projections: the projection from infinity from the Beltrami-Klein model to the upper hemisphere of the unit sphere, and stereo graphic projection (from the south pole) mapping the up per hemisphere to the Poincare disk (of radius 2) at z = 1. The effect of this is to transform the lines into circular arcs bending away from the origin (for a more precise de scription see [Sp, pp. 23-25]). Thus, one can think of the figures close to the boundary as being viewed from close to their edge. What made the Poincare model useful for Escher is that the Poincare disk model is conformal; that is, the angle between two "lines" is the angle the circular arcs make in Euclidean space.
© 2000 SPRINGER-VERLAG NEW YORK, VOLUME 22, NUMBER 3, 2000
39
One important property of hyperbolic space is that given
x and y of hyperbolic space, there exists a hyperbolic isometry sending x to y. Hence, while the cen any two points
ter of the Poincare disk inhabits a special position to the
Euclidean eye, to the hyperbolic eye it
is no different from
any other point. Of course, the reason for using two picture!> lli Lllat Ute Euci.IUeaH r-uLaUum; un.lle tm:;R
joint subsets
h1. . . . , h.n B
=
can be reassembled into a copy of E, and likewise for B.
For nresent numoses T will liSP an PmJivaiPnt definition for a set to be G-paradoxical. The condition is that th r
trie , so, as a result, the Euclidean eye
of E and sub ets
can see these hyperbolic isometries.
1993, Stan
E has two proper disjoint subsets A = UAi, UBJ, so that A can be broken into pieces which
In words, and
about the center are also hyperbolic isome-
In
A,, . . . , An, B1. . . . , Bm of E and g1, . . . , gn, G such that E = Ugi(Ai) and E = Uhj(Bj).
E
1.
\HI
1
2.
and
m, 1K'
=
n,
and
h1, h2 have ht(A) n h-z(A)
ircle Limit IV woodcut This provide
=
2. For distinct
half-third paradox on Escher's
figures
m * n; EHw
=
0 (and
similarly for K); and
These
3.
a new and
We
and
artistic way of looking at the
have
E
=
ukEKk(A) .
E=
UhEnh(A)
H Um of
In this case, we say that
Banach-Tar ki paradox: that a ball can be cut up into infi
demonstrates that A is
nitely many pi ces that can be
M (and similarly
assembled into two balls
strat
the size of th
A
that
Wagon posed the
problem of exhibiting Hausdorff's
I do in figure
exist a subset
H and K of G such
ach
that
we ar
original. The
A is
K demon
1/n
of M); if
fortunate to have
H
purpose of thi pap r is to ex
cyclically generated by h, then
plain these figur
we ay
and how to
is
create them. The paper i .
.
..
-
e how this definition
To
broken into .
h demonstrates that A
1/m of M.
...
I
three sections. In the first
implies the usual definition,
Figure 1
explain the idea of a para
note that condition
3
essen
doxical coloring. In the next I describe the algorithm pro
tially allows us to replace any set of m (disjoint) images of
ducing the coloring, justify its workings, and explain the
A with n (disjoint) images of A. This allows the creation of more than 2n (disjoint) images of A from E. Condition 3 for
programming difficulties. Finally I relate my coloring to the
may be skipped by the reader who wants a general under
H then implies these can be rearranged into two copies of E. In actuality, each of these "replacements" forces splitting the initial copy of A into mn disjoint subsets (correspond
standing.
ing to
Banach-Tarski paradox. For the sake of completeness, I in clude technical details in the constructions below, but they
Paradoxical Colorings
Intuitively, a m tric space M will b paradoxical if a
M (with
m
*
n)
und
r
on id
gers under th
m tti
a
*
b; th
v
th
finite
=
on -
half of all of the integ r
(nat
difficult. J. Mycielski and S. Wacton ([MW], ( W 1 ] , and [W2]) pro du eel paradoxical col01ing
w·ally) and as one-third of the integers ending
by
a
u ing th
permutation
2a to 3a.
chall nge was to create a sim ilar coloring of Escher's Circle
ctiven in
Limit IV woodcut. His idea
[W 1 ] :
was
Definition: L t G be a group acting on a E � X. E i
uppo
e
om
itiv
in
gers
and
n there are pairwise
THE MATHEMATICAL INTELLIGENCER
Figure 2
uch
a
coloring
an atti tic way
to look at th
Banach-Tarski
ing this qu
m
dis-
that
hould pr vid paradox.
G-paradoxical if
for
40
p
t X and
Klein-Fricke tiling of
hyp rbolic space IHI . Wagon's
The standard definition of a paradoxical set i
ets and how they are cteo
metrically lin of the space i
n int g r
can be viewed both
nfortu
nately, trying to visualize the e in
inte
d(a,b)
decompo ition
for infinite unbounded sets.
isom tries.
r
of paradoxical
need not be complicated if one allow
ub et A can b
For exampl , 1 for
struction
called
of M and 1/m of
lin
viewed as both
gi(A) n hj(A)) and these intersections are the new
sets that can be rearranged in different ways. As the example on the integers show , the con
utiou ly, answer ti n I d to a para-
doxical coloring of hyperbolic
Figure 3a. Tesselation centered at x.
Figure 3b. Tesselation centered at y.
space using triangles without vertices at infinity. Below,
T(m,n,k) + . The latter group is then generated by rotations of order m, n, and k around the appropriate vertices of one of the trian gles in the tesselation. For the case n = 3, m = k 4, the
for the purposes of clearer exposition, I shall discuss the related problem of coloring the underlying triangular tiling of the Poincare disk (see figure 3a).
entation-preserving isometries is denoted
=
tiling (or tessellation) is shown in figure 3a using the Poincare disk model of hyperbolic space. This is somewhat
The Algorithm Given a triangle
(n,m,k
E
C with angles measuring 1rln, 1rlm, and 1rlk
N) such that the sum of the angles is less than
1r,
misleading in that the center of the disk is no different from any other point in hyperbolic space. Thus, while there is
one can tile hyperbolic space with triangles congruent to
an evident 6-fold rotational symmetry in figure 3a, by
C. The group of hyperbolic isometries preserving this tes
choosing
selation is denoted by
have an evident 4-fold rotational symmetry (see figure 3b).
T(m,n,k), while the subgroup of ori-
Figure 4a. Regions T, S, and S'.
y
as the "center" of the plane instead of
x,
we
Figure 4b. T, S, and S' centered at y.
VOLUME 22, NUMBER 3, 2000
41
Figure 5a. First g-pass centered at x.
Figure 5b. First g-pass centered at y.
These two figures are central to our discussion; accord ingly, at every stage I will be presenting two pictures of the hyperbolic plane, the first with x centering the Poincare disk and the second with y at the center. The Poincare disk also distorts the size of objects close to the boundary. The effect of this on our figure is that while every triangle of figures 3a and 3b has the same area in hyperbolic space, the triangles at the boundary appear smaller. Because of the enormous number of triangles close to the boundary and the memory limitations of my computer, in later fig ures I may include less detail in the tiling of the space. Denote by g the rotation in the counterclockwise di-
rection of IHI of order three centered at x, and denote by h the rotation of IHI of order two centered at y. In figures 4a and 4b, let S be the area shaded yellow, let S' be the area shaded purple, and let T denote the unshaded region. The region T plays the central role in the coloring. At the first step of the coloring we color the region T red. We now con tinue by coloring g(1) blue and g2(1) green as in figures 5a and 5b. At this point, g cyclically permutes the portions col ored red, blue, and green. Hence g demonstrates that one third of the colored regions are colored red (most obvious in figure 5a). A pass of this nature I term a g-pass. We next perform an h-pass. That is, we color new regions accord-
Figure 6a. First h-pass centered at x.
Figure 6b. First h-pass centered at y.
42
THE MATHEMATICAL INTELLIGENCER
Figure 7a. Second g-pass centered at x.
Figure 7b. Second g-pass centered at y.
ing to h so that if a region is colored blue or green then the image of that region under h will be colored red. Similarly, if a region is colored red, we color the image of that region under h blue or green. In terms of our picture, we color the region h(1) blue and the regions h(g(1)) and h(g2 (1)) red. Thus we obtain figures 6a and 6b. In figure 6b, it is ap parent to the Euclidean eye that h demonstrates that one half of the colored regions are colored red. The process is to continue inductively. For example, the next step will be a g-pass in which we color the regions g(h(1)), g2 (h(g(1))), and g2 (h(g2 (1))) green, the regions g(h(g(1))) and g(h(g2 (1))) blue, and the region g2 (h(1))
red, to obtain figures 7a and 7b. Again, g demonstrates (fig ure 7a) that one-third of the colored regions are red. The next step is an h-pass in which we color the regions
that one-half of the colored regions are colored red. Now continue this process ad infinitum (see figures 9a and 9b) . If a triangle R has been colored red, then it was done so at some fmite stage. Hence after the next g-pass it is guaran teed that g(R) is colored blue and g2 (R) is colored green. Since a similar statement holds true for both blue and
Figure Sa. Second h-pass centered at x.
Figure Sb. Second h-pass centered at y.
h(g(h(1))), h(g2 (h(g(1)))), h(g2 (h(g2 (1)))), h(g(h(g(1)))), and h(g(h(g2 (1)))) red and the region h(g2 (h(1))) blue. This leaves us with figures Sa and 8b, where h demonstrates
VOLUME 22, NUMBER 3, 2000
43
Figure 9b. The limit centered at y.
Figure 9a. The limit centered at x.
green, g demonstrates that one-third of the triangles are colored red in the limit. Of course, after the next h(R) is blue or green, and it follows that
h
h pass,
demonstrates
that one-half of the triangles are colored red. Therefore we have a paradoxical coloring of the space IHI.
There are two steps necessary to see that this algorithm
really performs as advertised. First, at each step we only colored regions that had not yet received a color, yet I claimed that g permuted the colors after a g-pass and h per muted the colors after an h-pass. Second, at the end I claimed the whole space was colored, but we really do not have a proof of this either.
I break the proof of the first part into two cases. Suppose R is colored on an h-pass. Then, of course h(R) must have been colored before this step, and by the algorithm, exactly one of R and h(R) is red as claimed. Since the same holds
Recall from above that h interchanges S' and its complement
(S')C, and g and g2 both map SC into S. Since T r:t. S U S' and
S n S' = 0, we have
hgtk(T) c S'. Again, since S' n S -=f. 0, it . . . hgtk(T) C S'. Hence c(T) is a
follows inductively that hgti
subset of either S' or S. Because T has non-empty intersec
tion with S U S', it follows that c(T) -=f. T. This "Ping-Pong" ar gument, which actually proves that (g) and
(h) generate a free
product, was first used by Klein [K], and in [M] Macbeath pre sented a general form of it. That this is a free product is ac
tually the key to this entire construction, and the construc
tion would fail if g and h did not generate such a free product.
To see that every triangle gets colored, let U r:t. T be a tri
angle, and consider a shortest path of triangles U = U1 , . . . , Uk = C where Ui and Ui 1 share a side. For some triangle Ui this path enters into T; that is, Ui + 1 C T but Ui r:t. T. Because Ui has an edge lying on the boundary of T, a simple check
true if we replace h by g (and h(R) by either g(R) or g2(R)),
shows that either h(Ui), g(Ui), or g2(Ui) lies in T. Since g and
an h-pass and g(R) or g2(R) is already colored, and the case
h send triangle paths to triangle paths, it follows that one of h(U), g(U), and g2(U) lies closer to C than U. Hence, we can
we need only worry about the case where R is colored on where R is colored on a g-pass and h(R) is already colored. Suppose R is the first region for which one of the above
holds. If R is colored on an h-pass then R = a(T) for
a=
k
-
hgm 1 hgm2 . . . hgmk, where mi E { 1,2) for i = 1, 2, . . . , 1, and mk is allowed to be 0. Let R' be the region al
ready colored, which we will assume to be g2(R) as the ar
gument for R' = g(R) is similar. Since R' was already col ored, R' = b'(T) for some b'
E (g,h). Since g(R')
=
R, and
R was not colored on a g-pass, it follows that R' was col ored on an h-pass too. Hence b'
{ 1,2) for i =
=
hgh
E
gb ', we obtain that R = b(T). Since R = a(T), we obtain a - 1 b(T) = T. I will
1, . . . , l - 1.
Letting b
. . . hgiz where ji
=
now show this to be impossible. Let c =
44
a- 1 b =
g-mk hg -mk-1 h . . . hg-ml hghgil h . . . hgiz = gtl hgt2 h . . . hgtr.
THE MATHEMATICAL INTELLIGENCER
apply induction to show that every triangle is the image a(C)
for some a E (g,h). It now follows that the algorithm colors every triangle. In technical terms, this argument shows that
T is a fundamental domain for the group
(g,h).
Analyzing this proof gives a general algorithm for pro ducing paradoxical colorings of triangle groups acting on IHI. First find two isometries g and
h of different orders sat
isfying the "Ping-Pong" property. More specifically, there must be disjoint sets
Sh and Su satisfying
for 0 < k < ihl (the order of h), hk(S�) c Sh, 2. for 0 < k < lgl (the order of g), gk(SJf) C Sg, and 3. the region T S� U S8 contains a triangle.
1.
=
Given these elements, follow a coloring algorithm of g passes and h-passes so that g and
h demonstrate different
percentages of the plane for some base color.
Visualization
All of the figures in this paper were produced using Mathematica. The algorithm for producing the triangular coloring is that given above. To produce the triangulation, save fifteen complex data points marking the boundary of C in the complex unit disk. Next write g and h ' as linear fractional transformations of the complex plane (where h = h'2). The element g is the transformation given by g : z .....,. eC277i13)z, and the map
h'
is given by
h , z .....,. · 0
where a =
V3 - 1
v'2
(i - aa)z + (1 - i) a (i - 1 ) az + (1 - i aa) ' 0
e'7T13
is the complex number corre-
sponding to the point y. Given g and h', simply iterate them to get all images of the base triangle and hence the tesse lation. To color, the region T is defined by 60 complex data points and one produces the coloring by the algorithm given above using g and h = h '2 . For the Escher picture, the situation is about 4 times as complex. To store the devil requires about 60 data points, and to store the region T re quires 204 data points (in fact, slight irregularities in the figures are still visible, but computer limitations make it difficult to store more data points). Producing a paradoxical coloring in which the angels and devils have different shadings is more complicated. In par ticular it requires keeping track of which devils lie in the re gion T, for these must now receive a different color from the angels in T. The algorithm used checks a point of each devil to see whether it lies in T. Once we have a set T' of the tri angles lying in T, we then can perform our paradoxical col oring algorithm on T' where we have used a darker shading. This coloring is placed over the paradoxical coloring of the region we already have to achieve figures 1 and 2. Relationship with the Banach-Tarski paradox
Loosely speaking, the Banach-Tarski paradox states that you can carve a ball into finitely many pieces and use Euclidean motions of the pieces to put them back together into two balls each of the same size as the original. The proof of the Banach-Tarski paradox is harder than what I did above, in that one needs to produce a free product in the group of orthogonal 3 X 3 matrices. However, one can use the above pictures and arguments to get a flavor of the proof of the Banach-Tarski paradox. If we think of each triangle C' as representing the element a of the triangle group G associated to the tessellation such that a(C) = C', the coloring of the triangles corresponds to a coloring of the elements of G. Conversely, given a paradoxi cal coloring of G, we could similarly transfer it to the hyper bolic plane IHJ. In fact, this is how the algorithm above was conceived. A paradoxical coloring of the elements of G corresponds to a partition G = K U J U L where gK = J, g2K = L, and hK = J U L. To transfer this to IHJ, choose a set M C IHl which contains exactly one element from every G-orbit, and color the set {k(x) I k E K, x E M} red, the set
{j(x) lj E J, x E M) blue, and the set {l(x) I t E L, x E M) green. Choosing M is relatively easy in this case: it is a tri angle. In my algorithm I replaced the tessellation group G by its subgroup generated by g and h, and T was the choice set in this case. For the Banach-Tarski paradox, G is replaced by S03(1R), the isometry group of the two-sphere S2. The first task is then to find rotations of S03(1R) to act as our g and h. Defining F = (g,h) as our free product, we remove the set D of points of S2 fixed by any element of F. By the Axiom of Choice there ex ists a set M consisting of one point from each of the orbits of S21JJ under the action of F. Again, one colors each of the sets {g(x) I g E Ai, x E M} to transfer the paradox on F to S21JJ. This is actually a sketch of the proof of the Hausdorff para dox. The Banach-Tarski paradox requires a further argument to eliminate the setD which is uncolored (much as the bound aries of some of our triangles were uncolored above). To see how the paradoxical coloring might lead to dou bling the ball, I return to the above example. If R denotes the set of red triangles at the end of the coloring, then h(R) U R = IHJ. IfB represents the set of blue triangles, then g2(B) = R in the construction. But this implies that R U h(g2(B)) = IHJ. Thus one can reform hyperbolic space using just the red and blue triangles, leaving the set Gr of green triangles left over. Since g(Gr) = R, we still have half of hyperbolic space left over. As g and h are hyperbolic isome tries, so are the motions of R and B required to assemble IHJ. Similarly, given the decomposition of S21JJ for the Banach-Tarski paradox, the motions used to create the new sphere are isometries of Euclidean space. This visualization of the Banach-Tarski paradox in IHl al lows one to avoid the Axiom of Choice, for one easily defines a set of representatives of each orbit of IHl under the tessella tion group G. The original use of hyperbolic space for this is due to J. Mycielski and S. Wagon [MW], and a Mathematica program for producing paradoxical decompositions using the group PSLz(Z) due to S. Wagon appears in [W2]. Wagon's pro gram runs significantly faster than mine because it does not require keeping track of triangles inside the fundamental do main. On the other hand, keeping track of these triangles seems to be exactly what is necessary to produce a para doxical coloring that does not have vertices of triangles at in finity. One must ask, is there a coloring algorithm that com bines the speed of Wagon's program with the lack of vertices at infinity in this coloring. Such a program would seem to re quire a quick way of determining which image of the funda mental domain any triangle from the tesselation is in. The reader interested in learning more about the Banach Tarski paradox and related issues is encouraged to look at [W1]. In particular, he shows that this paradox has measure theoretic consequences: there is no isometry-invariant mea sure in IHl on the Borel sets that has total measure 1 . Such measures, however, do exist in IR2. The author is deeply in debted to Stan Wagon for his help with this project. REFERENCES [B] Bool, F.H., Kist, J . R . , et al, M.G. Escher: His Life and Complete Graphic Work, Abradale, 1 981 .
VOLUME 22, NUMBER 3, 2000
45
A U T H O R
[C1 ] Coxeter, H.S.M., "Crystal Symmetry and Its Generalizations," in A Symposium on Symmetry, The Transactions of the Royal Society of Canada 51 , ser. 3, sec. 3 (June 1 957), 1 -1 3 .
[C2] Coxeter, H.S. M . , Introduction to geometry, Second edition. John Wiley and Sons, Inc., New York-London-Sydney, 1 969. [C3] Coxeter, H.S.M., Angels and Devils, in The Mathematical Gardner, Wadsworth International, Boston. 1 97-21 1 , 1 981 . [C4] Coxeter, H.S.M., The trigonometry of Escher's woodcut "Circle Limit Ill", Math. lntelligencer 18 (1 996), no. 4, 42-46. [C5] Coxeter, H.S.M., Erratum: The trigonometry of Escher's woodcut "Circle Limit Ill", Math. lntelligencer 1 9 (1 996), no. 1 , 79. [K] Klein, F., Neue Beitrage zur Riernannischen Funktionentheorie, Math. Ann. 21 , 1 41 -2 1 8 (1 883).
CURTIS D. BENNETT Department of Mathematics and Statistics
[M] Macbeath, A.M . , Packings, free products, and residually finite
Bowling Green State Un ivers ity
groups, Proc. Cambridge Phil. Soc. 59, 555-558 (1 963).
OH 43403
Bowl ing Green,
[MW] Mycielski, J . , and S. Wagon, Large free groups of isornetries and
USA
their geometrical uses, L'Enseignement Mathematique 30 (1 984),
e-mail: [email protected]
247-267. [R] Reynolds, W.F., Hyperbolic Geometry on a Hyperboloid, The Curtis Bennett
American Mathematical Monthly 1 00 (1 993), 442-455.
did
Chicago, and was a
[S] Schattschneider, D., Visions of Symmetry: Notebooks, Periodic
his
doctoral
NSF
work at the University
postcodtoral fellow
of
1 992-1 995. He
Drawings, and Related Works of M.G. Escher, W.H. Freeman and
is now an associate professor at Bowling Green. His research
Company, San Francisco, 1 990.
is in finite and infinite group theory, Lie algebras, Coexter g roups, and combinatorics. He
[Sp] Spivak, M., A Comprehensive Introduction to Differential Geometry,
is
also co-editor of Starting
Our Careers: A Collection of Essays and Advice on Profes
Vol. IV, Publish or Perish, Boston, 1 975.
sional Development from the Young Mathematicians Network.
[W1 ] Wagon, S., The Banach-Tarski Paradox, New York, Cambridge
His hobbies are squash,
Univ. Press, 1 985.
and trying to keep
up with
his two
children.
[W2] Wagon, S., A hyperbolic interpretation of the Banach-Tarski para dox, The Mathematica Journal 3 (4) (1 993), 58-61 .
EXPAN D YO UR MATHE MATICAL BO U N DARIES ! Mathematics Without Borders A
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1 897, the IMU has sponsored
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SASHO KALAJDZIEVSKI
Some Evi dent Su m mation Form u las 1 . Two Dimensions
Here we see a proof of 1 of 2(1
+2+
· · ·
+2
+
+ n) + 2(n + 1)
··· =
+ n + (n + (n
+
1)
=
(n
+ 1)(n + 2) 2
. , or, eqUivalently,
1)(n + 2)
+1
Could it be that the higher-order analogues of this formula have proofs that are equally plain to see?
© 2000 SPRINGER-VERLAG NEW YORK. VOLUME 22, NUMBER 3, 2000
47
2. Three Dimensions
n(n + 1)(2n + 1) or 12 + 2 2 + . . . + (n - 1)2 + n2 = 6 6 [ 12 + 22 + + (n - 1)2] + 6n2 = n(n + 1)(2n + 1) · · ·
[The last of the series of pictures suffices to see a proof of the statement.]
6(1 2 ) = (1)(2)(3)
6(f + 22 ) = (2)(3)(5)
n:-1 '
- - - - - . .£
3. Four Dimensions
The next formula is
(
r
n(n + 1) 1 3 + 23 + · · · + (n - 1)3 + n3 = or 2 4(13 + 23 + + (n - 1)3) + 4n3 = [n(n + 1)]2. · · ·
We exhibit a pictorial proof of the inductive step: [(n - 1)n]2 + 4n3 = [n(n + 1)]2. We construct a larger four-dimensional parallelepiped by arranging five smaller ones. The six illustrations which fol low all depict the same construction with each of the six 4D parallelepipeds emphasized in turn. The first 4D paral lelpiped has a volume of [(n - 1)n]2; the next four have vol ume of n3, while the fmal union (the last picture) is a 4D parallelepiped with a volume of [n(n + 1)]2. Note that the edges of a given color in any one parallelepiped are paral lel and of equal length.
48
THE MATHEMATICAL INTELLIGENCER
A U T H O R
SASHO KALAJDZIEVSKI Department of Mathematics
University of Manitoba Winnipeg R3T 2N2
Canada e-mail: [email protected]
Sasho Kalajdzievski lived in Skopje, Macedonia until he com pleted his Master's degree. He then moved to Canada to fur ther his studies, got his Ph.D. in Toronto, and a year later he moved to Winnipeg. He has been chess champion of Skopje, and he has been chess champion of Manitoba. His former hobbies of soccer and table tennis are now supplanted by basketball.
4. Epilogue
In the next dimension we have the identity
J:1 i4 n
=
1 n(n 30
+
1)(3n2 + 3n - 1).
The idea used in lower dimensions seems to be too natural to fail here: there should be a five-dimensional block made of five-dimensional parallelepipeds in such a way that the corresponding volumes yield the above identity. However, something new intervenes in five dimensions: the polyno mial 3n2 + 3n 1 is irreducible over the integers. I do not know if that significantly affects the procedure, but visu alization will certainly be obstructed by the technical prob lem of having to draw 5-dimensional objects in 2 dimen sions. -
VOLUME 22. NUMBER 3. 2000
49
Wi¥1(9-fH.M
.J e remy G ray, E d i t o r
I
The Emergence T of Nonlinear Programming: Interactions between Practical Mathematics and Mathematics Proper TI N N E HOFF KJELDSEN
Column Editor's address:
he beginning of the modem mathe matical theory of nonlinear pro gramming can be dated back to 1950quite precisely to the Second Berkeley Symposium on Mathematical Statistics and Probability held in Berkeley, California. At this meeting Albert W. Tucker, a mathematician from Prince ton, presented a paper with the title Nonlinear Programming which he had written together with another Prince ton mathematician, Harold W. Kuhn. After the meeting the paper was pub lished in the proceedings of the Symposium and for the first time the name nonlinear programming ap peared in the literature [Kuhn and Tucker, 1950]. In the paper Kuhn and Tucker de fined the nonlinear programming prob lem-or maximum problem as they called it-as follows:
To find an x0 that maximizes g(x) constrained by Fx � 0, x � 0. Here Fx is an m-vector (fi(x), . . . , fm(x)), where fl (x), . . . . fm(x) are dif ferentiable functions of x defined for x � 0, and g(x) is a differentiable func tion of x also defmed for x � 0. In words, a nonlinear programming prob lem is a finite-dimensional optimiza tion problem subject to inequality constraints. Beyond introducing the nonlinear programming problem, they also proved the main theorem of the the ory-the Kuhn-Tucker theorem which later became so famous. This theorem gives necessary conditions for the ex istence of an optimal solution to a non linear programming problem, and launched the theory of nonlinear pro gramming. On the first page of the paper Kuhn and Tucker also revealed their spon sor:
This work was done under contracts with the Office ofNaval Research.
Faculty of Mathematics, The Open University, Milton Keynes, MK7 6AA, England
50
1 See e.g., [Dupree, 1 986] and [Zachary, 1 997].
THE MATHEMATICAL INTELLIGENCER © 2000 SPRINGER-VERLAG NEW YORK
In fact the joint work of Kuhn and Tucker on nonlinear programming grew out of an Office of Naval Research (ONR) project on game theory and lin ear programming which Tucker had led since 1948. The ONR was very im portant in the emergence and further development of nonlinear program ming. It functioned as a bridge be tween practical problem-solving in the "real world" and university-based re search in mathematics proper. In what follows I will trace this in teraction. Mobilization of Scientists in the USA during World War II
Kuhn and Tucker's work on nonlinear programming took place within a proj ect-originally on linear programming and game theory-which was initiated and fmanced by the Office of Naval Research. The background for under standing the interest of the Navy in such a project is the mobilization of civilian scientists in the USA during the second world war. The financial structure of science changed radically in the USA during World War II. Under the leadership of the MIT electrical engineer Vannevar Bush, science statesmen like James B. Conant from Harvard University, Karl T. Compton from MIT, and Frank Jewett from AT&T's Bell Laboratories initiated and organized the mobiliza tion of civilian scientists for the war ef fort. Before the war basic science had been fmanced largely by huge private (often family) foundations such as the Rockefeller Foundation. There was a widespread scepticism towards gov ernment influence on university sci ence. There was a fear that govern ment money in the universities would mean government control over scien tific research. All this changed as a consequence of the war. 1 Bush's vision was to create a system that could make scientific research in
war methods, weapon development, and defense systems more efficient than they were before the war. He wanted to create an organization of civilian scientists who worked on these problems independent of the mil itary. He established a system where the civilian scientists worked on mili tary-related problems and research, bound by contracts not directly with the different military establishments but through the Office of Scientific Research and Development (OSRD), an organisation that came into being in May 1941. OSRD was directly under the Congress and as such it was inde pendent of the military. Never before had civilians had so much influence on military affairs. The system worked as follows: A scientist who was believed to be able to handle a desired project was con tacted by OSRD. A contract was set up with the scientist as principal investi gator. It was then up to the principal investigator to appoint additional staff to work on the project, typically his or her own graduate students. Most of the scientists did not relocate their work to military laboratories, instead they stayed where they were-at the uni versities and in industry. This way of organizing the scientific war effort proved very efficient. One of the really great successes was the de velopment of radar technology, which took place at MIT. It created of course also a lot of problems. There was ri valry between the armed forces, and not everybody in the military thought it was a good idea to have civilian sci entists working on war-related issues outside the control of the military. There were, for example, a lot of prob lems connected with the use of radar, which the Navy initially simply refused to have anything to do with.2 Practical Mathematics: Solving A
Some of the military establishments also had their own scientific staff. In 1941 the U.S. Air Force hired the math ematician George B. Dantzig, who worked on the so-called programming planning methods-a tool in the Air Force for handling huge logistic plan ning. An Air Force Program was a pro posed schedule for activities. Dantzig gave the following explanation of such a program in 1951:
The levels of various activities such as training, maintenance, sup ply, and combat had to be adjusted in such a manner as not to exceed the availability of various equipment items. Indeed, activities should be so . . . the minimization of a linearform carefully phased that the necessary subject to linear equations and in amounts of these various equipment equalities. [Dantzig, 1982, p. 44] items were available when they were supposed to be available, so that the This is nowadays known as a linear activity could take place. [Dantzig, programming problem; originally 1951, p. 18] Dantzig called it Programming in a Linear Structure. According to Dantzig these meth ods for planning programmes were slow, expensive, and ineffective; they were built on personal experience, and incorporated a lot of ad hoc ground rules issued by those in charge. It took more than seven months to set up a program. Dantzig's job during the war was to train members of the Air Force staff to compute Air Force programs [Dantzig, 1968, p. 4]. 3 After the war Dantzig returned to the Air Force, where, from 1946 until 1952, he functioned as mathematical advisor for the Headquarters Staff of the U.S.A.F. The assignment he was hired to work on was to
. . . develop some kind of analog de vice which would accept, as input, equations of all types, basic data, and ground rules, and use these to gener ate as output a consistent Air Force plan. [Dantzig, 1988, p. 12]
Logistic Problem
Not all the scientists who participated in the war were organized by OSRD.
a very profound influence on Dantzig's work The idea of an "analog device" was rejected. Instead the work took a turn towards the development of what is now called linear programming. In the spring of 1947 the project SCOOP (Scientific Computation Of Optimum Programs) was established. The purpose of this pro ject was twofold: to build a mathemati cal model for the programming problem, and to assist the development and con struction of computers.4 It initiated a very intensive working period which resulted in a model that was reflected in the following mathe matical problem:
Around this time rumors about the computer started to circulate. This had
From Problem Solving to Theory: Linear Programming and Duality
The next problem was how to solve this model. Dantzig was advised to seek help from von Neumann [Dantzig, 1988, p. 13]. During the war von Neumann held a lot of advisory and consulting jobs in the military, and several of these continued after the war, so it was very natural for a mathematician working on a mathematical problem in the Air Force to pay a visit to him.5 It turned out to be a very fruitful meeting. Von Neumann had recently completed a book on game theory with Oskar Morgenstern, and-according to Dantzig-von Neumann immediately recognized the relationship between a linear programming problem and a two-person zero-sum game.6 The most important thing here for the development of nonlinear pro gramming is that von Neumann pro vided Dantzig's problem-the Air Force problem-with a theory, game theory, which had a mathematical
2See [Zachary, 1 997]. 3For further readings on the origin of linear programming see [Dantzig, 1 963, 1 982, 1 988, 1 991] and [Dorfman, 1 984]. 4See [Brentjes, p. 1 77]. 51n [Uiam, 1 958, p. 42] there is an incomplete list of von Neumann's relationship with the military establishment. 6For a history of von Neumann's perception of the minimax theorem in two-person zero-sum games and the different mathematical contexts it gradually appeared in, see [Kjeldsen, 1 997, 1 999a, 1 993b].
VOLUME 22, NUMBER 3, 2000
51
foundation in the theory of convexity and linear inequalities. This was very important for the later development of nonlinear programming, because it broadened the subject of linear pro gramming and made it a subject for mathematical research. This widened the interest from a narrow militarily defined problem to research areas within mathematics [1\jeldsen, 1999b ]. The Significance o f ONR: A Project in Game Theory and Linear Programming
The Office of Naval Research (ONR), which was established by the Navy in 1946, was an after-effect of the mobilization of scientists during World War II. Bush's organization during the war-OSRD-was an emergency orga nization, and it had been clear right from the beginning that the OSRD would disappear when the war ended. There was a common concern that the scientists would go back to their uni versity duties after the war. There also was a strong belief that the US had to be strong scientifically in order to be strong militarily. A lot of people were concerned about the further financing of science after the war, military-related science as well as basic science. 7 The Office of Naval Research was created to fill the void left by the dis appearance of the OSRD. The ONR was organized after the model Bush created for the OSRD. The scientists continued to work in universities and industry. Their relationship with the ONR was based on contracts. Every project had a principal investigator, and the financial support from the ONR covered salaries during the sum mers, salaries for research assistants working on the projects, conferences, guests, etc. In this way the ONR func tioned as a bridge between the interest of the military and peacetime research at the universities. During the first four years of its existence, the ONR was the main sponsor for government-sup ported research in the USA.8 The possible applications ofDantzig's model in connection with the develop ment of the computer caused the ONR to set up a special logistic branch
within its mathematics program. Mina Rees, who was the head of the Mathematics Division of the ONR, has described it like this:
. . . when, in the late 1940's the staff of our office became aware that some mathematical results obtained by George Dantzig, who was then work ingfor the Air Force, could be used by the Navy to reduce the burdensome costs of their logistics operations, the possibilities were pointed out to the Deputy Chief of Naval Operations for Logistics. His enthusiasmfor the pos sibilities presented by these results was so great that he called together all those senior officers who had any thing to do with logistics, as well as their civilian counterparts, to hear what we always referred to as a ''pre sentation". The outcome of this meet ing was the establishment in the Office ofNaval Research of a separate Logistics Branch with a separate re search program. This has proved to be a most successful activity of the Mathematics Division of ONR, both in its usefulness to the Navy, and in its impact on industry and the uni versities. [Rees, 1977a, p. 1 1 1] The theoretical connection between the linear programming model and mathematics proper-the theory of games in applied mathematics and the theory of convexity and linear in equalities in pure mathematics-made it an obvious subject for a university based ONR project. Thus in the spring of 1948 Dantzig went back to Princeton, this time on behalf of the ONR, to discuss with John von Neumann the possibilities for a university-based project financed by ONR on linear programming, its rela tions to game theory, and the underly ing mathematical structure. In an interview Tucker has described how at this occasion he was introduced to Dantzig and gave him a ride to the train station. During this short car trip Dantzig gave Tucker a brief introduction to the linear programming problem. Tucker made a remark about a possible connection to Kirchoff-Maxwell's law of
7See [Rees, 1 977], [Schweber, 1 988] and [Dupree, 1 986]. 8See [Sapolsky, 1 979], [Schweber, 1 988], [Old, 1 961], [Zachary, 1 997].
52
THE MATHEMATICAL INTELLIGENCER
electric networks, and because of this remark Tucker was contacted by the ONR a few days later and asked if he would set up such a mathematics pro ject [Interview, Albers and Alexander son, 1985, p. 342-343] . Tucker agreed to become the princi pal investigator for the project, and that changed his research direction com pletely-until this moment he had been absorbed in research in topology. The same happened for Kuhn, who at the time was finishing a Ph.D. project on group theory. Kuhn went to Tucker to ask for summer employment in the sum mer of 1948, because he needed the ad ditional income [Kuhn, 1998, Interview] . Tucker hired him together with another graduate student, David Gale, to work with him on the ONR project. What ONR did here was to initiate research in the connection between game theory and linear programming and the underlying mathematics. They placed this research in a university context and staffed it with mathemati cians normally engaged in research in pure mathematics. Tucker and his group presented the results of their work within the project at the first conference on linear pro gramming which took place in Chicago in June 1949 [Koopmans, 1951]. They had developed the mathematical theory of linear programming. Most prominent among their results was the duality the orem for linear programming: To a lin ear programming problem one can for mulate another linear programming problem on the same set of data called the dual program. The duality theorem says that the original, or primal pro gram has a finite optimal solution if and only if the dual has a finite optimal so lution, and the optimum value will be the same [Gale, et al., 1951]. The connection between game the ory and linear programming provided the Air Force problem with a mathe matical theory, and in so doing changed the scientific status of linear programming. Linear programming was no longer just a practical problem the military wanted solved, but formed part of mathematical disciplines such as linear inequality theory, convex ,
analysis, and game theory. The con nection to game theory broadened the subject of linear programming and sug gested further mathematical problems [1\jeldsen, 1 999b ]. This enhanced linear programming's attractiveness as a potential mathemat ical research area. This change in sci entific status was crucial for the further development of mathematical program ming, including nonlinear programming. Until Tucker got involved, the dri ving forces behind the development were practical applications-the solv ing of the Air Force programming problem. Tucker, Kuhn, and Gale, on the other hand, worked within a uni versity context, and the duality theo rem for linear programming made it an interesting research area in mathemat ics. And this is where nonlinear pro gramming enters the picture, because what happened was that Kuhn and Tucker never really left the project. Nonlinear Programming
In the autumn of 1 949-that is, a few months after the first conference on linear programming-Tucker went to Stanford on leave. He had time to ex plore his first intuition about linear programming-the resemblance to Kirchoff-Maxwell's law for electrical networks. He perceived the underlying optimization problem of minimizing heat loss. The objective function was not a linear but a quadratic function. This suggested to Tucker that maybe the Lagrangian multiplier method could be adapted to optimization prob lems with inequality constraints [Kuhn, 1 976, p.12-13]. Tucker then wrote to Kuhn and Gale and asked if they were interested in continuing their work to extend the du ality theorem for linear programming to quadratic programs [Kuhn, 1976, p.13]. David Gale said no, Kuhn on the other hand said yes. Some way along the working process, Kuhn and Tucker changed the focus from quadratic pro grams to the general nonlinear case. The central idea underneath Kuhn and Tucker's development of nonlin ear programming was the saddle-point property of the associated Lagrangian
function. From the linear programming problem: n
maximize
g(x)
=
I cixi,
i=l
subject to n
fh(x)
=
bh - I ahixi � 0, i=l
h=
1,
...
Xi � 0,
, m, i = 1,. . . , n, ahi• bh E R,
Kuhn and Tucker formed the corre sponding Lagrangian function:
¢(x,u)
=
g(x) +
I uhfh(x),
They realized that :xfJ C:xfJ1 , . . . , xDn) will maximize g(x) subject to the given constraints if and only if there exists a vector u0 E Rm with non-negative com ponents (multipliers), such that (:xfJ, u0) is a saddle-point for the Lagrangian ¢(x,u) [Kuhn and Tucker, 1950, p. 481]. The really neat thing about this sad dle-point property was, as Kuhn and Tucker phrased it, =
The bilinear symmetry of ¢(x, u) in x and u yields the characteristic dual ity of linear programming. [Kuhn and Tucker, 1 950, p. 481] If :xfJ is a solution to a linear pro gramming problem and (:xfJ,u0) is the saddle-point for the corresponding Lagrangian function, then u0 will be an optimal solution to the dual program ming problem. If the object was to ex tend the duality theory for linear pro gramming to the more general case of nonlinear programs, it would seem nat ural to take the saddle-point property of the Lagrangian as a starting point. And this was exactly what Kuhn and Tucker did. They proved that a necessary condi tion that a point :xfJ E Rn solve a non linear programmi ng problem is the ex istence of a point (multipliers) u0 E Rm, with the property that (:xfJ,u0) satisfy the necessary conditions for being a saddle point for the corresponding Lagrangian function. This result came to be the cel ebrated Kuhn-Tucker theorem. By re quiring concavity and differentiability
of the functions involved, the objective as well as the constraint functions, Kuhn and Tucker obtained complete equivalence between solutions to a nonlinear programmin g problem l!Pd saddle points for the corresponding Lagrangian [Kuhn and Tucker, 1950]. The conditions for being a saddle point for the Lagrangian, that is, the necessary conditions for being a solu tion to a nonlinear programming prob lem, are now called the Kuhn-Tucker conditions. The theorem about neces sary conditions is called the Kuhn Tucker theorem-or more properly the Karush-Kuhn-Tucker theorem.9 That theorem launched the mathemat ical theory of nonlinear programming. Conclusion
It was the duality theorem for linear programming-that is, a purely theo retical result-that sparked the inter est of Kuhn and Tucker. It was the du ality theory they wanted to extend to the general (quadratic) nonlinear case. It is in this respect that I fmd the de velopment of the duality theorem in linear programming so crucial for the emergence of nonlinear programming. Even though nonlinear program ming originated in a context of linear programming, the driving force behind Kuhn and Tucker's development of nonlinear programming was indeed very different from the stimulus that started the development of linear pro gramming. Linear programmin g origi nated in the context of the concrete solving of a practical problem within the Air Force. Nonlinear programming on the other hand developed in accor dance with the inner rules for research in mathematics proper as it is typically done in a university setting. The ap pearance of a concrete logistic prob lem played a decisive role in the origin of linear programming, but Kuhn and Tucker's research into nonlinear pro gramming was not motivated by a prob lem of this kind. The ONR did not pro vide Kuhn and Tucker with a concrete practical problem that they required to be solved. What they did provide was fi nancial support for mathematical re search-applied as well as pure-as
9William Karush proved a version of the theorem that later got known as the Kuhn-Tucker theorem in his master's thesis from 1 939 [Karush, 1 939]. For a contextual ized historical analysis of the aspect of the multiple discovery in connection with the Kuhn-Tucker theorem see [Kjeldsen, 1 999c].
VOLUME 22, NUMBER 3, 2000
53
long as it was related to optimization. This covered research in areas like game theory, linear inequality theory, theory of convexity, and mathematical pro gramming. It was in that spirit that Kuhn and Tucker's work on nonlinear pro gramming emerged. Even though Kuhn in his daily work did not feel the presence of the mili tary, one must say that the Office of Naval Research had an enormous in fluence on the origin of nonlinear pro gramming. From the perspective of the ONR the potential applicability was de cisive for the origin of the project. Tucker's project was not a product of university-based research in mathe matics proper, but came into being on the initiative of ONR, who ordered the research in these areas. ONR's influ ence on mathematical research areas like game theory and mathematical programming remained strong. They continued to support Tucker's project until 1972, when the National Science Foundation took over.
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Dantzig, G. B. (1 982): "Reminiscences about
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University of Roskilde in 1 999. She is
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female Ph.D. students of mathematics, Lulea
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University, Sweden, 1 997, pp.63-67.
cal Society, 64, 1 958, pp. 1 -49.
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history of Nonlinear Programming.
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Production
Kuhn, H. W. (1 976): "Nonlinear Programming:
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THE MATHEMATICAL INTELLIGENCER
Kjeldsen, T. H. (1 999a): "A History of the Minimax Theorem: a journey through differ ent mathematical contexts," in 9. November-
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Vannevar Bush, Engineer of the American Century. New York: The Free Press, 1 997.
NIKOLAI V. IVANOV
A To po ag i st' s Vi ew of the D u nford-Sc hwartz Proof of th e Brouwer Fixed Po i nt Theorem To the memory of G. -C. Rota
In his recent book Indiscrete Thoughts, G.-C. Rota pas sionately lauds the famous treatise Linear Operators by N. Dunford and J. T. Schwartz. In particular, Rota writes (see the section "Linear Operators: The Past," p. 33):
Section twelve of Chapter Five, presenting a proof of the Brouwerfixed point Theorem, is remarkable. The proof was submittedforpublication in ajournal in 1954, but was rejected by an irate referee, a topologist who was miffed by the fact that the proof uses no homology the ory whatsoever. Instead, the proof depends on some de terminantal identities, the kind that are again becom ing fashionable. Being just another topologist, I had only heard about this proof before, but these remarks prompted me to study and analyze it. It turned out that this proof is essentially the
standard topological proof in disguise. The differences are the following: Instead of the usual simplicial homology the ory, the de Rham cohomology theory is used and, in fact, only implicitly-as is clear to every topologist, it is suffi cient to deal with cochains (exterior differential forms in this case), and there is no need to mention the cohomol ogy groups explicitly. Back in 1954, de Rham cohomology and even differen tial forms were not such common knowledge as they are now. It is indeed remarkable that the Dunford-Schwartz approach to the Brouwer fixed-point theorem, apparently having a totally different motivation from the usual one, turns out to be the de Rham cohomology version of it. Dunford and Schwartz were working with de Rham coho mology without realizing it. The goal of this article is to uncover the topology hidden behind the Dunford-Schwartz proof. The calculus of exte-
© 2000 SPRINGER-VERLAG NEW YORK, VOLUME 22, NUMBER 3, 2000
55
rior differential forms will be the main tool. In particular, it makes quite transparent the otherwise mysterious determi nantal identities alluded to in the above quotation. After these identities are understood, the rest becomes transpar ent (to a topologist) also. So, the exterior forms, praised in another section of Rota's book ("Herman Grassmann and Exterior Algebra"), turn out to be the key to this proof. Now let us state the Brouwer fixed-point theorem. In the following, I sometimes use notations which are stan dard by now, but slightly different from the Dunford Schwartz ones, like Rn instead of En.
The Dunford-Schwartz proof consists of four parts. The first part is the following lemma, presenting the determi nantal identities.
suiting expressions are all zero because Di(t, x) = 0 for x E sn-l (indeed, j(t, X) = X for X E sn-l and all t, and, hence, fi(t, X) = 0 for X E sn-l and all t). Therefore J'(t) = 0, and the contradiction completes the proof of the Brouwer fixed-point theorem. Without doubt, the lemma is the most mysterious part of this proof. However, as we will see in a moment, once it is translated into the language of (exterior) differential forms, it becomes both transparent and trivial. After this, it is not hard to recognize the (proof of the) Stokes theo rem in the last part of the proof, and this opens the way to the topological underpinnings of the proof. Here is my proof of the lemma. (The Dunford Schwartz proof may be viewed as a down-to-earth ver sion of it: no differential forms are mentioned.) Consider f as a collection (f1, . . . , fn) of n real-valued functions. Consider the volume form w = dx1 1\ 1\ dx on Rn and n the form
LEMMA. Let f be an infinitely differentiable function of n + 1 variables (x0 , , Xn) with values in Rn. Let Di de note the determinant whose columns are the n partial de rivatives fx0, . . . , fxi- P fxi+ P . . . , fxn· Then,
on the domain of defmition off (the hat (/") indicates the omitted term). Clearly,
THEOREM. Let f be a continuous mapping of the closed unit baU qJJn in Rn to itself. Then, f has a fixed point, i.e., there exists an x E qJJn such that f(x) = x.
.
•
· · ·
.
The second part uses the Weierstrass approximation theorem to reduce the problem to the case of infinitely dif ferentiable maps. Any proof based on differential topology uses the same argument; proofs based on algebraic topol ogy use the simplicial approximation (which is more ele mentary, but not so widely known) instead. The third part is a completely standard argument re ducing the Brouwer fixed-point theorem to the nonexis tence of a smooth retraction qj)n � sn-l (where qJJn is the closed unit ball in Rn, as in the Theorem, and sn-l is the unit sphere) smoothly homotopic to the identity map. The last part, like the first one, looks mysterious at first sight. Letf [0, 1] X qJJn � qJJn be a smooth homotopy fixed on sn- \ connecting the identity map with a retraction qJJn � sn-l. Denote the determinant whose columns are the vectors fx,(t, x), . . . , fxJt, x) by Do(t, x) (where t E [0, 1] and x = (x1, . . . , Xn) E qJJn), and consider the integral
+ =
n
I Di dxo 1\ . . . 1\ d:i;i 1\ . . --<"-
i =O
is clear that /(0) is the volume of qJJn and, hence, /(0) * 0. On the other hand, D0(1, x) is identically zero (because 1 j(l, x) E s n - for all x) and, hence, /(1) = 0. To reach a contradiction, it remains to prove that I'(t) = 0. To this end, differentiate under the integral sign and use the above lemma (with x0 renamed as t) in order to conclude that n
I ( - ly-l . L
iJ
- Di(t, x) dx1 · · · dx . n 2lln OXt i= l . Finally, one applies the fundamental theorem of calcu lus to the individual integrals in the last formula; the re-
I'(t) =
56
THE MATHEMATICAL INTELLIGENCER
+� fX; dxi +
1\ dxn
=
• • •
)
0.
Now, dw = 0 simply because dw is an (n + I)-form on an n-dimensional space. Hence dO = df*w = f*dw = 0; but, n
dO = I
i =o
i a ( - l) - Di dxo 1\ axi
· · ·
1\ dxn.
The lemma follows. Thus, from the point of view of the theory of differen tial forms, the lemma is simply the statement that df*w = 0, trivially following from dw = 0. Turning to the last part of the proof, we clearly have
I(t) = It
.
• • •
J.tX®nf*w
and
l'(t) =
.!!:_ J. f*w = J.tX®n Lrf*w, dt tX®n
where T is the vector field iJ/iJt and Lr is the Lie derivative along T. Now let us use the Cartan homotopy formula
Lr = dir + ird, where ir is the interior multiplication by T (it is valid for any vector field in the role of 1). It follows that
Lrf*w = dir f*w + ir df*w = dirf*w,
with support in the interior of
because df*w = 0. Note now that dirf*w = dirfl
(� (t
= dir =d =
1
Di dxo 1\ · · · 1\ rfci · · · 1\ dx n
Di dx1 1\ · · · 1\
� 1\ ·
·
)
· 1\ dxn
n
0
)
. a (- 1)'- 1 - Di dx1 /\ · · · /\ dx n axi i�1 =
IOX'i!iln w' = IOx'i!iln f*w' J1X0Jn f*w' = 0. =
Now, w' defmes a cohomology class in the relative de Rham cohomology group H:JR(Rn, Rn\
L
(remember that x0
<
t now), and hence
ACKNOWLEDGMENT
This is exactly the Dunford-Schwartz formula for I'(t). In fact, it follows immediately from the lemma, but our ap proach reveals the homotopy formula Lr = dir + ird be hind it. Now, the proof can be completed by appealing to the fundamental theorem of calculus and the fact thatflct, x) = 0 for all i and (t, x) E [0, 1] X s n - 1 . From our point of view, it is more convenient to use the multivariable version of the fundamental theorem of calculus, namely, the Stokes theorem, and bypass the last formula for I' (t). In fact,
ltx'i!iln Lrf*w ltx'i!iln dirf*w ltxsn _1 irf*w =l 0=0 txsn- 1
I' (t) =
=
=
This work was supported in part by NSF Grant DMS 9704817. BIBLIOGRAPHY N . Dunford and
=
I[0,1]X0Jn df*w Ja([0,1]X0Jn)f*w = IOX'i!ilnf*w - J1 X0Jn f*w
1 958.
G.-C. Rota, Indiscrete Thoughts, ed. by Boston,
F. Palombini, Birkhii.user,
1 997.
A U T H O R
by the Stokes theorem and the fact thatf does not depend on t on [0, 1] X sn- 1 (which immediately implies irf*w = 0). The argument given by Dunford and Schwartz for I' (t) = 0 is essentially a specialization of the standard proof of the Stokes theorem for our case. Now, when the mechanics of the proof is clear, it is only natural to replace the (infinitesimal) Cartan homotopy for mula and the n-dimensional Stokes theorem by the direct use of the homotopy f and the (n + 1 )-dimensional Stokes theorem; namely, because df*w = 0, we have 0
J.T. Schwartz, Linear Operators, Part 1: General Theory,
lnterscience Publishers, New York,
NIKOLAI V. IVANOV Department of Mathematics Michigan State Un iversity East Lansing, Ml 48824- 1 027, USA
=
e-mail: [email protected] =
I(O)
-
/(1),
because
Nikolai Ivanov graduated from the Leningrad State University in
1 976 and received his Ph.D. from the Steklov Mathematical 1 988,
Institute in 1 980 under the direction of V. A. Rokhlin. In
I[O,l] XSn-
he received the Doctor of Sciences degree (something simi
1 f*w
=
0
[since .ftt, x) does not depend on t for X E s n - 1 ] , and 0 X
lar to the Habilitation in Europe). again from the Steklov Mathematical Institute. Since
1 991 , he has been at Michigan
State University, first as a visitor, then as permanent faculty, with a half-year break spent at Duke University. He also kept his former position at the Steklov Institute for a while, but, not surprisingly, eventually lost it. His mathematical interests in clude
low-dimensional topology,
mapping-class
groups,
Teichmuller spaces, and quasiconformal maps- bridging, like this article, Topology and Analysis.
VOLUME 22, NUMBER 3, 2000
57
1i.J$Mfij,J§,£ih$ili.l!IQMI
The Grave of Henry Briggs Thomas Sonar
Does your hometown have any mathematical tourist attractions such as statues, plaques, graves, the cafe where the famous conjecture was made, the desk where the famous initials are scratched, birthplaces, houses, or memorials? Have you encountered a mathematical sight on your travels? If so, we invite you to submit to this column a picture, a description of its mathematical significance, and either a map or directions so that others may follow in your tracks.
Please send all submissions to Mathematical Tourist Editor, Dirk Huylebrouck,
Aartshertogstraat 42,
Dirk H uy l e b ro u c k , Editor
F
or quite some time I was looking for traces of Henry Briggs (156111630), inventor of logarithms of base 10, and one of the first creators of the finite difference calculus, which he employed to compute his logarithms. He started his career as a Reader at Cambridge University and was ap pointed the first professor of Geometry in 1596 in Gresham College. In 1619 he was elected to hold one of the Savilian chairs at Oxford University, where he lived and died in Merton College. Some years ago a friend of mine went to Oxford and chose Merton College to stay during his PhD studies, which gave me the opportunity to visit the college quite regularly. Every visit was of course accompanied by a visit of the nice chapel, but I never recog nised the grave of Henry Briggs. I am thankful to Prof. John Fauvel from the Open University, who drew my atten tion to the grave of Briggs in Merton College Chapel during an e-mail dis cussion on the early history of the dif ference calculus last year. Most famous Renaissance scientists were buried with a certain pomp, and even in Merton College Chapel there are fine examples of pompously styled tombstones of people scientifically not more important than Briggs. However, Briggs must have been cut out of an other kind of wood. His grave is marked by a plain stone on the ground with the Latin version of his name Henricus Briggius engraved. After John Fauvel drew my attention to this grave, my friend Gerald Warnecke from the Otto-von-Guericke University in Magdeburg, who was visiting Oxford at that time, took the photo of Briggs's -
I
last resting place. The plain design of his grave wonderfully fits into the de scription of Briggs in A Memoir on the
Life and Work of that most famous and most learned man Mr Henry Briggs written by Thomas Smith, 1707: Briggs was a man endowed with a quiet and holy character, no man's enemy, ready of access for aU, devoid of all pride, conceit, sullenness, envy, jealousy or greed, despising wealth, happy and contented with his lot; for he had sought and found quietude of mind in the Groves of Academe and the Temple of the Muses. At the same place we find the follow ing transcription from the College Register:
January 26, 1 630. Dies in College, Henry Briggs, Fellow, a man indeed of most spotless character and life. He had devoted himself from his youth onwards to geometry, first at Cambridge in the Society ofSt. Johns College. Then he had continued this study as public Reader in Gresham College, London, for many years. Then Sir Henry Savile summoned him, as the most learned man of his time, to Oxford to be the first to jill the chair of Geometry on his Foundation. . . . Unfortunately, there are large gaps in our knowledge of the life of Henry Briggs. Neither Gresham College in London, nor Merton College in Oxford keep original manuscripts of his. The Bodleian Library keeps a handful of private letters which do not contain
1 Goldstine in A History of Numerical Analysis from the 1 6th Through the 1 9th Century, Springer-Verlag, 1 977, gives 1 556 as the year of Briggs's birth. A referee pointed out that the parish register says 1 561 and that there is no reason to believe that the register is wrong. Also Rouse Ball in his A Short Account of the History of
8400 Oostende, Belgium
Mathematics, Dover Publications, 1 960 (Reprint of the first edition 1 908), gives 1 561 near Halifax as date and
e-mail: [email protected]
place of Briggs's birth.
58
THE MATHEMATICAL INTELLIGENCER © 2000 SPRINGER-VERLAG NEW YORK
Figure 1. Merton College viewed from the chapel's tower.
significant information. Of highest in terest would be the knowledge about the relation between Briggs and Thomas Harriot. Goldstine, in a discussion on the dating of a Harriot manuscript, writes, This dating suggests that
Briggs, upon his arrival at Oxford, references suggest that distinguished learned of this interpolation formula contemporaries like . . . Briggs knew from Harriot, since the two over something of Harriot's scientific lapped by about two years, and the work. However, it is not known Harriot entry in the Dictionary of Scientific Biography states, . . . scanty
whether such relations existed at all. Did interpolation by finite difference formulae emerge out of a collaboration of the two genii? Acknowledgment
The author would like to thank a ref eree for pointing out certain weak nesses of a former version of the man uscript. lnstitut fOr Analysis Technische Universitat Braunschweig PockelsstraBe 1 4 D-381 06
Braunschweig
Germany Figure 2. The grave of Henry Briggs.
e-mail: [email protected]
VOLUME 22, NUMBER 3 , 2000
59
ERIC BAINVILLE AND BERNARD GENEVES
Con structi o n s U s i n g Con i cs he classical problems of constructibility using ruler and compass (duplication of the cube, trisection of an angle, quadrature of the circle, construction of the regular polygons) have been solved through the works ofRene Descartes (1 637), Karl Friedrich Gauss (1 796), Pierre Laurent Wantzel (1837), and Ferdinand Lindemann (1882) (see [3, 8]). In a recent paper, Videla [ 1 1] characterizes the points constructible by ruler, compass, and a "conic drawing tool." In this article, we present constructions using these three tools. The effective realization of these constructions is possible using the Cabri-Geometry software, which in tegrates the conics as base objects. To begin, we will recall the definitions of "constructible" using different tools. Then we give some theorems char acterizing constructible objects. Next, we discuss con struction of regular polygons. We show some known con structions of the polygons with 5, 7, 9, 13, and 17 sides, and some new constructions of the polygons with 19, 37, 73, and 97 sides. We close with some remarks on the auto mated construction of regular polygons.
Constructibility Definitions
Given two distinct points a and b, they define a unique line (containing a and b) and a unique circle (centered in a and containing b). No line or circle can be defined by two iden tical points.
stands for "ruler & compass. " Let A c IR2 be a set of points. Let (lftCf6(A) be the smallest set containing A such that the inDEFINITION 1 ((lftCf6-constructible point). (!ftC€
60
THE MATHEMATICAL INTELLIGENCER © 2000 SPRINGER-VERLAG NEW YORK
tersection points of two primitives (line and circle) de fined from points of (lftCf6(A) are in (lftCf6(A). Trivial inter sections (when the number of intersf}ction points is in finite) are considered empty. (lftCf6(A) is called the set of points mCf6-constructible from A. A complex number x + iy is mCf6-constructible if the cor responding point (x, y) is mCf6-constructible from the set { (O,D), (1,0) }. DEFINITION
2
(mCf6-constructible
number),
(!ftC€
=
(lftCf6({ (O,D), (l,D)}) denotes the set ofmCf6-constructible num bers. We can now add conics as primitives. Five points are usually on a unique algebraic curve of degree 2 (conic). When more than one conic passes through five given points, we will say that these points define no conic. The defmitions of mCf6-constructibility can then be extended as follows.
Let A C IR 2 be a set of points. Let Cf6 2 (A) be the smallest set containing A such that the intersection points of two primitives (line, circle, conic) defined from points of Cf6z(A) are in Cf6z(A). Trivial intersections are considered empty. Cf6 2 (A) is called the set of points Cf6 2-constructible from A. DEFINITION 3 (C€2-constructible point).
DEFINITION 4 (C{;; 2-constructible number). C(;; 2 = C(;; 2 ({ (O,Q), (1,0)}) denotes the set ofC{i;2-constructible numbers. In the definition of "conic-constructible" by Videla [11], conics are defmed from a point F (focus), a line L (direc trix), and a number e (eccentricity). The conic is then the set of points M of the plane satisfying dist(M, F) = e dist(M, L). These definitions are equivalent (five distinct points of the conic can be \JlC{;i-constructed from these elements and, conversely, the elements of the conic can be \JlC{;i-con structed from five points defining the conic uniquely). Characterization
THEOREM 1 (Wantzel, 1832) . \JlC{;; is the smallest subjield of IC stable under conjugation and square root.
is the smallest subjield ojiC stable under conjugation, square root, and cube root.
THEOREM 2 (Videla, 1997). C(;; 2
These theorems can be proven using the tools of Galois theory, as e:xplained by Stewart in [ 10]. Constructions
Given two complex numbers x and y, \JlC{;i-constructions of x + y, xy, 1/x, -x, and x are well known. Given a positive real r, the \JlC{;i-construction of r112 is also well known. C{;; 2-construction of r113 was first presented by Menaechmus (350 Be): given two numbers a and b, he showed how to construct two numbers x and y such that a!x = xly = ylb using two parabolas (see [3, 1 1]). The trisection of an arbitrary angle using conics was first accomplished by Pappus (third century); see [ 1 1 ] . Sum and product, bisection and trisection of angles
Roots of polynomials of degree 2 and 3
Given reals s and p, the two real zeros o f P = X 2 - sX + p, whose sum is s and product is p, are \JlC{;i(s, p)-constructible. A simple con struction uses a Carlyle circle, named after Thomas Carlyle
Figure 2. Construction of the zeros of X 3 + X 2 - 2X - 1 as the in
tersection of the right hyperbola XY = 1 and a parabola with axis parallel to the x axis.
though found earlier by Descartes (see [4]). For A(O, 1) and let c be the circle of diameter [AB] (see Fig. 1). c intersects the x axis if and only if P has one or two real zeros, and in that case, the abscissas of the intersection points are the zeros of P [4]. Given reals a, b, and c, and P = X3 + aX 2 + bX + c. The real roots of P are constructible as the abscissas of the in tersection points between two conics defined from points \JlC{;i-constructible from {a, b, c). Several convenient choices of the pair of conics can be made, using either a fixed hy perbola (the right hyperbola XY = 1) or a fixed parabola (Y = X 2):
B(s, p),
•
•
•
Figure 1 . Real zeros x1, x2 of X 2 - sX + p constructed using the Carlyle method.
= 1 and cy2 + X + bY + a = 0. This parabola has Y = -b/2c as axis (dashed line) and passes through the points (-a + b - c, - 1), (-a - b - c, 1), and ( - a, 0) (black dots). See Figure 2. XY = 1 and X2 + aX + cY + b = 0, a parabola with axis X = -a/2. See Figure 3. Y = X2 and XY + bX + aY + c = 0. This hyperbola has
XY
axes parallel to the x and y axes (dashed lines) and the point (-a, -b) as center. Its equation can be rewritten
VOLUME 22, NUMBER 3, 2000
61
X2 + aX + cY + b ; 0
Figure 3. Construction of the zeros of X 3
+
X2
-
(prime numbers in boldface). This list is given by Gauss in [6] (item 366). Videla [ 1 1] has shown that the %-constructible regular m polygons have p = zn3 P lP2 " . .Pk sides, where m, n, k ;:::: 0 and Pi are distinct prime numbers of the form 2a3b + 1. Up to 300 sides, this corresponds to the 130 values 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, 1 7, 18, 19, 20, 21, 24, 26, 27, 28, 30, 32, 34, 35, 36, 37, 38, 39, 40, 42, 45, 48, 51, 52, 54, 56, 57, 60, 63, 64, 65, 68, 70, 72, 73, 74, 76, 78, 80, 81, 84, 85, 90, 91, 95, 96, 97, 102, 104, 105, 108, 109, Ill, 1 12, 1 14, 1 1 7, 1 19, 120, 126, 128, 130, 133, 135, 136, 140, 144, 146, 148, 152, 153, 156, 160, 162, 163, 168, 170, 171, 180, 182, 185, 189, 190, 192, 193, 194, 195, 204, 208, 2 1 0, 2 16, 218, 219, 22 1, 222, 224, 228, 234, 238, 240, 243, 247, 252, 255, 256, 257, 259, 260, 266, 270, 272, 273, 280, 285, 288, 291, 292, and 296. If Rp is known, R2p can be ffi'fi:-constructed from it by bisection of the sides, and R3p can be 'fi:2-constructed from it by trisection of the angles. If Rp and Rq are known, their superposition generates at least one side of Rm, where m is the least common multiple of p and q; this side can be replicated to obtain RmConsequently, we have only to give ffi'fi:-constructions for prime numbers of the form za + 1 , which are 3, 5, 17, 257, 65,537, . . . . There is very little probability that there exists another prime number of this form. For za + 1 to be
2X - 1 as the in
tersection of the right hyperbola XY = 1 and a parabola with axis parallel to the y axis.
(X + a)(Y + b) = ab - c; that is, X'Y' = ab - c in a co ordinate system with origin at the center. See Figure 4. Descartes used such methods involving circles and parabolas to find the roots of third-degree polynomials. The methods presented here can easily be defined as macro constructions and used as building blocks for complex fig ures. The constructions of the regular polygons with 73 and 97 sides presented at the end of this article would have been far more difficult to carry out without using these macroconstructions.
-
-
-
-
-
--:
-
-
I I I I I I I I I I I I I I I ...._ I I
_
_
Regular Polygons
Let Rp be the regular p-gon having the points (cos(2k7r/p), sin(2k 7T/p)) as vertices, k = 0, 1 , . . . , p - 1 . Gauss [6] has shown that the ffi'fi:-constructible regular polygons have p "" znPlP2" · ·pk sides, where n, k 2: 0 and Pi are distinct prime numbers of the form za + 1 (numbers known as Fermat primes). Up to 300 sides, this corre sponds to the 38 values 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 1 7, 20, 24, 30, 32, 34, 40, 48, 51, 60, 64, 68, 80, 85, 96, 102, 120, 128, 136, 160, 170, 192, 204, 240, 255, 256, 257, and 272
62
THE MATHEMATICAL INTELLIGENCER
Figure 4. Construction of the zeros of X 3 + X 2
-
2X - 1 as the in
tersection of the parabola Y = X 2 and a hyperbola with axes paral lel to the coordinate axes.
prime, a must be a power of 2. Numbers of the form 22" + 1 are called Fermat numbers. In June 1998, the smallest Fermat number 22" + 1 not yet checked for primality was 2224 + 1 [7); this number has more than 5 million digits. Similarly, we have only to give � 2-constructions for prime numbers of the form 2a3b + 1, with b > 0; there are 36 values up to 106: 7, 13, 19, 37, 73, 97, 109, 163, 193, 433,
487, 577, 769, 1 153, 1297, 1459, 2593, 2917, 3457, 3889, 10,369, 12,289, 1 7,497, 18,433, 39,367, 52,489, 139,969, 147,457, 209,953, 331,777, 472,393, 629,857, 746,497, 786,433, 839,809, 995,329. There are 8 more candidates in [106, 107), 8 in [ 107, 108], 7 in [ 108, 109], 7 in [ 109, 1010] , and a total of 231 values in [ 1, 1030]. Constructions of Rz, Rs, and R5 (and, consequently, Rn for n = 2k, n = 3 ·2\ n = 5 · 2k, n = 15·2k) were known in antiquity. For larger polygons, the only geometric con structions known are transpositions of algebraic solutions. The complex numbers corresponding to the vertices of Rn are zeros of the polynomial zn - 1. For n an odd prime,
the irreducible factors in Q[Z] of this polynomial are Z 1 and Pn = .zn-1 + .zn - 2 + · · · + Z + 1. The idea of the constructions is to decompose the ex tension of Q by the zeros of Pn into successive extensions of degrees 2 and 3, whose elements can be obtained by � constructions from the elements of the previous extension, starting with rationals. We will illustrate this in the fol lowing paragraphs. For the definition of field extensions, Galois groups, and their connection with geometric con structions, see [ 10]. Pn has no real zero and (n - 1)/2 pairs of cof\iugate zeros. The monic polynomial Qn whose zeros are 2 cos(2k7Tin) has degree (n - 1)/2 and integer coefficients. n It is defined by the equation zC - l)12Qn(Z + 1/Z) Pn. It is easier to construct the zeros of Qn. In that case, the poly gon obtained will be inscribed in the circle of radius 2 cen tered at the origin. Unless specified, this will be the case in all the following constructions. Gauss has given in [6) an efficient algorithm to build the =
Figure 5. Construction of R5 using a Carlyle circle. 0(0, 0), /(1 , 0),
Figure 6. Construction of R7• The parabola Y2 - X + 2Y - 1
J(O, 1), the circle c0 of center 0 and radius 2, and the axes are given.
tains the points (- 1 , 0), (-2, - 1 ), and (2, 1) (dark dots), which have
=
0 con
K(1/2, 0) is the middle of [0 /]; L(- 1/2, 0) is symmetric to K with re
integer coordinates and thus are very easy to construct. Lines par
spect to 0; c(L, Vs/2) is the Carlyle circle of center L passing through
allel to the y axis and passing through the three points of intersec
J; X1((-1 - Vs)/2, 0) and X2((-1 + Vs)/2, 0) are the intersections of
tion between the hyperbola and the parabola contain the six non
c and the x axis, and are the abscissae of four of the vertices of the
trivial vertices of R7.
pentagon inscribed in co.
VOLUME 22, NUMBER 3, 2000
63
sequence of equations defuting the zeros of Qn. In the follow ing paragraphs, we try to introduce this algorithm step by step, with some simplifications proposed later in the literature.
Rs Q5
With 13 sides, interesting things begin to happen. Q 3 = 1 X6 + X5 - 5X4 - 4X3 + 6X2 + 3X - 1 has degree 6. As we can only construct zeros of polynomials of degree 2 and 3, we have two choices:
=
X2 + X
-
1 has zeros ( - 1 :±: v5)/2. These zeros can
be constructed with the Carlyle algorithm, as shown in
5. The Carlyle circle has diameter [JB] and center with J(O, 1), B( - 1, - 1), and L( - 112, 0).
Figure
L,
R1a
R7
X3
1. Arrange the six zeros of Q13 in three groups of two val ues, each pair being the zeros of a polynomial of degree
2. In that case, the coefficients of the polynomials of de 2 are in an extension of I[) of degree 3. 2. Arrange the six zeros of Q1 3 in two groups of three val gree
ues, each triple being the zeros of a polynomial of de
+
X2 - 2X - 1 us ing one of the methods in the previous section. Figure 6 shows a construction using the method of Figure 2.
We can construct the zeros of
Q7
=
gree
3. In that case, the coefficients of the polynomials 3 are in an extension of I[) of degree 2.
of degree
There are 15 ways of grouping 6 zeros xk of Q 13 into 3 pairs, as enumerated in the fol lowing array [the pair (xi, Xj) is noted i.j]: Three groups of two values
Re
the
Although 9
is not a prime and can be constructed from R3 by
trisection, we can give a simple construction of it (Fig. 7).
Q9
= (X + l)(X3 - 3X +
1.2, 1.3, 1.4, 1.5, 1.6,
1). The zero - 1 corresponds to the
nontrivial vertices of R3. The three other zeros can be ob
1 and the parabola 0. The axis of this parabola is Y = 3/2, and it contains the points ( -4, - 1), (2, 1), and (0, 0) (dark dots).
tained by intersecting the hyperbola X¥ =
Y2 + X - 3 Y
=
3.4, 2.4, 2.3, 2.3, 2.3,
5.6; 5.6; 5.6; 4.6; 4.5;
1.2, 1.3, 1.4, 1.5, 1.6,
3.5, 2.5, 2.5, 2.4, 2.4,
4.6; 4.6; 3.6; 3.6; 3.5;
1.2, 1.3, 1.4, 1.5, 1.6,
3.6, 2.6, 2.6, 2.6, 2.5,
4.5 4.5 3.5 3.4 3.4
w be a primitive root of 1; in this array, k represents the = wk + w13 -k of Q13. If we consider, for example, the pair (xb x3) noted 1.3, its two values are zeros of X2 - sX + p, with S = X1 + X3 = W + w3 + w1 0 + w12 and p = X1X3 = w2 + w4 + w9 + w11 . By taking the successive powers of s, reduced modulo P 3( w), and looking for vanishing rational 1 Let
zero xk
linear combinations of these powers, we can fmd a polyno
mial of minimal degree with rational coefficients having s as
+ 2X 5 - 7X4 - 6X 3 + 5X2 + 5X + 1. 3 Let (} = e2 7ri1 1 . When w takes the 12 values { 0, 02, . . . , 01 2 } of the primitive 13th roots of 1, x1 + X3 takes only 6
a zero: X6
values. These values are the zeros of the polynomial we obtained.
3 w takes all its 12 possible values. We can restrict our search to the five pairs l.k for k = 2, 3, 4, 5, 6 (because the other pairs are obtained from these when w varies). It appears that only the pair 1.5 takes three values of s, which are (}1 2 + (}8 + (} 5 + (} for w E { (}, (} 5 , (}8, (} 1 2), 0 11 + 0 10 + ()3 + 02 for w E { ez, ()3, 0 1 0, 0 1 1 }, and 09 + (}7 + ()6 + (}4 for w E { 04 , (}6, 07, 09 }. The pairs corresponding to the last two values of s are, respectively, 2.3 and 4.6. Consequently, we have to select the pairs taking only
values of s when
The polynomial having these three values of
H = X3 + X2 - 4X + I.
s
as zeros is
We see that among the 15 possible choices, only 1.5, 2.3, 4.6 meets our needs. We can factor Q 13 in the extension of I[) by the zeros of H. Using Maple, one would type >
a l i a s ( a lpha=Root0 f ( _Z A 3 +_Z A 2 - 4 * _Z+ l ) ) :
>
f a c t o r ( X A 6 +X A 5 - 5 * XA 4 - 4 * XA 3 + 6 * XA 2 +
# 3 * X- l , a l pha ) ;
Figure 7. Construction of R9• The intersection between the parabola y2 + X - 3Y = 0 and the right hyperbola XY = 1 provides six vertices of R9• The other three vertices of R9 are the vertices of Ra.
64
THE MATHEMATICAL INTELLIGENCER
#
Th i s
Th i s
is
is
(X2 - aX + if + (l' 3), (X2 + (a2 + 2a - 2)X + a), (X2 + ( - if - a + 3) X - a2 - 2a + 2). -
H Q_l3
Figure 8. Construction of R1 3 grouping the six zeros in three groups of two. We first solve a polynomial equation of degree 3 using the dark gray parabola, and then three polynomial equations of degree 2 using the three circles, one for each zero of the first polynomial.
The last two factors are obtained from the first one
(X2 - aX + if + a - 3) when a takes its three possible values (the zeros of H ). We can now construct R13 (Fig. 8). First, we construct the zeros of H using the intersection of the hyperbola XY = 1 and the parabola
p 1 : y2 + X - 4Y + 1 = 0 (dark gray). Y 2 (dark gray dashed line) and
This parabola has axis
=
contains the points (2, 1),
( - 1 0), and (3, 2) (dark gray ,
points).
a, we have to com pute the zeros of X2 - aX + a2 + a - 3; that is, XZ - sX + p with s = a and p = if + a - 3. For this, we construct the parabola p : Y = XZ + X - 3 2 (light gray), giving Y = p from X = s. p has axis X = - 112 and z For each of the resulting values of
VOLUME 22, NUMBER 3, 2000
65
contains (0, -3), (1, - 1), and (2, 3) (light gray points). Using this parabola, we obtain the points (s, p = s2 + s - 3) used in Carlyle's method. The three Carlyle circles (lighter gray) give the six zeros of QI3 (lighter gray vertical dashed lines). Two groups of three values As we have seen earlier, we have to find a triple l.u.v of zeros whose associated sum XI + X + X = w + wiZ + wu + wiS-u + wv + wi S-v takes only u v two values when w takes all the values () k for k = 1, 2, . . . , 12. After computations, it appears that only the triple 1.3.4 satisfies this criterion. The two values it takes are the ze ros of H x z + X - 3. One factor of QI3 in the extension of II) by the zeros of H is X3 - aXz - X - 1 + a, =
where a is an arbitrary zero of H (taking the other zero gives the other factor). The construction of R is (Fig. 9) deduced from this de composition begins with the construction of the two zeros of H using the Carlyle circle centered at ( - 1/2, - 1) and containing J(O, 1). For each zero a of H, we then construct the three zeros of X 3 - aXz - X - 1 + a using the intersection of the hy perbola XY = 1 and the parabola ( - 1 + a)y2 + X - Y a = 0. This parabola contains the points (0, - 1), (0, a + 3), (2, 1), (2, a + 1), (a, 0), and (a, a + 2). The points (0, - 1) and (2, 1) are shared by the two parabolas and are shown in black in Figure 9; the other points used in the con struction of the parabolas are filled dots. Rn
Gauss established the 0tct:-constructibility of R 1 7 in 1796 (see [ 1, 4, 5, 9] for historical details about this discovery and other constructions proposed afterward). As we have seen for RI3, the construction is obtained by finding the zeros of three successive polynomials of de gree 2, the coefficients of a polynomial being constructed from the zeros of the previous polynomial. We have Q1 7 = X8 + X7 - 7X6 - 6X 5 + 15X4 + 10X3 10Xz 4X + 1. Only the pair 1.4 take four values: 1.4, 2.8, 3.5, and 6. 7. They are the four zeros of Hz = X4 + X 3 6Xz - X + 1. Continuing to the next level, we look for a pair of these pairs taking only two values. The only choices are 1.2.4.8 and 3.5.6. 7, the two zeros of HI = xz + X - 4. Let a be a zero of HI, we can factor H as Hz = (X2 + X + z aX - l)(Xz - aX - 1). Let f3 be a zero ofXZ - aX - 1; QI 7 can be factored in II) [ /3] , one of the factors being xz - {3X 3/2 + {3/2 + a{3/2 - a/2 [the four factors are obtained by sub stituting the four possible values of (a, f3) in this factor]. In the construction (Fig. 10), the values of a (1.2.4.8 and 3.5.6. 7) are obtained using the circle centered at ( - 112, -312) containing J(O, 1) (dark gray). The values of f3 (1.4, 2.8, 3.5, and 6. 7) are constructed using two circles, centered at (a/2, 0) and passihg through J (medium gray). The eight roots of QI7 are obtained by four circles (light gray). To simplify, instead of constructing -3/2 + {312 + a{312 - a/2, we can remark that XIX4 X3 + X5, XaXs = X + X7, X&r5 = Xz + Xg, and X5X7 = 6 XI + x4. Gauss used sign tests to match the sums with the dif ferent roots. We have used numerical approximations. -
=
66
THE MATHEMATICAL INTELLIGENCER
R1s
Q 19 = X9 + X8 - 8X 7 - 7X6 + 2 1X5 + 1 5X4 - 20X3 10Xz + 5X + 1. As in the case of 17 sides (8 = 2·2·2), we do not have a choice of the decomposition (9 = 3·3). We have to find the zeros of a third-degree polynomial, and then the zeros of other third-degree polynomials whose co efficients depend on the first three zeros. Gauss proposed a way to find directly the suitable triples or pairs, which we did by systematic search in the last para graphs. We first find a number p in 1, 2, . . . , 18 whose suc cessive powers modulo 19 generate a permutation of 1, 2, . . . , 18; for example, p = 2 can be chosen because the suc cessive powers of 2 modulo 19 are 1, 2, 4, 8, 16, 13, 7, 14, 9, 18, 17, 15, 1 1, 3, 6, 12, 5, 10. Then, for a divisor m of 18, the successive powers of pm take only 18/m values; for example, with m = 3, p3 = 8 and the powers of 8 modulo 19 are 1, 8, 7, 18, 1 1, 12. We can associate to this sequence the polynomial Z + ZS + z7 + zi8 + zu + ziZ, which takes only three values when z varies among the 19th roots of 1. More generally, let n be a prime number and p a prim itive nth root of 1 in the multiplicative group 1, 2, . . . , n - 1. For a divisor m of n - 1, and d satisfying 0 :s=: d < (n - 1)/m, we defme the sequence [m, d] as the polyno mial [m, d] = Ik =O,I, ... , m- I z(pd+k(n-IJ!m) mod n . This notation is slightly different from Gauss's notation in [6]. As an example, with n = 19 and p = 2, we have the fol lowing sequences (the sequences of even length can be identified as the sums previously defined): [ 18, 0]
1.2.3.4.5. 6.7.8.9 zi + zz + z4 + zs + zi6 + zis + z7 + zi4 + z9 + zis + zi7 + zi5 + zu + z s + z6 + ziz + z5 + z10, = zi + z4 + zi6 + z 7 + z9 + zi 7 + zu + z6 + z5, = z z + zs + zi s + zi4 + zis + zi5 +ZS + ziz + z10, = 1.7.8 = zi + z s + z 7 + zis + zu + ziz , = 2.3.5 = zz + zi6 + zi4 + zi 7 + z s + z5 , 1 = 4.6.9 = Z4 + zis + z9 + zi5 + z6 + z 0, i z zu + z7 + = , =
=
[9, 01 [9, 11 [6, OJ [6, 11 [6, 21 [3, OJ [3, 11 [3, 21 [3, 31 [3, 4] [3, 5] [2, 0]
zz + zi4 + zs, z4 + z9 + z 6, = z s + zi s + ziz , zi6 + zi 7 + z5, zis + zi5 + z10, = = XI = zi + zis , z 2 + Z1 7, [2, 1] = x 2 [2, 2 1 = x4 = z4 + z1 7, =
=
=
=
[2, 8J
=
[1, 0]
=
x9 = z9 + z1 0, Zl,
[1, 11
=
z z,
[ 1, 17]
=
Z1 0 .
Figure 9. Construction of R1 3 grouping the six zeros in two groups of three. We first solve a polynomial equation of degree 2 using the dark gray circle, and then two polynomial equations of degree 3 using the parabolas.
When Z varies among the nth roots of 1, [m, d] takes only (n - 1)/m values, which are [m, 0], . . . , [m, (n - 1)/ m - 1]. For n = 19, it follows that the sums 1. 7.8, 2.3.5, and 4.6.9 3 are the zeros of H1 = X + X2 - 6X - 7. Let a be a zero of H1; one of the three factors of Q 19 in Q [ a) is H2 = X3 aX2 + ( a2 - 5)X + a2 - 6.
The zeros of H1 are found using the parabola 7Y 2 - X + 6Y - 1 = 0. Its axis is Y = - 317 and it contains the points ( - 1, 0), (0, - 1), and (15/4, 1/2). The zeros of H2 are found using the parabola (a2 - 6) Y2 + X + (a2 - 5)Y - a = 0. Its axis is Y = - a/2 - 1 and it contains the points (a, 0), ( - a, a), and (a + 1, - 1). The corresponding construction is given in Figure 11.
VOLUME 2 2 . NUMBER 3 . 2000
67
R37
Q37 has degree 18, and we have the choice between three de compositions: 2·3·3, 3·2·3, and 3·3·2. We will present a con struction using the first one. With the notations of the previ ous section, we take n = 37 andp = 2. The 2 sequences [18, *] are zeros of H1 = Z2 + Z - 9, the 6 sequences [6, *] are ze ros of H2 Z6 + Z5 - 15Z4 - 28Z3 + 15Z2 + 38Z - 1, and the 18 sequences [2, *] are zeros of Q37· Let a be a zero of H1 ; the zeros {3 of H are zeros of Z3 - aZ2 + ( -4 - 2a)Z + 2 (7 + 2a); the zeros of Q37 are zeros of 11Z3 - 1 1 {3Z2 + (4{35 - 2f34 - 57{33 - 21{32 + 97{3 - 2 1)Z + (8{35 - 4f34 1 14� - 53{32 + 194{3 + 2). The coefficients of this last poly nomials are not easy to construct. Fortunately, they can be expressed using linear combinations of longer sequences. We have the following relations, which allow us to construct Ra7: =
[ 18, 0], [18, 1] zeros of Z 2 + Z - 9, [ 18, 1] < [18, 0] , [6, 0] , [6, 2], [6, 4 ] zeros of z s - [18, 0] Z 2 + ( -4 - 2 [ 18, O])Z + (7 + 2[18, 0]), [6, 4 ] < [6, 5] < [6, 1] < [6, 3] < [6, 0] < [6, 2], [ 2, 0] , [2, 6], [ 2, 12] zeros of za - [6, OJ z 2 + ([6, OJ + [6, 4]) z + c -2 - [6, 1]), [2, 1 7] < [2, 7] < [2, 4 ] < [2, 15] < [2, 13] < [2, 11] < [2, 10] < [2, 12 ] < [2, 6] < [2, 16] < [2, 3] < [2, 14] < [2, 9] < [2, 5] < [2, 2] < [2, 8] < [2, 1] < [2, 0] .
The other relations are obtained by "shifting" the se quences: [ m, d] is replaced by [m, (d + 1) mod (n - 1)lm] . In the proposed construction (Fig. 12), [ 18, *] are con structed using a circle (dark gray) of center ( - 1/2, - 4). [6, *] are constructed using two parabolas (medium gray). [2, *] are constructed using the six other parabolas (light gray). R73 and Rs7
As suggested by Bishop in [2], the construction can be re stricted to give only one of the zeros of Qn. All the vertices of a regular polygon with a prime number of sides can be obtained by reflections from any pair of vertices. The re flections correspond to products of nth roots of 1. For n = 73, we can choose p = 5 and the decomposi tion 36 = 2·2·3·3. This leads to the following equations: zeros of Z2 + Z - 18, zeros of z 2 - [36, OJ z + 4 [36, OJ + 5 [36 , 1 1 , zeros of [6, 0 ] , [6, 4], [6, 8] z3 - [18, OJ z2 - (2 + [18, OJ + [18, 3]) z + 3 + 2 [ 18, 0] - 2 [18, 3] , [2, 0] , [2, 1 2 ] , [2, 24] zeros of z a - [6, OJ z 2 + ([6, OJ + [6, 9]) z - 3 - [6, 8 ] . [36, 0] , [36, 1] [18, 0] , [ 18, 2]
I
\
Figure 10. Construction of R17 using three levels of Carlyle circles.
68
Tl-IE MATl-IEMATICAL INTlELLIGENCER
II I I I
I VI
I I I
I
= = =I = = = I I I I I
I I I I I I I I I I -
-
-
-
I -� -
I I
-
I
-
I - -I I - �-
T
I
-
I
�
-
-
I I I
I
Figure 1 1 . Construction of R19 using two levels of parabolas. The three roots of H1 are constructed using the parabola about Y = -3n. Each of these three zeros is used to construct a parabola giving three zeros of Q19•
Using these equations, "shifted" when needed, we can con struct (Fig. 13) the following 15 values in 6 steps (each line corresponds to the construction of the zeros of a second or third-degree polynomial): [36, 0] [18, 0] [ 18, 1] [6, 0] [6, 1] [2, 0] 48
= 3. 772, = 5.397, = 0.047, = 4.966, = - 1.580, = 1.992,
[36, 1] [ 18, 2] [18, 3] [6, 4] [6, 5] [2, 12]
= - 4.772, = - 1. 625, = - 4.819, = - 1.967, = - 1.538, = 1.429,
[6, 8] [6, 9] [2, 24]
= 2.398, = 3.166, = 1.544,
For n = 97, we can use p = 5 and the decomposition = 2·2·2·2·3, which corresponds to the equations
From these equations, we can deduce the following con struction (Fig. 14) in 1 1 steps involving 23 values: [48, 0 ] = 4.424, [24, 0] = 1. 189, [24, 1 ] = 2.104, [ 12, 0] = 2.493, [12, 1] = 5.304, [12, 2 ] = 0.079, [ 12, 3] = - 3.318, [6, 0] = -0. 666, [6, 2] = 1.531, [6, 3] = - 0.441, [2, 0] = 1.995,
[48, 1] = - 5.424, [24, 2 ] = 3.234, [24, 3] = - 7.529, [12, 4 ] = - 1.303, [ 12, 5 ] = -3.199, [12, 6] = 3. 155, [12, 7] = -4.210, [6, 8] = 3. 159, [6, 10] = - 1.452, [6, 1 1 ] = - 2.877, [2, 16] = - 1.379,
[2, 32]
=
- 1.282.
[48, 0] [24, 0], [24, 2]
Toward Automatic Construction
[2, 0 ] [2, 16] [2, 32] zeros of z3 - [6, OJ z2 + ( [6, OJ + [6, 1 1]) z - 2 - [6, 21.
In the previous sections, we have presented more and more optimized ways to build the relations leading to construc tions of the regular polygons. The final version can be sum marized as follows. Let n be an odd prime number of the form 2a3b + 1. For example, n = 28 · 3 + 1 = 769. The first step is to find p such that the powers of p modulo n generate the set { 1, 2,
zeros of [48, 1 ] Z2 + Z - 24, zeros of z2 - [48, OJ z + 2 [48, OJ - 5, [12, 0] , [ 12, 4] zeros of z2 - [24, OJ z + 2 [48, 11 + 3 [ 24, 21 - [24, 1], zeros of [6, 0], [6, 8] z2 - [12, OJ z + [24, 11 + [12, 7] ,
VOLUME 2 2 , NUMBER 3, 2000
69
. . . , n - 1 ). For example, p = 1 1 for n 769. Then, choose the decomposition of (n - 1)/2 into an ordered product of 2's and 3's. For example, for n 97 = 25·3 + 1, we have five choices: 2·2·2 ·2 ·3, 2·2·2·3 ·2, 2·2·3·2·2, 2·3·2·2 ·2, and 3·2·2 ·2·2. In the general case, we have CU+g-1) possible choices. It turns out to be more convenient to first solve second-degree polynomial equations. The next step is to find the second- and third-degree polynomials whose zeros are the sequences corresponding to the previous decomposition. For n = 433 = 24·33 + 1, with p = 5 and the decomposition 216 = 2·2·2·3·3·3, the lengths of the sequences are 216, 108, 54, 18, 6, and 2. We have to find the polynomials whose zeros are { [216, 0], [216, 1 ] ), { [ 108, 0] , [ 108, 2] 1 , ( [54, 0], [54, 4] ) , { [18, 0], [ 18, 8] , [ 18, 16J l, ( [6, OJ , [6, 24], [6, 48] }, and ( [2, OJ , [2, 72], [2, 144] ). The coefficients of these polynomials can be expressed as linear combinations of longer sequences, with integer co efficients. The first polynomial has integer coefficients. The whole polygon can be deduced from the knowledge of only one of the sequences [2, *l. say [2, 0]. We have to find the smallest set (or at least a reasonably small set) of sequences allowing the computation of [2, 0]. See the con structions of R73 and R97 for example (in the construction of R37, we constructed aU the [2, *] sequences, without this simplification). =
=
Finally, use the constructions of the zeros of second and third-degree polynomials to build the successive se quences and, eventually, [2, 0]. This value gives a second vertex-we already have the point (2, 0)-of the polygon, which can be used to build all the others using reflections. Conclusion
After recalling definitions and results about the con structibility of a geometric object, we have shown by more and more efficient methods how the works of Gauss, com puter algebra systems (Maple), and dynamic geometry soft ware (Cabri-Geometry, distributed by Texas Instruments) could be used together to construct regular polygons, us ing ruler, compass, and simple conics. In particular, we have given the list of small �2-constructible polygons, and presented new � 2-constructions of the regular polygons with 19, 37, 73, and 97 sides. The ancient Greeks gave precedence to constructions us ing only ruler and compass, not because they did not know about the other curves (they invented a number of mechani cal devices drawing some algebraic curves of degrees 2, 3, 4, and more), but for the neatness, perfection of reasoning, and the simplicity of the shapes involved (circle and straight line). Today's tools such as Cabri-Geometry enlarge the no tion of geometric simplicity by allowing the manipulation I
[ ) o] I
2 11 [2 ?I �2 5) [ , 0) [2 2) I [2 8
[lsi, 0)
I
I
[6)2)
I I
I I
I
I I
I
I I
I
I
I I
-------i I I
1__-
I I I
�
-·
I
Figure 1 2. Three-level construction of R37· The rst level is the construction of the two zeros of H1 using the large circle. Each of these zeros is used to construct a second-level parabola, giving the six zeros of H2•
70
THE MATHEMATICAL INTELLIGENCER
Figure 13. Construction of R73 in six steps. The first three steps use Carlyle circles to solve second-degree polynomial equations, and the
last three steps use the method of Figure 4 to solve third-degree polynomial equations.
of algebraic expressions (the sequences defmed by Gauss) and complex geometric objects (the conic sections). Some generalizations of the questions treated here may be considered:
3. J.-C. Carrega, Theorie des corps. La regie et !e compas, Hermann,
Paris, 1 981 . 4. D.W. DeTemple, "Carlyle circles and the Lemoine simplicity of poly
gon construction," Am. Math. Monthly 98 (1 99 1 ), 97-1 08. See The Geometry of Rene Descartes with a facsimile of the first edition,
1. What does the set of constructible numbers become if we consider algebraic curves of higher degrees? 2. What is the asymptotic distribution of the primes of the form 2a3b + 1? 3. Can the � 2-constructions of the regular polygons be fully automated? 4. Given n, what is the most efficient way of � 2-con structing Rn, in terms of number of steps and in terms of precision of the intersections involved (avoiding in tersection between near-tangent curves)?
8. F. Klein, Famous Problems of Elementary Geometry, 2 nd ed. ,
REFERENCES
9 . L.L. Smith, "A construction of the regular polygon of seventeen
Dover, New York. 5. J . P. Friedelmeyer, "Emergence du concept de groupe," Bull. Assoc. Professeurs Math. Enseign. Public (,A.PMEP), 83 (1 99 1 ) . 6. K.F. Gauss, Recherches mathematiques, Courcier Paris, 1 807
(French translation by A.C.M. Poullet-Delisle of Oisquisitiones arith meticae, Leipzig, Germany, 1 801 ). 7. W. Keller, "Prime factors k2n + 1 of Fermat numbers Fm and complete
factoring status," http://ballingerr.xray.ufl.edu/proths/fermat.html, July 1 998.
Chelsea, New York, 1 962. 1 . R.C. Archibald, "Gauss and the regular polygon of seventeen
sides," Am. Math. Monthly 27 (1 920), 323-326. 2. W. Bishop, "How to construct a regular polygon," Am. Math. Monthly 85 (1 978), 1 86-1 88.
sides, " Am. Math. Monthly 27 (1 920), 322-323. 1 0. I. Stewart, Galois Theory, 2nd ed. , Chapman & Hall, London, 1 989. 1 1 . C.R. Videla, "On points constructible from conics," Math. lntelli gencer 1 9(2) (1 997), 53-57.
VOLUME 2 2 , NUMBER 3 , 2000
71
48.1 .
24.3 .
48.0 24.2
12.1
12J 12.3 12.5 ,.
Figure 14. Construction of R97 in 11 steps. The first 10 steps use Carlyle circles to solve second-degree polynomial equations, and the last step uses the method of Figure 4 to solve a third-degree polynomial equation.
A U T H O R S
ERIC BAINVILLE
BERNARD GENEVES
laboratoire Leibniz, projet Cabri
Laboratoire Leibniz, projet Cabri
46, Avenue Felix Viallet
46 Avenue Felix Viallet
38000 Grenoble
38000 Grenoble
France
France
e-mail: [email protected]
e-mail: [email protected]
Eric Bainville was born in Paris in 1 969. He graduated from the Ecole Normale Superieure de Lyon, and has a PhD in com puter science from Grenoble. He has been working since 1 997
Bernard Geneves, born in Paris in 1 948, studied at Jussieu.
getting his diploma in 1 975. Starting as a teacher of mathe
matics in lycee in 1 972, and of computer science in 1 985, he
for the Cabri-Geometry project on algebraic curves and nu
has also worked on applications of computing in education,
merical algorithms. He enjoys mountain biking, squash, and
and on education of teachers. He has been with the Cabri
black and white photography.
group since 1 997. He is married with two teen-aged sons; his hobbies are reading, brico/age, biking, and classical music.
72
THE MATHEMATICAL INTELLIGENCER
iii§IH§I.Ifj
.J et Wi m p ,
Ed itor
I
Two· and Three-Dimensional Patterns of the Face
by Peter W. Hallinan, Gaile G. Gordon, A. L. Yuille, Peter Giblin and David Mumford Feel like writing a review for The Mathematical Intelligencer? You are welcome to submit an unsolicited review of a book of your choice; or, if
NATICK, MASSACHUSETTS, A K. PETERS, LTD., 1 999, viii+ 262 pp . US $48, ISBN 1 -56881 -087-3
REVIEWED BY JAN J. KOENDERINK
you would welcome being assigned a book to review, please write us, telling us your expertise and your predilections.
roughly defmed by the Venetian mask: brow, cheeks and the upper part of the chin, with holes at the eyes, nostrils, and mouth. The limitation to the face mask is necessary because of practical reasons (hairs are hard to scan be cause of their structure, eyes because of their mobility and optical structure, nostrils and ears because of geometri cal occlusion), as well as application oriented constraints (wigs and sun glasses are often used in disguise). When you
abstract
from
the
mi
crostructure and the optical properties
T
his is a book by five authors so well known that I reveal no secret when
I identify A.
of the skin, you are left with a smoothly curved, connected surface with four
L. Yuille as Alan. Unfortu
holes. The "curvature landscape" is
reads like a multi-au
quite intricate, making the face mask
thor book. I found it almost impossible
an attractive example of a generic
nately, the book
to suppress the urge to mumble to my
smooth surface patch with elliptic and
self, "Now I'm reading David," and so
hyperbolic areas. It has interesting lo
forth. The book is more like a pro
cal and global structures. Thus the face
ceedings than a well integrated per
mask makes a great setting for the nu
spective on the subject.
merical differential geometry of sur
As might be expected, there are
faces in three-dimensional space. That
many facial images in the book, and I
is what a large part of the book is
found several of the authors portrayed
about.
on its pages (David Mumford actually
From a physiological and anatomi
made the cover). The book is modem
cal point of view, the structure of the
in the sense that much of the effort has
face is due to the underlying bony
applications in computer vision, image
structure
processing and coding, but it hardly
structure, the superficial layer of fatty
(the skull), the muscular
refers to the history of the subject, ei
tissue, and the skin cover [ana] . Of
ther in general or mathematically. The
these the skull is to be considered (at
structure of faces is of great impor
any given age) a rigid body, whereas
tance in forensic science, surveillance,
the muscles may contract and thus
identification (borrowing books from
change the superficial geometry under
a library, authorized access), and con
conscious and unconscious control of
ference television, and considerable
the possessor of the face. Since there
funding is available for such studies.
are only few facial muscles and their
This contrasts starkly with the early
attachments to the bony structure are
roots of the subject, which lie in visual
fixed, the state of the musculature can
aesthetics and the study of human
be specified fully through fewer than a
character, areas of applications that
dozen parameters. (This is exploited in
will hardly help to keep the scientist's
conference television over phone lines,
pot boiling. This may explain the hap
for instance. One needs to send the ba
hazard collection of topics. The subject matter of the book is
sic facial structure only once, and sub sequently one only sends the para
Column Editor's address: Department
the geometry ("two- and three-dimen
metric changes of musculature and po
of Mathematics, Drexel University,
sional patterns") of the human face,
sitional and orientational changes of
Philadelphia, PA 1 91 04 USA.
that is to say, that part of the head
the skull. This suffices to reconstruct
74
THE MATHEMATICAL INTELLIGENCER © 2000 SPRINGER-VERLAG NEW YORK
the face at the receiving end. The view
the book as a whole. The chapter is fol
mensions. Klein's work is quite rele
ers look at a model, rather than a true
lowed by an "Overview of Approaches
vant to chapters
image.)
to Face Recognition" which succinctly
and
ridges
on
6 ("Parabolic curves surfaces") and 7
Artists working in the academic
covers the current trends in the com
("Sculpting a surface"). It would have
vein have to acquire a working knowl
puter vision of faces. After that the
been easy enough to scan Klein's bQst
edge of the anatomical and physiolog
chapters are (only loosely connected)
and finally check on its famous para
ical facts in order to be able to strike
discussions on various specific topics.
bolic curves (one wonders how the student managed at the task).
a likeness, or construct a facial ex
Again, I regret the absence of (short
pression in the absence of a model.
and introductory) pieces on anatomy/
There exists much more interesting
These topics are well understood and
physiology, (human) perception, and
material on facial structure that could
seem to me a necessary background of
especially-a historical setting.
The
have been used here to good advan
knowledge for work on the structure
work on faces in the western world is
tage. For instance, John Willats' data
of faces. Regrettably, though, this book
no doubt Lavater's [lav]. He managed
on
gives no summary review of the phys
to capture the interest of both the sci
London museum
an
early
Greek face from the
iology and anatomy, and if you want to
entific and the general public for a con
made a great example. There also ex ist series of sculptures on which ex
[will
would have
obtain a working knowledge of facial
siderable period. Most of his work is
structure from scratch, you'll have to
on silhouettes of the face in profile or
tensive
read through other literature first.
on frontal line drawings containing pri
speculations exist. Methods such as
literature
and
considerable
Faces are of interest for social rea
marily contour, edges of features, and
discussed in the book could have been
sons, thus the "stimulus" must be stud
creases, not at all on the surface struc
quite effective in settling a number of
ied along with the observer. Indeed,
ture. However, a bridge from surface
open issues. Examples are the Roman
there exists an extensive literature on
structure
crease
copies in marble of Greek bronze orig
face perception. Primates apparently
structure (ridges and courses of the
inals [wit] . It is speculated that the
reserve part of their brains especially
depth map) is definitely possible and
marked differences in the original and
for the recognition of faces and facial
would have been a welcome addition
the copies are due to attempts to coun
to
silhouette
and
expressions. The book doesn't men
to the book. Lavater's work was mainly
teract the effects of different optical
tion this perceptual side at all. This is
focussed on human character (which
properties of the materials on the vi
a pity: it is so central to the special rel
explains the wide interest), whereas al
sual shape. Another obvious target
evance of faces. Most topics treated in
most all of the other historical mater
would have been the set of dozens of
the book could equally well have been
ial centers on aesthetic issues. Interes
busts (actually self-portraits) of the
done on potatoes or peppers if grant
tingly, much of this has (different from
Austrian sculptor Franz Xavier Mes
money were forthcoming from the
Lavater's
relations
serschmidt [mes], produced over many
agricultural field. Faces are special be
to mathematics, thus it would have
years at the end of his career when he
work)
intimate
cause our brains are especially tuned
been most relevant to the present
apparently became somewhat of a
to them [vic].
book. Early work was done by artist
mental case. The variety of facial shape
The "bottom lines" of the various parts of the book relate primarily to is sues of computer vision and geometry,
per se.
mathematicians such as Piero della
in these busts is staggering, though the
Francesca
Diirer
portraits are all of a single person. The
[due], who were obsessed with the sci
modem methods, such as discussed in
[pie]
or Albrecht
Of
ence of proportion. Later Felix Klein
this book, allow novel attacks to be
course this is fine, but the buyer should
did some differential geometrical work
made on problems that have worried
not to the topic of faces
be aware of it and not be led astray by
much
book.
people for ages. It is as though the au
its title.
Although Klein does not seem to have
thors grind their knives but never use
written on the topic (except for asides
them, at least not on topics that might
Foundations
in some of his books), he apparently
arouse the interest of the academic
The book starts with a kind of credo
had a student
minded person. After all, who's really
(indeed called "manifesto") which I
in
the
spirit
of the
draw the parabolic
curves on a bust of the Apollo of
interested
take to be by David Mumford. He in
Belvedere (then an undisputed pinna
forensic purposes, say? Well, no doubt
troduces the notion of "pattern theory"
cle of male facial beauty). The result is
some people are, but I'm not.
due to Ulf Grenander.
still in the collection of the mathemat
work [gre]
Grenander's
in
recognition
rates for
is generally considered
ical institute at Gottingen, and is illus
Illumination, Images, and
"deep" (that is to say, hard to fathom),
trated in Hilbert and Cohn-Vossen's
Recognition
but, especially with our current per
well known popular book [hilb] as well
Part of the book is on the effect of var
spective,
as in the recent collection of pho
ious illuminations on faces. The topic
of
tographs of mathematical models by
is of interest to the computer vision
Bayesian inference by visual observers
Fischer [fis]. Klein apparently general
community and is largely independent
on the basis of optical input. Although
ized what William Hogarth [hog] called
of the specific topic of faces. It is
the chapter is regrettably short, it is
the "Line of Beauty" (the generic in
known from the visual arts that "shad
well worth reading and important for
flection of a planar curve) to higher di-
ing" can be a very effective tool in en-
Mumford
it makes a lot of sense. explains
it
in
terms
VOLUME 22, NUMBER 3, 2000
75
coding three-dimensional structures in
one can do quite well in such a re
though. For instance, consider an "oval
flat images.
stricted setting. Since the physics is so
with a dent". The dent will involve a bi
Moreover,
human
ob
servers manage to perceive shape quite
complicated, one skips it by process
tangent, two inflection points, and a
well despite the fact that retinal images
ing a large number of examples and
vertex (of the opposite curvature to the major part). The configuration is
are co-determined by the position of
"learning" the relevant models from a
light sources, not simply the perspec
training set. Although disappointingly
not arbitrary, either; for instance, the
tive of the fiducial object (so called
simplified from the perspective of the
pair of inflections must lie between the points of tangency of the bitangent.
"shape constancy"). It is not so easy to
physicist, the results are of interest to
understand how this comes about, for
the biologist and psychologist because
Such "feature clusters" are certain to
the relevant physics is quite compli
human and animal observers have to
be important in practice, yet they don't figure in the literature. One way to get
cated. Solutions have been formulated
function on a similar basis.
only for very simplified conditions,
chapters offer a snapshot of what goes
a handle on this is to view the dent as
such as rarely pertain to portraits.
on in computer vision, a rapidly devel
a result of some operation on an oval.
Yet successful recognition seems to
These
oping field.
When you "put the dent in" you must pass through a stage where the oval de
imply at least partial shape constancy. Although the general problem is a com
Differential Geometry
plicated one (in fact, too complicated
Chapter
to expect forthcoming solutions), the
Ridges on Surfaces") essentially con
of the flat point into inflection points,
field of face recognition may be suffi
siders modem solutions to the problem
etc.
ciently restricted to
one to
Felix Klein put to his student. Parabolic
contained in a single point, they really
frame workable solutions. Even in this
curves and ridges are curves on the sur
belong together! Similar analyses can
enable
6
("Parabolic
velops a flat point, and the features are Curves
and
seen to "originate" from an explosion Since the features were "once"
face that are invariant with respect to
be done in surface theory. The authors
order to arrive at some level of suc
Euclidean congruences. They may not
consider any surface as the end result
cess. The authors deal mainly with
be particularly relevant to the visual
of the deformation of an initially sim
more or less frontal views of the un
perception of faces because human vi
ple surface such as an ovoid.
obstructed face mask The faces are in
sual perceptions appear to be invariant
surface "develops," feature clusters are
setting you need many constraints in
As
the
a natural state (no stage cosmetics,
under more general transformations.
created, and by noting the history one
sun-glasses, and so forth), and are il
But Klein's conjecture that the pattern
may build a hierarchical order of fea
luminated by a uniform beam. The au
of parabolic curves might be a key to
tures. It is like noting the changes put
thors use the fact that faces are on the
the beauty in a face remains relatively
by a sculptor on an originally shape
whole quite similar to each other. For
unexplored.
less blob of clay that made it into a sim ile of a human face.
instance, it is easily possible to map
The authors offer a nice introduc
one face on the next. After all, that's
tion into the theory of ridges in the line
The one aspect that I fmd question
what we all do when we say that "so
of Ian Porteous's elegant text (por] .
able here is the specific way the se
and so has a long nose," "eyes wide
Much of this work is not widely known
quence is constructed. The authors
apart," and so forth: All such remarks
and deserves study. The chapter is well
simply interpolate between a face and
relate deviations from some canonical
worth reading by anyone with an in
a (best-fitting) triaxial ellipsoid. This
example.
terest in a modem discussion of the
seems really arbitrary; moreover, it
For computer-vision-savvy people there are some interesting tidbits to find in this part. I liked the section on
classical differential geometry of sur
doesn't recognize the fact that the fa
faces.
cial mask is not just a surface in three
Chapter
7 ("Sculpting a Surface")
dimensional space, but the boundary
fmding the parts of the face in shadow.
uses the material covered in the previ
of a head. What is of primary interest
One uses an enormously simplified
ous chapter to perform some really
is the three-dimensional region that is
model of the physics to approximate
novel geometry on intricately curved
the head, not its boundary. When the
the image in order to find outliers
surfaces. This chapter (an easy guess
face is to be embedded in a series of
which are likely to be specularities
of its
Peter
progressively blurred replicas, it seems
(too bright) and shadows (too dark).
Giblin) for me makes the book (I may
to me that the only procedure that
authorship
would
be
With the outliers identified, one ap
be prejudiced here because the au
makes sense is to blur (change the res
proximates once again and considers
thors put a tool to good use that I
olution of the observation of the head)
the outliers. When this process is iter
helped forge myself [koe ]. ) The issue
the characteristic function. Of course
ated, one arrives at a credible repre
is the relation between "features."
this
sentation of the illuminated and shad
Features are local singularities often
(which might be defined as the level
will
also
change
the
surface
used to describe an object in qualita
surface at 0.5 say), but in an essentially
tive terms. For instance, a planar curve
different way from the surface inter
step towards recognition. Indeed, suc
might be characterized by its inflec
polation procedure. For instance, it
cessful
tions, vertices
owed parts of the face. Such methods are used as a first
(curvature extrema),
would not necessarily conserve the
among a large number of others is the
and bitangents. It is obvious that not
topological structure of the surface.
holy grail of this field. It turns out that
just any set of features is appropriate,
That this makes physical sense is ob-
76
recognition
of
THE MATHEMATICAL INTELLIGENCER
individuals
vious when you consider that-at a
sider the course of future develop
very high resolution-the head is ac
ments. I fully agree with them that to
tion", Cambridge University Press, 1 994.
tually made up of atoms, thus the "sur
develop powerful statistical models of
[vas] Vasari, G . , "The Lives of the Painters,
face" has myriads of mutually discon
shape will certainly be a major line of
nected components.
endeavor
in
computer
vision.
[por] Porteous, I. R., "Geometric Differentia
Sculptors, and Architects", tr. A Hinds,
I'm
London, 1 927.
somewhat disappointed with their pre
[wit] Wittkower, R., "Sculpture: Processes and
possible with the advent of powerful
dictions as a whole, though. They con
Principles, " Harper and Row, New York,
computers
graphics.
sist mainly of a short list of topics cur
1 977.
This is really innovative (numerical)
rently being explored widely in the
Such feats as shown here are only and
computer
I said, this
field, several of which (from my per
Buys Ballot Laboratory
part alone would justify the expense of
spective) can already be considered
Princetonplein 5
differential geometry.
As
the book for me. The figures are stun
cui-de-sacs. I hope the future will turn
PO Box 80000
ning and repay close study.
out to be more exciting and hold some
3508 TA Utrecht
true surprises for us.
The Netherlands [email protected]
Range Images Most of the work in the book is done ei
REFERENCES
ther on images or image sequences, or
[ana] Bell, C., "The Anatomy and Philosophy of
on three-dimensional scans of faces.
Expression", 3rd ed. , George Bell, London,
This latter type of data
1 844.
is quite useful
and an innovation that was not avail
[vic] Bruce, V. and A Young, "In the Eye of the
able to the earlier authors. One nowa
Beholder. The Science of Face Perception",
days scans a head in a few seconds with
Oxford University Press, Oxford, 1 998.
sub-millimeter accuracy. The machines
[gre] Grenander, U . , "Lectures in Pattern
are used to produce similes for grave
Theory", 3 Vols . , Springer-Verlag, 1 976-
sculpture (without incurring the cost of
1 981 .
by Peter Hilton, Derek Holton, and Jean Pederson NEW YORK: SPRINGER-VERLAG, INC., 1 996. xvi + 351
pp .
a professional artist) and to scan peo
[pie] Francesca, Piero della. See: "Petrus Pictor
ple like Arnold Schwarzenegger for spe
Burgensis. De Prospectiva Pingendi ", ed. C.
cial effects in science-fiction movies.
Winterberg, Strassbourg, 1 899.
The book ends with chapters on
Mathematical Reflections: In a Room with Many Mirrors
[due] Durer, A, "Underweysung der Messung
such "range images." These are simply
mit dem Zirckel un Richtscheyt, in Linien
the raw results of scanning a face
Ebnen und gantzen Corporen", Nuremberg,
frontally. You obtain a rectangular ar
1 525.
US $39.95. Hardcover, ISBN
0-387-94770-1
REVIEWED BY JOHN MALITO
A
lthough this book is one within an extensive series of undergraduate
texts in mathematics, I remain unsure
as to whom the book is directed. Is it
ray of depth values, like a topographi
[lav] Lavater, J. C., "Essays on Physiogomy:
cal map of a landscape. Such "range
For the Promotion of the Knowledge and
aimed at secondary or tertiary level
images" are of interest because they
Love of Mankind", Robinson, London, 1 793.
students? Is it aimed at students or
clearly contain all the geometry, yet
[will Dubery, F. and Willats, J . , "Drawing
teachers? Indeed, is it aimed at anyone
The Herbert Press, London,
in particular? In their long preface the
[fis] Fischer, G. (ed.), "Mathematical Models"!
book is (at least) twofold. First, we
are not contaminated with the effects of light and dark as regular images are. Thus
range
respects
images
much
are
"cleaner"
in
many
input
to
recognition algorithms than regular (ra
Systems",
1 972.
authors say that "the purpose of this
Friedrich Vieweg & Sohn, Braunschweig/
want to show you what mathematics
Wiesbaden, 1 986.
is, what it is about, done-by those who
diance) images are. You may apply
[hilb] Hilbert, D. and S. Cohn-Vossen, "An
techniques you wouldn't apply to regu
schauliche Geornetrie", Dover, New York,
lar images. They are also of interest for
1 944.
academic reasons. The well-known aes
[hild] Hildebrand, A von, "Das Problem der
and how it is do it success
fully." Their second purpose is "to at tract you-and, through you, future readers-to mathematics."
thetic theory of the German sculptor
Form in der bildenden Kunst", Strassburg,
Their approach is to present eight
Adolf Hildebrand [hild] departs from
1 893, (Trs. M. Meyer and R. M. Ogden, " The
topics, one per chapter, drawn from
the frontal range image. It has a history
Problem of Form in Painting and Sculpture",
both "pure" and "applied" mathemat
that at least goes back to Vasari's [vas] account of "Michelangelo's method."
G. E. Stechert & Co. , New York, 1 907.)
ics. The chapter topics, which include spirals,
quilts,
and paperfolding as
The authors discuss the calculation
[hog] Hogarth, W., " The Analysis of Beauty", London, 1 753.
lead-ins to polar coordinates,
of surface features like ridges and
[koe] Koenderink, J. J . , "Solid Shape", The
geometry, number theory, and symme
courses: Nose ridge and bridge, eye
M . I .T. Press, Cambridge, Massachusetts,
try, are esoteric choices justified by the
sockets, and so forth. The idea is to
1 990.
authors' stressing that "mathematics is
do,
solid
build a map of features that can be
[mes] Messerschmidt, Franz Xaver (1 736-
used as the input to statistical recog
1 784). See: E. Kris, "Psychoanalytic Explora
we read and try to learn." Of course,
nition procedures (the final chapter).
tions in Art", George Allen & Unwin Ltd,
they do also adhere to the more tradi
London, 1 953.
tional method of setting problems for
In the conclusion the authors con-
something we
not just something
VOLUME 22, NUMBER 3, 2000
77
the reader to solve (Breaks and Final Breaks). Apart from the conspicuous absence of calculus, statistics, and lin ear algebra, many ideas are presented, culminating in a discussion of fractal geometry, in by far the most challeng ing chapter of the book Having said that, however, I suggest that the ninth
Some of you may recognize Figure 1 as polar graph paper but we won't worry about thatfor a moment or two. What we are interested in is that you go off and find a rectangular piece of cardboard. You'll need a pencil too. We'll wait here while you go and get them. (Chapter 1, page 2)
and final chapter, "Some of Our Own Reflections, " may be the most inter esting. In this last chapter, the authors of fer two sets of principles: one is of a general nature and refers to "the over all approach we should take to doing mathematics," while the other is more specific, dealing with things like effec tive notation. Following these is what the authors call an "Appendix" despite the fact that it is part of the final chap ter. This appendix outlines their prin ciples
of "Mathematical
Pedagogy,"
which hark back to their primary pur pose as given in the preface. They say that "there is no merit in the teacher completing the syllabus unless the stu dents complete it too." Who can argue with that? It all comes down to tech nique; things like using real world examples,
conceptual
proofs,
the
non-separation of intimately related topics like geometry and algebra, the generating of questions, and the incor poration of historical aspects and/or personages.
Peter
Hilton,
Derek
Holton, and Jean Pedersen have tried to do all these things, but their at tempts are inconsistent. The authors also stress that one should never be pedantic or unneces sarily precise. From this they develop what they call
their
"Principle
of
Licensed Sloppiness, " which is clearly the basis for the reader-friendly style in which this book is written. In fact, if it is read out loud it will sound like a collection of lecture transcripts. For some this may be a good thing, but as far as I am concerned, for a mathe matics text, this book is too imprecise in its language. At the same time, how ever, it often ironically flogs a dead horse, belaboring points already well enough made. Worse still, the presen tation is sometimes condescending:
78
THE MATHEMAllCAL INTELLIGENCEA
Surely, this book
will appeal to some
readers, but I was left feeling somewhat dismayed,
like
that
Douglas Adams
character unnerved by the printed in structions on a box of toothpicks. The mathematics presented are presented in an accessible but tedious manner. For example, everyone will come away from "Mathematical Reflections" with a bet
Are there two women in the world with the same number ofhairs on their heads? . . . Now an unintelligent way of approaching this hairy question is to put an ad in all the papers in the world, asking every woman to count (or have counted) all the hairs on her head, and then getting them to send back their answers. When all of these numbers are assembled, you could go through them and see if two were pre cisely the same. (Chapter 7, page 249)
ter understanding of polar coordinates, but
will they want to continue holding
Hilton's, Holton's, or Pedersen's hand into the next chapter? One more thing: why did the authors choose polar co ordinates as the subject for their first chapter anyway?
Department of Chemistry Cork Regional Technical College Rossa Avenue, Bishopstown Cork, Ireland
1¥Jflrrl .ii·h•l§i
R o b i n Wilson
Mathematics and Art I l l Florence Fasanelli and Robin Wilson
Delaunay's La joie de vivre
Mondrian's Broadway Boogie-Woogie
F RA NCE
POST ES 1971
I One can make out geometrical mathematical systems of arrangement in almost any work of art. -Rudolf Arnheim, Art and Visual Perception
At the end of the 19th century and dur ing the first quarter of the 20th century, artists, musicians, novelists, and poets were fascinated by non-Euclidean geometry and the fourth dimension and how to portray these concepts. Their in terest in mathematics, as well as in re cent scientific discoveries such as elec tromagnetic waves (confirmed in 1888), X-rays (1895), and electrons (1897), pro voked artists to look at the world in a new way and to record their observa tions in paintings and sculpture. Robert Delaunay (1885-1941) re joiced in the visual impact of colors, as is apparent in his famous 1931 paint ing Rhythme: Joie de Vivre repro duced on this stamp. The circles, bro ken and unbroken, represent the halos around glowing electric street lamps. Interpreting this work, Delaunay re marked "Tout est halo," adding that he had never seen a straight line in his life. Earlier he painted many versions of the Eiffel tower, celebrated by Parisians as the location for the transmission of Hertzian waves that made long-range communication possible. In addition to this interest in radio waves, Delaunay explored the wave nature of light and color. Delaunay and his contemporaries were aware of the new geometries, the work of Victor Schlegel, and the writ ings of Jules Henri Poincare, in part through the actuary Maurice Princet,
Vasarely's Trimensional design
who was deeply involved with Parisian artists at the beginning of the twenti eth century. In 1912, Princet wrote the catalog for Delaunay's first exhibition, which was held that year. Broadway Boogie-Woogie of 1943 is one of the last three paintings of jazz loving Piet Mondrian (1872-1944), in which he creates spatial ambiguity by colored (no longer black) lines. For years, Mondrian shared the enthusi asms of other artists in portraying the fourth dimension. He wrote that in his work, the negative or white space represented the fourth dimension. Mondrian attempted to portray the uni versal mathematical harmonies behind visual phenomena. Eventually, he re jected 3-dimensional visual reality as a superficial illusion. He believed that two dimensions represented objects clearly but that perspective represen tation, implying three dimensions, weakens them. Consequently, he made sure that all overlapping shapes and di agonals, which could be construed as 3-dimensional, were eliminated from his work Optical art stimulates and confuses our perception. In Vega-chess, Victor Vasarely (1908--97) utilised this illu sion by repeating the same motif over and over, making it difficult for us to see the picture. The painting, which on close scrutiny appears to have no background, can be hung with any side at the top because the picture has the eight symmetries of a square. His Tri mensional design is similarly based on symmetry. Vasarely said that he was in fluenced by isochromatic lines in physics and by the isobar maps he studied in school geography.
Please send all submissions to the Stamp Corner Editor,
Florence Fasanelli
Robin Wilson, Faculty of Mathematics,
Mathematical Association of America
The Open University, Milton Keynes,
1 529 1 8th Street NW
MK7 6AA, England
Washington, D.C. 20036 USA
e-mail: [email protected]
e-mail: [email protected]
80
THE MATHEMATICAL INTELLIGENCER © 2000 SPRINGER-VERLAG NEW YORK
