No title

•:enw

Se lf-Si m i l ar Hex-Sums of Squares R. BREU

eading the very stimulating article by A. van der Poorten, et al. in the Math­ ematical Intelligencer [1], I realized that self-similar sums of squares have a nice analogue in what I call self-similar hex-sums of squares, just as de­ composing primes of the form 4q + 1 into sums of squares a2 + b2 or factoring 4q + 1 in the ring of Gaussian integers has an analogue in the ring of Eisenstein integers, that is, decomposing primes of the form 3q + 1 into "hex-sums" of squares a2 + b2 ::!:: ab. ("Hex," as the fundamental domain in the Eisenstein lattice is a hexagon, versus a square in the Gaussian case.) Eisenstein integers have the form a ::!:: bw, where w ( - 1 + \1'=3)/2. Their norm a2 + b2 ::!:: ab corresponds to the norm a2 + b2 of Gaussian integers. Both signs ::!:: can be chosen, as the "upper sign" version is just the mirrored version of the "lower sign," but then have to be used consistently throughout, that is, al­ ways the lower or always the upper sign. A norm equation of Gaussian integers, say =

translates into a similar norm equation of Eisenstein integers (a2 + b2

::!::

ab) (c2 + d2

::!:: cd) (ac + bd

::!::

ab) (c2 + d2 ::!:: cd) (ac + bd

::!::

=

ad)2 + (be - ad)2

::!::

(ac + bd

::!::

ad) (be - ad)

or (a2 + b2

::!::

=

bc)2 + (ad - bc)2

::!::

(ac + bd

::!::

be) (ad - be).

It now turns out that all the formulas of self-similar sums of squares apply mu­ tatis mutandis to self-similar hex-sums of squares: If for a given k � 1, a and b form a self-similar hex-sum of squares, that is, a2 + b2

::!::

ab

=

lOk a + b,

then a' and b with a'= lOk- a+

b

also form a self-similar hex-sum a'2 + b2

::!::

a'h

=

1Qk a' + b;

and in the lower case in general we have two additional solutions (assuming a > b; one of a or a' is always >b) A

=

a and B'

=

a-b + 1

and A'

=

lOk

-

A + B'

=

l Ok - b + 1 and B'

I f p is a suitable factor of

with a hex-decomposition of then p is also a factor of i.e., a factor generating self-similar hex-sums, a1 ' lOk + a1 or a1 lOk + a 1 '

4

THE MATHEMATICAL INTELLIGENCER © 2008 Springer Science+Business Media, Inc.

=

a - b + 1.

(in the latter case a1 and a1', or in all formulas the primed and unprimed symbols need to be switched). Choose such a k0 and (p, a 1 , a/). For the "lower sign" there are in general (if p =F 3) 2*4 families of self-similar hex-sums (with A, A', B, B', a, a', b, b' all dependent on k):

10k 10k lOk lOk

a+h = a i 2( 102k +1Qk+ 1 )/p a'+ h = (a/ 10k+a 1 )2/p A+B' (AI 10k +A/)2/p A' +B' = Al ' 2 (102 k +10k + 1)/p u = 0, 1, . with k uk1+k0 =

=

and (if p

=F

3) 10k a+h' = (a1 10k+a/)2/p 10k a' + h' = a1'2 (102k+ 10k + 1 )/p 10k A+B A 1 2 (102k + 10k+ 1 )/p 10k A'+ B (A 1 ' 10k+A,)Z/p u = 0, 1, with k uk 1 +2k0 =

=

=

.

where k1 is the smallest solution to and A1 = a 1 , A1' = a 1 - a/ , if a 1 > a 1' A1 a1' - a1 , A1' = a1', if a1' > a1 =

But for the "upper sign" only 2 families exist:

lOk a+h = al2 (102k - 10k+ 1)/p lOk a'+h (a1' 10k +a1?/p =

u = 0, 1 , . .

with k=uk1+k0

The families are nice, that is, the b's never have leading zeros, whenever

10 a 1 a1'

�

p

As a final surprise it turns out that the Hermite-Serret algorithm for computing the square decomposition

a2+])2

=

p = 4q

+ 1

has an analogous "modified Cornacchia" type of algorithm [2] for computing the hex­ decomposition a2 +h2

±

ab = p = 3q

+

1

One simply solves x2 +3y 2

=

4p

by aePiying the Euclidian algorithm to 4p and 2*10k 0, else a=y b= ± (x - y)/2. EXAMPLES

ko

=

1, k1

=

3, a1

=

7, a1'

=

3, "lower sign" version

49 oo + 4)2/37 = 148, 132'447'568, 132'432'447'567'568,. =

© 2008 Springer Science+Business Media, Inc., Volume 30, Number 3, 2008

5

16 oo 0, then every point on the circle l z l = R is a limit point of zeros of the polynomials Pn(Z). Another theorem gives quantitative information on the location of those zeros. For e > 0, let (n) denote the number of zeros of Pn(z) that lie in the disk l z l < R + e. Jentzsch proves that lim supn--.oo (n)ln = 1 . Nothing better can be expected, because it may happen that lim inf,_.oo (n)/ n = 0, as Jentzsch shows by the example 1 + L�= I z"!. These theorems admit various generalizations. For instance, Jentzsch remarks without proof that the Faber polynomial expansion associated with any analytic Jordan curve displays similar behavior. All of these results are contained in Jentzsch's dissertation []4], which is reproduced almost verbatim in his paper []7]. A later paper USl continues the investigation. Jentzsch proved his main theorem as follows. Without loss of generality, suppose ao =I= 0. Let j(z) denote the sum of the series, so that Pn(Z) __,. j(z) in l z l < R. If the zeros of Pn(z) do not cluster at some point zo with lzol = R, then PnCz) =I= 0 for some e > 0 and all points z in the disk l z - zol :::::; e. Hence VP;JZ5 has a single-valued branch there. If Pn has degree v :::::; n, denote its zeros by z1 1, . . .

Zvn and write

Pn(z) = ao Since j(O) 0. Thus

=I=

(

1 -

� �n

)(

1 -

� �n

0, the points Zkn satisfy

)...(

)

1 - � .

�n

l zknl 2:: b for some b >

IVP;iJ51 :::::; � ( 1 + 1�1 ) :::::; C ( 1 +

R

; e)

in the disk lz - zol :::::; e. On the other hand, Pn(z) __,. j(z) and so � __,. 1 for l z - zol < e and lzl < R. By Vitali's theorem, VP;JZ5 __,. 1 uniformly on compact subsets of l z - zol < e. For l z - zol :::::; e/2, it follows that

l an z "l = IPn(Z) - P11- J(z) l :::::; 2(1

+ 8) ''

for each 8 > 0 and all n sufficiently large. Now choose 8 = ..£. and l z l = R + E to conclude that lim supn--.oc VlaJ i anl < 1/R, 4R 2 which says that the power series has radius of convergence larger than R, a contradiction. In earlier work []3], Jentzsch developed a continuous analogue of the celebrated Perron-Frobenius theorem on

22

THE MATHEMATICAL INTELLIGENCER

the existence of positive eigenvectors for matrices with pos­ itive elements. Specifically, Jentzsch showed that if a ker­ nel K(s, t) is continuous and positive in the square a :::::; s :::::; b, a :::::; t :::::; b, then the integral operator K 1 1 , contain only irrational points, {x E IR : :i2 = 51,

etc. Indeed, this can easily be strength­ ened: Semialgebraic sets have arbitrary homotopy types, singularities, or need points from large extension fields of IQ. But can realization spaces for com­ binatorial structures be so complicated and "wild"? It is a simple exercise to see that the realization space for a convex k-gon P C IR2 has a simple structure (equiva­ lent to [R Zk-6) . Moreover, Steinitz [29, 30] proved in 1910 that the realization space for every 3-dimensional polytope is equivalent to [Re-6 , where e is the number of edges of P. In particular, it contains rational points. A similar result was also stated for general polytopes [24]-but it is not true. A universality theorem now mandates that the real­ ization spaces for certain combinatorial structures are as wild/complicated/in­ teresting/strange as arbitrary semialge­ braic sets. A blueprint is the universality theo­ rem for oriented matroids by Nikolai Mnev, from which he also derived a universality theorem for d-polytopes with d + 4 vertices: 1 2 (Mnev 1986 [17,18]). For every semialgebraic set S C [RN, there is for some d > 2 a d-polytope P C [R d with d + 4 vertices whose realization space ffi.(P) is "stably equivalent" to S.

THEOREM

Such a result of course implies that there are nonrational polytopes, that there are polytopes that have realiza­ tions that cannot be deformed into each other (counterexamples to the "isotopy conjecture"), etc. (Here we consider the realization space of the whole poly­ tope, not only of its boundary, that is, we are considering convex realizations only.) To prove such a result, a first step is to find planar configurations that en­ code general polynomial systems; the starting point for this are the "von Staudt constructions" [31 , 2. Heft] from the 19th century, which encode addi-

tion and multiplication into incidence configurations. This systematically pro­ duces examples such as the pentagon configuration that we discussed. Then one has to show that all real polyno­ mial systems can be brought into a suit­ able "standard form" (compare Shor [27]), develop a suitable concept of "stably equivalent" (compare Richter­ Gebert [23]), and then go on. Since the mid-1980s, a number of substantial universality theorems have been obtained, each of them technical, each of them a considerable achieve­ ment. The most remarkable ones I know of today are the universality the­ orem for 4-dimensional polytopes by Richter-Gebert [23] (see also Gi.inzel [1 1]), a universality theorem for sim­ plicial polytopes by Jaggi et a!. [12], universality theorems for planar me­ chanical linkages by Jordan and Steiner [13] and Kapovich and Millson [14], and the universality theorem for polyhedral surfaces by Brehm (to be published [6]).

Four Problems Since the mid-1960s, there have been amazing discoveries in the construction of nonrational examples, in the study of rational realizations, and in the de­ velopment of universality theorems. However, great challenges remain-we take the opportunity to close here by naming four. Small coordinates

According to Steinitz, every 3-dimen­ sional polytope can be realized with ra­ tional, and thus also with integral ver­ tex coordinates. However, are there small integral coordinates? Can every 3polytope with n vertices be realized with coordinates in {0, 1 , 2, . . . , p( n)l, for some polynomial p(n)? Currently, only exponential upper bounds like 2 p(n) :S; 533n are known, thanks to Onn and Sturmfels [19], Richter-Gebert [23, p. 143], and finally Rib6 Mor and Rote; see [22, Chap. 6]. The blpyramldal 720-cell

It may well be that nonrational poly­ topes occur "in nature. " A good candi­ date is the "first truncation" of the reg­ ular 600-cell, obtained as the convex hull of the midpoints of the edges of the 600-cell, which has 600 regular oc­ tahedra and 1 20 icosahedra as facets. This polytope was apparently studied

by Th. Gosset in 1897; it appears with notation li, 5 1 in Coxeter [7, p. 162]. Its dual, which has 720 pentagonal bipyra­ mids as facets, is the 4-dimensional bipyramidal 720-cell of Gevay [8,20]. It is neither simple nor simplicial. Does this polytope (equivalently: its dual) have a realization with rational coordinates? Nonratlonal cubical polytopes

As argued previously, it is easy to see that all types of simplicial d-dimen­ sional polytopes can be realized with rational coordinates: "Just perturb the vertex coordinates." For cubical poly­ topes, all of whose faces are combina­ torial cubes, there is no such simple ar­ gument. Indeed, it is a longstanding open problem whether every cubical polytope has a rational realization. This is true for d = 3, as a special case of Steinitz's results. But how about cubi­ cal polytopes of dimension 4? The boundary of such a polytope consists of combinatorial 3-cubes; its combina­ torics is closely related with that of im­ mersed cubical surfaces [26]. On the other hand, if we impose the condition that the cubes in the bound­ ary have to be affine cubes-so all 2faces are centrally symmetric-then there are easy, nonrational examples, namely the zonotopes associated to nonrational configurations [32, Lect. 7l. Universality for simplicial 4-polytopes

There are universality theorems for sim­ plicial d-dimensional polytopes with d + 4 vertices, and for 4-dimensional poly­ topes. But how about universality for simplicial 4-dimensional polytopes? The realization space for such a poly­ tope is an open semialgebraic set, so it certainly contains rational points, and it cannot have singularities. One specific "small" simplicial 4-polytope with 10 vertices that has a combinatorial sym­ metry, but no symmetric realization, was described by Bakowski, Ewald, and Kleinschmidt in 1984 [3]; according to Mnev [17, p. 530] and Bakowski and Guedes de Oliveira [4] this example does not satisfy the isotopy conjecture, that is, the realization space is discon­ nected for this example. Are there 4-di­ mensional simplicial polytopes with more/arbitrarily complicated homotopy types?

© 2008 Springer Science+ Business Media, Inc., Volume 30, Number 3, 2008

41

[1 0] B. Grunbaum, Convex Polytopes, Inter­

ACKNO�EDGEMENTS

Thanks to Volker Kaibel for the discus­ sions and joint drafts on the path to this article, to Nikolaus Witte for many com­ ments and some of the pictures, to John M. Sullivan and Peter McMullen for care­ ful and insightful readings, to Ravi Vakil and Michael Kleber for their encourage­ ment and guidance on the way toward publication in the Mathematical Intelli­ gencer, and in particular to Ulrich Brehm for his permission to report about his mathematics "to be published." This work was supported by DFG via the Research Group "Polyhedral Surfaces and a Leibniz grant.

sis, FU Berlin, 2005, 23 + 1 67 pages.

prepared by V. Kaibel, V. Klee, and G. M .

[23] J. Richter-Gebert, Realization Spaces of

Ziegler, Graduate Texts in Mathematics

Polytopes, Lecture Notes in Mathematics 1 643, Springer, 1 996.

221 , Springer, New York, 2003. [1 1 ] H. Gunzel, On the universal partition the­

[24] S. A. Robertson, Polytopes and Symme­

orem for 4-polytopes, Discrete Comput.

try, London Math. Soc. Lecture Note Se­ ries 90, Cambridge University Press, Cam­

Geometry, 1 9 (1 998) 521 -552. [1 2] B. Jaggi, P. Mani-Levitska, B. Sturmfels, and N. White, Uniform oriented matroids without the isotopy property,

Discrete

Comput. Geometry 4 (1 989) 97-100.

THE BOOK, 3rd ed. , Springer-Verlag, Hei­ delberg, 2004.

[25] T. R6rig, Personal communication, August 2007. [26] A. Schwartz and G. M. Ziegler, Construc­ tion techniques for cubical complexes,

spaces of mechanical linkages, Discrete

odd cubical 4-polytopes, and prescribed

Comput. Geometry, 22 (1 999) 297-31 5.

dual manifolds,

[ 1 4] M . Kapovich and J. J . Millson, Universality theorems for configuration spaces of planar

[ 1 ] M. Aigner and G. M . Ziegler, Proofs from

bridge, 1 984.

[1 3] D. Jordan and M . Steiner, Configuration

linkages, Topology 41 (2002) 1 051-1 1 07.

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Experimental Math.

13

(2004) 385-41 3. [27] P. W. Shor, Stretchability of pseudolines in Applied Geometry and

is NP-hard,

Discrete Mathematics-The Victor Klee Festschrift, P. Gritzmann and B. Sturm­

[1 6] S. Maclane, Some interpretations of ab­

fels, eds. , DIMACS Series Discrete Math.

[2] L. J. Billera and B. S. Munson, Polarity and

stract linear algebra in terms of projective

Theor. Computer Science 4, Amer. Math.

inner products in oriented matroids, Euro­

geometry, American J . Math. 58 (1 936)

pean J. Combinatorics, 5 (1 984), 293-308.

236--240.

[3] J.

Bokowski, G. Ewald , and P. Klein­

Soc., 1 991 , pp. 531 -554. [28] J.

[1 7] N. E. Mnev, The universality theorems on

Simutis,

Geometric

toroidal maps,

realizations

of

PhD thesis, UC Davis,

1 977, 85 pages.

schmidt, On combinatorial and affine au­

the classification problem of configuration

tomorphisms of polytopes, Israel J. Math.

varieties and convex polytopes varieties,

[29] E. Steinitz, Polyeder und Raumeinteilun­

in Topology and Geometry - Rohlin Sem­

gen, in Encyklopadie der mathematischen

[4] J. Bokowski and A. Guedes de Oliveira,

inar, 0. Y. Viro, ed., Lecture Notes in

Wissenschaften, mit Einschluss ihrer An­

Simplicial convex 4-polytopes do not have

Mathematics 1 346, Springer, Berlin 1 988,

wendungen,

pp. 527-544.

111 . 1 . 2 . , Heft 9, Kapitel Ill A B 1 2, W. F.

47 (1 984) 1 23-130.

the isotopy property, Portugaliae Math. 47 (1 990) 309--3 1 8.

[ 1 8] N.E. Mnev, The universality theorem on

Dritter

Band:

Meyer and H. Mohrmann, eds. , B. G. Teubner, Leipzig, 1 922, pp. 1 -1 39.

[5] U. Brehm, A universality theorem for real­

the oriented matroid stratification of the

ization spaces of maps. Abstracts for the

space of real matrices, in "Discrete and

Conferences 20/1 997 "Discrete Geome­

Computational Geometry," J . E . Good­

gen

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man, R. Pollack, and W. Steiger, eds. , DI­

Springer-Verlag ,

tisches Forschungsinstitut Oberwolfach.

MACS Series in Discrete Math. Theor.

Springer-Verlag, 1 976.

[6] U . Brehm, A universality theorem for real­ ization spaces of polyhedral maps,

in

preparation, 2007. [7] H. S.

M. Coxeter, Regular Polytopes,

Macmillan, New York 1 963; second, cor­ rected reprint, Dover, New York, 1 973. [8] G. Gevay, Kepler hypersolids, in "Intuitive geometry," Szeged, 1 991 , Colloq. Math. Soc. Janos Bolyai 63, North-Holland, Am­ sterdam, 1 994, pp. 1 1 9--1 29. [9] P. Gritzmann, Polyhedrische Realisierun­ gen geschlossener 2-dimensionaler Man­ nigfaltigkeiten im !R3, PhD thesis, Univ. Siegen, 1 980, 86 pages.

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THE MATHEMATICAL INTELUGENCER

Computer Science, vol. 6, Amer. Math. Soc., 1 991 , pp. 237-243. [1 9] S. Onn and B. Sturmfels, A quantitative Steinitz' theorem,

Beitrage zur Algebra

und Geometrie 35 (1 994) 1 25-129. [20] A. Paffenholz and G. M . Ziegler, The Er construction for lattices, spheres and poly­ topes,

Discrete Comput. Geometry 32

[30] E. Steinitz and H . Rademacher, Vorlesun­ Ober

die

Theorie Berlin,

der Polyeder, 1 934.

plices in

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Annals of Discrete Math. 20

(1 984) 253-254. [22] A. Rib6 Mor, Realization and Counting Prob­ lems for Planar Structures: Trees and Link-

Reprint,

[31 ] G. K. C. von Staudt, Beitrage zur Geome­ trie der Lage,

Verlag von

Bauer und

Raspe, Nurnberg, 1 856--1 860. [32] G.

M. Ziegler, Lectures on Polytopes,

Graduate

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Springer, New York, 1 995. Revised edi­ tion,

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seventh updated printing,

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(2004) 601 -624. [2 1 ] M . A. Paries, At most 2d + l neighborly sim­

Geometrie,

Institute of Mathematics MA 6-2, TU Berlin D-1 0623 Berlin Germany e-mail: [email protected]

The P rob l e m of the B roken Stick Reconside red GERALD S. GOODMAN

� very year, starting in the 1700s until 1910, Cambridge

--, University held examinations on Pure and Applied L....., Mathematics, which lasted for several days. They were originally called the "Senate-House Examinations," after the name of the building in which they took place, and later they became known as the "Mathematical Tripos." After­ wards, the best solutions were published. The ensuing vol­ umes can be found in the Rare Book Collections of the British Library in London and the University Library in Cam­ bridge. In the morning session of January 18, 1854, there was posed [13, pp. 49-52] an elementary problem in geometrical probability that was destined to become a classic. The British call it "The Problem of the Broken Rod," whereas Americans refer to it as "The Problem of the Broken Stick." It says, "A rod is marked at random at two points, and then divided into three parts at these points; shew [sic] that the probability of its being possible to form a triangle with the pieces is 1/4." The exact value of the probability is of little interest, ex­ cept possibly to numerologists. What is interesting is to see how various later authors, apparently unaware of the orig­ inal formulation of the problem, reinterpreted what it means to break a stick "at random," and developed fresh methods to solve it. Be advised that I am employing the word "random" in the narrow technical sense used in probability and statis­ tics to refer to any chance phenomenon that is governed by the uniform distribution over a suitable sample space. This usage was recommended by de Finetti [ 1 , p. 1 52] and [2, p. 62], who was writing on the very topic of random di­ vision. However, even with such a restriction, there is more than one way to interpret what is meant by the term "ran­ dom," depending on the identity of the sample space, just

as in Bertrand's Paradox [3], [ 1 1 ] . I shall examine two of these interpretations in detail. Although the problem originated in England, it found its way to France, possibly with the aid of John Venn, who was enrolled as an undergraduate at Gonville and Caius College at the time. Presumably, he had taken the exam and done well on it, for he was awarded the title of "Math­ ematical Scholar" at his college later in the year. The first journal publication was in the founding volume of the Bull. Soc. Math. de France in 1875, written by Emile Lemoine [6]. Lemoine formulated a discrete version of the problem by considering the rod as a measuring stick divided into equally spaced intervals and allowing the breaks to occur only at their endpoints. This gives rise to a finite number of outcomes, and he treated them in the traditional way, interpreting the word "random" as meaning that all trisections of the rod at a given scale are equally likely. He made tables and calcu­ lated the ratio of the number of favorable cases to the num­ ber of possible ones. Then he passed to the limit as the scale decreased to zero. In this way, he found the answer to be 1/4, in agreement with the Cambridge Examiners, whom he does not cite. Subsequently, several French mathematicians, including Lemoine himself [7], showed that the same answer could be obtained by formulating the problem directly in terms of the continuum and solving it by use of geometry. Ref­ erences can be found in [7] and [12]. They took as their sample space an equilateral triangle and interpreted the tri­ linear coordinates of a sample point as the lengths of the broken pieces, as in Figure 1 . Relative area provides a uni­ form distribution of probability on the space. Thus, for them, random trisection of a rod amounted to choosing a point "at random" in the triangle.

© 2008 Springer Science+ Business Media, Inc., Volume 30, Number 3, 2008

43

A

Because of this, the Problem of the Broken Stick, often snubbed as a mere mathematical diversion by those who forget that probability theory had its origins in mathemati­ cal diversions, deserves to occupy a more dignified place in the hierarchy of mathematical thought.

Lemoine's Combinatorial Approach In the formulation of Lemoine [6], the problem goes as fol­ lows. Figure

I.

Trilinear coordinates.

Finally, Henri Poincare took up the question in his text­ book, Calcul des Probabilites [9], and presented the geo­ metrical approach just cited. There is, however, a differ­ ence. Whereas previous authors took for granted that the uniform distribution was the appropriate one to use, Poin­ care raised the trenchant question as to how that choice of probabilities could be justified. His own answer is inge­ nious, but I shall not go into it here. Instead, I propose a new answer. My motivation lies in the remarkable fact that, although these authors held vastly different conceptions of what is meant by a random tri­ section, and they used contrasting methods for computing probabilities-one combinatorial, and the other geometri­ cal-they nonetheless came up with the same answer. To explain this, I demonstrate how the authors' ap­ proaches are connected by giving a representation of the data set arising in Lemoine's discrete model as nodes of a triangular lattice and showing that Poincare's and his own later setup emerge from it by rescaling and passing to the limit. To do so, I employ a tool that was available to them at the time, namely, the calculation of areas by quadrature, and I prove that, as the subdivisions get finer and finer, nonnalized counting measure on the rescaled lattice converges on Euclidean figures-and, therefore, weakly-to the rela­ tive area of the figure.

We thus have an example where the assumption, com­ monly made in geometrical probability ever since its in­ ception in 1777 by Buffon [ 1 1 , p. 502], that regarding prob­ ability as relative area captures the essence of randomness, finds its justification by tracing it back to the classical no­ tion of "equally likely cases" in finite problems.

A rod is broken into three pieces. What is the probability that the pieces can be made into a triangle?

The obstacle is that the lengths of the pieces must satisfy all three triangle inequalities. In Lemoine's approach, the rod is calibrated by dividing it up into an even number of units, and each break is as­ sumed to occur at one of the points of division. The num­ ber of favorable configurations is counted and compared to the number of all possible ones. Taking their ratio and passing to the limit as the size of the unit vanishes yields the answer 1/4. Here are the details. Let us divide the rod into 2m equal parts, and suppose that the three pieces contain, respectively, x, y and z of these parts. We shall then have x + y + z = 2 m. In order for them to fonn a triangle, it is necessary that x ::s y + z, y ::s x + z, z ::::; x + y. Eliminating z with the aid of the previous equation, these inequalities become x ::5 m, y ::5 m, x + y ;::: m. Let

us

Table

find the number offavorable cases:

1.

X

m

1

m, m - 1

2

m, m - 1 , m - 2

m

GERALD S. GOODMAN studied probability with Marl< Kac in a special course at Haverford College, before going to Stanford. There at Stan­ ford, he did his doctorate in control and conformal mapping under the supervision of Charles Loewner. Although Goodman has done indus­ trial worl< and has published in analysis and probability, some consider his finest mathematical achievement to have been landing a permanent job in the beautiful Renaissance city of Florence. After thirty years of teaching and research at the University there, he is now retired. via Dazzi , I I 1 -50 1 4 1 Firenze Italy e-mail: [email protected]

44

THE MATHEMATICAL INTELLIGENCER

y can take on the values

0

m, m - 1 , m - 2,

. . .

,0

Tbere are thus a total of 1 +2 +3+

· · ·

+ ( m + 1)

( m + 1 )( m + 2) 2

:. =-::.c..: .:. --':__ = ..o.:.:..:_

favorable cases. Let us find the number ofpossible cases: Table

2.

X

y can take on the values

0

2m, 2m - 1 , . . . , 2, 1 , 0 2m - 1 , . . . , 2, 1 , o

2m

0

Tbere are thus a total of 1 + 2 + 3 + . . . + (2m + 1 )

=

(2 m + 1 )(2m + 2) 2

cases possible. Tbe ratio of the number offavorable cases to the num­ ber ofpossible ones is ( m + 1 )( m + 2) (2m + 1)(2m + 2) ' which, for m = oo, yields 1/4. Tbus, the probabili�y sought

is

114.

To arrive at these tallies, the piece associated with x is tacitly assumed to have an endpoint in common with one end of the rod. The discretization occurs when the lengths of the pieces have been rounded off to take only fractional values. The roundoff error wears off when m is large, and vanishes in the limit.

A Graphical Representation of the Data We can represent the tabular data graphically by introduc­ ing a triangular grid on an equilateral triangle ABC of height 2 m. The trilinear coordinates x, y and z of a point in the triangle are its distances from the three sides see Figure 2 . The entries i n the above two tables are then counts of grid points, where x, y and z assume integral values. A

Figure 2. Graphical representation in the case m = 4.

Table 1 represents the counts of the white grid points, stratified according to their distances from the base, which we regard as their x-coordinate. They lie in the triangle whose vertices are the bisectors of the sides of the original triangle.

Table 2 gives the counts of all the grid points, both black and white. When m increases, the triangle becomes larger and larger, but the ratio of the two counts tends to a finite limit, as we have seen. The foregoing representation enables us to understand the role of the assumption of equally likely cases. It means that each grid point has the same weight. Hence, the prob­ ability assigned to each one is simply the reciprocal of the number of nodes in ABC.

What Happens in the Scaling Limit? With increasing m, the size of the triangle ABC approaches infinity. To avoid this, we can assume the rod has unit length, and we can rescale the triangle so that its height remains equal to one. With that normalization, the coordi­ nates x, y, and z of the nodes assume the values k/2 m, where k = 0, 1 , . . , 2 m. Each coordinate is then the sum of the lengths of the parts belonging to the corresponding piece. In other words, they are the lengths of the pieces of a broken rod, measured with precision 1 /2m. Now, as m increases, the mesh tends to zero, and the nodes become more and more dense. Since they are equally spaced, the proportion of them falling in a particular zone ought to be comparable to its area. Consequently, if the counting measure on ABC is normalized so as to make it into a probability measure, we would expect that, in the continuum limit, the measure of the zone actually converges to its relative area. That assertion can be proved by using a weak con­ vergence argument, as in Lalley [5], who treats a more general case. To prove it within the paradigm of nine­ teenth-century mathematics requires a more elementary ap­ proach. It turns out that the possibility of approximating integral expressions for areas by appropriate finite sums does the job. Here is the idea. Let E be a Euclidean figure having pos­ itive area, contained in the triangle ABC. Given a triangu­ lar grid on ABC, choose a rhombus whose sides belong to adjacent lines in the grid, and assign to each node in E the translate of the rhombus that has this node as its up­ per left vertex. Summing the areas of the rhombi provides an approximation to the area of E, and it is proportional to the number of nodes being counted. Doing the same for ABC and taking their ratio gives the proportion of nodes falling in E. It is expressed as the quotient of sums ap­ proximating the definite integrals that represent the areas in question. Provided the inclinations of the sides of the chosen rhombi do not change as the mesh of the grid goes to zero, the relative node count will converge to the ratio of the in­ tegrals, and thus to the relative area of E. Hence, in the scaling limit, the uniform distribution emerges. It is the ul­ timate expression of the notion of "equality likely cases, " based o n the "principle of insufficient reason, " that pre­ vailed in the discrete case. .

Scholium

The scaling limit of Lemoine's discrete model is an equi­ lateral triangle ABC of height 1 , endowed with the uniform distribution, whose points represent trisections M of the unit

© 2008 Springer Science+ Business Media, Inc., Volume 30, Number 3, 2008

45

interval and its trilinear coordinates the lengths of the con­ stituent pieces.

Poincare's Account in his "Calcul des Probabilites" Poincare dedicated a course of lectures at the Sorbonne in 1893 to probability. The notes were published in textbook form a few years later, and a revised edition is still in print [9]. He included in his chapter on continuous probability a treatment of the Problem of the Broken Stick. We are un­ able to say whether he was familiar with Lemoine's earlier work or not, because he habitually omitted references. Let us examine his text [9, 1 st ed. , pp. 81-83, 2nd ed., pp. 1 23-126] for clues. It starts off in this way: Let us break a stick of length 1 into three pieces, x + y + z = 1 . Here, x, y, z represent both the lengths of the pieces and the pieces themselves. Considered as lengths, they match what we have seen in the rescaled lattice case, but now they assume continuous values. After some preliminaries, he states the problem: What is the probability that x, y, z form a triangle?

Poincare introduces an equilateral triangle ABC as his sample space and uses the lengths x,y,z as the trilinear co­ ordinates of a point representing the way the stick can be broken, just as we did. He then appeals to Viviani's theo­ rem, which asserts that the sum of the distances of a point M in ABC from the sides equals the altitude of the trian­ gle. To prove it, join M to the vertices and compare the ar­ eas of the resulting triangles with the area of ABC Let us draw an equilateral triangle of height 1 . From a point M in its interior, drop perpendiculars to the three sides. 1be sum of their lengths will be equal to the height of the triangle, which is 1 : They represent the three pieces x, y, z of the stick. 1be point M may be thought ofas representing the way in which the stick has been divided. What is the proba­ bility that this point belongs to a certain part of the tri­ angle?

To answer the question, Poincare identified probability with relative area, and thus with the uniform distribution. How does he justify this choice of probability? Poincare's treatment makes no mention of combinatorics. As a result, he had dispensed with the motivation employed above for choosing the uniform distribution. As de Finetti [1], [2] later put it, without any heuristics, the uniform distribution has lost its probabilistic meaning. Poincare was .well aware of that loss and proposed the following expedient. He would formulate assumptions con­ cerning the distribution of pairs of lengths of the broken pieces, and then show that they implied that the distribu­ tion of M is uniform on ABC His reasoning is not obvious. It involves the use of an ad hoc conditioning argument, made well before its time, and considerable effort is re­ quired to deconstruct it. I shall not pursue the matter here because we already know how the uniform distribution arises as the continuum limit of the rescaled discrete model. I shall add further mo­ tivation below by establishing that it is enough to assume that the breakpoints are uniformly distributed and statisti-

46

THE MATHEMATICAL INTELLIGENCER

cally independent, as the Cambridge Examiners did, to en­ sure that the joint distribution of x, y, z, which is the dis­ tribution of M, is uniform on ABC Once it is accepted that the phrase "random trisection" is synonymous with the assertion that the distribution of M is uniform on ABC, the rest of Poincare's proof goes through smoothly. He proceeds as follows. Let us join the midpoints A ',B ', C' of the sides of ABC by line segments. M must belong to the interior ofA 'B 'C' for x,y,z to form the sides of a triangle. IfM belongs to a side ofA 'B 'C', one of thefollowing equations is satiified: z = x + y, x = y + z, y = z + x.

lf M lies outside of A 'B 'C', one of the magnitudes x,y, z

is larger than the sum of the other two. 1be probability that one canform a triangle with x,y,z is thus 1/4. The triangle A 'B'C' in Figure 3 is the continuum limit

of the locus of white points in the lattice of Figure 2. In view of the equation x + y + z = 1 , the triangle inequali­ ties become x, y, z ::::; 1/2. In other words, no piece can have length greater than half the length of the stick. As Poincare notes, equality occurs along the bisectors of the sides of ABC Identifying probability with relative area yields the final result. Poincare's idea of introducing ABC as a sample space and endowing it with a probability distribution is an early example of what is now a standard practice. However, as I have suggested, it may have stemmed from Lemoine's 1883 work, or even his work of 1875, elaborated in the way described above, and then discarded. [9] If so, it affords a precious insight into his working style.

Geometry of the Space of Trisections As we have seen, the triangle ABC can be thought of as a sample space whose points M represent the different ways of breaking a stick of unit length into three pieces. The tri­ angle has height 1 , and its sides have length A The trilinear coordinates of M fix its position in ABC, as in Fig­ ure 1 . They are regarded as the lengths of the pieces mak­ ing up the corresponding trisection, conventionally ranged from left to right and denoted successively by x, y, z. As an alternative, the trisection M can be described by specifying its breakpoints u and v. When v < u, the re=

A

Figure

3.

(after Poincare)

fV3.

lation between the two formulations is given by the for­ mulas X

=

V,

y= U

-

V, Z

I introduce the identification map ( U V)'

= 1 - U.

'

·

=

{(

U, V), if V :5 U, ( V, U), otherwise

Permuting u with v yields the formulas when u < v. These two representations of a trisection M can be vi­ sualized in the following way. Draw the triangle ABC and construct the point D sym­ metric to C with respect to the side AB. Take B as the ori­ gin and introduce oblique axes A U and A V, oriented as in Figure 4, where distances are measured in units of A. By definition, the oblique coordinates of M are the mag­ nitudes of the projections of M on the basis vectors. In Fig­ ure 4, they are labelled Au and A v, respectively. The trilinear coordinates of M are its distances x, y, z from the sides of ABC, labelled as in Figure 1 . The following result legitimizes our use of the symbols x, y, and z to denote the distances of M from the sidelines, and Au and Av to denote its oblique coordinates, by sup­ plying them with the appropriate semantics.

Geometrically, the action of the identification map is to fold the triangle ADB over the diagonal AB of ADBC Conse­ quently, if M were in ADBC, its image would lie in ABC When M belongs to ADB, the Duality Theorem applies to its image under the identification map. Doubling undoes the action of the identification map and unfolds ABC onto ADBC, so that the inverse image of any figure in ABC is the figure itself and its mirror image with respect to the diagonal AB. The identification map as­ signs to the original figure the probability of their union. When the rhombus ADBC is endowed with the uniform distribution, probability reduces to relative area. Since the identification map is rigid, and the two preimages of ABC are congruent, the probability assigned to any figure in ABC is twice its area relative to ADBC. Hence, the image of the

DUALITY THEOREM Let M he a point in ABC determined

the Duality Theorem.

hy the trilinear coordinates x, y, and z. Suppose that the breakpoints u and v in the corresponding partition of [0, 1] are labelled in such a way that v < u. Then the oblique co­ ordinates of M are u and v multiplied hy A, and x, y, and z are the lengths of the corresponding pieces. The converse also holds. PRooF. Let A u and Av be the oblique coordinates of

M.

In Figure 4 , the length of the hypotenuse of the right tri­ angle with vertex M and side x perpendicular to BC is A v, so x/Av = sin 60° = 1/A, and thus x = v. Similarly, z= 1 u. As y = 1 - (x + z) = u v, or directly from the figure, the formulas displayed above show that u and v can he identified with the breakpoints of the partition whose lengths are x, y, and z, and vice-versa. 0 To proceed further, "double" the sample space ABC to form the rhombus ADBC. The rhombus has height 1 , and its sides have length A. It is a product space made up of pairs (U, V) with 0 :5 U, V:5 1 , rescaled by A. The zone ABC represents the case in which V < U, whereas ADB represents the opposite one, and y is negative when M lies there. -

Figure 4.

-

Oblique coordinates and breakpoints.

uniform distribution on the rhombus is the uniform distri­ bution on ABC This implies the following consequence of

v are in­ dependent and uniformly distributed on [0,1 ) . Then the cor­ responding trisection M bas a uniform distribution on ABC

COROLLARY Suppose that the break-points u and

The hypotheses imply that Au and A v are also in­ dependent and uniformly distributed, and therefore their joint distribution is uniform on ADBC. It is then uniform on ABC As Au and Av are the oblique coordinates of the sample point M, we conclude that M is uniformly distributed on ABC. 0 PROOF.

In other words, assuming that the breakpoints are ran­ dom implies that the trisection M arises by sampling from the uniform distribution in ABC That result was stated, without proof, by Paul Levy [8, p. 1 47) in 1939, who noted that it generalizes to the ran­ dom division of an interval into n pieces, n ;::: 3. The sample space is now a regular n-simplex. See Kendall and Moran [4, pp. 28-31 ) and de Finetti [ 1 ) , [2) for de­ tails. Suppose we assume that items such as the temporal or­ der in which the breakpoints were labeled do not matter. Then, as recognized by de Finetti (1), [2), the variables u and v will be exchangeable, meaning that their joint d istri bu ti o n on ADBC is unchanged when u and v are permuted. Con­ sequently, any figure in ABC that is defined in terms of u, v will have the same probability as its mirror image with re­ spect to the diagonal AB of ADBC Assume now that M has a uniform distribution on ABC Then, as seen above, the probability relative to ABC of a fig­ ure in ABC is twice its area relative to ADBC Since that prob­ ability is the sum of the probabilities relative to ADBC of the figure and that of its mirror image, and they are the same when exchangeability prevails, the probability of each must agree with its area relative to ADBC. Consequently, the dis­ tribution of Au and A v is uniform on ADBC, and therefore u and v are independent and uniformly distributed on [0, 1]. That is the converse to what Levy found. 0

© 2008 Springer Science+Business Media. Inc . . Volume 30. Number 3. 2008

47

CONVERSE OF LEVY's THEOREM Suppose that M is uni­ formly distributed on ABC and that its breakpoints u and v are exchangeable. Then they are randomly distributed on [0, 1] and statistically independent.

It is a folklore result that an easy way to generate ran­ dom points uniformly distributed on ABC is to make use of the identification map. First generate a random point in the rhombus by choosing, independently, each of its co­ ordinates Au and Av at random from the uniform distribu­ tion on [O,A] . Then apply the identification map. The result will be a point uniformly distributed on ABC SCHOLIUM To generate a random point un!formly dis­

tributed on ABC, double ABC by reflecting it on the side AB to form a rhombus. Generate a point, Au, uniformly dis­ tributed on the bottom of the rhombus, and another one, lt v, on the adjacent side. Applying theforegoing map to (Au, lt v) will produce a point uniformly distributed on ABC

The above scheme is an alternative to the one in which points are generated at random in the rhombus, and those that do not fall into the triangle ABC are discarded. It can be adapted to any 2-simplex, since relative areas are pre­ served under affine maps. It can also be generalized to sim­ plices in higher dimensions, where its advantage over the acceptance-rejection procedure is more apparent. See Ru­ binstein [10] for further material on this subject. Here is another procedure that will generate a pair of random breakpoints on a stick, due to de Finetti [ 1 , p. 156] . It displays the quality of his probabilistic instincts. De Finetti's random algorithm

Break the stick at a random point u, uniformly distributed in [O,ll. Then choose one of the two pieces with probabil­ ity equal to its length. Now break the chosen piece at a point v, uniformly distributed on it. Then the points u and v are uniformly distributed on [0, 1] and statistically inde­ pendent. This works because it expresses the distribution of v, conditional on u, as a mixture of two uniform distributions, one on [O, u] and the other on [u, 1 ] , employing u and 1 u as their respective weights. The resulting distribution is uniform on [0, 1 ] , and, since it does not depend on u, u and v are statistically independent. REFERENCES

[1] de Finetti, B. Alcune osservazioni in tema di "suddivisione casuale", Giomale lstituto Italiano deg/i Attuari, anno XXVII, n.1 (1 964), 1 5 1 1 73. (2] de Finetti, B. Sulla suddivisione casuale di un intervallo: spunti per riflessioni. Rend. Sem. Mat. Fis. Milano 37 (1 967), 51-68. [3] Holbrook, J. and Kim, S. S. Bertrand's paradox revisited. Math. lntelligencer 22 (2002), no. 4, 1 6-1 9. [4] Kendall, M . and Moran, PAP. Geometrical Probability. Griffin's Sta­ tistical Monographs 1 0, London, 1 963. [5] Lalley, S. The packing and covering dimensions of some self-sim­ ilar fractals. Indiana Univ. Math. J. 37 (1 988), 699-709.

48

THE MATHEMATICAL INTELLIGENCER

[6] Lemoine, E. Sur une question de probabilities. Bull. Soc. Math. de France 1 (1 875), 39-40. (7] Lemoine, E. Quelques questions de probabilites resolues geo­ metriquement. Bull. Soc. Math. de France 1 1 (1 883), 1 3-19. [8] Levy, P. Sur I a division d ' u n segment par des points choisis au hasard. C. R. Acad. Sci. Paris 208 (1 939), 1 47-149. [9] Poincare, H. Calcul des Probabilites. George Carre, Paris, 1 896; 2 nd ed., Gauthier-Villars, 1 91 2; repr. Jacques Gabay, Paris, 1 981 . [1 0] Rubinstein, R. Y. Generating random vectors uniformly distributed inside and on the surface of different regions. Eur. J. Op. Res. 1 0

(1 982), 205-209. (1 1 ] Seneta, E. , Parshall, K. H . , and Jongmans, F. Nineteenth-century developments in geometrical probability. Arch. Hist. Exact Sci. 55

(2001), 501 -524. [1 2] Seneta, E. and Jongmans, F. The problem of the broken rod and Ernesto Cesaro's early work in probability. Mathematical Scientist 30

(2005), 67-76.

[1 3] University of Cambridge. Solutions of the Problems and Riders Proposed in the Senate-House Examinations for 1 854, by the Mod­

erators and Examiners [William Walton and Charles F. Mackenzie]. With an appendix, containing the examination papers in full. Macmillan and Co. , Cambridge, 1 854.

Appendix: The Original Cambridge Text Here is the way that the problem was formulated by the Cambridge Examiners, along with the solutions they elected to publish [ 13]. A rod is marked at random at two points, and then divided into three parts at these points; shew that the probability of its being possible to form a triangle with the pieces is 1/4. Let AB be the rod, C its middle point, D, E, the mid­ dle points of AC, CB. In order that it may be possible to form a triangle, each of the pieces must be less than the sum of the other two, or in other words, each must be less than half the rod. To secure this it is clear that the two points of divi­ sion P, Q, must lie on opposite sides of C: the proba­ bility of their doing so is 1/2. Let x be the probability that two points lying on op­ posite sides of the middle point of a line contain be­ tween them less than half the line: the required proba­ bility will be x/2. Now there are four classes of ways in which the points may fall, all equally likely, the chance of each is therefore 1/4. In the first of these classes, viz. when the points of division lie in DC, CE, success is certain, in the second, viz. when the points lie in AD, EB, success is impossible; in the third, viz. when the points lie in AD, CE, the probability of suc­ cess is x, for success depending on the distance be­ tween the points being less than AC, the probability is the same as if DC were removed, and success depended on the distance between the points being less than AD, and this probability is x by supposition; lastly in the fourth class, viz. when the points lie in DC, EB, it may be shewn by similar reasoning that the probability of success is x.

being greater than y. Then the lengths of the three pieces are y, x y, a - x. And the conditions of the problem give, as above shown,

y

-

B

y 5040. In 1984 Guy Robin3 showed that, in fact, this statement is true if and only if the Riemann hypothesis is true. This is Robin's the­ orem. We think that it is delightful that the property of this particular integer is mentioned for the first time in Plato's "Laws. "

Diethnous Kyprologikou Synedriou (Lefkosia, 1 6--20 Apriliou, 1 996) To­ mos A, APXAION TMHMA, Ekdosi Georgiou loannidi kai Steliou Xatjistyli , Horigia ldrimatos A. G. Leventi, Lefkosia 2000, pp. 63-1 36. [4] Andreas Zachariou and Eleni Zachariou. Logos kai Texni, Aristotle University of Thessaloniki, Polytechnic School, Department of Math­ ematics, Volume dedicated to Professor Emeritus loannis D. Mittas, Thessaloniki, 1 999--2000.

3http://blogs. msdn.com/devdev/archive/2007/07/1 6/robin-s-theorem .aspx http://mathworld .wolfram.com/RobinsTheorem.html.

� Springer

the I ngu•g of s =

I xdi

.fA = axA + bx13 /B = CXA + dxn

i=O

A mutation changes a sequence from i to j by changing the nucleotide base that occupies a position in the genome sequence. All mutations are encom­ passed in the mutation matrix Q whose elements are the probabilities qif (q;; is the probability that a mutation does not happen). This leads to the Eigen-Schus­ ter Equation, the master equation of Nowak's book: xi =

tion). Since .fi is simply the net repro­ ductive rate of individuals, the Eigen­ Schuster Equation then reduces to the replicator equation of Hofbauer and Sig­ mund [6] . This replica tor equation is for­ mally equivalent to the Lotka-Volterra equations of competition, where

x1

j�()

The first ( summation) term on the r.h.s. is the input of genome sequences to genome i by mutations from all other j given their fitness j (this also includes the probability q1i that individuals with genome i are represented in the next generation as determined by their fit­ ness i). If the fitness of genome i and mutation rate into i exceeds the aver­ age fitness of the population () and mutation ( q1; ) . Additional terms can be added to cover immigration into and emigration from the target population. The equation can also encompass the random sampling required to describe small populations. But the Eigen-Schuster Equation also links evolutionary theory to population theory and game theory. If Q is the identity matrix, then q,i = 1 (no muta-

The payoff matrix which then multi­ plies the vector x is:

f=

[� �]

We can now find a Nash Equilibrium for the Eigen-Schuster Equation, which evolutionary biologists call an evolu­ tionarily stable strategy (ESS). Thus, beginning with the definition of information in DNA as a sequence of nucleotide bases and generalizing that to a sequence space, Nowak arrives at a dynamical systems theory that syn­ thesizes the information content of a genome with evolutionary and ecolog­ ical dynamics and evolutionary game theory. Nowak then uses this theory to shed fresh light on some classic evolu­ tionary games such as rock-paper­ scissors, hawk-dove, and the Prisoner's Dilemma. A later chapter generalizes these games to spatial domains. Nowak also uses the theory to lay the founda­ tion of the evolution of disease in sev­ eral chapters, showing with almost ef­ fortless ease why the HIV vims is able to persist by mutation, a behavior which is one of the main problems impeding the development of a vaccine and cure for HIV/AIDS. One chapter also presents the basis of evolutionary graph theory. Assume all individuals in a population are la­ beled i = 1 , . . . , N Denote the prob­ ability that an offspring of individual i replaces j with wi/. The individuals form the vertices of a graph and, if Wy > 0, then there is an edge from vertex i to j weighted by wu. Thus, the population structure and its dynamics can be rep­ resented by a weighted digraph. A wide

© 2008 Spnnger Science+ Business Media, Inc .. Volume 30, Number 3, 2008

65

variety of graphs ensue, including com­ plete graphs given by the Moran process of any individual being re­ placed by any other with equal proba­ bility, to cyclic graphs, and to a wide variety of other graphs of great aesthetic and mathematical beauty. However, perhaps because of the newness of this approach to evolutionary theory, the biological implications are not as de­ tailed as with the chapters that build on dynamical systems and game theo­ ries. Perhaps this chapter will stimulate research into the application of this po­ tentially powerful technique to partic­ ular ecological and evolutionary situa­ tions. One frontier in evolutionary theory, which has barely begun to be explored, is its interface with ecosystems ecology. Ecosystems ecologists study the biolog­ ical and physical processes that control the flux of nutrients in forests, lakes, prairies, oceans, and other ecosystems. The growth of populations is often lim­ ited by the availabilities of nutrients. However, individuals also affect their availabilities through such traits as up­ take rates and the release of nutrients during the decay of leaf litter and car­ casses when they are returned to the soil. It would seem logical to suppose that the ways in which individuals af­ fect resource availability, through their uptake and through the decay of dead material, could be powerful selective forces. But evolutionary biologists have ignored ecosystem processes because the resources themselves do not evolve (and are therefore assumed to remain fixed), whereas ecosystem ecologists have ignored individual variation in or­ ganismal traits that could affect nutrient cycling, preferring to concentrate on the larger differences between species. But the approach outlined by Nowak could he used to unify evolutionary and ecosystems theory. For example, the chemistry of leaf litter, which often de­ termines decay rates and the rate of re­ lease of nutrients, is partly under ge­ netic control [7]. Different sequences, may therefore have different decay rates and influence soil nutrient avail­ ability differently. Assume also that the fitness, /;, depends on nutrient avail­ ability, perhaps by means of a Michaelis-Menten or other suitable function. We can now propose a dy­ namical system in which an equation

xi,

66

THE MATHEMATICAL INTELLIGENCER

for the dynamics of the nutrient re­ is coupled to the Eigen­ source, Schuster Equation:

R,

J xi = I xif/R)qii - rpxi l :0+ ( � c x ) Ji

k; ;m;

g

�

;

,

· · ·

ford, UK, 1 930. (4]

xi;

qii

J.

B. S. Haldane,

A

mathematical theory of

natural and artificial selection, Part I. Trans.

eR

R

R

1 980. [3] R . A Fisher, The Genetical Theory of Nat­ ural Selection. Oxford University Press, Ox­

where IR and eR are inputs and out­ puts, respectively, of the nutrient R from and to the external environment. The function g describes the total re­ lease of nutrient from dead individ­ uals of based on mi, the mortality rate of the concentration of nu­ trient in the dead individuals or dead shed parts of ki, the release rate of the nutrient during decomposition; and the ellipsis denotes other factors, such as climate, that also affect decay rate. This coupled system is a framework for examining how mutations affect the cycling of R and conversely how the cycling of affects the trajectory of x in sequence space by means of the fit­ ness function The entire system is constrained by the mass balance im­ posed by IR and eR. One can therefore also investigate how increased nutrient inputs, such as the increased phos­ phorus loading of a lake that causes al­ gal blooms, affects the evolutionary dy­ namics of x. Depending on the forms of ./j(R) and g( . . . ) there could be some interesting bifurcations in this sys­ tem. Nowak's book is a readable, hand­ somely illustrated, and thought-provok­ ing guide to modern evolutionary the­ ory. Some beautiful mathematics is used to illuminate difficult evolutionary ideas, and evolutionary biology is used to mo­ tivate the synthesis of dynamical sys­ tems theory, game theory, and graph theory. The book is suitable for gradu­ ate classes in evolution, ecology, or bio­ mathematics. In my view, it would be best used in a seminar class where stu­ dents are encouraged to develop one or more of these techniques further to shed some light on their thesis prob­ lems.

xi xi; ci,

Press, Cambridge, Massachusetts, USA,

Cambridge Philosophical Soc. 23 (1 924), 1 9-4 1 . [5] S . Wright, Evolution in Mendelian popula­ tions. Genetics 16 (1 93 1 ) , 97-1 59.

[6]

J.

Hofbauer and K. Sigmund, Evolutionary

Games and Population Dynamics. Cam­

bridge University Press, Cambridge, UK, 1 998. [7] J. A Schweitzer, et al, Genetically based trait in a dominant tree affects ecosystem processes. Ecol. Lett. 7 (2004), 1 27-134.

Department of Biology University of Minnesota Duluth Duluth, MN 558 1 2 USA e-mail: [email protected]

f/R).

REFERENCES

( 1 ] G. H. Hardy, Mendelian proportions in a mixed population.

Science 28 (1 908),

49-50. (2] E. Mayr and W. B. Provine, editors, The Evolutionary Synthesis. Harvard University

The Trouble with Physics: The Rise of String Theory, the Fal l of a Science, and What Comes N ext by

Lee Smolin

BOSTON, MASSACHUSETIS, HOUGHTON MIFFLIN, 2006, 416

pp,

HARDCOVER U S $26.00. ISBN-10:

0618551050, ISBN-13: 978-0-8218-3933-1. 2007, PAPERBACK, US $15.95. ISBN-13EAN: 9780618918683, ISBN-10: 061891868X

REVIEWED BY JOHN HARNAD

uperstring Theory has been the subject of intense study by a sub­ stantial segment of the theoretical physics community for over two decades. The theory's goal is extremely ambitious, to say the least: nothing less than a uni­ fied quantum framework for all the fun­ damental interactions of matter, a "the­ ory of every1hing" [ 1 , 2] . Its mathematical intricacies, however, are barely under-

standable even to a majority of physi­ cists, and it has yet to prove itself a valid physical theory. Nevertheless, in The Trouble with Physics, Lee Smolin tries to give an overview, including the background, history, motivation, and content, as well as a detailed critique, at a level accessible to a general read­ ership. To help understand why a physicist might want to address an audience that cannot be expected to comprehend such a subject's hermetic details, it should be mentioned that only about a third of the book is concerned with String Theory per se. The remainder consists of an earnest plea for two things. The first is greater attention to alternative approaches to the funda­ mental questions of theoretical physics, in particular the line of research con­ cerning quantum gravity that comprises the author's own main interests; the sec­ ond is a critique of the assumptions and social pressures of the research com­ munity in which he works, with sug­ gestions for improvements. Nearly as much of this book is devoted to the so­ ciology of science - more specifically, the String Theory community - as to science itself. In view of the exceptionally high level of mathematical preparation that a genuine understanding of the subject would require, only a selective, non­ technical description of the ingredients can be presented in such a work. To make this at all meaningful is a very dif­ ficult task. Smolin succeeds in giving the interested general reader an idea of the flavor of the subject, and a vantage point from which to make some sense of the critiques that follow. It is hard to disagree with the gen­ eral socio-academic critiques, but many are far from unique to this particular field; in fact, they are rather common­ place within the broader academic com­ munity: the pressures upon young researchers to fit into an already estab­ lished "niche," the lack of incentive or reward for independent thought, the tendency towards "tribalism" and "group thought," the tendency by more senior scientists, who can provide or deny op­ portunities to younger researchers, to judge the merits of candidates by how well they agree with their own outlook and interests, etc. To these general sociological cri-

tiques, Smolin adds some that are more Nobel prizes to eight theoretical physi­ specific to the String Theory commu­ cists, and six more to experimentalists nity. This is an exceptional one, both who discovered or confirmed many of in the incomparable· ambitiousness of its observable consequences. The theory its scientific agenda and, according to seems at least to have no intrinsic de­ the author, its particular susceptibility to fects other than those shared by any rel­ the "group thought" phenomenon. ativistic quantum field theoretic model. Apart from this, many would agree that These include, of course, one un­ String Theory suffers from a central de­ usual feature that physicists have fect that sets it apart from nearly all learned to live with for decades; other scientific pursuits: it is unable as namely, the fact that perturbation the­ yet to provide anything that may be ory, based on successive approxima­ subjected to the test of experimental tions, leads at first to infinite quantities verification, beyond what is already ad­ that must be eliminated through a equately provided by more conven­ scheme of infinite renormalization be­ tional frameworks, such as the "Stan­ fore arriving at anything that may be dard Model. " From the viewpoint of the compared with experiment. However, Scientific Method this is anathema, this is generally not seen as an essen­ putting the subject into a difficult posi­ tial defect but rather a necessary feature tion to defend. Is it really physics, or of the perturbative approach, and the just mathematical conjecture that is too final results do agree with experiment incomplete to stand up as a physical to a high accuracy - at least in the theory? All depends on promises of weak and electromagnetic case. The things to come. fact that perturbation theory is not re­ The Standard Model has been very ally applicable to the Strong interactions successful in accounting for observable is partially mitigated by the fact that cal­ phenomena involving the electromag­ culations valid to all orders, using the netic and Weak nuclear forces as parts "renormalization group" approach, of a unified theory of electroweak in­ demonstrate the existence of "Asymp­ teractions. It also includes a consistent totic Freedom"; that is, the Strong in­ framework, Quantum Chromodynam­ teractions become arbitrarily weak at ics (QCD), for the Strong nuclear sufficiently short distances. forces, which hold the atomic nucleus No known quantum field theoretical together, although these are not yet framework exists, however, that in­ fully understood at a sufficiently de­ cludes gravitation and is consistent with tailed level to be able to account for General Relativity. The incompatibility the huge quantity of strong interaction lies in the fact that when trying to treat data accu-mulated over decades in gravitation as a quantum field, infinities high-energy physics laboratories. More­ persist in the perturbation theoretic cal­ over, the short-range nature of the culations that cannot be eliminated Strong interactions remains to be satis­ through renormalization. This problem factorily explained as a direct conse­ has never been overcome in any quan­ quence of the model. This is all as­ tum field theory setting. The desire to cribed to the complicated collective include gravitation within a quantum effects that are necessarily present in framework unifying all the fundamen­ such a relativistic many-body setting, in tal interactions of physics has been the which any number of particles may be main justification for the huge effort to created or destroyed within extremely develop String Theory over the past short time and space intervals. twenty-five years. This alternative to The discovery of this framework and quantum field theory conceives of all proof of its consistency were hailed as particles, including the graviton, as a great breakthrough in our under­ quantum excitations of strings. This is standing of the laws governing the in­ very different from the usual quantum teractions of elementary particles. It fol­ field theory framework, but the latter is lowed four decades of struggle, was expected to be recoverable in a suitable consolidated within the short period "low energy" regime (which includes all 1970-1974, and confirmed to be in energies accessible to high energy lab­ agreement with experiment in the years oratories to date). that followed. It led within the subse­ Smolin's own research priorities are quent two decades to the award of three spelled out in the first chapter, entitled

© 2008 Springer Science+Business Media, Inc., Volume 30, Number 3 . 2008

67

"The Five Great Problems in Theoreti­ cal Physics". In his view, these are: (1) To combine General Relativity and Quantum Theory in a single, complete theory of nature; (2) To resolve what he regards as problems in the founda­ tions of Quantum Mechanics. These in­ clude finding a better epistemological explanation of quantum theory, or an alternative theory that is not based on the necessity of involving the observer as part of the measurement process; (3) To come up with a unified theory that combines all the known forces of na­ ture (a bit redundant, when compared with Great Problem 1); (4) To explain all the apparently arbitrary constants ap­ pearing in the "Standard Model"; (5) To account for the existence (or nonexis­ tence) of "Dark Matter" and "Dark En­ ergy . " (This is a conjectured explana­ tion of the apparent inconsistency between the amount of observed mat­ ter in galaxies and the rates at which stars and galaxies are moving, which supposes the existence of matter and energy that is "invisible," adding up to as much as 96% of the total energy in the universe.) These may all be worthy goals, but not everyone would agree on their pri­ macy or achievability. In fact, there are many other outstanding problems that are at least as worthy of attention, but do not appear on this list. For example, the so-called "Mass Gap Problem," which was one of the "Millennium Prize Problems" [3] announced as a challenge in the year 2000 by the Clay Institute in its list of outstanding, mainly mathe­ matical problems (one of which - the Poincare conjecture - has since been solved), that would gain for the suc­ cessful solver, besides fame and glory, a modest prize of $ 1 ,000,000. For a physicist, the most concrete part of this problem consists of explaining, as a consequence of QCD - or some vari­ ant - why it is that the Strong nuclear interactions are short range. Although the short-range nature of the Weak in­ teractions follows directly from the Stan­ dard Model through the mechanism of "spontaneous symmetry breaking," an essential ingredient, the existence of the massive, spinless "Higgs Boson" re­ mains to be confirmed experimentally. QCD, however, does not allow for a similar breaking of the underlying gauge symmetry governing the strong

68

THE MATHEMATICAL INTELLIGENCER

interactions. Phenomenological expla­ nations, based on "flux tubes, " and "quark confinement" (neither of which have actually been demonstrated to fol­ low directly from the model) have been put forward. But the fact remains that the Weak and Electromagnetic interac­ tions can be dealt with directly; that is, the scattering amplitudes, decay rates, and correlations can be computed and compared, successfully, with experi­ ment, whereas the same cannot as yet be done for the Strong interactions us­ ing just QCD as a starting point. No one expects to improve on the accuracy of the results of perturbative calculations, but it would be nice to have them on a logically complete mathematical footing. Some (though not all) theoreticians see it as desirable that the foundations of quantum field the­ ory be revamped so as to make the ex­ istence of interacting quantum fields ac­ cord with our current level of mathematical understanding. (This is, roughly, the other part of the Millen­ nium Problem referred to previously.) These goals are no less compelling than the ones listed by Smolin, and probably several more could be added. Perhaps the reason for his particular choices was to keep within a range that could possibly be addressed by String Theory (although he concedes that (2) is not one of these). Or perhaps, the others are not listed because they con­ sist of improving or completing our un­ derstanding of an already existing the­ ory, and therefore do not figure as "revolutionary" (cf. T. S. Kuhn [4]). Although Smolin repeatedly empha­ sizes his preference for independent approaches, and abhorrence of "group thought," it seems that some assump­ tions of the community he is criticizing have also been adopted by him. In par­ ticular, little mention is made in The Trouble with Physics of approaches other than String Theory, except for his own specialty of Loop Gravity, which is given nearly equal prominence and space, although it only represents the interests of a rather small sector of the research community. The main scientific critiques of String Theory elaborated in the book are: (1) The necessity for introducing unob­ servable "higher dimensions" that are understood as spontaneously "curled up" to such small sizes that they escape

detection, without any dynamical mech­ anism implying such a process, or its stability, and no physical explanation of why one or another of these "back­ grounds" (in ten or eleven dimensions) is preferred. The vast multiplicity of pos­ sible background geometries available in the "string landscape" seem to make it impossible to arrive at definite pre­ dictions since they introduce a huge number of additional parameters (the "moduli") whose values the theory is in­ capable of determining; (2) The lack of any experimental evidence for the dis­ tinct consequences of string theory. In particular, the essential ingredient of "supersymmetry" (from which derives the name "Superstrings") is required to assure finiteness. At the very least, this means a matched pairing of all the fermionic particles in the universe, hav­ ing 1/2 integer spins (such as electrons, neutrinos, quarks) with bosonic part­ ners, sharing similar properties, but hav­ ing integer spins. Nothing remotely like this exists in the currently observed spectrum of elementary particles. The usual reply to this is: "Yes, but it is a spontaneously broken symmetry", which just suggests sweeping an essential fea­ ture under the rug because it is incon­ venient to face up to its observable implications; (3) The absence of a "com­ plete" version of string theory. In par­ ticular, there is no "Superstring Field Theory" that would allow for processes involving the creation and annihilation of strings. Beyond these broad critiques, there are a number of doubts raised regard­ ing more technical points, such as the interpretation of the various types of "duality" that enter into the theory, which help to reduce what might be five distinct versions to one, and the sign of the cosmological constant, which has direct implications for the be­ haviour of matter in the large. Whether right or wrong, these are best left to the specialists to debate, since they do not in themselves imply fundamental trou­ ble with the theory, simply differences of view regarding its implications. (See, e.g. , [5] for a discussion and rebuttal of some of these points.) Smolin also regrets that an entire generation of theoretical elementary particle physicists has been raised to work with such a highly esoteric math­ ematical model, founded on far-fetched

speculation, without seeing the neces­ sity for pinning down their conjectures by precise mathematical demonstration, and without adequate consideration of possible alternatives. Finally, there are the further sociological-psychological critiques: the arrogance with which many string-theorists have vaunted the relative importance of their work, and the pressure that has been put upon up­ coming researchers to work along cur­ rently accepted lines, at risk of achiev­ ing nothing of any real significance (and getting no employment). The criticism of immodest swagger, however, comes from an author who ends his book by stating, with no hint of irony or self­ mockery, that he will now return to his "real job," which is to "finish the revo­ lution that Einstein started.'' Although some of the most brilliant and talented researchers of our day have devoted themselves to String The­ ory, it is undeniable that advances have largely been driven by far-reaching, but incompletely demonstrated conjectures. New ideas have had an esoteric, self­ contained character, independent of the traditional checks based on careful comparison with experiment. There is no doubt that Einstein, who always was concerned with testing theoretical pre­ dictions against observation, would be astonished at such a situation persisting after more than two decades of research in the subject. The most important con­ cern for String Theory, in the end, may be that the legacy of the current gen­ eration of theorists risks being purely mathematical in nature if no contact is made within a reasonable time with ex­ perimentally verifiable phenomena. Whatever the validity of the argu­ ments in this book as a critique of String Theory, they do not justify the sweep­ ing conclusion implied in the title. The Trouble with String Theory might have sold fewer copies, but it would have been a more accurate characterization of the content of this book. It is a poor service to potential young physicists, and an injustice to the rest of physics,

to dismiss as unworthy of mention un­ der the title The Trouble with Physics the many wonderful discoveries, both experimental and theoretical, that have been made over the past three decades in other areas of physics. The most interesting of these, which have merited not a few Nobel Prizes in recent years. have tended to concern the "other" frontier of experimental and theoretical physics: the physics of tem­ peratures near to absolute zero and macroscopic quantum phenomena. The latter largely concern quantum effects and states of matter that can be ob­ served at or near macroscopic scales, both at very low temperatures, and at temperatures at which it was never pre­ viously imagined such collective quan­ tum effects could occur. These include many of the really remarkable achieve­ ments of physics of the past few decades. For example, there was the creation, in plasma physics laboratories, of a ·'Bose-Einstein Condensate," an en­ tirely new state of matter, achievable only at the lowest temperatures, pre­ dicted by Einstein in the 1920s, but not observed in a laboratory until the 1990s. 1 Related to this is the remarkable use of "laser cooling" and optical or other methods for reducing the motion of individual molecules to such a slow point that they can be individually stud­ ied and controlled, and together with this, the creation of purely optical grids, and traps 2 There have also been im­ portant theoretical advances in the un­ derstanding of disordered systems, liq­ uid crystals, and polymer dynamics 5 There have been the observation and theoretical explanation of the "fractional quantum Hall effect" , a phenomenon that brings to light the beautiful role played by topology in the electromag­ netic properties of ordinary matter in a suitable quantum regime. '' The discov­ ery of high-temperature superconduc­ tivity has led to fundamental challenges to explain associated recently-discov­ ered phenomena. Another dramatic de­ velopment in theoretical physics has

concerned the precise understanding of scaling properties in critical phenomena in two dimensions, and related phe­ nomena such as percolation, and criti­ cal wetting. In the high-energy domain it should also be mentioned that the prediction of "asymptotic freedom", s which was an essential step in making sense of a quantum field theory of the strong interactions, obtained its experi­ mental verification within the past two decades. There are numerous other examples that could be cited. Perhaps none of the above can be compared in ambition or scope with seeking a "Theory of Every­ thing" , nor to the impact of the revolu­ tionary developments of the early twen­ tieth century: Relativity Theory and Quantum Mechanics. They nevertheless represent very exciting advances in our understanding of the physical world, and provide abundant evidence that much more is healthy and robust in the physics of the past three decades than what might, by narrowly focusing on the shortfalls of String Theory, lead to the conclusion that physics of our time is in trouble. REFERENCES

[ 1 ] Brian R. Greene, The Elegant Universe: Su­ perstrings, Hidden Dimensions, and the Quest for the Ultimate Theory, Norton,

1 999. [2] Paul C. W Davies, Julian Brown (eds.), Su­ perstrings: A Theory of Everything?, Cam­

bridge University Press ( 1 988). [3] A Jaffe, E. Witten, "Yang-Mills and the Mass Gap," Clay Institute Millenium Prize Problem. URL: http://www.claymath.org/ millennium/Yang-Mills_Theory. [4] T. S. Kuhn, The Structure of Scientific Rev­ olutions, 3rd. ed. , University of Chicago

Press, 1 996. [5] J. Polchinski, "All strung out," The Ameri­ can Scientist online, Jan-Feb 2007.

Centre de Recherches Mathematiques Montreal, Canada e-mail: harnad@crm umontreal.ca

1 Nobel prize to E. Cornell, W. Ketterle, and C. Wiemann (200 1 ) . 2Nobel prize t o S . Chu, C . Cohen-Tannoudji, and W . O . Phillips (1 997). 3Nobel Prize to Pierre-Gilles de Gennes ( 1 99 1 ) . 4Nobel Prize to R. B. Laughlin, H. L . Stormer, and D . C. Tsui (1 998). 5Nobel Prize to some of the theoreticians who derived it: D. Gross, D. Politzer, and F. Wilczek (2004).

© 2008 Spnnger Science+ Business Media, Inc., Volume 30, Number 3 , 2008

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The Shoelace Book: A Mathematical G uide to the Best (and Worst) Ways to Lace Your Shoes by Burkard Polster PROVIDENCE, RHODE ISLAND, AMERICAN MATHEMATICAL SOCIETY, 2006, 125 PP., ISBN· 10: 0-8218-3933-0, ISBN-13: 978-0-8218-3933-1 SOFTCOVER, EQUIPPED WITH A PAIR OF SHOELACES FOR EXPERIMENTATION, US $29.00.

REVIEWED BY JOHN H. HALTON

S

ome time early in 1992, my wife asked me whether, if our child had broken her shoelace near one end, it might be possible to relace the shoe so as to still be able to tie it. Being a mathematician, I first ab­ stracted the problem to what Polster calls a mathematical shoe (see Fig. 1 ; I have slightly adapted my original nota­ tion to conform with Polster's.). I then defined a "lacing" as a polyg­ onal line starting at the eyelet A1 and finishing at B1 , and applied reasonable

conditions to such lacings, limiting them to those with the line passing through every eyelet just once (as does Polster), and alternating between the column of eyelets A1, A2, . . . , An on one side of the shoe and the column of eyelets B1 , Bz, . . . , En on the other side (Polster calls such an arrangement a dense lac­ ing-one having no vertical segments, joining two eyelets in the same col­ umn). The problem was then to find the shortest such lacing. The rationale of limiting oneself to dense lacings is that every segment of lace (between eyelets) would contribute something to the force pulling the sides of the shoe together (see Chapter 7 of Polster's book). Finally, I proved a theorem that showed that a certain kind of lacing (see Fig. 2)-called an American lacing by me, and a crisscross lacing by Polster­ is uniquely of shortest length, under the conditions imposed here. I first published my result as a tech­ nical report (UNC#-92-032) in August of 1992. (Meanwhile-as is the way of most mathematicians' wives-my wife had long before gone to the store and bought a new pair of laces for our daughter.) After an extended search for a compliant journal, Tbe Mathematical Intelligencergave my results a wider au­ dience in 1995. Because mathematical research moves in a tree-like structure, it is a relatively rare distinction to originate a new line of enquiry (which I tentatively claim to have done in 1 992) or-as Burkard Pol­ ster has done, with admirable elegance, breadth, and style-to write the first

book about such a new field, gather­ ing together the results of several work­ ers. He has considerably widened the field of enquiry and rounded-off the known universe in this area with new and unexpected results, all in a friendly and engaging manner that is both en­ tertaining and interesting, unassuming and intriguing. One can only wonder whither the new-fledged bird will now fly. The book is organized into an in­ formative Preface (which includes a Summary of Contents) and eight chap­ ters, followed by two addenda, on Re­ lated Mathematics ( "Traveling Sales­ man Problems" and "The Shoelace Formula") and Loose Ends (covering such topics as "History," "Shoelace Su­ perstitions," "Questions of Style," "Fash­ ion," and "What Is the Best Way to Lace Your Shoes?"). In addition to dense lacings, Polster defines straight lacings as those con­ taining all n possible "horizontal" links, a superstraight lacing as a straight lac­ ing all of whose nonhorizontal seg­ ments are vertical, and a simple lacing as one in which the eyelets are first vis­ ited in downward sequence and then in upward sequence (i.e., there are no "backtracks"). Chapter 1 , "Setting the Stage," com­ pletes the definitions of terms used in the book and gives examples of com­ mon lacings used as illustrations. Chapter 2, "One-Column Lacings," considers the problem in which the two columns of eyelets are "pulled together" into one. This simplified problem, in­ volving a lacing that visits each of n eyelets just once, still has subtleties, and Polster solves many intriguing prob­ lems. He determines the number of one-column lacings (without loss of gen­ erality, for simplicity, Polster here takes h = 1): for n 2, just one; for n > 2, +( n - 1)! [Theorem 2 . 1] . For n > 2, the number of shortest such lacings is rela­ tively easily seen to be 2 n -3, each of length 2(n - 1) [Theorem 2.2]. The problem of finding the number and common length of the longest one-col­ umn lacings [see Chapter 6] turns out to be much more difficult to solve. Pol­ ster proves that, for even n :::::: 4, there 2) n ) ! longest lacings, ! are =

0 0 0

1(

Figure

70

I.

The mathematical shoe.

THE MATHEMATICAL INTELLIGENCER

Figure

2.

Crisscross [American] lacing.

�

(;

each of length n2 , while, for odd n :::::: 3, 2

there are

n

;1

(n;3) (n; 1 ) !

longest lacings, each of length

n2 -

!

1

---

2 (Theorem 2.3]. Some additional (later useful) special cases are also analyzed. Chapter 3, "Counting Lacings," very thoroughly determines the number of possible lacings of the ten possible dif­ ferent types. Write G for general, D for dense, S for straight, Tfor superstraight, and M for simple; then the types of lac­ ings are (i) GDcscMc, (ii) GDcscM, (iii) GDCSTCMc, (iv) cncs1Mc, (v) cncsrcM, (vi) GDcS7M, (vii) GDScMc, (viii) GDScM, (ix) GDSFMc, and (x) GDST'M The omissions are correct, because TC S and D T = 0. I may add, to remove any pos­ sible confusion, that Polster gives a summarizing "Venn diagram," in which types of lacings are denoted by the in­ teriors of (overlapping) oval figures, and in which a "disjoint region" (the in­ tersection of such ovals), in which the name of a lacing-type is shown, also shows the total number of lacings of that type; whereas a disjoint region, without an explicit name, shows the to­ tal number of lacings of the corre­ spondingly overlapped types (see, e.g., Theorem 3. 1 , on p. 20 of the book). Al­ though this notation is slightly non­ standard, it is consistently used in Pol­ ster's book, leading to no confusion. Chapter 4, "The Shortest Lacings," is closest to my own, relatively narrow re­ sult. (It is half of Theorem 4 . 1 1) Polster proves (sometimes in several ways) that the bowtie n-lacing is the shortest n­ lacing overall and that the crisscross n­ lacing is the shortest dense n-lacing (Theorem 4. 1 ) ; that, if n is even, the simple-and-superstraight n-lacings are the shortest straight n-lacings, whereas, if n is odd, the zigzag n-lacings are the shortest straight n-lacings (Theorem 4.2); and that the star n-lacings are the shortest dense-and-straight n-lacings (Theorem 4.3) . In Theorem 4.4, he com­ pares the lengths of several types of lac­ ings, pointing out that some of these re­ sults have already been proved by me. Altogether, this chapter is a formidable tour de force. A number of special types of lacings recur often in the book. A sampling of four of them is illustrated here (see Figs.

Figure

considers a number of natural general­ izations of the original "mathematical shoe" (see Fig. 1 ) . The columns of eye­ lets may not be parallel, or the spacing of eyelets in each column may be dif­ ferent, and it may be irregular. Perhaps the most amazing result is that the criss­ cross lacing remains the shortest dense lacing under rather arbitrary and even brutal changes in the shoe! Chapter 6, "The Longest Lacings," at­ tacks a different problem, the reverse of the "shortest lacing" problem. In many cases, one of the optimizations is very interesting, whereas the reverse is simple and lacks both usefulness and charm. This, surprisingly, is not the case with shoelaces; even Dr. Polster was as­ tonished by the richness and complex­ ity of the results he was able to prove, and the variety of conjectures that he

3-6).

In Chapter 5, "Variations on the Shortest Lacing Problem," the author

3. Star [European) lacing.

Figure

Figure

4.

Zigzag [shoeshopl

found too hard to prove (having seen the elaborate methods of proof that he has brought to bear on this work, I do not take lightly the difficulty of the un­ solved conjectures!). Chapter 7, "The Strongest Lacings," turns to an entirely different question. As a first approximation, when a lace passes through an eyelet, we may pos­ tulate that no friction occurs, so that a certain amount of tension along the lace is the same at all points, and that the component of force acting to pull the sides of the shoe together is subject to the vector, or cosine, law (see Fig. 7). One can derive the values of factors of the form sinO or cosO from the positions of the eyelets involved (e.g., tan a = 2 h and tan f3 = h). Thus, relative to any tension T, Polster was able to obtain the strength of any given lacing from the

2

2

n

n

5. Bowtie lacing.

lacing.

Figure

6.

Zigsag

B, lacing.

© 2008 Springer Science+Business Media, Inc., Volume 30, Number 3. 2008

71

that this book might well be the only one in the history of printing whose dedication ("For Dudu and joujou"­ presumably the author's young chil­ dren) is illustrated by a photograph of two differently-laced pairs of tiny sneak­ ers; this in itself should ensure its en­ during fame! Department of Computer Science University of North Carolina at Chapel Hill Sitterson Hall, CB 3 1 75 Chapel Hill, NC 27599-31 75

Figure

7.

Forces of pulley effect.

USA e-mail: [email protected]

total horizontal forces, what he calls the pulley sum. For example, defining C(n, h) to be the pulley sum of the crisscross lacing and Z(n, h) to be the pulley sum of the zigzag lacing, he ob­ serves that there is a unique h n > 0, such that C(n, hn) Z(n, h n ); from this, he is then able to infer Theorem 7 . 1 , which asserts that (i) if h < h n, then the crisscross n-lacing is the strongest joran Friberg n-lacing; (ii) if h > hn, then the zigzag is the strongest; and (iii) if h = hn, then SINGAPORE: WORLD SCIENTIFIC PUBLISHING CO. XII + 294 PP., 2005, US $64.00, both of these lacings are equally the ISBN 981-256-328-8, ISBN 13 978-981-256-328-8 strongest. Similarly, Theorem 7.2 deter­ mines the strongest straight n-lacings. REVIEWED BY LEO DEPUYDT Thus Polster has defined an entirely novel, interesting, and potentially use­ his book has much to offer. Al­ ful area of enquiry in this field, which most every facet of Egyptian and is itself new. By arguments that are orig­ Babylonian mathematics is disinal and by no means trivial, he man­ cussed. Most of what has been accom­ ages to solve quite a few of the prob­ plished before in this field is surveyed; lems that arise. By analogy to the leap the hook is a good introduction to the from Chapter 4 to Chapter 6 (from subject. But most of what is useful in it shortest to longest lacings), he goes bears no relation to the title. from Chapter 7 to Chapter 8, "The History is, by definition, the period Weakest Lacings." Here again, the prob­ for which we have written sources. In lems that arise turn out to be, on the that sense, the history of Western civi­ whole, tougher than the earlier ones, lization begins roughly about 3000 Bc. and Polster is able to deduce only some In its 5000-year history, different nations of the answers. For the rest, he has al­ have over time occupied center stage lowed computer experiments to guide by virtue of intense displays of complex him to some interesting conjectures. activity. From 3000 BC until about 600/ This completes the author's remarkably 500 Be, and preceding the rise of Greece comprehensive analysis of some major and Rome, two Near-Eastern cultures problems in the theory of lacings. stood out: Egypt and Mesopotamia. Burkard Polster has managed to give Babylon was the main cultural center in the world a new, large, and varied field Mesopotamia, and the two largest bod­ of enquiry, which it did not have be­ ies of sources from that long span of fore, beginning with a simple question. time are in hieroglyphic writing and What is more, he has written a book, cuneiform writing. The shift came in 500and it is hard to imagine any mathe­ 300 Be and is epitomized in the epic matician not finding this hook irre­ conflict, about whose early decades sistible. By the way, it would appear Herodotus famously wrote, between

U nexpected Links

between Egyptian

=

and Babylonian Mathematics

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THE MATHEMATICAL INTELLIGENCER

Greece on the one hand and Persia as principal player of Near-Eastern nations including Egypt and Mesopotamia, on the other. The Persian king Cambyses conquered Egypt around 525 BC, after his predecessor Cyrus took Babylon in 539 BC. Thus was created the largest em­ pire ever seen. The ancient world grad­ ually became a more interconnected place. At the beginning of the period 500-300 BC, Persia could still challenge Greece on its own territory at Marathon (490 sc) and Salamis (480 sc), even if suffering crushing defeat. At the end of that period, the Macedonian Alexander the Great (356 Bc-323 Be) conquered the vast Persian empire. Also in that pe­ riod, Greek culture became preeminent throughout the inhabited world. Mean­ while, Rome had begun its ascent and, after eliminating its rival Carthage in the Western Mediterranean, would take its own turn on the world stage. Ancient science and mathematics evolved against this macropolitical background. The central theme of the book under review is links between Egyptian and Babylonian mathematics. In the search for links between Egyp­ tian and Mesopotamian culture in gen­ eral, a distinction is necessary between the time before and after Camhyses's conquest, 525 Be. Undoubtedly, there was much contact and mutual influence between the two cultures after 525 BC: Babylon and Memphis now belonged to the same Persian empire. Also, after Egypt regained its political indepen­ dence from Persia around 400 BC, and later, when the Persian empire fell apart with Alexander's conquests, Egypt and Mesopotamia remained in close con­ tact. The influence between the two cultures extended to mathematics and science. At the same time, a third fac­ tor played a crucial role in the interre­ lation between Babylonian and Egypt­ ian math and science, complicating it and rising above it, namely the scien­ tific approach of Greek rationalism. Alexander's conquests contributed much to its spread. As to the 2500 years before 525 BC, we need to distinguish the era before about 1 500 Be and the era after that. In the Egyptian New Kingdom, from about 1 500 BC onward, contacts between Egypt and West Asia vastly increased. The likelihood of cultural contact of any form or shape is therefore much more

likely. In sum, when it comes to as­ sessing the probability of intellectual in­ fluence between the two cultures, there are fundamental qualitative differences between three main periods: (1) before 1 500 BC; (2) from 1 500 BC to 525 BC; ( 3 ) after 5 2 5 BC. The fundamental claim of the work under review is the existence of links between Egyptian and Babylonian mathematics. The book does not pre­ sent new texts. Friberg defends his claim by new interpretations of known texts. The importance of the claim is made clear by printing Links in reel in the title on the cover. But what about the book's central claim? Let"s look at three general issues of method: the time-frame of the sources and the words Links and Unexpected in the title. First, the time-frame. In light of the macropolitical history outlined previ­ ously, one would expect the earliest manifestations of mathematics to appear in Mesopotamia and Egypt, as they in fact do. These manifestations exhibit three striking characteristics. First, they are dated to the centuries before 1 500 BC when contacts between Egypt and West Asia were much more restricted than they would become later. Second, they are concentrated in high quantity in a relatively short time period, the early second millennium BC. Third, this short time period happens to he roughly the same for both Egyptian and Baby­ lonian mathematics. As regards Egypt­ ian mathematics, two hieroglyphic pa­ pyri dating to that time preserve most of what we have, the Moscow Mathe­ matical Papyrus and the Papyrus Rhine! of the British Museum . The cuneiform mathematical texts of the same time pe­ riod are preserved in a large number of clay tablets. After these early manifestations, there is a noticeable drought of mathe­ matical texts in both Mesopotamia and Egypt that ceases only when, with the rise of Greece in 600/500 nc, the world became a more international place, and cultural links are no longer a matter of speculation. The links postulated in the hook under review concern not only sources dated to before 1500 BC hut also after 500 BC. There is a gap of more than a millennium between the two halves of the book. In light of the macropolitical context, I consider the existence of links highly

improbable before 1500 BC and proven beyond a doubt after 500 BC. This sharp contrast between the earlier and later sets of sources is undeniable, although not articulated explicitly anywhere in the book. Because of this absence, the undeniable links in the later set of sources may prejudice the assessment of the earlier set. Although the earlier Egyptian and Babylonian sources date to about the same time, I do not know what to make of the links. Writing emerged around the same time in both Mesopotamia and Egypt, and there has been much speculation about what this means for the relationship between the two writing systems. But there is no ev­ idence to contradict the notion that the two systems came into existence inde­ pendently. The second issue of method con­ cerns the concept of links. One can think of two types of cultural links: or­ ganic and typological. Typological links between two items are nothing more than similarities. In fact, there is a case to he made for not calling them links at all. Only organic links are links in the strict sense of the word. They concern not just similarity of knowledge but ac­ tual transmission of knowledge. Impor­ tantly, such transmission presupposes movements of people and contacts be­ tween people as historic events. In the case of Egyptian and Babylonian math­ ematics, there is no direct knowledge from the sources about the lives and travels of mathematicians, let alone about actual encounters between math­ ematicians from Egypt and from Baby­ lon. All evidence about possible con­ tacts must he inferred indirectly from the texts themselves. J . F. Quack has advocated distinguishing carefully be­ tween ''typological juxtaposition'' and "genetic ties" as a prerequisite for a ·'pure methodological foundation" in the comparison of ancient wisdom texts ( Die Lehren des Ani, 1994, p. 206). The same distinction should be valid in com­ paring mathematical texts. Nowhere in the book under review is there any mention of the difference between these two types of links. Instances of both types appear to be mixed together, and it is often not clear to which type a certain instance is supposed to be­ long. It would have been useful if Friberg hac! assembled his strongest evidence

in one place, so the reader could know which evidence was the firmest and most undeniable. He does begin with a comparison between the mathematical cuneiform text M. 7857 from Mari and problem no. 79 in the hieroglyphic mathematical Papyrus Rhind; this com­ parison triggered his entire investiga­ tion. Both texts concern a geometric progression, the relationship between 1 , 5 , 25, 125, etc. The similarity between the texts cannot be denied. But there are also differences. The central num­ ber in the Mari text is 99; in the Egypt­ ian text, it is 7. Egyptian mathematical problems often revolve around the number 7. Multiplication was achieved by doubling, and the Egyptians were also comfortable with 3 and with 10 and with 5 as half of 10. The numbers 4, 6, 8, and 9 have 2 or 3 as factors. That makes 7 the most challenging number from 1 to 10, the odd man out as it were. The attention to 7 is characteris­ tic of Egyptian mathematical exercises. It comes as no surprise that they often practiced with it. The third issue of method concerns the term "unexpected." What it means is not immediately clear. As a historian of antiquity, I knew "unexpected" as a description of what kind of death Cae­ sar wanted for himself (aprosdoketos in Plutarch, aiphnidios in Appian, in­ opinatus in Suetonius) . But how does "unexpected" designate a book's con­ tents? And since "unexpected" negates "expected," what is expectation? The mind stores impressions of the world outside itself by contact with reality through the senses. The order is: phys­ ical contact first, mental impression sec­ ond. However, the mind is able to in­ vert this order. Past impressions can, independently from reality, spawn men­ tal impressions that cause the mind to look out for the occurrence or nonoc­ currence of future physical contacts be­ tween the senses and reality. The resulting state of mind is called expec­ tation (from late Latin expectare "look out"). When what happens contradicts what the mind is looking out for, some­ thing unexpected has occurred. The need for the mind to adjust leaves a strong impression in its own right. The word "new" is repeated through­ out the book, apparently as confirming "unexpected." But not evetything that is new is necessarily unexpected. All

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depends on the state in which the mind places to have forced itself onto the finds itself when it is affected by some­ book's contents. This can lead to awk­ thing new. As a reader of the book and ward developments in the argument. It potential addressee of its title, I had the may be useful to illustrate this rhetori­ uneasy sensation that something was cal phenomenon because it occurs assumed about my mind. Why would I more than once. Consider the sequence take everything new to be either ex­ that begins as follows (p. vii): pected or unexpected? The book does My observation that there seems to identify two readers who would prob­ exist clear links between Egyptian ably deem the links in question unex­ and Babylonian mathematics is in pected. According to J. Hoyrup and J. conflict with the prevailing opinion Ritter, nothing in the surviving evidence in formerly published works on suggests that the kind of links proposed Egyptian mathematics, namely that by Friberg existed (p. 26). The author's practically no such links exist. eminent expertise in the book's subject The author then describes a goal that matter is beyond question. But the same differs from that of the title: can be said about Hoyrup and Ritter. However, in view of the mentioned dynamic (p. viii) character of the his­ When the expectations of some who are among the best qualified to know what tory of Mesopotamian mathematics, not least in the last couple of to expect and what not to expect stand in direct contradiction, the stakes have decades, it appeared to me to be been raised, and nonspecialists may be high time to take a renewed look at confused. Egyptian mathematics against an up-to-date background in the history Just as in the case of "unexpected," I felt that something was assumed about ofMesopotamian mathematics! That someone else's state of mind when is the primary objective of this book. Hoyrup's and Ritter's views are de­ [Author's italics.] scribed as "pessimistic" (p. 26). Why But then the author reaffirms that prov­ would the fact that the sources suggest ing the existence of links is his main absence of contact between two cul­ aim (p. viii): My search for links between Egyp­ tures, say ancient China and ancient Egypt, make anyone either pessimistic tian and Babylonian mathematics has been unexpectedly successful, or optimistic? Why would the absence of links between Egyptian and Baby­ in more ways than one. Not only has lonian mathematics be cause for gloom? the search turned up numerous pos­ One more concern: aside from sible candidates for such links, but whether the similarities between Egypt­ the comparison of Egyptian and ian and Babylonian mathematics iden­ Babylonian mathematics has in many cases led to a much better un­ tified in this book are similarities or ev­ idence of true historical links, calling a derstanding of the nature of impor­ similarity "unexpected" is more difficult tant Egyptian mathematical texts and with mathematics than with any other of particularly interesting exercises expression of culture. Deductive that they contain. thought differs from inductive thought Yet this statement too exhibits a twist. and activity such as literature, religion, The term "possible candidates" blunts and art. Mathematics is by definition the impact of the title. Are there no universal; one expects it to be essen­ "certain" candidates? The shifts I would tially the same everywhere. That does perceive in this succession of three not mean that modes of expression may statements recur often in the line of ar­ not differ, leaving opportunity for de­ gument. The result is a loss of focus. tecting historical links. Still, the burden True, publishers may pressure authors of proof in deriving historical links from to exhibit a Big Theme. But the author's similarities is on the whole more oner­ large collection of observations do not ous in the case of mathematics than it yield one. Although the pressure for is with just about any other type of hu­ finding a theme is understandable, the man activity. need for it to generate a certain mea­ The book does not, in my opinion, sure of excitement for its own sake is do what its title says it will. A false im­ less so. pression of unity and coherence is the But again, my overall impression of result. The title theme seems in many the book is hardly unfavorable. This is

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a highly sophisticated, greatly knowl­ edgeable, and thoroughly accurate study of Egyptian and Babylonian math­ ematics. Nor can it be denied that there is similarity when the author says there is, and that this similarity may, indi­ rectly, inspire attempts to enrich the in­ terpretation of these ancient scientific texts. But in my opinion the similarities do no more than confirm that mathe­ matics is universally true and is there­ fore bound to be the same everywhere within certain parameters. The book is a long string of discrete case studies of individual problems of Egyptian and Babylonian mathematics, supported by extensive bibliography and copious references to other work accomplished in the field. Perhaps it could have been organized by putting the rich surveys of the sources at the beginning and then numbering the case studies from 1 to 100 and beyond. An extended review would need to consist of as many subdivisions. There is no space here to address in­ dividual matters of interpretation, which number in the hundreds. For example, I would beg to differ with the charac­ terization of problems 28 and 29 of the Papyrus Rhind, which B.L. van der Waerden once called "the climax of Egyptian arithmetic" (Science Awaken­ ing, 196 1 , p. 29), as incomplete exer­ cises. My own impression that they are fully complete would require a line of argument of some considerable length. Nor is there room here for general mat­ ters of method. I am still waiting for a monograph on Egyptian mathematics that nowhere mentions the words "mul­ tiplication" or "division." In my opinion, Egyptian mathematics had no such things. But this review is not the place to discuss it. In conclusion, although the focus of my reflections has been narrow, I strongly urge anyone seeking to advance the cause of the history of mathematics to keep the present book close at hand for consultation, along with the other standard monographs on the subject. Department of Egyptology and Ancient West Asian Studies Brown University Box 1 899 Providence, Rl 029 1 2- 1 899 USA e-mail: [email protected]

The Volterra Chronicles: The Life and Times of an Extraordinary Mathematician,

1860-1940 by judith

R. Goodstein

AMERICAN MATHEMATICAL SOCIETY, LONDON MATHEMATICAL SOCIETY, 2007,

xxvi

+

310 PP., ISBN-10: 0-8218-3969-1, ISBN-13: 978-0-8218-3969-0, US $59

REVIEWED BY GIORGIO ISRAEL

he proclamation of the Kingdom of Italy in 1861 marked the be­ ginning of an extraordinary effort of cultural unification and development of education and scientific research. One oft-cited example is that of three eminent Italian mathematicians-Enrico Betti, Francesco Brioschi, and Felice Ca­ sorati-who, in the years that followed, undertook a journey to study the mod­ els presented by the more advanced Eu­ ropean countries, above all Germany and France. These were the models that would later influence the institutional and scientific development of Italian mathematics. This development was promoted with such vigour that by the end of the 19th century Italian mathe­ matics ranked second only to the two leading countries in world mathematics. It would take too long and be too com­ plicated to shed light on all aspects of these influences here. In a word, the German model may be said to have ex­ erted greater influence on the organi­ zational and institutional aspects of education. It also encouraged the estab­ lishment of a school of geometry with an interest in both "pure" algebraic geometry and differential geometry. The French model, on the other hand, stim­ ulated interest in mathematical physics and mathematical analysis, which were deemed to be closely linked. France also represented a point of reference at the level of "general" scientific culture. These early steps led to the creation of the renowned Italian school of alge­ braic geometry. After toning down the

"purist" excesses of Luigi Cremona, a central figure not only in the formation of the mathematics community but also of engineers, the leaders of this school­ Federigo Enriques, Guido Castelnuovo, and later Francesco Severi-placed at the focus of research such themes as that of the classification of algebraic sur­ faces. They made a brilliant and pro­ found contribution, albeit in an intuitive and aristocratic approach that was un­ mindful of rigour and that still today is a source of inspiration for research. Vito Volterra ( 1860-1940) was the top rep­ resentative of the school of mathemat­ ical physics, which was closer to the French view of Henri Poincare or Emile Picard. A separate case was that of Tul­ lio Levi-Civita, who expressed a syn­ thesis between the French influence and that of the German mathematics: a brilliant heir to the differential geome­ try tradition of Luigi Bianchi, he built Gregorio Ricci-Curbastro's research up into a rigorous foundation of tensor cal­ culus. His solid background, both in mathematical physics and in differential geometry, allowed him to set up a rig­ orous mathematical foundation of gen­ eral relativity. Keenly aware of the lat­ est developments, such as applied mathematics research in the field of tur­ bulence, Levi-Civita was perhaps the most profound and brilliant Italian mathematician of that period. And yet it was Vito Volterra who was considered abroad the main representative and am­ bassador of Italian mathematics in the 20th century, to the point of being nick­ named "Mr. Italian Mathematics." The reasons underlying this special status are not dependent solely on Volterra's scientific prestige and the fact he was older than the other main fig­ ures in Italian mathematics. There were two other decisive factors. Volterra played a very important role in the es­ tablishment of scientific institutions in the country. He was also a man of cul­ ture in the full sense of the word. Only Enriques can be compared with him. Volterra was in any case the most ef­ fective in promoting a true cultural pol­ icy. Following Cremona's example, Volterra realized that the country was in need of suitable institutions if it was to become a true scientific "power." He principally took the French model as his inspiration, which proved to be a limi­ tation, as the German model was cer-

tainly more innovative. He founded and refounded a large number of institu­ tions, such as Societa Italiana di Fisica, Consiglio delle Ricerche, and the Comi­ tato Talassografico. He embraced the extraordinary idea that science, in order to gain prestige, needed cultural dis­ semination and the involvement of all sectors of society: teachers, engineers, economists, and men of general culture. The Societa Italiana per il Progresso delle Scienze (SIPS), which he reestab­ lished in 1906, played a decisive role in this direction. Until the Second World War, the SIPS congresses were a meet­ ing place for the whole of Italian sci­ entific culture. Voterra turned the Ac­ cademia dei Lincei, of which he was a long-time president, into one of the cen­ tres of dissemination of scientific cul­ ture. He was inspired by a moderate progressive conception. He was a de­ mocrat with enlightened views, pro­ foundly convinced of the value of sci­ entific and technological progress. At the scientific level, he expressed a view that may he encapsulated in the for­ mula: to defend and extend the scope of classical reductionism based on a de­ terministic and differential type of math­ ematical approach. This led him to take an interest in and contribute actively to the application of mathematics to biol­ ogy and economics, which were viewed with scepticism by many of his con­ temporaries. The figure of Volterra is rich, fasci­ nating and complex, full of contrasts. The collision between his democratic and enlightened view and fascist au­ thoritarianism was inevitable. In this clash Volterra demonstrated all his courage and nobility of character. Nev­ ertheless, the model he proposed dis­ played some weaknesses. This was clearly seen when he rejected the edu­ cational reforms in a humanist direction promoted by the idealist philosopher Giovanni Gentile, who was a minister in the fascist regime. He tried to com­ bat it with a proposal based on a par­ tial reform of the old and inadequate Casati law. He fought against the ten­ dency of the fascist regime to separate pure and applied research and thus ma­ terially opposed a trend that neverthe­ less possessed some aspects of moder­ nity. The fascist regime with its policy of autarky (of which anti-Semitic racism represented the extreme expression)

© 2008 Springer Science+ Business Media, Inc., Volume 30, Number 3 , 2008

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destroyed Italian science, although ideas such as that of setting up two sep­ arate institutes of Higher Mathematics (Istituto Nazionale di Alta Matematica, INDAM) and Calculus Applications (Is­ tituto Nazionale per le Applicazioni del Calcolo, INAC) were valid. Volterra's in­ terest in biomathematics was a wide­ ranging and modern intuition although his dislike of probability calculus and his view that the only serious tools for mathematics were differential equations represented a severe handicap. His pro­ French and bitterly anti-German atti­ tude-Volterra was a leader of the Latin Union-led him to boycott an Halo-Ger­ man applied mathematics congress or­ ganized by Levi-Civita. In this way, he severed the relations of Italian science with the new schools of research in the field of turbulence. He was also rela­ tively insensitive to the new develop­ ments in physics and opposed the cre­ ation of a chair of theoretical physics for Enrico Fermi. Vito Volterra had high status as a sci­ entist, an intellectual, and a cultural or­ ganizer who played a decisive role in raising Italy to among the first scientific ranks at a world level and who was in­ spired by a form of enlightened ratio­ nalism that was as lofty and profound as, in certain respects, it was backward look­ ing. It is to this figure that judith Good­ stein addresses her book. By means of a thorough and systematic use of the pri­ mary sources (including, in particular, the Volterra Archive at the Accademia Nazionale dei Lincei in Rome, and nu­ merous other archives in Italy and the United States) and through the painstak­ ing collection of many oral histories, the author has provided a detailed and ex­ haustive reconstruction of the life of Volterra, and of his scientific, institutional, and personal relations. Her achievement patently involved a great workload and the patient sifting of documents, some of them hitherto unknown, as well as a high degree of archival skills. Henceforth those interested in the figure of Volterra have at their disposal a valuable tool that provides an illustration down to the smallest details of the life of the great sci­ entist. Among other things, the book con­ tains a fascinating collection of pho­ tographs. Having said this, the book also has a number of flaws. The first consists in

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the almost complete absence of any substantial treatment of Volterra's sci­ entific work. The author correctly states that this was not her purpose: the title refers to the "life and times" and not the "life and works" of Volterra. The book essentially follows a time line and is mainly devoted to personal events. However, it is difficult to convey a sat­ isfactory image of the figure of Volterra without reference to his scientific pro­ duction. This omission makes it difficult to describe the significance of his cul­ tural project, which was part and par­ cel of his scientific project, and to shed light on the mixture of conservatism and innovation that represents the most sig­ nificant and complex characteristic of the scientist's figure. The most significant of Volterra's many contributions are focused on functional analysis, the theory of elas­ ticity, integral and integra-differential equations, and biomathematics. The close links among all these topics rep­ resent both the strength and the weak­ ness of Volterra's program. He has been called the founder of modern functional analysis. However, several historio­ graphic schools have challenged this role, as the concept of "function of lines" is restrictive and insufficient as far as developing a general foundation of the theory is concerned. Volterra him­ self sought to ward off early criticism by claiming that he had never had a purely mathematical theory in mind, and thus his research should be viewed against the background of the problems of viscoelasticity. This shows that the close links he established between mathematics and applications placed him outside the ongoing early axiomatic developments. These aspects are ex­ tremely important for a deeper under­ standing of the figure of Volterra. A similar situation arises in the case of biomathematics. Goodstein rightly states that eve1y contemporary biomath­ ematics textbook recognizes Volterra as the founder of the discipline. However, she does not mention that most of Volterra's program has fallen into com­ plete oblivion. It is not only a matter of the essentially unsuccessful attempt to create a rational and analytical me­ chanics of biological associations [1]. It is also a matter of the difficulties of pro­ viding an empirical and experimental

foundation of his mathematical theories, for which he sought support all over the world. Volterra never resigned him­ self to adopting an abstract mathemati­ cal modelling approach. He wanted to give biomathematics a foundation sim­ ilar to that of classical mathematical physics. In this, he clashed with his son­ in-law, Umberto D'Ancona. The world of scientists in which Volterra endeav­ oured to defend his view was doomed to disappear [2]. No mention of this is made in the book, thus omitting an as­ pect of fundamental importance in un­ derstanding the position occupied by Volterra at the time. Another set of observations refer to the political, institutional, and cultural aspects of his times. Also here, the de­ scription of Volterra's activity, and in particular his courageous opposition to fascism, is oversimplified. No one can afford to have an indulgent attitude to fascism and to underestimate its disas­ trous effect on Italian science. However, the matter cannot simply be wrapped up by using adjectives. Mussolini was not just "a bull and a brute." History must explain why the vast majority of Italian intellectuals, including many sci­ entists and colleagues of Volterra (also jews), sided with fascism [3,4]. On this point, the author should have sifted through the vast Italian bibliography on the topic which, despite the different viewpoints, shed light on the innova­ tive aspects of fascism that account for the approval it succeeded in arousing. I mentioned the clash between Volterra and Gentile on educational reform; at­ tentive examination and the subsequent developments show that the Gentile re­ form-which was supported by mathe­ maticians such as Enriques-was more advanced and modern than the pro­ posals made by the commission set up by Volterra at the Accademia dei Lin­ cei. Moreover, Gentile changed his orig­ inally hostile attitude to scientific cul­ ture that he had adopted at the beginning of the century and estab­ lished a close relationship with Federigo Enriques. Gentile gave Enriques the di­ rection of the scientific section of the Treccani Enciclopedia Italiana, which during that period published articles that still today may be considered a model of scientific dissemination. In this connection it seems inappro-

priate to speak of a "Volterra circle" made up of mathematicians such as En­ riques and Castelnuovo, as Goodstein does continually. Volterra was certainly a point of reference and a pole of Ital­ ian mathematics, but it is hard to speak of him as the centre of a "circle. " Also the figures closer to him, such as En­ riques, Castelnuovo, or Levi-Civita, were scientifically, culturally, and even politically different from Volterra. In the case of persons like Severi the differ­ ence is abyssal. It must also be borne in mind that Volterra was the only Ital­ ian mathematician who did not swear allegiance to the regime and that, after 193 1 , his presence was considered in­ creasingly disconcerting. He no longer attended the evening meetings with En­ riques, Castelnuovo, or Levi-Civita, and Enrico Fermi, who did not fail to wear the fascist black shirt at the meetings of the Accademia d'Italia of which he was a member. It is even less appropriate to speak of a "Jewish circle of Italian mathe­ maticians." No such body has ever ex­ isted, except in the mind of the cham­ pions of fascist anti-Semitism. This is a highly delicate point that the author would have been well advised to treat in depth, considering the vast existing bibliography available. Goodstein is quite right to call the Introduction "The Jewish Mathematician." Among other things, this expression was already in use at the time with different meanings and intentions, sometimes with racist connotations. However, if we revive the term today we must define the mean­ ing we intend it to have. The only meaning in which it is possible to speak of Volterra as a Jewish mathematician is on the strength of his genealogical membership in the Jewish community, which the book reconstructs in great de­ tail. As confirmed by the numerous anecdotes in the book, it is possible to speak of the persistence of membership bonds that do not seem to extend be­ yond the tendency to mix and to arrange marriages inside the group. But as soon one attempts to discover a trace of "jewishness" of any kind in Volterra's life, writings, and letters, as in Levi­ Civita, Enriques, Castelnuovo, and many others, the disappointment is to­ tal. There is not a single reference that justifies the existence of a sense of be-

longing and of Jewish identity of a re­ ligious or cultural nature. One impor­ tant clue is the fact that many jewish scientists-including Volterra him­ self-attempted in 1938 to avail them­ selves of the "discrimination" proce­ dure, which afforded them exemption from the consequences of the anti­ Jewish racial laws if they were able to demonstrate their special service to the nation. The attitude held by Volterra in 1938, quite different from the vig­ orous one displayed in 1 93 1 on the is­ sue of the oath of allegiance, shows that his jewish nature was felt more as a problem than as something to be claimed and defended. This is a complex matter, and the au­ thor ought to have treated it in greater detail as it too is of decisive importance in the construction of an appropriate im­ age of the figure of Volterra. In-depth analysis shows that the Jewish intellec­ tuals, and scientists in particular, were highly integrated into the surround­ ing culture and society, and their rela­ tionship with Jewish identity, if any, was reduced to a vague reminiscence. Volterra-like Enriques, Levi-Civita, and many others-was painfully surprised when according to the racial laws of 1938, he found himself identified as a member a "race": they all believed they were now Italians to all intents and pur­ poses and were completely integrated into the social, political, and cultural re­ ality of the country. Fascism was dif­ ferent from Nazism and did not have anti-Semitic racism as one of its consti­ tutive projects. In my opinion the gen­ eral tendency towards a racial and eu­ genic policy had set the stage for the adoption of anti-Semitic policies. Other historians consider that, on the contrary, the 193B laws were passed merely to gratify Hitler. However this may be, there is no justification for attributing a specific hostility toward fascism or to­ ward Mussolini to a "jewish circle of Italian mathematicians. ·· Enriques was a fascist until 1938. Practically no mathematicians migrated, unlike the physicists. Indeed the community of physicists was projected toward an in­ ternationalist dimension of the scientific undertaking and had no difficulty in transporting its activities elsewhere. In contrast, figures such as Volterra were too closely linked to the national cui-

rural context to be able to move else­ where with ease. For him and a large number of the jewish mathematicians in Italy the racial laws came as an in­ comprehensible and unexpected tragedy. Their Jewish identity was so unsubstantial that they were unable to lay claim to it with pride; they rather suffered their identification as belong­ ing to the "Jewish race" almost as some­ thing shameful. This is an essential is­ sue if we are to understand fully the figure of Volterra [4,5). A final remark concerns the bibliography, which is somewhat incomplete. In conclusion, the book represents an important milestone in the recon­ struction of Volterra's life, but an ex­ haustive and comprehensive scientific and cultural biography of the scientist remains to be written. REFERENCES

[ 1 ] G. Israel, "Volterra's 'analytical mechanics'

of biological associations," Archives lnter­ nationales d'Histoire des Sciences 41 (1 26, 1 27) (1 99 1 ) pp. 57-1 04, 306-351 ; G. Is­

rael, "Vito Volterra, Book on Mathematical Biology (1 93 1 )," in Landmark Writings in Western Mathematics, 1 640-1 940, I . Grat­

tan-Guinness (ed.) Amsterdam, Elsevier, 2005, pp. 936-944. [2] G. Israel, A. Millan Gasca, The Biology of Numbers. The Correspondence of Vito Volterra on Mathematical Biology, Basel­

Boston-Berlin, Birkhauser Verlag, Science Networks - Historical Studies, 26 (2002) pp.

X +

406.

[3] G. Israel, L. Nurzia, "Fundamental trends

and conflicts in Italian Mathematics be­ tween the two World Wars," Archives ln­ ternationales d'Histoire des Sciences 39

(1 22) (1 989) 1 1 H 43.

[4] G. Israel, " Italian Mathematics, Fascism and Racial Policy," in Mathematics and Culture I.

M.

Emmer (ed .),

Berlin-Heidelberg,

Springer-Verlag, 2004, pp. 2 1 -48. [5] G. Israel, "Science and the Jewish Ques­

tion in the Twentieth Century: The case of Italy and what it shows," Aleph, Historical Studies in Science and Judaism 4 (2004) 1 91 -261

Dipartimento di Matematica Universita di Roma "La Sapienza" Piazzale A. Moro 5 - 00 1 85 Rome, Italy e-mail: giorgio.israel@uniroma1 .it

© 2008 Spnnger SCience+ Business Media, Inc., Volume 30, Number 3, 2008

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M athematics at Berkeley by Calvin C Moore WELLESLEY, MASSACHUSETTS, A.

K.

PETERS, 376

PP., 2007 ISBN: 978-1-56881-302-8, HARDCOVER

us

$39.00

REVIEWED BY T- W. GAMELIN

W

ho would ever want to read a history of the Berkeley mathematics department? It could be a crashing bore. But having spent my graduate student years (19601963) there, and having had occasional contact with the Berkeley scene while teaching at its young sister university in Los Angeles 0968- ), I was curious about the history. As I read, I became more and more drawn to the story as it relates to policy and hiring issues that have concerned me at one time or an­ other as a department administrator and citizen. Initial curiosity eventually turned into enthusiasm and a recogni­ tion that this case study is relevant to a wide spectrum of mathematicians. It will be particularly informative for mathematics faculty in large state uni­ versities involved in designing strategy and making programmatic and hiring decisions. It will have special appeal to Berkeley graduates, postdocs, visitors, and MSRI program participants. From a broader point of view, the book can be viewed as a case study of a single high­ profile mathematics department that sheds light on the development of mathematics in America.

Thumbnail Sketch of the History The University of California was created by merging the College of California in Oakland, which needed money, and the Agricultural and Mechanical College, which the California legislature had es­ tablished on paper in 1866, and which needed faculty and land. The merger was consummated in 1868, and the University of California was born. Classes opened in 1869, and the uni­ versity moved to a newly constructed campus in Berkeley in 1873. From its founding, the University of California aspired to academic excel­ lence. Most founding department chairs

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THE MATHEMATICAL INTELLIGENCER

had strong academic credentials. How­ ever, as founding chair of the mathe­ matics department, the UC Regents se­ lected a West Point graduate, William Welcker, who had never taught a math­ ematics course in his life. Welcker was a capable organizer, and he imported a strong mathematics curriculum from West Point, but he was miscast as math­ ematics department chair, and his ap­ pointment reflected the perception of mathematics as a service enterprise. The struggle between the views of mathe­ matics as a service provider and math­ ematics as an intellectual pursuit is one of the recurring themes of the history. In organizing the history, the author identifies three dramatic changes in the direction of the mathematics depart­ ment since its initial founding. Each change strengthened the view of math­ ematics as a scholarly endeavor valu­ able in its own right, although each change was precipitated by a different catalytic agent. The first dramatic change occurred in 1871-1872, when the UC Regents ef­ fectively fired Welcker and replaced him by a bona fide mathematician, Irv­ ing Stringham, with the goal of raising the level of scholarship in the depart­ ment. The second dramatic change oc­ curred in 1933-1934, during a period when the chair of the mathematics de­ partment was about to retire. The math­ ematics department had become in­ grown, focusing primarily on its teaching and service role. Other science departments had already risen to promi­ nence on the national scene, and they recognized the importance of changing the direction of the mathematics de­ partment. At their instigation, the search for a new mathematics department chair was removed from the hands of the mathematics department and was placed in the hands of university sci­ ence leaders. The eventual outcome was the importation of a mathematician who was highly respected on the na­ tional scene, Griffith Evans, to take the helm. Under the direction of Evans, the mathematics department focused on re­ tooling itself as a research department with high aspirations and with a broad view of mathematics. The third dramatic change was pre­ cipitated in 1957 by the clamor raised by the mathematics faculty for more re-

sources corresponding to its nstng stature and its expanding role in the uni­ versity. Among other things, the de­ partment called for an aggressive hiring strategy that ran against preferred UC hiring practices by recruiting several dis­ tinguished mathematicians in mid-ca­ reer. The leading figure in this effort was John Kelley (a UCLA alumnus), whose 1957 white paper served to crys­ tallize department sentiment. At this crit­ ical point, the mathematics department had strong administration support, par­ ticularly from a remarkable educator, Clark Kerr, who served as Berkeley Chancellor from 1952 to 1958. Kerr had become convinced that if a modern university "were to have one preemi­ nent department in modern times, it should be mathematics." The science dean responsible for the mathematics department was also coming around to the view, however belatedly, that math­ ematics was evolving to something more than a service department. The upshot was that the chairmanship was thrust upon Kelley, who presided over the rapid move of mathematics from a good department to an excellent de­ partment over the tenure, 1957-1960, of his service. The successful recruiting ef­ fort included the signal high-level ap­ pointments of S. S. Chern and E. Spanier. The author includes a chapter on the founding of the NSF-funded Mathemat­ ical Sciences Research Institute (MSRI) at Berkeley in the early 1980s; The au­ thor was one of the principal architects of the institute and was its first Deputy Director. Though established as an in­ dependent nonprofit corporation, the MSRI received substantial support from UC Berkeley, including a building site on the upper reaches of the UC cam­ pus that commands a spectacular view of the San Francisco Bay area. The MSRI successfully navigated dangerous shoals and survived renewal cycles to become now firmly established with substantial outside funding (due in no small part to the generosity of James Simons). The MSRI has contributed to establishing Berkeley as a principal focus of math­ ematics in America, covering a wide spectrum of mathematical endeavors. An interesting thread that runs through the history is the dependence of the university, thus the mathematics de­ partment, on state and national politics

and on the public perception of the uni­ versity. McCarthyism was reflected by the oath controversy on the statewide scene, which impeded hiring and which sent several faculty members into tem­ porary exile. This period in the early 1950s was followed by a period of growth and optimism, in which educa­ tion in California flourished under the leadership of Kerr and friendly political leaders such as Governor Pat Brown. The high point of this period was the adoption of the California Master Plan for Higher Education in 1960. The de­ partment limped through the free speech movement, which arose abruptly in the mid 1960s. Curiously, the free speech movement did not impede hiring, but it did lead to a loss of public support and the attendant budgetary problems dur­ ing the governorship of Ronald Reagan. This was followed by the more severe budgetary problems posed by the frugal Jerry Brown governorship. Support re­ bounded in the 1980s under Governor George Deukmejian. However, two ma­ jor economic downturns, in the early 1990s and in 2001 , have led to belt tight­ ening and have accelerated the increased dependence of the university on alter­ native revenue sources such as extra­ mural grants and student fees. UC Berkeley typifies many large state universities that are evolving from in­ stitutions with full state support to pri­ vate enterprises with some state assis­ tance. In the current political and economic climate, a department must generate extramural funding and donor support in order to thrive. Currently less than 30o/o of the UC Berkeley operating budget is derived from the state. The mathematics department is adapting to the new reality by aggressively seeking donor support and building endow­ ment to fund programs and activities that the state will no longer support.

Lessons Derived from the Case Study What lessons can be learned from this case study? What strategies can one glean to improve one's own depart­ ment? Perhaps the most important lesson is the importance of the hiring of profes­ sorial faculty. The author drills this into the reader through a relentless focus on hiring decisions, including sketches of each of the newly appointed faculty

members. Flexible hiring strategies are used to fill out the ranks of professor­ ial faculty with quality appointments that maintain a balanced department. Targeted mathematicians may be courted over a period of years, with the aid of short-term visits and long-term visiting positions. One can also refer to this case study to learn how one mathematics depart­ ment has resolved the problem of lo­ cating various mathematics and math­ related fields, such as statistics and applied mathematics, within the uni­ versity. At Berkeley, statistics attained de­ partmental status in 1955, thanks in large part to the efforts of Jerzey Ney­ man. The statistics department houses a number of probabilists, and it has close ties to the mathematics depart­ ment through joint appointments. (A rule of thumb for university organiza­ tion is that statistics benefits from de­ partmental status though probability may suffer.) Applied mathematics, on the other hand, has evolved to a loosely defined entity within the mathematics depart­ ment, which has interests in a number of different directions and which main­ tains links to departments such as elec­ trical engineering, physics, biology, and economics. A number of Berkeley math­ ematics faculty have joint appointments in other departments, and even more view their research as lying at least par­ tially within the realm of applied math­ ematics. The lesson that emerges is that pure and applied mathematics are inex­ tricably linked, and both can flourish in a symbiotic relationship operating from the same departmental base. The evolution of computing at Berkeley, as recounted by the author, b more complicated. The upshot is that most computing research is now housed in engineering, though the mathematics department maintains a significant presence in the area through faculty working in numerical analysis and in computational aspects of alge­ bra. Since the arrival of Alfred Tarski in 1942, mathematical logic has flourished with a series of strong appointments to become a powerhouse within the math­ ematics department. The question of balance between logic and other areas was resolved, according to the author,

through a historical rule of thumb that allocates logic to roughly lOo/o of the professorial faculty. The Berkeley mathematics depart­ ment has not hired in the nascent field of mathematics education, though sev­ eral of the professorial faculty have be­ come involved in mathematics educa­ tion issues as an adjunct to their mathematics research careers. In the 1950s, the department appointed a fac­ ulty member focused on teacher prepa­ ration, but he did not fit into the de­ partment, and after being denied tenure, he moved on to a successful ca­ reer at San Francisco State University. Since the adoption of the master plan, teacher preparation in California has been the province of the California State University system. We note that UC has recently become interested in the preparation of mathematics and science teachers, in response to the public per­ ception of a critical shortage of highly qualified teachers. It remains to be seen what role the traditional research math­ ematics department will play in teacher preparation.

Why did UCB Rise to Prominence? The Berkeley mathematics department rose early to prominence. A claim can be made that UCB was considered to be among the top ten mathematics de­ partments in the country as early as 1899, even though it did not award its first PhD until 190 1 . It is difficult to com­ pare UCB to private universities such as the Ivy League schools; it is easier to rank UCB among large state universi­ ties, which form a relatively homoge­ neous group with similar goals and par­ allel funding sources. Within this group, the great land-grant institutions of Wis­ consin and Illinois, and their distin­ guished predecessor Michigan, have ceded ground to Berkeley. In fact the Berkeley mathematics department now stands head and shoulders above other mathematics departments at major state universities. Was it preordained that Berkeley should rise to prominence? What were the ingredients that allowed Berkeley to compete so successfully with its peers? One can point to Berkeley's early start, an outcome of the gold rush and the attendant rapid economic develop­ ment of the San Francisco area. The

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economic and cultural base provided by the thriving local economy has played a role over the years. One can also point to the moderate climate, the proximity to ocean beaches, and the karma of San Francisco and the Golden Gate. But a main ingredient emerging from the case study is an unrelenting search for scholarly excellence in making ap­ pointments. Even in the earliest days in Oakland, university leaders demanded scholarly excellence of its appointees, at least in areas other than mathemat­ ics. There was a recurrent tendency to view mathematics as a service depart­ ment rather than an independent field of scientific research. However, when the same standards of high quality were applied to mathematics as were being applied to other areas, the stage was set for mathematics to make significant ad­ vances. Evans brought to the department a broad view of mathematics that over­ lapped with areas of application. This vision of mathematics within the uni­ versity, reinforced by the extensive use of fractional joint appointments to reach out to other parts of the university, has been an important ingredient of suc­ cess.

UCLA mathematics department and the university as a whole. I particularly en­ joyed the discussion related to the long­ range planning for Berkeley in the years 1955-1957, including Kerr's calculations for an optimal size for the student body and the considerations entering into planning for a building for mathemat­ ics. As a case study, this history has much to say to mathematicians and to academic leaders of today's university. Mathematics Department UCLA Los Angeles, California 90095-1 555 USA e-mail: [email protected]

An Imaginary Tale: The Story of v=I by Paul j. Nahin

REPRINT OF THE 1998 EDITION AND FIRST PAPERBACK PRINTING, WITH A NEW PREFACE AND APPENDIXES BY THE AUTHOR, 2007. PRINCETON, NEW JERSEY, PRINCETON UNIVERSITY PRESS, 269 PP. , 2007, U S $16.95, PAPERBACK ISBN: 0·691-12798-0

Concluding Comments Moore has a clear writing style, some­ what reminiscent of a departmental let­ ter in support of a personnel action. He depends heavily on department admin­ istration documents such as department letters in support of appointments, re­ ports of visiting committees, and arti­ cles memorializing deceased faculty. There is a steady flow of facts, facts, facts, and these are used to buttress oc­ casional summarizing assessments. The author focuses on painting the big pic­ ture. There is very little offered about the operational details of the function­ ing of a mathematics department. The reader will find neither gossip nor much insight into the personalities of the mathematicians in the department (though one is left with no doubt that Berkeley deans regarded Neyman as a pain in the neck). The history is very well written. I found it an immensely enjoyable read, particularly when set against the back­ drop of my own connections with Berkeley. The history has given me in­ sight into the development of my own

80

THE MATHEMATICAL INTELUGENCER

REVIEWED BY GENEVRA NEUMANN

ne of the first things welcoming a reader to this history of v=1 is a Calvin and Hobbes cartoon about imaginary numbers. Throughout this book, biographical details and anecdotes abound, along with detailed calculations and examples. The book covers an amazing variety of topics re­ lated to complex numbers. This "tale" is written in the first person, in a con­ versational tone with lots of figures in­ cluded to help the reader follow the dis­ cussion. The author is not shy about expressing his opinions. The stories give the impression that mathematics is a lively and human enterprise, warts and all. Almost all of the mathematics is presented at a level that should be accessible to a student who has com­ pleted freshman calculus, although par­ tial derivatives and multiple integrals are used in a few places. The introduction presents a prehis­ tory of v=l, from ancient Egypt to In­ dia to the introduction of the term

"imaginary. " The first three chapters de­ scribe how v'=1 evolved from being regarded as an impossible expression appearing in a solution of a cubic equa­ tion to an honest-to-goodness, legiti­ mate number. The next two chapters discuss a variety of applications of com­ plex numbers, ranging from a puzzle of G. Gamow's concerning buried treasure to examples of electronic circuits. The sixth chapter focuses on results of Euler. The last chapter provides a glimpse into function theory. There are six appen­ dixes, three of which are new in the paperback edition. The preface to the paperback edition discusses comments the author has received concerning pos­ sible errors in the text, but the errors mentioned are not corrected in the body of the text. The book has a section of detailed notes at the end, a name in­ dex, and a subject index. In addition to references, the notes often contain ad­ ditional discussion. The table of con­ tents also includes a short summary of each chapter. This book is a "tale" that interweaves mathematics with history and applica­ tions. I don't have sufficient expertise to comment on the correctness of the historical and biographical material nor the material on physics and engineer­ ing. As with any good tale, there are heroes and villains. The historical dis­ cussions include feuds about priority, people who helped the careers of oth­ ers, and those who made things diffi­ cult for others-all told in a chatty and opinionated way. As this is not a text­ book, there is no table of symbols and readability trumps rigor. Arguments are often computational, similar in flavor to those found in a typical freshman cal­ culus book, and there are plenty of helpful figures and drawings. The cal­ culations are broken into easily digested steps; one complicated calculation is deferred to Appendix E. Also, some nonstandard notation is used (rLO rep­ resents a complex number in polar form). Readers who are comfortable let­ ting the equations wash over them while enjoying the historical details or who wish to follow along with their cal­ culators will be comfortable with the mathematical presentation. Some readers will be put off by the use of rounded values in equations; for example, the last equation on page 59 is

u =

-1

::t:::

2

Vs

=

0.618034 and

-

1 .618034.

A more significant example is the calcu­ lation on page 58, which is another ver­ ification that one of the cube roots of z = 2 + v=T21 is 2 + i. De Moivre's formula is used with arg(z) = tan - J ( 1 1/2), which i s then set equal to 79.69515353° This rounded calculation gives the desired answer, but it's a bit misleading. It would have been nice to use instead of = when rounded val­ ues are used. This is difficult material to present at such an elementary level. The author does point out some of the more egre­ gious sleights of hand (for example, substituting ix for x in the power series for e'. on pages 144-145). However, this is not always the case. A reader might need more reminders than given that ( 1 ) the polar representation of a com­ plex number isn't unique and that ( 2 ) extra care must b e taken with the com­ plex logarithm. For example, this is a source of confusion in Section 6.7's dis­ cussion of i i (for which some correc­ tions are discussed on page xi.x in the new preface). Also, in Appendix B ( sec­ ond paragraph on page 230), a reader may be misled by the suggestion that the transcendental function under con­ sideration will have an infinite number of zeros, because its power series has all powers of z. An infinite power se­ ries is not a polynomial as suggested on page 195 when justifying the analytic­ ity of ez. Because not all readers will have previous sections firmly in mind, it would have been nice to remind the reader of assumptions made in previ­ ous sections when stating equations. For example, the gamma integral is de­ fined in Section 6 . 1 2 on page 175, along with the assumption n > 0. The reason for this assumption is not given (the in=

tegral is infinite for n :S 0) and the reader is not reminded of this assump­ tion when the integral is reintroduced in Section 6.13 on page 182. The gamma function is extended to negative real numbers by using the recursion relation mentioned on page 1 76 and not by this integral; no mention is made of the fact that the gamma function is undefined for integers n :s: 0. Moreover, the expres­ sions for n n)f(l - n) and for ( n)!( n!) on page 184 involve dividing by sin( mr), and no restriction on n is given. It would have been nice to have included graphs for some of the special functions introduced in Chapter 6, as well as a sketch of a Riemann surface. This book might fmstrate readers who do not read the book linearly or who read the book in small chunks over a long time period. As the story of vC1 unfolds, the author revisits earlier topics. He doesn't include page numbers (or equation numbers) when referring to material elsewhere in the book. For ex­ ample, the calculation on page 58 is an­ other way of verifying that the positive solution of the irreducible cubic :x3 = 15x + 4 is given by Cardano's formula (even though it involves square roots of negative numbers) and was settled in an­ other manner in Chapter 1. Instead of referring the reader back to the specific page (page 18), the reader is asked to "recall from chapter 1 the Cardan for­ mula to the irreducible cubic considered by Bombelli" followed by the formula for the solution. For the most part, equa­ tions are not numbered; the exceptions are the equations in Appendix E and a boxed equation on page 53. Although this may make the book more welcom­ ing to a general reader, there are several more places where equation numbers on selected equations would make argu­ ments easier to follow. For example, in the second paragraph on page 145, the author points out that the series being -

discussed "provides the proof to a state­ ment I asked you just to accept back in section 3.2" and then gives a formula for sm x. The table of contents does not give X page numbers for specific sections, and Chapter 3 mns over 30 pages. Section 3.2 starts on page 60 and ends on page 65. Going back to section 3.2, the fact that limx--.o sin x = 1 is used to go from the equation for si� 0 on the bottom of page 63 to an infinite product formula for 2/ 1r on page 64. Near the top of page 64, the author says that he will derive this limit in Chapter 6 (again, no page number or equation number). Chapter 6 mns over 40 pages. It's clear that a great deal of research went into preparing this book. An amazing amount of mathematics has been presented in historical context at an elementary level. There are mistakes and careful readers will probably he un­ happy with the balance chosen be­ tween rigor and readability. I'm disap­ pointed that the corrections mentioned in the new preface weren't either cor­ rected in the text itself or at least listed in a separate section of errata for the sake of the unsuspecting reader who skips prefaces. Because of the lively presentation and variety of topics, I think this book has the potential to con­ vince a general reader that there's in­ teresting (and even useful) mathemat­ ics out there beyond freshman calculus. The author makes a strong case that complex numbers are useful and aren't just funny looking solutions from the quadratic formula; mathematics teach­ ers (high-school level and above) might find this book useful as a source of ex­ amples and anecdotes. University of Northern Iowa Department of Mathematics Cedar Falls, lA 5061 4-0506 USA e-mail: [email protected]

© 2008 Springer Soence + Business Media, lnc., Volume 30, Number 3 , 2008

81

Kifij I ;q nit§I •

.

.

R o b i n Wilso n

The P h i lamath's A l phabet - R

I

form motion. Einstein reconciled this ap­ parent discrepancy by postulating that the laws of physics were the same for all observers in uniform motion relative to one another. Ten years later, in his "general theory," he extended these ideas to accelerated motion and gravity.

Ricci Ramanujan Srinivasa Ramanujan (1887-1920) was one of the most intuitive mathemati­ cians of all time. Mainly self-taught, he left India in 1914 to work in Cambridge with G. H . Hardy, producing some spectacular joint papers in analysis, number theory, and the theory of par­ titions, before his untimely death at the age of 32.

Relativity In 1905 Einstein published his "special theory of relativity." Until then it had been assumed that Maxwell's equations were valid only in a particular frame of reference (the "ether" that carries the waves) and were thus unlike Newton's laws, which held for all observers in uni-

The first missionary in China, near the end of the Ming dynasty, was the Ital­ ian jesuit Matteo Ricci ( 1552-1610), who disseminated knowledge of west­ ern science, especially in mathematics, astronomy, and geography. His most important contribution was an oral Chi­ nese translation of the first six books of Euclid's Elements.

Riese Gutenberg's invention of the pnntmg press around 1440 enabled mathemati­ cal works to be widely available for the first time, and gradually vernacular texts in algebra, geometry, and practical cal­ culation began to appear at a price ac­ cessible for all. In Germany the most influential of the commercial arithmetics was by Adam Riese (ca. 1489-1559); it

Relativity

Ramanujan

proved so reputable that the phrase "nach Adam Riese" (after Adam Riese) came to indicate a correct calculation.

Rubik's cube Rubik's cube, invented in 1974 by the Hungarian engineer Erno Rubik, is a 3 X 3 X 3 colored cube whose six faces can be independently rotated so as to yield 43,252,003,274,489,856,000 differ­ ent patterns. Given such a pattern, the object is restore the original color of each face. In the early 1980s, when the Rubik's cube craze was at its height, over 1 00 million cubes were sold and public cube-solving contests were held in several countries.

Russell Bertrand Russell (1872-1970) was one of the outstanding figures of the 20th century, a Nobel prize-winner for liter­ ature and winner of the London Math­ ematical Society's prestigious De Mor­ gan medal. In 1913 he and A. N. Whitehead completed their pioneering three-volume work, Principia Mathe­ matica, on the logical foundations of mathematics.

Ricci

Riese

Please send al l submissions to the Stamp Corner Editor,

Robin Wilson,

Faculty of Mathematics,

Computing and Technology The Open University, Milton Keynes , M K 7 6 AA , England e-mail: r.j .wilson@open .ac.uk

84

Rubik's cube

THE MATHEMATICAL INTELLIGENCER © 2008 Springer Science+ Business Media. Inc.

Russell