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Acknowledgement n the Summer issue of The Mathematical Intelligencer, vol. 32, no. 2, we presented, with permission, a g...

Author: M.L. Senechal | C. Davis (Editors in Chief)

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Acknowledgement n the Summer issue of The Mathematical Intelligencer, vol. 32, no. 2, we presented, with permission, a geometric-combinatoric pattern due to Anthony Hill. This was an array of 66 six-segment graphs, whose significance, and extraordinary origin, were explained on p. 3 of the issue; the array appeared on the cover. Its creator, Anthony

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Hill, has pointed out to us that the colors in the cover design have no role in its status as solution of a certain combinatorial problem, and he asks that we apologize for departing from his concept by using color. We take full responsibility for this design decision, and we regret our failure to carry out his intentions in this respect. We hope Mr. Hill is comforted by the fact that every careful reader of the explanation we published will appreciate the meaning of his discovery and the irrelevance of the colors thereto. Chandler Davis and Marjorie Senechal

Ó 2010 Springer Science+Business Media, LLC, Volume 32, Number 4, 2010

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Letter to the Editors

Maria Teresa Calapso’s Hyperbolic Pythagorean Theorem The Mathematical Intelligencer encourages comments about the material in this issue. Letters to the editor

providing a short proof that appeals only to basic formulas of hyperbolic geometry, simpler than the proofs in [1, 2]. The author also asks for the proper axiomatic setting in which the above-stated Pythagorean theorem would hold. Given that areas of circles demand the full axiomatic import of the real numbers, the version referring to areas of circles falls outside elementary (first-order logic) axiomatic considerations. If a version regarding triangles satisfying A = B + C turns out to be true in Hilbert’s absolute geometry (axiomatized by the axioms I 1–3, II, and III of [5] or by the axioms A1–A9 of [6]), then it must involve areas of polygons on the sides a, b, c (perhaps right isosceles triangles with equal sides having the length of the triangle side on which they are erected), with the Hilbert definition of area equality as equivalence by completion (Erga¨nzungsgleichheit). Relevant for an absolute version of the Pythagorean theorem is also the absolute version of the Intersecting Chords Theorem (III.36 in Euclid’s Elements) in [4].

should be sent to either of the editors-in-chief, Chandler Davis or Marjorie Senechal.

REFERENCES

[1] Familiari-Calapso, M. T., Le the´ore`me de Pythagore en ge´ome´trie absolue. C. R. Math. Acad. Sci. Paris. Se´r. A-B. 263 (1966), A668– A670.

read with interest Paolo Maraner’s recent Mathematical Intelligencer note ‘‘A Spherical Pythagorean Theorem’’ (Vol. 32, No. 3, Fall 2010, 46–50, DOI:10.1007/s00283010-9152-9). In it, the author shows that a proper generalization of the Pythagorean theorem that would render it true in an absolute setting, cannot stay with the hypothesis that one angle of the triangle be right, but rather has to relax it to state that one angle, say A, should be the sum of the other two, say B and C. Also the Pythagorean theorem should state that the areas of the circles with sides b and c as radii should be equal to the area of the circle with side a as radius (let us denote the latter area by sa). I would like to point out that precisely this form of the Pythagorean theorem was stated and proved in the hyperbolic plane by Maria Teresa Calapso in [2], where it is shown that the converse holds as well, that is, that we have sa = sb + sc only in triangles in which A = B + C holds. In [8] it was shown that the generalized Pythagorean formula, valid in any hyperbolic triangle, is a ¼ b þ sinðABÞ sin C c, and [1, 3, 7] contain like-minded forms of the generalized Pythagorean theorem. As its title indicates, the main novelty in Paolo Maraner’s paper is the fact that this version of the Pythagorean theorem holds in the spherical setting as well. Even in the hyperbolic case, the paper has the merit of

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THE MATHEMATICAL INTELLIGENCER Ó 2010 SPRINGER SCIENCE+BUSINESS MEDIA, LLC

[2] Familiari-Calapso, M. T., Sur une classe di triangles et sur le the´ore`me de Pythagore en ge´ome´trie hyperbolique. C. R. Acad. Sci. Paris Se´r. A–B 268 (1969), A603–A604. [3] Calapso, M. T., Ancora sul teorema di Pitagora in geometria assoluta. Atti Accad. Peloritana Pericolanti Cl. Sci. Fis. Mat. Natur. 50 (1970), 99–107. [4] Hartshorne, R., Non-Euclidean III.36. Amer. Math. Monthly 110 (2003), 495–502. [5] Hilbert, D., Grundlagen der Geometrie, 12. Auflage. Teubner, Stuttgart, 1977. [6] W. Schwabha¨user, W. Szmielew, and A. Tarski, Metamathematische Methoden in der Geometrie. Springer-Verlag, Berlin, 1983. [7] Vra˘nceanu, G., Sopra la geometria noneuclidea. Atti Accad. Peloritana Pericolanti Cl. Sci. Fis. Mat. Natur. 50 (1970), 119–123. [8] Vra˘nceanu G. G., Sur la trigonome´trie noneuclidienne. Rend. Circ. Mat. Palermo (2) 20 (1971), 254–262 Victor Pambuccian Division of Mathematical and Natural Sciences Arizona State University—West Campus Phoenix, AZ 85069-7100 USA e-mail: [email protected]

Note

Even Hilbert Nods… BOB LLOYD

e often describe a text as ‘‘authoritative’’ when we do not expect there to be much question about the content, so that students can safely be referred to it. Nevertheless, there is a downside to this concept. We expect to make our own mistakes, but in dealing with the work of an authority, there can be a reluctance to question, so mistakes can persist. In mathematics the supreme example of this is probably Aristotle’s claim that space can be completely filled by packing cubes or tetrahedra. Two millennia passed before it was pointed out that this is not i true for tetrahedra. I would like to draw attention to another mistake which has persisted, though only for three-quarters of a century. Anschauliche Geometrie, by David Hilbert and Stephan Cohn-Vossen,ii appeared in 1932, was published in English as Geometry and the Imagination in 1952,iii and reissued in 1999. A second German edition came out in 1995.iv Despite its age, the book has clearly been in demand, and the comment that ‘‘many of us for years have been pushing the classic Geometry and the Imagination (to graduate or advanced undergraduate students)’’v suggests that it is a text with authority and influence. The following note is not intended to be critical; rather, the sense is that, ‘‘If it can happen to him, then there’s hope for the rest of us!’’ The problem comes in the discussion of the symmetries of the Platonic solids, and concerns the diagram of a cube within a dodecahedron; this may be older than Aristotle,vi though the first description is in Euclid.vii Figure 1 reproduces two diagrams from Hilbert and Cohn-Vossen.viii These diagrams were used to discuss the relationships between the point groups of the three different solid figures shown, using the pure rotation groups rather than the full point symmetries; the same approach will be used here. These rotation groups are frequently represented by the symbols T or 332 for tetrahedral symmetry, and O or 432 for octahedral, the symmetry of the cube. The dodecahedron has icosahedral symmetry, I or 532. In the left-hand diagram in Figure 1, the tetrahedra and the cube have different symmetries. I consider just one of the tetrahedra, though the argument is unaffected by working with two independent tetrahedra. Combining two objects of different symmetry often gives a lower symmetry; combinations with decreasing or increasing symmetry are

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discussed in Cromwell.ix The rotation axes of the tetrahedron/cube combination are the same as those of an isolated tetrahedron, and the 4-fold axes of the cube are absent in the combination. The symmetry of this combination is thus T (332), and it can be constructed as shown because T is a subgroup of O (432). This is essentially the argument presented by Hilbert and Cohn-Vossen,x though they do not mention the symmetry of the combination. Hilbert and Cohn-Vossen extend this argument to the right-hand diagram in Figure 1. In the English version,xi they claim that: ‘‘Similarly it turns out that the octahedral group is a subgroup of the icosahedral group. This is the reason why a cube can be inscribed in a dodecahedron in the same way as the tetrahedron can be inscribed in a cube.’’ The English is a precise translation of the original German (see note I), and the German text has remained unchanged in the new edition. The first sentence of this extract is clearly erroneous. The operations of the group O include 4-fold rotations. The operations of I (532) include 5-fold and 3-fold rotations, but no 4-fold rotations, so O cannot be a subgroup of I. The second sentence contains a different error. Unlike the tetrahedron-cube combination, the inscription of a cube within a dodecahedron does not depend on the symmetry of one body being that of a subgroup of the other. A special case has been used to make a more general argument. Table 1 shows a correlation of the elements of the three groups concerned. Here, Cn denotes an axis of 360/n rotation symmetry, and the table gives the numbers of these rotation axes for each of the groups T, O, and I. It is evident that although O is not a subgroup of I, T is a subgroup of both I and O. The rotations transforming the cube/dodecahedron combination into itself are the operations of the four C3 axes through opposite corners of the cube, and of the three C2 axes through opposite face centers of the cube. There are no other rotation operations, so the combination of the two solids has symmetry T. The diagram can be constructed, not because the symmetry of one body is a subgroup of that of the other, but because the bodies separately have O and I symmetry, and these have a subgroup, T, in common.xii The diagram of a tetrahedron in a cube is a special case of this, where the combination happens to have the same symmetry as one of the two bodies being combined, but this is not always the case. The combinations of an octahedron with a cubexiii are even more special, since here the symmetry groups of the two solids are identical with that of the combination. A remarkably similar mistake occurs in a much more recent work.xiv This also discusses the cube-tetrahedron and dodecahedron-cube diagrams, and claims that: 2010 Springer Science+Business Media, LLC, Volume 32, Number 4, 2010

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Figure 1. Two different ways of inscribing a tetrahedron in a cube, and a cube inscribed in a dodecahedron. (Reproduced from ii.) REFERENCES

Table 1. A correlation of elements in the groups I, T, and O

i

I (532)

6C5

10C3

–

–

15C2

T (332)

–

4C3

–

–

3C2

O (432)

–

4C3

3C2

3C4

6C2

‘‘…every symmetry of the cube is also a symmetry of the dodecahedron.’’ This is argued from the observation that the vertices of the cube are a subset of those of the dodecahedron. However, as above, the 4-fold rotation axes (‘‘symmetries’’) of the cube have disappeared in the combination. The book does not reference Hilbert and Cohn-Vossen here, and the argument is expressed in the language of the full groups rather than that of the rotation groups (see note II), so it seems that the same error has occurred independently. Readers who are involved with the teaching of geometry might consider warning students about this problem.

Heath, T. L., Mathematics in Aristotle, Oxford, Clarendon Press, 1949, pp. 177–178.

ii

Hilbert, D. and Cohn-Vossen, S., Anschauliche Geometrie, Die Grundlehren der Mathematischen Wissenschaften Band XXXVII, Berlin, Julius Springer, 1932. Hilbert, D. and Cohn-Vossen, S., Geometry and the Imagination,

iii

translated by P. Nemenyi. New York, Chelsea Publishing Co. 1952. iv

Hilbert, D. and Cohn-Vossen, S., Anschauliche Geometrie, mit einem Geleitwort von Marcel Berger (2. Aufl.), Berlin, Springer, 1995.

v

Banchoff, T., Bulletin of the American Mathematical Society, 34, 1, January 1997, p. 34.

vi

Altmann, B., Euclid–The Creation of Mathematics, New York, Springer-Verlag, 1999, p. 285.

vii

Ref. vi, Euclid, Book XIII; see ref. vi, p. 294.

viii

Ref. ii, p. 83.

ix

Cromwell, P. R., Polyhedra, Cambridge, Cambridge University Press, 1996, pp. 359–385.

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ACKNOWLEDGMENT

I thank Springer Science+Business Media for permission to reproduce the two diagrams.

Ref. ii, p. 83; Ref. iii, p. 92.

xi

Ref. iii, p. 92.

xii

Ref. ix, pp. 361–362. Ref. ii, p. 82.

xiii

Notes I. The original reads, ‘‘Ebenso erweist sich nun die Oktaedergruppe als Untergruppe der Ikosaedergruppe. Aus diesem Grunde kann man einen Wurfel in eine Dodekaeder in gleicher Weise hineinstellen wie ein Tetraeder in einen Wurfel.’’ II. Hilbert and Cohn-Vossen’s book is claimed as ‘‘an inspiration’’ in the bibliography, and is referenced at other points. In the full groups, the symmetries are Oh, Ih, and Th for the combination.

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xiv

Smith, J. T. Methods of Geometry, New York, Chichester, John Wiley & Sons, Inc., 2000, p. 404.

School of Chemistry Trinity College Dublin 2 Ireland e-mail: [email protected]; [email protected]

The Bilinski Dodecahedron and Assorted Parallelohedra, Zonohedra, Monohedra, Isozonohedra, and Otherhedra BRANKO GRU¨NBAUM

ifty years ago Stanko Bilinski showed that Fedorov’s enumeration of convex polyhedra having congruent rhombi as faces is incomplete, although it had been accepted as valid for the previous 75 years. The dodecahedron he discovered will be used here to document errors by several mathematical luminaries. It also prompted an examination of the largely unexplored topic of analogous nonconvex polyhedra, which led to unexpected connections and problems.

F

Background In 1885 Evgraf Stepanovich Fedorov published the results of several years of research under the title ‘‘Introduction to the Study of Figures’’ [9], in which he defined and studied a variety of concepts that are relevant to our story. This book-long work is considered by many to be one of the milestones of mathematical crystallography. For a long time this was, essentially, inaccessible and unknown to Western researchers except for a summary [10] in German.1

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The only somewhat detailed description of Fedorov’s work available in English (and in French) is in [31]. Fedorov’s book [9] was never translated to any Western language, and its results have been rather inadequately described in the Western literature. The lack of a translation is probably at least in part to blame for ignorance of its results, and an additional reason may be the fact that it is very difficult to read [31, p. 6].

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Several mathematically interesting concepts were introduced in [9]. We shall formulate them in terms that are customarily used today, even though Fedorov’s original definitions were not exactly the same. First, a parallelohedron is a polyhedron in 3-space that admits a tiling of the space by translated copies of itself. Obvious examples of parallelohedra are the cube and the Archimedean six-sided prism. The analogous 2-dimensional objects are called parallelogons; it is not hard to show that the only polygons that are parallelogons are the centrally symmetric quadrangles and hexagons. It is clear that any prism with a parallelogonal basis is a parallelohedron, but we shall encounter many parallelohedra that are more complicated. It is clear that any nonsingular affine image of a parallelohedron is itself a parallelohedron. Another new concept in [9] is that of zonohedra. A zonohedron is a polyhedron such that all its faces are centrally symmetric; there are several equivalent definitions. All Archimedean prisms over even-sided bases are

AUTHOR

......................................................................... ¨ NBAUM received his PhD BRANKO GRU

from the Hebrew University in Jerusalem in 1957. He is Professor Emeritus at the University of Washington, where he has been since 1966. His book ‘‘Convex Polytopes’’ (1967, 2003) has been very popular, as was the book ‘‘Tilings and Patterns’’ (coauthored by G. C. Shephard) published in 1986. He hopes that ‘‘Configurations of Points and Lines’’ (2009) will revive the interest in this exciting topic, which was neglected during most of the twentieth century. Gru¨nbaum’s research interests are mostly in various branches of combinatorial geometry. Department of Mathematics University of Washington 354350 Seattle, WA 98195-4350 USA e-mail: [email protected]

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zonohedra, but again there are more interesting examples. A basic result about zonohedra is: Each convex zonohedron has a center. This result is often attributed to Aleksandrov [1] (see [5]), but in fact is contained in a more general theorem2 of Minkowski [27, p. 118, Lehrsatz IV]. Even earlier, this was Theorem 23 of Fedorov ([9, p. 271], [10, p. 689]), although Fedorov’s proof is rather convoluted and difficult to follow. We say that a polyhedron is monohedral (or is a monohedron) provided its faces are all mutually congruent. The term ‘‘isohedral’’—used by Fedorov [9] and Bilinski [3]—nowadays indicates the more restricted class of polyhedra with the property that their symmetries act transitively on their faces.3 The polyhedra of Fedorov and Bilinski are not (in general) ‘‘isohedra’’ by definitions that are customary today. We call a polyhedron rhombic if all its faces are rhombi. It is an immediate consequence of Euler’s theorem on polyhedra that the only monohedral zonohedra are the rhombic ones. One of the results of Fedorov ([9, p. 267], [10, p. 689]) is contained in the claim: There are precisely four distinct types of monohedral convex zonohedra: the rhombic triacontahedron T, the rhombic icosahedron F, the rhombic dodecahedron K, and the infinite family of rhombohedra (rhombic hexahedra) H. ‘‘Type’’ here is to be understood as indicating classes of polyhedra equivalent under similarities. The family of rhombohedra contains all polyhedra obtained from the cube by dilatation in any positive ratio in the direction of a body-diagonal. These polyhedra are illustrated in Figure 1; they are sometimes called isozonohedra. The polyhedra T and K go back at least to Kepler [23], whereas F was first described by Fedorov [9]. I do not know when the family H was first found — it probably was known in antiquity. An additional important result from Fedorov [9] is the following; notice the change to ‘‘combinatorial type’’ from the ‘‘affine type’’ that is inherent in the definition. Every convex parallelohedron is a zonohedron of one of the five combinatorial types shown in Figure 2. Conversely, every convex zonohedron of one of the five combinatorial types in Figure 2 is a parallelohedron.4

Minkowski’s theorem establishes that a convex polyhedron with pairwise parallel faces of the same area has a center; the congruence of the faces in each pair follows, regardless of the existence of centers of faces (which is assumed for zonohedra). 3 The term ‘‘gleichfla¨chig’’ (= with equal surfaces) was quite established at the time of Fedorov’s writing, but what it meant seems to have been more than the word implies. As explained in Edmund Hess’s second note [21] excoriating Fedorov [10] and [11], the interpretation as ‘‘congruent faces’’ (that is, monohedral) is mistaken. Indeed, by ‘‘gleichfla¨chig’’ Hess means something much more restrictive. Hess formulates it in [21] very clumsily, but it amounts to symmetries acting transitively on the faces, that is, to isohedral. It is remarkable that even the definition given by Bru¨ckner (in his well-known book [4, p. 121], repeating the definition by Hess in [19] and several other places) states that ‘‘gleichfla¨chig’’ is the same as ‘‘monohedral’’ but Bru¨ckner (like Hess) takes it to mean ‘‘isohedral.’’ Fedorov was aware of the various papers that use ‘‘gleichfla¨chig,’’ and it is not clear why he used ‘‘isohedral’’ for ‘‘monohedral’’ polyhedra. In any case, this led Fedorov to claim that his results disprove the assertion of Hess [19] that every ‘‘gleichfla¨chig’’ polyhedron admits an insphere. Fedorov’s claim is unjustified, but with the rather natural misunderstanding of ‘‘gleichfla¨chig’’ he was justified to think that his rhombic icosahedron is a counterexample. This, and disputed priority claims, led to protests by Hess (in [20] and [21]), repeated by Bru¨ckner [4, p. 162], and a rejoinder by Fedorov [11]. Neither side pointed out that the misunderstanding arises from inadequately explained terminology; from a perspective of well over a century later, it seems that both Fedorov and Hess were very thin-skinned, inflexible, and stubborn. 4 In different publications Fedorov uses different notions of ‘‘type.’’ In several (e.g., [10, 12]) he has only four ‘‘types’’ of parallelohedra, since the rhombic dodecahedron and the elongated dodecahedron ((c) and (b) in Figure 2) are of the same type in these classifications. Since we are interested in combinatorial types, we accept Fedorov’s original enumeration illustrated in Figure 2.

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THE MATHEMATICAL INTELLIGENCER

(a) T

(b)

F

(c) K

H

Figure 1. The four isozonohedra (convex rhombic monohedra) enumerated by Fedorov. Kepler found the triacontahedron T and the dodecahedron K, whereas Fedorov discovered the icosahedron F. The infinite class H of rhombic hexahedra seems to have been known much earlier.

(d)

(e)

Figure 2. Representatives of the five combinatorial types of convex parallelohedra, as determined by Fedorov [9]. (a) is the truncated octahedron (an Archimedean polyhedron); (b) is an elongated dodecahedron (with regular faces, but not Archimedean); (c) is Kepler’s rhombic dodecahedron K (a Catalan polyhedron); (d) is the Archimedean 6-sided prism; and (e) is the cube.

Fedorov’s proof is not easy to follow; a more accessible proof of Fedorov’s result can be found in [2, Ch. 8].

Bilinski’s Rhombic Dodecahedron Fedorov’s enumeration of monohedral rhombic isohedra (called isozonohedra by Fedorov and Bilinski, and by Coxeter [7]) mentioned previously claimed that there are precisely four distinct types (counting all rhombohedra as one type). Considering the elementary character of such an enumeration, it is rather surprising that it took three-quarters of a century to find this to be mistaken.5 Bilinski [3] found that there is an additional isozonohedron and proved: Up to similarity, there are precisely five distinct convex isozonohedra. The rhombic monohedral dodecahedron found by Bilinski shall be denoted B; it is not affinely equivalent to Kepler’s dodecahedron (denoted K) although it is of the same combinatorial type. Bilinski also proved that there are no other isozonohedra. To ease the comparison of B and K, both are shown in Figure 3. Bilinski’s proof of the existence of the dodecahedron B is essentially trivial, and this makes it even more mysterious

K

B

Figure 3. The two convex rhombic monohedra (isozonohedra): Kepler’s K and Bilinski’s B.

how Fedorov could have missed it.6 The proof is based on two observations: (i) All faces of every convex zonohedron are arranged in zones, that is, families of faces in which all members share parallel edges of the same length; and (ii) All edges of such a zone may be lengthened or shortened by the same factor while keeping the polyhedron zonohedral.

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This is a nice illustration of the claim that errors in mathematics do get discovered and corrected in due course. I can only hope that if there are any errors in the present work they will be discovered in my lifetime. 6 A possible explanation is in a tendency that can be observed in other enumerations as well: After some necessary criteria for enumeration of objects of a certain kind have been established, the enumeration is deemed complete by providing an example for each of the sets of criteria––without investigating whether there are more than one object per set of criteria. This failure of observing the possibility of a second rhombic dodecahedron (besides Kepler’s) is akin to the failure of so many people that were enumerating the Archimedean solids (polyhedra with regular faces and congruent vertices, i.e., congruent vertex stars) but missed the pseudorhombicuboctahedron (sometimes called ‘‘Miller’s mistake’’); see the detailed account of this ‘‘enduring error’’ in [13].

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In particular, all such edges on one zone can be deleted (shrunk to 0). Performing such a zone deletion—a process mentioned by Fedorov—starting with Kepler’s rhombic triacontahedron T yields (successively) Fedorov’s icosahedron F, Bilinski’s dodecahedron B, and two rhombohedra, the obtuse Ho and the acute Ha. This family of isozonohedra that are descendants of the triacontahedron is shown in Figure 4. The proof that there are no other isozonohedra is slightly more complicated and is not of particular interest here. The family of ‘‘direct’’ descendants of Kepler’s rhombic dodecahedron K is smaller; it contains only one rhombohedron H*o (Fig. 5). However, one may wish to include in the family a ‘‘cousin’’ H*a—consisting of the same rhombi as H*o, but in an acute conformation. One of the errors in the literature dealing with Bilinski’s dodecahedron is the assertion by Coxeter [7, p. 148] that the two rhombic dodecahedra—Kepler’s and Bilinski’s—are affinely equivalent. To see the affine nonequivalence of the two dodecahedra (easily deduced even from the drawings in Fig. 3), consider the long (vertical) body-diagonal of Bilinski’s dodecahedron (Fig. 3b). It is parallel to four of the faces and in each face to one of the diagonals. In two faces this is the short diagonal, in the other two the long one. But in the Kepler dodecahedron the corresponding diagonals are all of the same length. Since ratios of lengths

F

T

B

Ho

Ha

Figure 4. The triacontahedron and its descendants: Kepler’s triacontahedron T, Fedorov’s icosahedron F, Bilinski’s dodecahedron B, and the two hexahedra, the obtuse Ho and the acute Ha. The first three are shown by .wrl illustrations in [25] and other web pages. 8

THE MATHEMATICAL INTELLIGENCER

K

H*o

H*a

Figure 5. Kepler’s rhombic dodecahedron K and its descendant, rhombohedron H*o. The rhombohedron H*a is ‘‘related’’ to them since its faces are congruent to those of the other two isozonohedra shown; however, it is not obtainable from K by zone elimination.

of parallel segments are preserved under affinities, this establishes the nonequivalence. If one has a model of Bilinski’s dodecahedron in hand, one can look at one of the other diagonals connecting opposite 4-valent vertices, and see that no face diagonal is parallel to it. This is in contrast to the situation with Kepler’s dodecahedron. By the theorems of Fedorov mentioned previously, since Bilinski’s dodecahedron B is a zonohedron combinatorially equivalent to Kepler’s, it is a parallelohedron. This can be easily established directly, most simply by manipulating three or four models of B. It is strange that Bilinski does not mention the fact that B is a parallelohedron. In this context we must mention a serious error committed by A. Schoenflies [30, pp. 467 and 470] and very clearly formulated by E. Steinitz. It is more subtle than that of Coxeter, who may have been misguided by the following statement of Steinitz [34, p. 130]: The aim [formulated previously in a different form] is to determine the various partitions of the space into congruent polyhedra in parallel positions. Since an affine image of such a partition is a partition of the same kind, affinely related partitions are not to be considered as different. Then there are only five convex partitions of this kind. [My translation and comments in brackets]. How did excellent mathematicians come to commit such errors? The confusion illustrates the delicate interactions among the concepts involved, considered by Fedorov, Dirichlet, Voronoi, and others. A correct version of Steinitz’s statement would be (see Delone [8]): Every convex parallelohedron P is affinely equivalent to a parallelohedron P0 such that a tiling by translates of P0

Figure 6. An affine transform of the lattice of centers at left leads to the lattice of the tiling by regular hexagons. The Dirichlet domains of the points of the lattice are transformed into the hexagons at right, which clearly are not affinely equivalent to regular hexagons.

coincides with the tiling by the Dirichlet-Voronoi regions of the points of a lattice L0 . The lattice L0 is affinely related to the lattice L associated with one of the five Fedorov parallelohedra P00 . But P0 need not be the image of P00 under that affinity. Affine transformations do not commute with the formation of Dirichlet-Voronoi regions. In particular, isozonohedra other than rhombohedra are not mapped onto isozonohedra under affine transformations that are not similarities. As an illustration of this situation, it is easy to see that Bilinski’s dodecahedron B is affinely equivalent to a polyhedron B0 that has an insphere (a sphere that touches all its faces). The centers of a tiling by translates of B0 form a lattice L0 such that this tiling is formed by Dirichlet-Voronoi regions of the points of L0 . The lattice L0 has an affine image L such that the tiling by Dirichlet-Voronoi regions of the points of L is a tiling by copies of the Kepler dodecahedron K. However, since the Dirichlet domain of a lattice is not affinely associated with the lattice, there is no implication that either B or B0 is affinely equivalent to K. A simple illustration of the analogous situation in the plane is possible with hexagonal parallelogons (as mentioned earlier, a parallelogon is a polygon that admits a tiling of the plane by translated copies). As shown in Figure 6, the tiling is by the Dirichlet regions of a lattice of points. This lattice is affinely equivalent to the lattice associated with regular hexagons, but the tiling is obviously not affinely equivalent to the tiling by regular hexagons. It is appropriate to mention here that for simple parallelohedra (those in which all vertices have valence 3) that tile face-to-face Voronoi proved [38] that each is the affine image of a Dirichlet-Voronoi region. For various strengthenings of this result see [26].

Nonconvex Parallelohedra Bilinski’s completion of the enumeration of isozonohedra needs no correction. However, it may be of interest to

examine the situation if nonconvex rhombic monohedra are admitted; we shall modify the original definition and call them isozonohedra as well. Moreover, there are various reasons why one should investigate—more generally— nonconvex parallelohedra. It is of some interest to note that the characterization of plane parallelogons (convex or not) is completely trivial. A version is formulated as Exercise 1.2.3(i) of [16, p. 24]: A closed topological disk M is a parallelogon if and only if it is possible to partition the boundary of M into four or six arcs, with opposite arcs translates of each other. Two examples of such partitions are shown in Figure 7. Another reason for considering nonconvex parallelohedra is that there is no intrinsic justification for their exclusion, whereas—as we shall see—many interesting forms become possible, and some tantalizing problems arise. The crosses, semicrosses, and other clusters studied by Stein [32] and others provide examples of such questions and results.7 It also seems reasonable that the use of parallelohedra in applications need not be limited to convex ones. It is worth noting that by Fedorov’s Definition 24 (p. 285 of [9], p. 691 of [10]) and earlier ones, a parallelohedron need not be convex, nor do its faces need to be centrally symmetric. Two nonconvex rhombic monohedra (in fact, isohedra) have been described in the nineteenth century; see Coxeter [7, pp. 102–103, 115–116]. Both are triacontahedra, and are self-intersecting. This illustrates the need for a precise description of the kinds of polyhedra we wish to consider here. Convex polyhedra discussed so far need little explanation, even though certain variants in the definition are possible. However, now we are concerned with wider classes of polyhedra regarding which there is no generally accepted definition.8 Unless the contrary is explicitly noted, in the present note we consider only polyhedra with surface homeomorphic to a sphere and adjacent faces not coplanar. We say they are of spherical type. There are infinitely many combinatorially different rhombic monohedra of this type—to obtain new ones it is enough to ‘‘appropriately paste together’’ along common faces two or more smaller polyhedra. This will interest us a little bit later. The two triacontahedra mentioned above are not accepted in our discussion. However, a remarkable

Figure 7. Planigons without center have boundary partitioned into 4 or 6 arcs, such that the opposite arcs are translates of each other.

7

Recent results on crosses and semicrosses can be found in [14]. Many different classes of nonconvex polyhedra have been defined in the literature. It would seem that the appropriate definition depends on the topic considered, and that a universally accepted definition is not to be expected.

8

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nonconvex rhombic hexecontahedron of the spherical type was found by Unkelbach [37]; it is shown in Figure 8. Its rhombi are the same as those in Kepler’s triacontahedron T. It is one of almost a score of rhombic hexecontahedra described in the draft of [15]; however, all except U are not of the spherical type. For a more detailed investigation of nonconvex isozonohedra, we first restrict attention to rhombic dodecahedra. We start with the two convex ones—Kepler’s K and Bilinski’s B—and apply a modification we call indentation. An indentation is carried out at a 3-valent vertex of an isozonohedron. It consists of the removal of the three incident faces and their replacement by the three ‘‘inverted’’ faces—that is, the triplet of faces that has the same outer boundary as the original triplet, but fits on the other side of that boundary. This is illustrated in Figure 9, where we start from Kepler’s dodecahedron K shown in (a), and indent the nearest 3-valent vertex (b). It is clear that this results in a nonconvex polyhedron. Since all 3-valent vertices of Kepler’s dodecahedron are equivalent, there is only one kind of indentation possible. On the other hand, Bilinski’s dodecahedron B in Figure 10(a) has two distinct kinds of 3-valent vertices, so the indentation construction leads to two distinct polyhedra; see parts (b) and (c) of Figure 10. Returning to Figure 9, we may try to indent one of the 3-valent vertices in (b). However, none of the indentations produces a polyhedron of spherical type. The minimal departure from this type occurs on indenting the vertex opposite to the one indented first; in this case the two indented triplets of faces meet at the center of the original dodecahedron (see Fig. 9c). We may eliminate this coincidence by stretching the polyhedron along the zone determined by the family of parallel edges that do not intrude into the two indented triplets. This yields a parallelogram-faced dodecahedron that is of spherical type (but not a rhombic monohedron); see Figure 9(d). A related polyhedron is shown in a different perspective as Figure 121 in Fedorov’s book [9].

Figure 8. Unkelbach’s hexecontahedron. It has pairs of disjoint, coplanar but not adjacent faces, which are parts of the faces of the great stellated triacontahedron. All its vertices are distinct, and all edges are in planes of mirror symmetry. 10

THE MATHEMATICAL INTELLIGENCER

(a)

(c)

(b)

(d)

Figure 9. Indentations of the Kepler rhombic dodecahedron K, shown in (a). In (b) is presented the indentation at the vertex nearest to the observer; this is the only indentation arising from (a). A double indentation of the dodecahedron in (a), which is a single indentation of (b), is shown in (c); it fails to be a polyhedron of the spherical type, since two distinct vertices coincide at the center; hence it is not admitted. By stretching one of the zones, as in (d), an admissible polyhedron is obtained—but it is not a rhombic monohedron.

It is of significant interest that all the isozonohedra in Figures 9 and 10—even the ones we do not quite accept, shown in Figures 9(c) and 10(e)—are parallelohedra. This can most easily be established by manipulating a few models; however, graphical or other computational verification is also readily possible. To summarize the situation concerning dodecahedral rhombic monohedra, we have the following polyhedra of spherical type: Two convex dodecahedra (Kepler’s and Bilinski’s); Three simply indented dodecahedra (one from Kepler’s polyhedron, two from Bilinski’s); One doubly indented dodecahedron (from Bilinski’s polyhedron). We turn now to the two larger isozonohedra, Fedorov’s icosahedron F and Kepler’s triacontahedron T. Since each has 3-valent vertices, it is possible to indent them, and since the 3-valent vertices of each are all equivalent under symmetries, a unique indented polyhedron results in each case (Fig. 11). The icosahedron F admits several nonequivalent double indentations (see Fig. 12); two are of special interest, and

(a)

(b)

(d)

(c)

(e)

Besides a brief notice of this possibility by Fedorov, the only other reference is to the union of two rhombohedra mentioned by Kappraff [22, p. 381].9 For an example of this last construction, by attaching two rhombohedra in allowable ways one can obtain three distinct decahedra, one of which is shown in Figure 13. Another is chiral, that is, comes in two mirror-image forms. This construction can be extended to arbitrarily long chains of rhombohedra; from n rhombohedra there results a parallelohedron with 4n + 2 faces; see Figure 13 for n = 3. For another example, from three acute and one obtuse rhombohedra of the triacontahedron family, that share an edge, one can form a decahexahedron E. It is chiral, but it has an axis of 2-fold rotational symmetry. By suitable unions of one of these decahexahedron with a chain of n rhombohedra (n C 2), one can obtain isozonohedra with 4n + 16 faces. All isozonohedra mentioned in this paragraph happen to be parallelohedra as well. Hence there are rhombic monohedral parallelohedra for all even k C 6 except for k = 8. The isozonohedra just described show that there exist rhombic monohedral parallelohedra with arbitrarily long zones. However, there is a related open problem: Given an even integer k C 4, is there a rhombic monohedral parallelohedron such that every zone has exactly k faces? The cube has k = 4, the rhombic dodecahedra K and B have k = 6, and the doubly indented icosahedra D1 and D2

Figure 10. Indentations of the Bilinski dodecahedron shown in (a). The two different indentations are illustrated in (b) and (c), the former at an ‘‘obtuse’’ 3-valent vertex, the latter at an ‘‘acute’’ vertex. The double indentation of (a), resulting from a single indentation of (b), is presented in (d); (e) shows an additional indentation of (c) which, however, is not a polyhedron in the sense adopted here, since two faces overlap in the gray rhombus.

we shall denote them by D1 and D2. There are many other multiple—up to sixfold—indentations; their precise number has not been determined. An eightfold indentation of the triacontahedron T is shown in [39, p. 196]; it admits several additional indentations. The double indentations D1 and D2 of F shown in Figure 12 are quite surprising and deserve special mention: They are parallelohedra! Again, the simplest way to verify this is by using a few models and investigating how they fit. This contrasts with the singly indented icosahedron, which is not a parallelohedron. None of the other isozonohedra obtainable by indentation of F or T seems to be a parallelohedron. A different construction of isozonohedra is through the union of two or more given ones along whole faces, but without coplanar adjacent faces; clearly this means that all those participating in the union must belong to the same family of rhombic monohedra—either the family of the triacontahedron, or of Kepler’s dodecahedron, or of rhombohedra (with equal rhombi) not in either of these families.

(a)

(b)

(c)

(d)

Figure 11. (a) Icosahedron F and (b) its indentation; (c) Triacontahedron T and (d) its indentation.

9

In carrying out this construction we need to remember that adjacent faces may not be coplanar. This excludes the ‘‘semicrosses’’ of Stein [32] and other authors, although it admits the (1,3) cross. For more information see [33].

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(a) F

Figure 14. A nonconvex parallelohedron without a center of symmetry.

(b) D1

(a)

(c) D2

(b)

Figure 12. (a) The Fedorov rhombic icosahedron F; (b) A double indentation of the F yields a nonconvex rhombic icosahedron D1 of the spherical type that is a parallelohedron; (c) A different double indentation D2 is also a parallelohedron.

Figure 13. Isozonohedra with 10 and 14 faces.

are examples with k = 8. No information is available for any k C 10. Although the number of examples of nonconvex isozonohedra and parallelohedra could be increased indefinitely, in the next section we shall propose a possible explanation of which isozonohedra are parallelohedra.10

Remarks (i) The parallelohedra discussed previously lack a center of symmetry, which was traditionally taken as present in parallelohedra and more generally—in zonohedra. Convex zonohedra have been studied extensively; they have many 10

interesting properties, among them central symmetry.11 However, the assumption of central symmetry (of the faces, and hence of the polyhedra) amounts to putting the cart before the horse if one wishes to study parallelohedra— that is, polyhedra that tile space by translated copies In fact, the one and only immediate consequence of the assumed property of polyhedra that allow tilings by translated copies is that their faces come in pairs that are translationally equivalent. For example, the octagonal prism in Figure 14 is not centrally symmetric, and its bases have no center of symmetry either. But even so, it clearly is a parallelohedron. The dodecahedra in Figures 9(b) and 10(b),(c) have no center of symmetry although their faces are rhombi and have a center of symmetry each. On the other hand, the doubly indented polyhedron is Figure 10(d) has a center. As mentioned before, each of these is a parallelohedron. We wish to claim that central symmetry is a red herring as far as parallelohedra are concerned. The reason that the requirement of central symmetry may appear to be natural is that studies of parallelohedra have practically without exception been restricted to convex ones. Now, for convex polyhedra the pairing of parallel faces by translation implies that they have equal area, whence by a theorem of Minkowski (see Footnote 2) the polyhedron has a center, which implies that the paired faces coincide with their image by reflection in a point—that is, are necessarily centrally symmetric, and therefore are zonohedra. But this argument is not valid for nonconvex parallelohedra, hence such polyhedra need not have a center of symmetry. In his first short description of nonconvex parallelohedra, Fedorov writes (§83 in [9, p. 306]): The preceding deduction of simple [that is, centrally symmetric polyhedra with pairwise parallel and equal faces] convex parallelohedra is equally applicable to simple concave [that is, non-convex] ones, and hence we bring here only illustrations. We do not show the concave

Crystallographers are interested in parallelohedra far more general than the ones considered here: The objects they study in most cases are not polyhedra in the sense understood here, but object combinatorially like polyhedra but with ‘‘faces’’ that need not be planar. The interested reader should consult [29] and [24] for more precise explanations and details. 11 It is worth mentioning that Fedorov did not require any central symmetry in the definition of zonohedra ([9, p. 256], [10, p. 688]). However, he switched without explanation to considering only zonohedra with centrally symmetric faces. As pointed out by Taylor [36], this has become the accepted definition.

12

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tetraparallelohedron [the hexagonal prism] since this is simply a prism with a concave par-hexagon as basis. Fig. 121 presents the ordinary, and Fig. 122 the elongated concave hexaparallelohedron [the rhombic dodecahedron and the elongated dodecahedron]; Fig. 123 shows the concave heptaparallelohedron [the truncated octahedron]. Obviously, there exists no concave triparallelohedron [cube]. (My translation and bracketed remarks) Fedorov’s parallelohedron in Figure 121 of [9] is isomorphic to the polyhedron shown in our Figure 9(d). A monohedral rhombic dodecahedron combinatorially equivalent to it is shown in our Figure 10(d) and is derived from the Bilinski dodecahedron. However, Fedorov does not provide any proof for his assertion, and in fact it is not valid in general. For example, his Figure 123 does not show a polyhedron of spherical type, since one of the edges is common to four faces. This can be remedied by lengthening the short horizontal edges, but shows the need for care in carrying out the construction. (ii) The study of nonconvex parallelohedra necessitates the revision of various well-established facts concerning convex parallelohedra. For example, one of the crucial insights in the enumeration of parallelohedra (and parallelotopes in higher dimensions) is the property that every zone has either four or six faces. This is not true for nonconvex parallelohedra. For example, the double indentation D1 of Fedorov’s F shown in Figure 12(b) is a parallelohedron— even though all zones of D1 have 8 faces. For another example, in some cases changing of the lengths of edges of a zone has limitations if the spherical type is to be preserved. At present, there seems to be no clear understanding of the requirements on a polyhedron of spherical type to be a parallelohedron. As mentioned earlier, the three indented polyhedra in Figures 9(b) and 10(b),(c) are parallelohedra; They can be stacked like six-sided prisms. In fact, with a grain of salt added, starting with suitably chosen six-sided prisms, they may be considered as examples of Fedorov’s second construction of nonconvex polyhedra [9, p. 306]: If we replace one or several faces of a parallelohedron, or parts of these, by some arbitrary surfaces supported on these same broken lines, in such a way that a closed surface is obtained, and observing that precisely the same [translated] replacement is made in parallel position on the faces that correspond to the first ones or their parts, then, obviously the new figure will be a parallelohedron, though without a center…. It seems clear that Fedorov did not consider this construction important or interesting, since he did not provide even a single illustration. But it does lead to parallelohedra with some or all faces triangular, in contrast to the convex case; an example is shown in Figure 15. A more elaborate example of a nonconvex parallelohedron with some triangular faces, that does not admit a lattice tiling, is described by Szabo [35]. Another difference between convex and nonconvex parallelohedra is that the convex ones can be decomposed into rhombohedra; this is of interest in various contexts— see, for example, Hart [18] and Ogawa [28]. In general, such

Figure 15. A monohedral parallelohedron with triangles as faces.

decomposition is not possible for nonconvex parallelohedra. For example, the doubly indented dodecahedron in Figure 10(d) is not a union of rhombohedra. (iii) Examination of the various isozonohedra that are— or are not—parallelohedra, together with the observation that questions of central symmetry appear irrelevant in this context, lead to the following conjecture:

Conjecture Let P be a sphere-like polyhedron, with no pairs of coplanar faces. If the boundary of P can be partitioned into pairs of non-overlapping ‘‘patches’’ {S1, T1}; {S2, T2}; …; {Sr, Tr}, each patch a union of contiguous faces, such that the members in each pair {Si, Ti} are translates of each other, and the complex of ‘‘patches’’ is topologically equivalent as a cell complex to one of the parallelohedra in Figure 2, then P is a parallelohedron. Conversely, if no such partition is possible then P is not a parallelohedron. As illustrations of the conjecture, we can list the following examples: (a) The three singly indented dodecahedra in Figures 9 and 10 satisfy the conditions, with the patches S1, T1 formed by the triplet of indented faces and their opposites, and the other pairs formed by pairs of opposite faces. Then this cell complex is topologically equivalent to the cell complex of the faces of the six-sided prism (Fig. 2d). As we know, these dodecahedra are parallelohedra. Note that the fact that they are combinatorially equivalent to the convex dodecahedra K and B is irrelevant, since the complex of pairs of faces of the indented polyhedra is not isomorphic to that of the un-indented ones: Some pairs {Si, Ti} of parallel faces are separated by only a single other face, whereas in K and B they are separated by two other faces. (b) The doubly indented dodecahedron in Figure 10(d) complies with the requirements of the conjecture in a different way: Each pair {Si, Ti} consists of just a pair of parallel faces; the complex so generated is isomorphic to the one arising from Kepler’s K. (c) The doubly indented icosahedron D1 of Fedorov’s F, shown in Figure 12(b), provides additional support for the conjecture. Two of the pairs—say {S1, T1} and {S2, T2}—are formed by the indented triplets and their 2010 Springer Science+Business Media, LLC, Volume 32, Number 4, 2010

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these rhombi do not have the correct shape and cannot be folded to form any polyhedron with planar faces. (Since the angles of the rhombi are, as close as can be measured, 60 and 120, the obtuse angles of the shaded rhombus would be incident with two other 120 angles—which is impossible.) An Internet discussion about the net mentioned the possibility that the engraver misunderstood the author’s instructions; however, it is not clear what the author actually had in mind, since no text describes the polyhedron. The later edition of [6] mentioned by Hart [17] was not available to me.

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[1] A. D. Aleksandrov, Elementary deduction of the theorem about the center of a convex parallelohedron in 3 dimensions [In Russian]. Trudy fiz.-mat. Inst. Akad. Nauk im. Steklov 4 (1933), 89–99. [2] A. D. Alexandrov, Convex Polyhedra. Springer, Berlin 2005. Russian original: Moscow 1950; German translation: Berlin 1958. [3] S. Bilinski, U¨ber die Rhombenisoeder. Glasnik Mat. Fiz. Astr. 15 (1960), 251–263. The quite detailed review by J. J. Burckhardt in Zentralblatt v. 99, p. 155 #15506, does not mention that this contains a correction of Fedorov’s claim. Coxeter in MR 24#A1644

Figure 16. Cowley’s net for a rhombic dodecahedron.

mentions that ‘‘…the ‘second’ has never before been noticed’’ – but does not mention Fedorov.

opposites. The other pairs {Si, Ti} are the remaining four pairs of parallel faces. The complex they form is isomorphic to the face complex of the elongated dodecahedron shown in Figure 2(b). The same situation prevails with the doubly indented icosahedron D2 of Figure 12(c). Other double indentations of the icosahedron F, as well as the single indentation of F, fail to satisfy the assumptions of the conjecture and are not parallelohedra. (d) No indentation of the rhombic triacontahedron satisfies the assumptions of the conjecture, and in fact none is a parallelohedron. (e) The decahexahedron E mentioned previously has a decomposition into pairs {Si, Ti} that is isomorphic to the complex of the faces of the cube. The same situation prevails with regard to the chains of rhombohedra mentioned previously. (iv) The present article leaves open all questions regarding parallelohedra that are not rhombic monohedra. In particular, it would be of considerable interest to generalize the above conjecture to these parallelohedra. Such an extension would also have to cover the results on ‘‘clusters’’ of cubes such as the crosses and semicrosses investigated by S. K. Stein and others [32, 33, 14]. One can also raise the question of what are analogues for suitably defined ‘‘clusters’’ of rhombohedra, or other parallelohedra. (v) There just possibly may be a prehistory to the Bilinski dodecahedron. As was noted by George Hart [17, 18], a net for a rhombic dodecahedron was published by John Lodge Cowley [6] in the mid-eighteenth century; see Figure 16. The rhombi in this net appear more similar to those of the Bilinski dodecahedron than to the rhombi of Kepler’s. However, 14

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[4] M. Bru¨ckner, Vielecke und Vielflache. Teubner, Leipzig 1900. [5] J. J. Burckhardt, U¨ber konvexe Ko¨rper mit Mittelpunkt. Vierteljschr. Naturforsch. Ges. Zu¨rich 85 (1940), Beiblatt. Festschrift R. Fueter, pp. 149–154. [6] J. L. Cowley, Geometry Made Easy: A New and Methodical Explanation of the ELEMNENTS [sic] of GEOMETRY. Mechell, London 1752. [7] H. S. M. Coxeter, The classification of zonohedra by means of projective diagrams. J. de math. pures et appliq. 41 (1962), 137– 156. Reprinted in: Twelve Geometric Essays, Southern Illinois Univ. Press, Carbondale, IL, 1968 = The Beauty of Geometry. Twelve Essays. Dover, Mineola, NY, 1999. [8] B. N. Delone, Sur la partition re´gulie`re de l’espace a` 4 dimensions. Izv. Akad. Nauk SSSR Otdel Fiz.-Mat. Nauk 7 (1929), 79–110, 147–164. [9] E. S. Fedorov, Nachala Ucheniya o Figurah [In Russian] (= Elements of the theory of figures) Notices Imper. Petersburg Mineralog. Soc., 2nd ser., 24 (1885), 1–279. Republished by the Acad. Sci. USSR, Moscow 1953. [10] E. S. Fedorov, Elemente der Gestaltenlehre. Z. fu¨r Krystallographie und Mineralogie 21 (1893), 679–694. [11] E. v. Fedorow (E. S. Fedorov) Erwiderung auf die Bemerkungen zu E. v. Fedorow’s Elemente der Gestaltenlehre von Edmund Hess. Neues Jahrbuch fu¨r Mineralogie, Geologie und Paleontologie, 1894, part 2, pp. 86–88. [12] E. S. Fedorov, Regula¨re Plan- und Raumtheilung. Abh. K. Bayer. Akademie der Wiss. Vol. 20 (1900), pp. 465–588 + 11 plates. Russian translation with additional comments: ‘‘Pravilnoe Delenie Ploskosti i Prostranstva’’ (Regular Partition of Plane and Space). Nauka, Leningrad 1979. [13] B. Gru¨nbaum, An enduring error. Elemente der Math. 64 (2009), 89–101.

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Mathematically Bent

Colin Adams, Editor

Group Therapy COLIN ADAMS

The proof is in the pudding.

Opening a copy of The Mathematical Intelligencer you may ask yourself uneasily, ‘‘What is this anyway—a mathematical journal, or what?’’ Or you may ask, ‘‘Where am I?’’ Or even ‘‘Who am I?’’ This sense of disorientation is at its most acute when you open to Colin Adams’s column. Relax. Breathe regularly. It’s mathematical, it’s a humor column, and it may even be harmless.

â Column editor’s address: Colin Adams, Department of Mathematics, Bronfman Science Center, Williams College, Williamstown, MA 01267, USA e-mail: [email protected] 16

THE MATHEMATICAL INTELLIGENCER 2010 SPRINGER SCIENCE+BUSINESS MEDIA, LLC

r. Stew: Hello, come on in. You’re Hank, right? I’d like you to meet the group. This is Karen, and Bill, and Amanda and Sylvia. I’m Dr. Johnson, but you can call me Dr. Stew. Is this your first experience with group therapy? Hank: Uh, yes, it is. Dr. Stew: Well I think you will find it very helpful. Why don’t you take a seat over there between Karen and Bill. They’re married, but they don’t mind sitting apart. Hank: Married? You allow two people who are married in the same group therapy session? Dr. Stew: Why not? Since they have a child, we know they can multiply. (Laughter from the group. Hank laughs along nervously.) Dr. Stew: Now, the basic idea of group therapy is to associate our issues and problems with concepts in group theory. Hank: What? Dr. Stew: You know, the properties of abstract groups. Hank: I’m confused. Dr. Stew: You are here for group therapy, aren’t you? Hank: But group therapy refers to the fact that there is a group of us here in the room. (Everyone laughs.) Dr. Stew: Hardly. It refers to a group. You know, a set of elements with a multiplicative operation, inverses, etc. You have had Abstract Algebra, haven’t you? Hank: Yes, but … Dr. Stew: Good. Well, let’s get started. Sylvia, you were talking last week about how your mother favors you over your sister. Do you think she has put a partial ordering on your whole family? Sylvia: No question. Often the poset structure becomes more important than the algebraic structure. Dr. Stew: Does that bother you? Sylvia: Yes, it seems misguided. Why should a set-theoretic construct supersede an algebraic one? Over many years, we have built up a sophisticated set of relations that have allowed us to interpret our family relationships as a group. To give up all that structure seems counterproductive. Hank: Wait a minute. How sophisticated an algebraic object can it be? After all, your family is a finite set. Sylvia: Are you implying that finite groups can’t be interesting? What about the general linear group of dimension n over a finite field? What about the Weyl groups? I find your attitude quite condescending. Hank: I’m sorry. I didn’t mean to …

D

Sylvia: My family happens to be isomorphic to the quasidihedral group. And I am proud of that. Dr. Stew: Please everyone. Let’s try to be there for each other. Hank, we have a rule here. ‘‘Our support should always have measure 1.’’ Hank: Sorry, I just … Dr. Stew: Okay, let’s turn to Karen. How are things with you this week? Karen: Well, we had family over for the holidays, and what a disaster that was. My younger sister Emily, who was recently divorced, showed up with her new boyfriend Frank. Oh, was he obnoxious. He made lewd jokes, insulted my grandmother to her face, and accused my mother of intentionally giving him the smallest pork chop. Then, after dinner, my older sister Claire showed up with her new beau and it was none other than Emily’s ex-husband Craig. Everyone was stunned. Especially since we all couldn’t stand Craig when he was married to Emily. Sylvia: What happened? Karen: Amazingly enough, Frank and Craig hit it off. They took turns insulting the de´cor, the food and members of the family. They were awful. Dr. Stew: Well, the subgroup generated by Craig and Frank seems to be a problem. Hank: All families have subgroups like this. Sounds normal enough to me. Karen: What? Are you kidding? There is no way this subgroup is normal. Just conjugate it by Emily, and you don’t get the same subgroup. Bill: Hank, where exactly did you take group theory? Sylvia: Are you confusing conjugate with conjugal? Hank: That wasn’t what I meant. Dr. Stew: If the subgroup generated by Frank and Craig were normal, then the family could quotient out by the subgroup and they would have a perfectly functioning family group again. But unfortunately, the subgroup is not by any means normal. Hank: I didn’t mean normal in the group-theoretic sense. You’re making this all so complicated. It seems to me it should be simple. Bill: Well, it’s one thing to acknowledge that the subgroup generated by Frank and Craig is not normal, but to claim none of the proper subgroups is normal, well that’s another matter. Karen: (Angry.) What makes you think that there isn’t a single proper subgroup in my entire family that is normal? You don’t even know my family. It’s incredibly presumptuous on your part. Hank: No, I didn’t mean … Dr. Stew: Okay, I think we had better move on. Hank, please try to be considerate of other group member’s situations. Let’s do some free association. I say a word, you say what you think of. We’ll go around the room, starting with Amanda. Blue. Amanda: Carrot. Dr. Stew: Good. Sylvia, weasel. Sylvia: Chocolate. Dr. Stew: Good. Bill, clarinet. Bill: Horse. Dr. Stew: Good. And for Hank, wingnut.

Hank: Excuse me, but I’m confused. These associations don’t seem to have anything to do with the words you are saying. Dr. Stew: That’s right. Hank: But then I don’t understand why people are saying them. Dr. Stew: Well, Hank, you have to remember. Everything’s associative in a group. Hank: Oh, come on … Dr. Stew: Anyway, enough of that. Amanda, what’s been going on with you? Amanda: Well, my father has been commuting back and forth between Boston and New York. He’s just home on the weekends. Dr. Stew: By himself? Does he drive? Amanda: He usually rides with my uncle and the neighbor down the street. They have the same situation. Karen: That must be tough for you. Bill: You must miss him a lot. Hank: Isn’t someone going to say something about the commutator subgroup, or the group being abelian because everyone commutes? Dr. Stew: No. What does that have to do with anything? Hank: I am so confused. Dr. Stew: Let’s talk about that, Hank. It sounds like you are having an identity crisis. Hank: It does? Dr. Stew: Yes, it’s unclear who is the identity in your family group. Hank: I’m not following you. Dr. Stew: Who is it when multiplied by any other member of the family yields that same member of the family? Hank: You know, I am having some trouble interpreting this analogy with a group. What exactly is multiplication of two people? Dr. Stew: Well, what would you like it to be? Hank: How about something I can understand, not just some nebulous ill-defined concept created so this piss poor analogy can be sustained ad nauseam. Dr. Stew: I sense some hostility from you. Hank: Well, yes, I am a little frustrated. Dr. Stew: Maybe we can figure out where this hostility is coming from. It probably goes back a ways. Do you have siblings? Hank: Yes. I have two older brothers, Jeff and Tom, and an older sister Caroline. And then a much younger sister Liz. She was really brought up more by my three older siblings than by my parents. Dr. Stew: I see. Now, tell us. If you were going to give a word in the generators that are your mother, father, and siblings that best describes you, what would it be? Hank: Excuse me? Dr. Stew: You know. Let M denote your mother, F your father, J for Jeff, T for Tom, C for Caroline and L for Liz. Then make a word from these generators and their inverses that best describes you, that encompasses what parts of you come from these generators. We are all a product of our families. Hank: I would say you’re kidding, but I am guessing you are not. Okay, I’ll play along, How about … um … 2010 Springer Science+Business Media, LLC, Volume 32, Number 4, 2010

17

M F C-1JTC L-1C F -1M -1? Bill: Ha! Hank: What? What now? Bill: That’s the trivial word. Hank: No it’s not. Bill: Yes it is. You said before that Liz was a product of Jeff, Tom and Caroline so L equals JTC, meaning JTCL-1 is trivial. That was in the middle of your word. Once we trivialize that, the remaining generators and inverses cancel and the word collapses. So yes that is the trivial word.

18

THE MATHEMATICAL INTELLIGENCER

Dr. Stew: Hank, I find it indicative of your feelings of self loathing that you would pick a trivial word to describe yourself. Clearly an identity crisis, as you see yourself as the identity element. Hank: (Stands up.) That’s it. I am out of here. You people are crazy. Dr. Stew: We don’t use the word ‘‘crazy,’’ Hank. We say ‘‘topologically mixing.’’ But, you know, perhaps it is best if you leave. You aren’t ready for group therapy. I think you need one-on-one help, probably on a continuous basis. You should see an analyst.

Years Ago

David E. Rowe, Editor

Two Great Theorems of Lord Brouncker and His Formula bðs1Þbðsþ1Þ ¼ s 2; bðsÞ ¼ s þ

12 2s þ

ð1Þ

32 52 2sþ2sþ

..

Spring of 1655 Google search for ‘Spring of 1655’ returned a list of events, including the Insurrection of March 1655 against Cromwell [30], the discovery of a satellite of Saturn by Christiaan Huygens [16, pp.14–16], and, by the way, a reference to my own paper on Brouncker’s continued fraction [19]. These events, with the exception of my paper, took place in March of 1655, when Brouncker made his greatest discovery in mathematics. My paper had the modest purpose of restoring historical justice to William Brouncker, one of the brilliant minds in England in those times. Cromwell’s contribution reduced to suppression of Royalists, which on the one hand kicked Brouncker out of Big Politics for about 15 years, to which he finally returned only in 1660, and on the other hand promoted John Wallis to the position of Professor in Oxford, since he belonged to the opposite political camp and, according to historical records, even helped the Parliamentarians decode Royalist messages. Since Brouncker achieved nothing more in mathematics after 1660, I have the strong conviction that without Cromwell

â

Send submissions to David E. Rowe, Fachbereich 08, Institut fu¨r Mathematik, Johannes Gutenberg University, D-55099 Mainz, Germany. e-mail: [email protected]

Arithmetica Infinitorum

.

SERGEY KHRUSHCHEV

A

there would be no Brouncker’s formula, especially because there would, possibly, be no Wallis’s formula either. Christiaan Huygens didn’t believe Brouncker’s formula and asked one question. We discuss his role in more detail a bit later. To clarify the role of Saturn whose symbol is opposite that of Jupiter I cite a website on astrology: ‘‘In astrology, Saturn is associated with restriction and limitation. Where Jupiter expands, Saturn constricts. Although the themes of Saturn seem depressing, Saturn brings structure and meaning to our world. Saturn knows the limits of time and matter. Saturn reminds us of our boundaries, our responsibilities, and our commitments. It brings definition to our lives. Saturn makes us aware of the need for self-control and of boundaries and our limits.’’1

By the end of 1654, John Wallis, a good friend of Brouncker, had almost finished his book Arithmetica Infinitorum. A few words to explain the title: The area under the graph of y = x over the segment [0, 1] of the real line can be obtained by approximating the graph from below with inscribed rectangles with bases [k/n, (k + 1)/n], k = 1, ..., n - 1. Their total area 1 2 n1 1 nðn 1Þ 1 1 þ þ ... þ ¼ ¼ n n n n 2n2 2 2n approaches 1/2 as n? +?, which is the area of the triangle. This method obviously extends to ‘parabolas’ y = x p, p being a positive integer, by the arithmetic formulas n1 X k¼1

kp ¼

n pþ1 þ Oðn p Þ: pþ1

So, the title Arithmetica Infinitorum meant a new Arithmetics of Infinities. At the beginning of 1655, Wallis’s main problem with this book was that he couldn’t complete it with an arithmetical pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ formula for the area under the arc of the circle y ¼ 1 x 2 over [0, 1]. He knew, of course, Vie`te’s formula qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃ pﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃ 2þ 2þ 2 2 2 2þ 2 ¼ ...; ð2Þ 2 p 2 2 but it didn’t fit his understanding of Arithmetics of Infinities, since the number of radicals in (2) increases with every new step to the right. Moreover, Wallis had another restriction. He planned to obtain this new formula using his method of interpolation, presented for the first time in his book. In modern notations 1

http://www.cafeastrology.com/saturn.html

Ó 2010 Springer Science+Business Media, LLC, Volume 32, Number 4, 2010

19

Wallis wanted to find an arithmetic expression for the integral Z 1 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ p ¼ 1 x 2 dx: 4 0 Motivated by Vie`te’s formula, he introduced a family I(p, q) of reciprocals to the integrals, which he was able to compute: I ðp; qÞ ¼ R 1

1

ð1 x 1=p Þq dx 0 pþq : ¼ p

¼

ðp þ 1Þðp þ 2Þ. . .ðp þ qÞ 1 2 ... q ð3Þ

Here p and q are positive integers. For p = 1/2, q = n we have 1 1 3 . . .ð2n þ 1Þ 1 ¼ : I ;n ¼ ð4Þ 2 2 4 . . . ð2nÞ vn Clearly, vn decreases with increasing n. Since this sequence is obtained by a very simple law, one may hope that it may be naturally interpolated to positive numbers and, in particular, to n = 1/2. The value of v0 is 1 and of v1 is 2/3 = 0.66. . . The value of interest v1/2 = p/4 = 0.78. . . is regularly placed between v0 and v1. In other words, the results of this numerical experiment can only be explained by some simple formula for v1/2. But what is the formula? This question bothered Wallis a lot and, being convinced that he was only one step away from the solution (which was, in fact, the case), he continued his tremendous efforts to find the formula. Finally, good luck entered on his side (possibly under the influence of Saturn) and Wallis not only found the infinite product

AUTHOR

......................................................................... was born in Leningrad, USSR, graduated from the. Leningrad State University, Department of Mathematics and Mechanics, in 1972, and received his D.Sc. degree from the Steklov Institute. (Leningrad branch) in 1982. From 1987 to 1995 he directed the creation of the Euler International Mathematical Institute in Leningrad/St.Petersburg. After seven years as full professor at Atilim University in Ankara, Turkey, he is now full professor at the Eastern Mediterranean University in North Cyprus. His research interests include classical analysis, operator theory and probability theory. In his spare time he climbs mountains.

SERGEY KHRUSHCHEV

Department of Mathematics Eastern Mediterranean University North Cyprus via Mercin 10 Gazimagusa Turkey e-mail: [email protected] e-mail: [email protected] 20

THE MATHEMATICAL INTELLIGENCER

1 I ð1=2; 1=2Þ 2 2 13 35 57 ð2n 1Þ ð2n þ 1Þ ... . . .; ¼ ¼ p 22 44 66 2n 2n

ð5Þ

now bearing his name, he also proved its convergence to 2/p. On the one hand, according to Stedall [33], pp. xviii-xix, formula (5) appeared after February 28, 1655. On the other hand, already at the beginning of April, 1655, Wallis responded to Hobbes’s threats to reveal a quadrature of the unit circle by publishing some excerpts from [35]. It follows that (5) was most likely proved at the beginning of March, 1655. Now put yourself in Wallis’s place. What would you do if one way or another you obtained such a brilliant result? Yes, Wallis did the same thing. He wrote a letter to his friend in mathematics and music, William Brouncker, see [22]. Soon, Brouncker responded with the formula (1) and mentioned that b(1) = 4/p. In modern literature, the latter formula is named for him. This is not completely correct, since the identity b(1) = 4/p can be obtained quite elementarily [10]. It is much more difficult to prove that b(s - 1)b(s + 1) = s2, which actually was Brouncker’s great contribution. However, there are exceptions: See, for instance, [5], where one can find a version of the story of Brouncker’s formula quite close to that promoted by this paper.

Wallis’s Puzzle Things are not so simple with Brouncker’s proof. In the last section 191 of [35], Wallis writes that he tried several times to convince Brouncker to publish the proof, but all his attempts were in vain. One can only guess why Brouncker didn’t want to publish it. It is quite possible that—bearing in mind his position as a true Royalist, especially in view of the Insurrection of March 1655—Brouncker just didn’t want to be involved in the severe controversy between Wallis and Hobbes [3], which also started in March of 1655. The truth in this controversy was, of course, on Wallis’s side, since Hobbes was a man completely unable to understand mathematics. However, very often such public disputes, especially with educated people, result in big trouble. Being a great patriot of British mathematics, Wallis couldn’t leave what he got from Brouncker unpublished. Therefore, he undertook an attempt to present this in his book, providing explanations of his own. I think that he understood that his comments were not complete and required some further study. But as is clear from the last words of his book, he hoped that later this would be explained in full detail. As further developments showed, it was a very good decision which, in fact, doubled the value of [35]. The puzzle Wallis left didn’t escape the careful attention of Euler, who even took his copy of Arithmetica Infinitorum to St. Petersburg. In [12, 17] Euler writes: This theorem, which explicitly presents values of the continued fraction as integral formulas, deserves mention the more as it be less obvious. . . . Therefore, for quite a long time I have undertaken great efforts to prove this Theorem so that its proof a priori can be related to this function; this research, in my opinion, is

more difficult, but I believe it may result in great benefits. While any such research was condemned to failure, I regret most of all the fact that Brouncker’s method was nowhere present and most likely fell into oblivion. Euler couldn’t find a solution to Wallis’s puzzle, but his proofs of (1) led him to the discovery of the theory of the Gamma and Beta functions [20].

b0

Q2kþ1 Q2k \p\b0 ; P2kþ1 P2k

where b0 ¼ 4

22 42 62 82 102 122 32 52 72 92 112

¼ 78:602424992035381646. . .;

Huygens’s Question When [35] was finally published in 1656, Wallis distributed a number of copies of the book among working mathematicians. One copy of [35] reached Huygens, who found Brouncker’s formula in its last section. There is no doubt that Huygens understood nothing from Wallis’s comments, since he demanded a numerical confirmation. This was, by the way, not such an easy task since, as Euler later showed in [10], the convergents to b(1) are nothing but n 1 12 32 ð2n 1Þ2 Qn X ð1Þk p ¼ ! ; ¼ 1þ 2 þ 2 þþ 4 2 Pn 2k þ 1 k¼0

ð6Þ

where to save paper we use for continued fractions Roger’s notations Pn 12 32 ð2n 1Þ2 : ¼1þ Qn 2 þ 2 þþ 2 P Similar to the case of sums nk=1, continued fractions can be also written as ! n Pn ð2k 1Þ2 : ¼1þ K k¼1 Qn 2

Here K stands for German ‘‘Kettenbru¨che’’. The alternating series in (6) converges to p4 ¼ arctanð1Þ but not very quickly. In addition, if one does not know even formula (6), which was the case with Huygens, the prospect of evaluating 2the continued fraction with, say 20 simple called partial fractions, does not look very terms ð2k1Þ 2 encouraging. Needless to say, Wallis redirected Huygens’s question to Brouncker. And Brouncker shortly found an ingenious solution. We can rewrite Brouncker’s formula as follows: bðsÞbðs þ 2Þ ¼ ðs þ 1Þ2 :

ð7Þ

On the one hand, a look at Brouncker’s continued fraction (1) shows that its evaluation for large s requires fewer partial fractions. On the other hand, formula (7) suggests a way to relate b(1) with, say, b(4n + 1) and b(4n + 3): 22 42 ð2nÞ2 4 ... ð2n þ 1Þ; 13 35 ð2n 1Þð2n þ 1Þ p 13 35 ð2n 1Þð2n þ 1Þ bð4n þ 3Þ ¼ 2 2 . . . ð2n þ 1Þp: 2 4 ð2nÞ2

bð4n þ 1Þ ¼

ð8Þ If n = 6, then the continued fraction b(4 6 + 1) = b(25) has partial denominators 2 25 = 50, which considerably improves its convergence. Thus, we obtain the following bounds for p:

and P /Q are convergents to b(25). Putting k = 0, 1, 2 in the above formula, we find that k¼0

3:14158373269526\p\3:14409699968142

k¼1

3:14159265194782\p\3:14159274082066

k¼2

3:14159265358759\p\3:14159265363971 :

Notice that already the first convergent to b(25) gives four true places of p. The fifth convergent without tedious calculations gives 11 true places. This was the first algebraic calculation of p. Vie`te in 1593 couldn’t use his formula (2) and instead applied the traditional method of Archimedes to obtain 9 decimal places. The same, by the way, is true for Wallis’s infinite product (5). In 1596, Ludolph van Ceulen obtained 20 decimal places by using a polygon with 60 9 229 sides. The cumbersome calculations made by Ludolph are incomparable with Brouncker’s short and beautiful calculations. A detailed historical account of Brouncker’s calculations can be found in [31]. It seems that this achievement of Brouncker’s remained unnoticed, and even his formulas (8) were later rediscovered by Euler. In 1654 Huygens published a book [18] in which he presented his geometric method, which considerably improved that of Archimedes. However, although Huygens got from Brouncker the numerical confirmation he demanded, he definitely didn’t realize its importance. As is clear from above, one can find with Brouncker’s method as many true decimal places of p as necessary. In [27, pp.75–77] there is a table of achievements in finding true decimal places of p. I have no doubt that Brouncker with his formula could get all the places obtained in the period of 1596–1793 in just a couple of evenings. Still, no book includes him in the lists of winners . . .

How Was It Done? Now the time has come to reveal the secrets of Brouncker’s proof. To begin with, it is important to realize, as I confirm in the next section, that Brouncker developed the theory of continued fractions with positive terms. Next, we know that Brouncker got formula (5) from Wallis. from § What would a person like Brouncker who developed a beautiful theory do in this case? First, he would rewrite the partial products of Wallis’s infinite product as a continued fraction: ð2n 1Þ ð2n þ 1Þ 13 35 57 ... 2n 2n 22 44 66 13 22 35 ð2n 1Þ ð2n þ 1Þ ð2nÞ ð2nÞ : ¼ þ 0 þ 0 þ 0 þ...þ 0 1 ð9Þ

Ó 2010 Springer Science+Business Media, LLC, Volume 32, Number 4, 2010

21

The problem with the continued fraction in (9) is that its zero partial denominators cannot be extended ad infinitum, since if one does this, the convergents will alternate between 0 and ?. (Wallis, in his comments, refers to such a continued fraction as oscillating.) Another look at (9) shows that the odd numerators are products of the form s(s + 2) = s2 + 2s = (s + 1)2 - 1, where s is odd. Now, at the very beginning of §191 in [35] we find: ‘‘The Noble Gentleman noticed that two consecutive odd numbers, if multiplied together, form a product which is the square of the intermediate even number minus one. . . . He asked, therefore, by what ratio the factors must be increased to form a product, not equal to those squares minus one, but equal to the squares themselves’’. This suggests increasing s to b(s) and s + 2 to b(s + 2) for odd s in the right-hand side of (9) so that they satisfy (7). Then, to keep (9) valid, odd zero partial denominators in the right-hand side of (9) must become positive. That is exactly what we need to complete the proof. The fact that s + 1 is even is also helpful, since it may provide necessary cancellations. Now, using (7) repeatedly, we may write bð1Þ ¼ ¼

into a simple fraction, and, by the Euclidean algorithm, can finally be developed into a regular continued fraction: 31415926535 10000000000 1 ¼3þ 88514255 7þ 1415926535 1 ¼3þ 1 7þ 88212710 15þ 88514255 1 1 1 301545 ¼3þ 7 þ 15 þ 1 þ 88212710 1 1 1 1 161570 : ¼3þ 7 þ 15 þ 1 þ 292 þ 301545

3:1415926535 ¼

This gives us the first seven true terms [3; 7, 15, 1, 292, 1, 1] of the regular continued fraction for p. The later one cuts off the decimal expansion of p, the more true terms of the continued fraction are obtained. Motivated by (12), we can similarly represent b(s) as a sum:

22 22 22 62 22 62 ¼ 2 bð5Þ ¼ 2 ¼ 2 2 bð9Þ ¼ ... bð3Þ 4 4 bð7Þ 4 8

bðsÞ ¼ s þ c0 þ

bðsÞ ¼ s þ

12 32 ð2n 1Þ2 ¼ 2 2 ... bð4n þ 1Þ 2 4 ð2nÞ2

s2 þ 2s þ 1 ¼ bðsÞbðs þ 2Þ ¼ s2 þ 2s þ 2c1 þ oð1Þ; s ! þ1; ð10Þ

implying that c1 = 1/2. It follows that bðsÞ ¼ s þ

ð11Þ

ð12Þ

ð13Þ

bðsÞ ¼ s þ

8s4 18s2 þ 153 ¼ 16s5

¼

p ¼ 3:1415926535. . . 1 4 1 5 9 2 6 ¼ 3 þ þ 2 þ 3 þ 4 þ 5 þ 6 þ 7. . . : 10 10 10 10 10 10 10

¼

22

THE MATHEMATICAL INTELLIGENCER

8s4 18s2 þ 153 c7 þ 7. . . : 16s5 s

Cutting the above formula at c6 s6 0 and applying the Euclidean algorithm to the quotient of polynomials, we have

It remains to find a formula for b(s). Let us analyze how one obtains the regular continued fraction for a real number, say p. First, the number is expanded into an infinite decimal fraction

Then this infinite decimal fraction is cut at, say, the 10th place. After that, the decimal fraction obtained is converted

1 c2 þ ... : 2s s2

Similarly, c2 = 0, c3 = -9/8, c4 = 0, c5 = 153/16, c6 = 0 and, therefore,

where o(1) denotes a function approaching 0 at ?. Since s + 2 \ b(s + 2) and b(s)b(s + 2) = (s + 1)2, we have

which together with (11) give 2 bð4n þ 1Þ 4 bð1Þ ¼ lim þ oð1Þ lim ¼ : n n p ð2n þ 1Þ p

c1 þ ... s

and then determine c1 from the equation

13 35 57 ð2n 1Þð2n þ 1Þ bð4n þ 1Þ : ¼ 2 2 2 ... 2 4 6 ð2n þ 1Þ ð2nÞ2

s2 þ 2s þ 1 1 ¼sþ ; s\bðsÞ\ sþ2 2þs

ð14Þ

The coefficients c0, c1, c2, . . ., can be found inductively using (7). By (12), c0 = 0. To find c1 we assume that

22 62 102 ð4n 2Þ2 ... bð4n þ 1Þ 42 82 122 ð4nÞ2

Combined with Wallis’s formula, this implies 2 bð4n þ 1Þ bð1Þ ¼ þ oð1Þ ; p ð2n þ 1Þ

c1 c2 c3 þ þ þ ... : 2s s2 s3

1 9ð4s3 34sÞ 2s þ 4 8s 18s2 þ 153

1 2s þ

8s4 18s2 þ 153 4s3 34s 1

2s þ

9

1

¼

9

9

2s þ

2

2s þ

25ð2s þ 153=25Þ 4s3 34s

:

25 2s þ 2s þ ...

A remarkable property of these calculations is that 12 = 1, 32 = 9, 52 = 25, etc., appear automatically as common

divisors of the coefficients of the polynomials in Euclid’s algorithm. The fraction 153/25 appears only because we didn’t find the exact value of c7. Increasing the number of terms in (14), we naturally conclude that bðsÞ ¼ s þ

12 32 52 72 ð2n 1Þ2 : þ... 2s þ 2s þ 2s þ 2s þ ... þ 2s

and define Pn and Qn formally by ascendant continued fractions. Then fPn gn>0 and fQn gn>0 satisfy

ð15Þ

If you think that it was already enough to finish the proof for a mathematician working in 1655, when even Newton’s calculus was not available, then you make a big mistake, because following Wallis’s writings you discover the following formulas: P0 ðsÞ P0 ðs þ 2Þ ðs þ 1Þ2 ¼ sðs þ 2Þ ðs þ 1Þ2 ¼ ð1Þ ; Q0 ðsÞ Q0 ðs þ 2Þ P1 ðsÞ P1 ðs þ 2Þ 4s4 þ 16s3 þ 20s2 þ 8s þ 9 ðs þ 1Þ2 ¼ Q1 ðsÞ Q1 ðs þ 2Þ 4s2 þ 8s

Pn a1 a2 an b0 þ ; Qn b1 þ b2 þ þ bn

4s4 þ 16s3 þ 20s2 þ 8s 9 ¼ 2 ; 4s2 þ 8s 4s þ 8s

Pn ¼ bn Pn1 þ an Pn2 ;

Pn Qn1 Pn1 Qn ¼ ð1Þn1 a1 . . .an where P1 ¼ 1;

Pn Pn1 ð1Þn1 a1 . . .an ¼ ; Qn Qn1 Qn Qn1

ð16Þ b0 ¼

One can find these very formulas in [33, pp. 169–170], where Wallis writes after the last formula: ‘. . . which is less than the square F2 + 2F + 1. And thus it may be done as far as one likes; it will form a product which will be (in turn) now greater than, now less than, the given square (the difference, however, continually decreasing, as is clear), which was to be proved.’ [In Wallis’s notations s = F.] To clarify these comments of Wallis, we sketch briefly Brouncker’s theory of continued fractions with positive terms.

Q0 ¼ 1:

To evaluate a finite continued fraction, for instance the fourth convergent to the regular continued fraction for p: 1 1 1 1 ; 3þ 7 þ 15 þ 1 þ 292 one must rewrite it as an ascendant continued fraction 1

þ3¼ þ7

355 ; 113

ð19Þ

Since Wallis included (17) in [35], and Euler wrote a chapter on continued fractions in [11], formulas (17) are now called the Euler-Wallis formulas. Formulas (19–20) lead to a simple criterion for the convergence of a continued fraction with positive terms.

C OROLLARY 2 A continued fraction with positive terms converges to a finite value if and only if a1 a2 . . . an ! 0: Qn Qn1

ð21Þ

T HEOREM 3 Brouncker’s continued fraction (1) converges for every s [ 0. P ROOF . To remove factorials, let Qn = (2n + 1)!!Dn. By Corollary 2, the continued fraction (1) converges if and only if a1 a2 . . . an ð2n 1Þ!!2 ¼ Qn Qn1 ð2n þ 1Þ!!ð2n 1Þ!!Dn Dn1

þ 15

and make all arithmetic operations in the direction from the bottom to the top. The result is Metius’s approximation for p. It is obvious that such a method leads to tremendous numerical difficulties. Brouncker found another way to do this. For every n we write

n ¼ 1; 2; . . .;

P0 P2k P2kþ1 P1 P1 \\ \\ \\ \ ¼ þ1: Q0 Q2k Q2kþ1 Q1 Q1 ð20Þ

The Euler-Wallis Formulas

1 þ1 292

P0 ¼ b0 ;

positive, then

225 : 16s4 þ 64s3 þ 136s2 þ 144s þ 225

1

Q1 ¼ 0;

As soon as formulas (17) and (18) are stated they can be easily proved by induction. If you ask why P-1 = 1 and Q-1 = 0, then the answer is given by the following theorem (which again was definitely known to Brouncker).

16s6 þ 96s5 þ 280s4 þ 480s3 þ 649s2 þ 594s ðs þ 1Þ2 ¼ 16s4 þ 64s3 þ 136s2 þ 144s þ 225

1

ð18Þ

T HEOREM 1 (B ROUNCKER , 1655) If ak and bk are all

P2 ðsÞ P2 ðs þ 2Þ ðs þ 1Þ2 Q2 ðsÞ Q2 ðs þ 2Þ

¼

ð17Þ

Qn ¼ bn Qn1 þ an Qn2 ;

¼

1 ! 0: ð2n þ 1ÞDn Dn1

By (17) Dn ¼

2s 2n 1 Dn1 þ Dn2 ; 2n þ 1 2n þ 1

Ó 2010 Springer Science+Business Media, LLC, Volume 32, Number 4, 2010

ð22Þ

23

which implies that ð2n þ 1ÞDn Dn1 ¼

2 2sDn1

þ ð2n 1ÞDn1 Dn2 :

ð23Þ

Since D0 = 1, D-1 = 0, the iteration of (23) leads to a nice formula, 2 : ð24Þ ð2n þ 1ÞDn Dn1 ¼ 2s D02 þ D12 þ . . . þ Dn1 For even n we have Dn2 [

1 : nþ1

ð25Þ

It is true for n = 2, since Dn(0) = 3/5 by (22) and 9/25 [ 2 [ ðn 1Þ1 , then by (22) 1/3. Now, if Dn2 2n 1 2 2 2n 1 2 1 1 Dn2 [ [ ; Dn2 [ 2n þ 1 2n þ 1 n 1 n þ 1 as elementary algebra shows. Now, by (24) and (25), 1 1 1 ; ð2n þ 1ÞDn Dn1 [ 2s 1 þ þ þ þ 3 5 2m þ 1 where m is the greatest number satisfying 2m 6 n 1. Since the series 1 X

1 ¼1 2m þ1 m¼0 diverges, Brouncker’s continued fraction converges for s [ 0. Let us observe that for the case of s > 1; which is all that required for the evaluation of p, the proof of Theorem 3 can be completed in a very elementary way. Indeed, if s = 1, then, by (22), Dn lies between Dn-1 and Dn-2, implying that the whole sequence lies between D0 = 1 and D1 = 2/3. If s [ 1, then Dn(s) [ Dn(1), since by (22) all coefficients of the polynomials Dn are positive. It follows that for s>1 Pn ðsÞ Pn1 ðsÞ 3 Q ðsÞ Q ðsÞ \4n þ 2: n n1

The Functional Equation Recall that we derived Brouncker’s continued fraction from the single assumption that it satisfies the functional equation (7). This, of course, hints that b(s) indeed satisfies (7), but how can we prove it? Wallis’s notes at the end of § explain this. Since by Theorem 3 Brouncker’s continued fraction converges, one should only compare the values of the convergents at s and s + 2. Passing to the polynomials, one may notice that 2

Pn ðsÞPn ðs þ 2Þ ðs þ 1Þ Qn ðsÞQn ðs þ 2Þ ¼ bn

ð26Þ

P2k ðsÞ P2k ðs þ 2Þ P2kþ1 ðsÞ P2kþ1 ðs þ 2Þ \ðs þ 1Þ2 \ : Q2k ðsÞ Q2k ðs þ 2Þ Q2kþ1 ðsÞ Q2kþ1 ðs þ 2Þ By Theorems 1 and 3, then, b(s) satisfies (7). It remains to prove (26). By (19) Pn ðsÞQnþ1 ðsÞ Pnþ1 ðsÞQn ðsÞ ¼ ð1Þn ½ð2n þ 1Þ!!2 ¼ bn ; ð27Þ implying that the polynomials in the left-hand sides of (26) and (27) coincide. It follows that Pn ðsÞfPn ðs þ 2Þ Qnþ1 ðsÞg ¼ Qn ðsÞfðs þ 1Þ2 Qn ðs þ 2Þ Pnþ1 ðsÞg

is equivalent to (26). By (27), the polynomials Pn and Qn cannot have common factors. It follows that Pn(s + 2) -Qn+1(s) = ln(s)Qn(s), where ln(s) is a linear function. To find ln let us observe that the polynomial part of Qn+1/Qn is 2s by (17). It follows also from (17) that Pn(s) = 2nsn+1 + asn-1 + ... and Qn(s) = 2nsn + bsn-2 + .... Hence, the polynomial part of Pn(s + 2)/Qn(s) is s + 2n + 2, implying that ln(s) = 2n + 2 - s. The proof now is completed by a technical lemma.

L EMMA 4 Let Pn(s)/Qn(s) be the nth convergent to Brouncker’s continued fraction (15). Then ðs þ 1Þ2 Qn ðs þ 2Þ ¼ Pnþ1 ðsÞ þ ð2n þ 2 sÞPn ðsÞ; Pn ðs þ 2Þ ¼ Qnþ1 ðsÞ þ ð2n þ 2 sÞQn ðsÞ:

24

THE MATHEMATICAL INTELLIGENCER

ð29Þ ð30Þ

P ROOF . The Euler-Wallis formulas (17) for convergents P/Q look as follows Pn ðsÞ ¼ 2sPn1 ðsÞ þ ð2n 1Þ2 Pn2 ðsÞ; P0 ðsÞ ¼ s; P1 ðsÞ ¼ 1; Qn ðsÞ ¼ 2sQn1 ðsÞ þ ð2n 1Þ2 Qn2 ðsÞ; Q0 ðsÞ ¼ 1; Q1 ðsÞ ¼ 0;

and can be used to establish (29) and (30) for n = 0, 1, 2. Let us check, for instance, (30). A natural idea is to show that the right-hand side of (30) satisfies the Euler-Wallis equation for Pn (s + 2). Then assuming that (30) is true for every n \ k, we can write Pk ðs þ 2Þ ¼ 2ðs þ 2ÞPk1 ðs þ 2Þ þ ð2k 1Þ2 Pk2 ðs þ 2Þ ¼ 2ðs þ 2ÞQk ðsÞ þ 2ðs þ 2Þð2k sÞQk1 ðsÞ þ ð2k 1Þ2 Qk1 ðsÞ þ ð2k 1Þ2 ð2k 2 sÞQk2 ðsÞ ¼ 2sQk ðsÞ þ ð2k þ 1Þ2 Qk1 ðsÞ ðð2k þ 1Þ2 ð2k 1Þ2 ÞQk1 ðsÞ þ 4Qk ðsÞ þ 2ðs þ 2Þð2k sÞQk1 ðsÞ þ ð2k 1Þ2 ð2k 2 sÞQk2 ðsÞ ¼ Qkþ1 ðsÞ þ ð2k þ 2 sÞQk ðsÞ 8kQk1 ðsÞ

is a constant at least for the first values of n = 0, 1, 2; see formulas (16). If we know that bn does not depend on S, then it is not difficult to find it. Putting s = -1 in (26), we obtain bn = Pn(-1)Pn(1). By (17), polynomial Pn(s) is odd for even n and is even for odd n. Moreover, Pn(1) = (2n + 1)!! It follows that bn ¼ ð1Þn1 Pn ð1Þ2 ¼ ð1Þn1 ð2n þ 1Þ!!2 . Assuming that (26) holds for every n with bn = -(-1)n [(2n + 1)!!]2, we obtain, for s [ 0,

ð28Þ

ð2k 2 sÞQk ðsÞ þ 2ðs þ 2Þð2k sÞQk1 ðsÞ þ ð2k 1Þ2 ð2k 2 sÞQk2 ðsÞ ¼ Qkþ1 ðsÞ þ ð2k þ 2 sÞQk ðsÞ þ ð2k þ 2 sÞf2sQk1 ðsÞ þ ð2k 1Þ2 Qk2 ðsÞ Qk ðsÞg ¼ Qkþ1 ðsÞ þ ð2k þ 2 sÞQk ðsÞ:

A similar calculation proves (29).

Z

Formula (30) can be used to estimate the accuracy of Brouncker’s algebraic method for the evaluation of p; see § . It follows from (30) that Pn ðsÞ [

Qnþ1 ðs 2Þ Pnþ1 ðs 2Þ; Pnþ1 ðs 2Þ

Z

s [ 2:

p=2

0 p=2

p 1 3 5 ... ð2n 1Þ p ¼ un ; sin2n hdh ¼ 2 2 4 6 ... 2n 2

sin2nþ1 hdh ¼

0

Observing that Qn ðsÞ/Pn (s) ? 1/b(s) as n??, we obtain that

0

0

2n

Wallis’s Product

ðs þ 1Þ2 ðs þ 1Þ2 ðs þ 5Þ2 bðs þ 4Þ ¼ bðs þ 8Þ bðsÞ ¼ ðs þ 3Þ2 ðs þ 3Þ2 ðs þ 7Þ2 ðs þ 1Þ2 ðs þ 5Þ2 ðs þ 4n 3Þ2 bðs þ 4nÞ 2 2 ... ðs þ 3Þ ðs þ 7Þ ðs þ 4n 1Þ2 ðs þ 1Þðs þ 5Þ . . . ¼ ðs þ 1Þ ðs þ 3Þ2 ðs þ 4n 3Þðs þ 4n þ 1Þ bðs þ 4nÞ : ðs þ 4n þ 1Þ ðs þ 4n 1Þ2 ¼

Multipliers are grouped in accordance to the rule of Wallis’s formula: ðs þ 4n 3Þðs þ 4n þ 1Þ 4 ¼1 ; ðs þ 4n 1Þ2 ðs þ 4n 1Þ2

T HEOREM 5 Let y(s) be a function on (0, +?) satisfying (7) and the inequality s \ y(s) for s [ C, where C is a constant. Then 1 Y ðs þ 4n 3Þðs þ 4n þ 1Þ

¼sþ K

n¼1

ð31Þ

for every positive s. If we put s = 1 in (31), we obtain Wallis’s product (5). Nowadays, the proof of Wallis’s formula can be shortened to a few lines. Integration by parts shows that

2

implying (5). This now standard proof is, in fact, a small improvement (use of the inequalities (33) was Euler’s idea [14, Ch. IX, § 356]) over Wallis’s original arguments. Notice that in 1654– 1655, when Wallis worked on his book, neither integration by parts nor the change of variable formula were available. Instead, Wallis made his discoveries using a simple relation of integrals with areas as well as his method of interpolation. One can also observe that there is no direct relation between Wallis’s and Brouncker’s proofs. Therefore, it is unlikely that Brouncker consulted Wallis when he tried to find his own proof. Moreover, this shows that Wallis’s proof was the first.

Daniel Bernoulli and Goldbach posed the problem of finding a formula extending the factorial n ? n! = 1 2 . . . n to real values of n. In his letter of October 13, 1729 to Goldbach, Euler solved this problem. There are no doubts that Euler’s solution was motivated by Wallis’s interpolation method [35]. Arguing by analogy with Brouncker, one can seek an extension CðxÞ for Cðn þ 1Þ ¼ n! as a solution to Cðx þ 1Þ ¼ xCðxÞ;

x [ 0:

ð35Þ

If 0 \ x \ 1, then iterating (35) we obtain that

ðs ! þ 4n 1Þ2

ð2n 1Þ2 2s

13 35 57 ð2n 1Þ ð2n þ 1Þ 2 3 ... \ ; 22 44 66 2n 2n p pð2n þ 2Þ ð34Þ

Ramanujan’s Formula2

which provides the convergence of the product, at least for s [ -3. Since by Theorem 3 the continued fraction (1) converges, we can combine Brouncker’s ideas to obtain the following theorem which is fair to attribute to him.

1

h on (0, p/2)), we

which shows that

The functional equation (7), which finally resulted in (15), can easily be used to develop b(s) into an infinite product:

n¼1

h[ sin

2nþ2

un 2 un 1 [ [ 1 ; 2n þ 2 vn p vn

0\

yðsÞ ¼ ðs þ 1Þ

2nþ1

(observe that sin h[ sin immediately obtain

for some c [ 0. Now, by (18) Qn b0 ð25Þ p \b0 Qn ð25Þ Qn1 ð25Þ P P Pn1 n n 2 ½ð2n 1Þ!! 1 ¼ b0 ¼ O 25 : n Pn ð25ÞPn1 ð25Þ

ð32Þ

Combining (32) with the trivial inequalities Z p=2 Z p=2 Z p=2 2n 2nþ1 sin hdh[ sin hdh[ sin2nþ2 hdh ð33Þ 0

Pn ð25Þ [ cPnþ12 ð1Þ ¼ cð2n þ 25Þ!!

2 4 6 ... 2n ¼ vn : 3 5 7 ... ð2n þ 1Þ

CðxÞ ¼

Cðx þ n þ 1Þ ; xðx þ 1Þ. . .ðx þ nÞ

n > 0:

ð36Þ

Now with convexity arguments one can easily obtain Euler’s formula which leads to Euler’s definition of the Gamma function: nx n! : n!1 xðx þ 1Þ. . .ðx þ nÞ

CðxÞ ¼ lim

ð37Þ

See [29], noticed by Hardy.

Ó 2010 Springer Science+Business Media, LLC, Volume 32, Number 4, 2010

25

Euler’s formula is not of Wallis’s type, but it can be easily rearranged into the Newman-Schlo¨milch formula 1 Y 1 x x=j cx ¼ xe 1þ e ; ð38Þ CðxÞ j j¼1 c = 0.577215... being the Euler-Mascheroni constant. See [17] for details. Counting the zeros and poles of b and C, we arrive at the Ramanujan formula:

T HEOREM 6 (R amanujan) For every s [ 0 1

bðsÞ ¼ s þ K

n¼1

ð2n 1Þ2 2s

!

¼4

2 Cð3þs 4 Þ ¼ RðsÞ: Cð1þs 4 Þ

As soon as Ramanujan’s formula is found, it is easy to prove it. By Theorem 5, it is sufficient to check that R(s) [ s for big s. This is equivalent to C2 ðs þ 12Þ 1 [s : 4 C2 ðsÞ Stirling’s formula implies Cðs þ 1=2Þ pﬃﬃ 1 1 ¼ s 1 þ þ . . . ; CðsÞ 8s 128s2

ð39Þ

which proves the required inequality if s? +?.

Fermat In 1657, Arithmetica Infinitorum reached Pierre de Fermat in Toulouse, France. Fermat, interested mostly in number theory, didn’t read the book carefully (see [33, pp. xxvixxvii]). However, Fermat challenged Wallis to solve the Diophantine equation x 2 ¼ 1 þ y2 D

ð40Þ

in positive integers x and y if D is not a perfect square. If D = P2 with integer P, then (40) obviously does not have positive integer solutions since x + yP, which is an integer factor of x2 - y2P2, cannot divide 1. I omit the details of the initial misunderstanding of this problem on the part of Brouncker and Wallis. (They may be found in [32] and in [6].) Instead, I mention a mystery here. Although Fermat never looked in [35, 191], one of his two challenges was the problem (40), which can be solved by the method presented in [35, 191]. In the fifth century BC, the Pythagorean Hippasus of Metapontum solved an important geometry problem. Namely, he showed that if AB?AD , then x1 = |AB| = |AD| and x0 = |BD| are not commensurable. Hippasus’s geometrical construction is remarkably similar to the construction of regular continued fractions (see Fig. 1). Indeed, x0 [ x1 [ x2 = |ED|, where E is defined so that |AB| = |BE|. Computations with angles in DABE; DAEF and DFED show that |AF| = |FE| = |ED|. Hence x0 ¼ 1 x1 þ x2 ; x1 ¼ 2 x2 þ x3 ; jA1 Dj ¼ x3 \x2 :

26

THE MATHEMATICAL INTELLIGENCER

Figure 1. x1 = 2x2 + x3.

Observing that DABD DEFD, we have x2 = 2 x3 + x4. The construction can now proceed by induction, and it will never stop (notice that An never equals D). The result is that x0/x1 can be represented by an infinite continued fraction pﬃﬃﬃ x0 1 1 1 1 2¼ ¼1þ : ð41Þ 2 þ 2 þ 2 þ 2 þ... x1 Since rational numbers are the values of finite regular continued fractions and the development into pﬃﬃaﬃ regular continued fraction is unique, this shows that 2 ¼ jBDj= jADj is an irrational number. In the 17th century, Descartes’s method of coordinates was very popular, and the above arguments make a good illustration of its algebraic nature. As we see later, Brouncker translated Descartes’s paper on musical scales into English and even wrote an addendum to it [1]. Taking for granted Brouncker’s skills demonstrated in the proof of (1) and his interest in Descartes, it is natural to assume that Brouncker knew the periodic continued fraction (41). Let D = 2. The first solutions to (40) can be found by inspection: x ¼ 3 17 99 ð42Þ y ¼ 2 12 70 To begin with, let us observe that equation (40), at least formally, looks very much like equation (26), solved by Brouncker to find his formula at Wallis’s request. This observation hints that continued fractions can pﬃﬃﬃ possibly be used here too. The table of convergents to 2 is this: 1

2

1 1 ; ; 0 1

2

2

3 7 ; ; 2 5

2

2

17 ; 12

41 ; 29

2

2

...

99 239 ; ; . . .: 70 169

ð43Þ

It is easy to see that the quotients x/y from (42) are the first odd convergents in (43). Using (17), we easily find the next pair in (43): x ¼ 2 239 þ 99 ¼ 577 y ¼ 2 169 þ 70 ¼ 408 and by a direct calculation obtain that 1 þ 2 4082 ¼ 332929 ¼ 5772 :

The only conclusion which one may derive from this is that the solutions to (40), at least for D = 2, are given bypthe ﬃﬃﬃﬃ numerators and denominators of odd convergents to D. There is no direct evidence that Brouncker argued this way. However, the form of the solution he sent to Wallis (see [32, pp. 321–322]) indicates that most likely he found it following arguments similar to those he used to prove (1): 1 5 29 2 Q : 2 5 ¼ 12; 12 5 ¼ 70 ; 70 5 ¼ 408. . . 1 6 35 ð44Þ 1 5 29 169 2Q:25 5 5 5 ...: 1 6 35 204

For first values of n, the combination in the third parentheses is P2n1 P2n3 2Q2n1 Q2n3 ¼ 3:

ð47Þ

Compare, by the way, (47) with (26). So, we may incorporate (47) into the induction hypotheses and obtain that P2nþ1 P2n1 2Q2nþ1 Q2n1 ¼ ð6P2n1 P2n3 ÞP2n1 2ð6Q2n1 Q2n3 ÞQ2n1

ð45Þ

To break the code p ofﬃﬃﬃ(45), let Qn be the denominator of the nth convergent to 2 . Then Q1 = 2, Q3 = 12, Q5 = 70, .... Clearly, (44) relates Qn to Qnþ2 ¼ 2. Then (45) represents the solutions y as partial products of the infinite product of Wallis’s type: Q1

2 P2nþ1 2Q22nþ1 ¼ ð6P2n1 P2n3 Þ2 2ð6Q2n1 Q2n3 Þ2 ¼ 1 þ 36 12ðP2n1 P2n3 2Q2n1 Q2n3 Þ:

¼ 6 ðP2n1 P2n3 2Q2n1 Q2n3 Þ ¼ 3; which completes the construction. For D = 3, Brouncker gives the following solution: 3Q:

1 3 11 41 1 3 3 3 3 ... : 1 4 15 56

We easily find that

Q3 Q5 Q7 Q9 ... : Q1 Q3 Q5 Q7

The repeating constant 5 in (45) is explained by an elementary lemma.

pﬃﬃﬃ 1 1 1 1 1 1 ; 3¼1þ 1 þ 2 þ 1 þ 2 þ 1 þ 2 þ... and the convergents with odd indexes x¼ y¼

L EMMA 7 The recurrence Qn+2 = 6Qn - Qn-2 holds for

2 7 26 97 1 4 15 56

ð48Þ

n>1 .

again satisfy the equation x2 - y2D = 1. In this case,

P ROOF .

pﬃﬃﬃ Qn 1 1 ¼4 ! 2 þ 1 3 ¼ 3:732050807568877. . . 44... Qn2

8 > < Qnþ2 ¼ 2Qnþ1 þ Qn þ 2Qnþ1 ¼ 4Qn þ 2Qn1 > : Qn ¼ 2Qn1 Qn2

ð46Þ

The proof follows by adding the three formulas in (46). Adding the first two equations in (46) results in Qn+2 = 5Qn + 2Qn-1, which together with Lemma 7 imply that 5 \ Qn+2/Qn \ 6, as is clearly indicated in (45). Now, Lemma 7 hints that Qnþ2 1 1 1 1 ¼6 ! a; 6 6 6 6 Qn where a [ (5, 6) is the solution to the quadratic equation pﬃﬃﬃ 1 X ¼ 6 ; that is, X ¼ 3 þ 2 2 ¼ 5:82842712474619. . .: X Notice that 3 = x and 2 = y is the minimal solution to (40) with D = 2, whereas the decimal values of the fractions in (45) are 5 5 ¼ 5:83. . . ; 6

29 5 ¼ 5:82857. . . ; 35

169 5 ¼ 5:828431. . .: 204

Using pﬃﬃﬃ Lemma 7, we now can prove that odd convergents to 2 give solutions to equation (40) if D = 2. Let us assume that this is true for all indexes 2k - 1 with k 6 n. Then, by Lemma 7

and 3 11 41 3 ¼ 3:75 ; 3 ¼ 3:7333. . . ; 3 ¼ 3:73214285714. . .; 4 15 56 since Qn+2 = 4Qn - Qn-2. For D = 7, this law must be modified, since x/y = 3/1 = P1/Q1 is not a solution to x2 - y2D = 1. However, P3/Q3 = x/y = 8/3 is a solution. In general, if x1, y1 is a solution to (40), then xn and yn in pﬃﬃﬃﬃ pﬃﬃﬃﬃ ð49Þ xn þ yn D ¼ ðx1 þ y1 DÞn pﬃﬃﬃﬃ are also solutions. Indeed, since D is irrational, (49) is still valid with + replaced by -. Then pﬃﬃﬃﬃ pﬃﬃﬃﬃ xn2 yn2 D ¼ ðx1 þ y1 DÞn ðx1 y1 DÞn ¼ ðx12 y12 DÞn ¼ 1 : These formulas however are not so convenient for practical computations of the solutions starting with the minimal one. Here is a simple theorem on continued fractions which solves this problem. We put x0 = 1, y0 = 0, which is also a solution to (40).

T HEOREM 8 The solutions fðxn ; yn Þgn>1 to equation (40) satisfy xnþ1 ¼ ð2x1 Þxn xn1 ; ynþ1 ¼ ð2x1 Þyn yn1 ;

Ó 2010 Springer Science+Business Media, LLC, Volume 32, Number 4, 2010

ð50Þ

27

and the fractions fyn =xn gn>0 are the convergents to the continued fraction 1 y1 1 1 1 pﬃﬃﬃﬃ ¼ : D x1 2x1 2x1 2x1 ...

pﬃﬃﬃﬃ D is irrational. Then, by the first equation of (52), Pn Qn1 ¼ kn þ : Qn Qn

ð51Þ

ð54Þ

Let [x] be the greatest integer not exceeding x. Then, by (53) and (54), we obtain that

pﬃﬃﬃﬃ Pn kn ¼ ¼ ½ D: ð55Þ Qn

P ROOF . By (49) for n = 0, 1, ... xnþ1 ¼ x1 xn þ y1 Dyn ;

since

ynþ1 ¼ y1 xn þ y1 yn :

Iterating these formulas, we obtain

Therefore, the problem of finding the minimal odd n such that Pn2 Q2n D ¼ 1 reduces to the search of the minimal odd n, satisfying pﬃﬃﬃﬃ ð56Þ Pn ¼ ½ DQn þ Qn1 :

xnþ1 ¼ x1 xn þ y12 Dxn1 þ x1 y1 yn1 D ¼ x1 xn þ x12 xn1 þ x1 y1 yn1 D xn1 ¼ x1 xn þ x1 ðx1 xn1 þ y1 Dyn1 Þ xn1 ¼ ð2x1 Þxn xn1 ;

Application of (50), as in the case of D = 2, leads to the same conclusion. Brouncker’s formulas (50) conveniently lists infinitely many solutions, provided one is known. By (51), y1 divides every yn. The problem of the minimal solution to (40), for a given value D, can be solved similarly to the proof of formula (26). Let us observe that if Pn2 Q2n D ¼ 1, then, by (20), the integer n is odd. Hence, Pn2 Q2n D ¼ 1 if and only if (see (18))

Later, Euler [13] proved that the minimal solution to (56) (equivalently to (40)) in thepform ﬃﬃﬃﬃ of (Pk, Qk) exists if the regular continued fraction of D is periodic with period d. If d is even, then k = d - 1. For instance, if D = 7, then d = 4, implying that (P3, Q3) is the minimal solution. If d is odd, then k = 2d - 1. For instance, if D = 2, then d = 1, and, therefore, (P1, Q1) is the minimal solution. This paper, written in 1765, appeared only in 1767. After that, Lagrange 2 2 proved in [23] and [24] pﬃﬃﬃﬃ that if x - y D = 1, then x/y is an odd convergent to D and also that the regular continued fraction of any quadratic irrationality is periodic. This completed the proof that Brouncker’s method lists all the solutions, as well as that each Pell’s equation has infinitely many solutions. Later, Euler presented his results in his book [15], which was translated into French by Lagrange. Lagrange included his theory as an addendum to this translation. In contrast to the case of Wallis’s product, this time Wallis, taking Brouncker’s hints, found his own solution to Fermat’s problem. It is now called the English method; see [6] and [32] for details. In spite of his comments on continued fractions in [35, §191], Wallis didn’t follow the lines indicated above. It was Euler who named equation (40) after Pell, in his first papers on this subject (see, for instance, [8, §15] or [7]). Most likely, the correspondence between Brouncker and Wallis was unavailable in St. Petersburg. Therefore, Whitford’s opinion [36] that Euler mentioned Pell, because Pell included the Diophantine equation x = 12y2 - z2 in the English translation of Rahn’s Algebra [28, p.134], looks very convincing. There is some evidence that the first appearance of this problem goes back to Archimedes’s Cattle Problem, which reduces to Pell’s equation

Pn2 Q2n D ¼ Pn Qn1 Pn1 Qn

x 2 4729494y2 ¼ 1:

which proves the first identity in (50). Similar calculations prove the second. Now, (50) implies that yn/xn are the convergents to the continued fraction (51), which converge pﬃﬃﬃﬃ to 1= D as n ! 1, by 1 yn 1 1 pﬃﬃﬃﬃ pﬃﬃﬃﬃ ¼ pﬃﬃﬃﬃ !0: ¼ x D xn ðxn þ yn DÞ xn ðx1 þ y1 DÞn n Analyzing the correspondence of Wallis and Brouncker on Fermat’s question, Whitford presents in [36, p.52] exactly the same formulas as in (50). This makes it natural to conjecture that Brouncker, in fact, used Theorem 8 for his solution to Fermat’s question. Theorem 8 shows that Brouncker’s method works not only for particular values of D such as D = 2, 3, 7, but also for any D, provided a minimal solution (x1, y1) exists. Indeed, we may write DQ:

y1

y2 y3 y4 y5 ... : y1 y2 y3 y4

By (50), yn+1 = x1yn + y1xn, which implies that pﬃﬃﬃﬃ ynþ1 xn ¼ x1 þ y1 ! x1 þ y1 D: yn yn

or equivalently,

By (18), the greatest common divisor of Pn and Qn is 1. It follows that Pn ¼ kn Qn þ Qn1 ; ð52Þ Qn D ¼ kn Pn þ Pn1 :

Due to Hippasus’s problem and Brouncker’s solution of Fermat’s Challenge, this may look reasonable. But the minimal solution to the Pell’s equation has thousands of places. Therefore, it is not clear how Archimedes could have written the minimal positive solution himself. See [6], [21, p.3], [34] and [36] for the history of Pell’s equation and a recent paper [25] for history and applications.

By Theorem 1 (recall that n is odd), Pn pﬃﬃﬃﬃ P1 pﬃﬃﬃﬃ 0\ D\ D\1; Qn Q1

The Weber-Fechner law says that a human being’s response to physical phenomena obeys a logarithmic law (see [26,

Pn ðPn Qn1 Þ ¼ Qn ðQn D Pn1 Þ:

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THE MATHEMATICAL INTELLIGENCER

Harmony ð53Þ

pp. 111–112). By converting exponential growth to a linear scale, this ability makes people less sensitive to changes in the outside world and reduces their reactions to the most significant ones. We cannot control too many parameters at the same time, and the Weber-Fechner law reflects this fact. In particular, our ear compares not the heights of pitches but the logarithms of their quotients. The main problem of a musical scale is to arrange a system of quotients of pitches creating an impression of harmony under the logarithmic law of response. In practice, this means finding a step of linear scale such that the logarithms of the quotients chosen can be well approximated by integer multiples of this step. The human ear can normally hear pitches in the range 20 Hz to 20 kHz. Notice that 20 24 and 20000 214 . Applying the logarithms, we see that 29 = 512 Hz corresponds to the center of the logarithmic scale. If a string of length l creates a pitch of the frequency x = 512 Hz, then the string of length l/2 doubles the frequency to 2x. The logarithmic base a is chosen so as to normalize the following number as a unit: 2x ¼ loga 2 ¼ 1; loga x which implies that a = 2. The ratio 2x : x = 2 determines the interval (x, 2x) called the octave. The ratio 3x/2 : x corresponding to the half of the interval (x, 2x) (the frequency 3x/2 is generated by the string of length 2l/3), is called the perfect fifth. The ear hears this ratio as 3 log2 x : x ¼ log2 3 1: 2 Our ear hears the perfect fifth the best, and, therefore, it must be approximated the best possible way. The convergents to the continued fraction log2 3 1 ¼ 0; 584962500721. . . 1 1 1 1 1 1 1 1 1 ¼ 1 þ 1 þ 2 þ 2 þ 3 þ 1 þ 5 þ 2 þ 23 þ...

approximation by a uniform scale is completely determined by those for log2 3/2, and log2 5/4 and cannot exceed the maximum of the two. Now, 5 ¼ 0:32192809488736234787. . . log2 4 1 1 1 1 1 1 ¼ 3 þ 9 þ 2 þ 2 þ 4 þ 6 þ... shows that 1/3 = 4/12 is a convergent to log2 5/4. This guarantees that the equal temperament system of 12 uniform semitones gives two good rational approximations 7/12 and 4/12 to two basic intervals 3/2 and 5/4, and, hence, to all seven consonant intervals. See [4] and [20] for a more detailed discussion. In the addendum to [1], which was published two years before his first great discovery in continued fractions, Brouncker analyzed the scale of 17 equal semitones from the point of view of the Descartes theory. He didn’t apply continued fractions then, but, as is clear from the above, continued fractions are important for the analysis of harmony. Simple calculations show that log2 3=2 ¼

7 10 þ 0:00162. . . ¼ 0:00327; 12 17

which implies that the scale of 17 equal semitones doubles the error of approximation for the perfect fifth compared with the scale of 12 equal semitones. As to the approximation of log2 5/4, the 17-based scale also almost doubles the error compared with the 12-based scale: log2 5=4 ¼

4 5 0:011. . . ¼ þ 0:027. . . : : 12 17

In my opinion, it was the study of problems of musical scales which finally led Brouncker to positive continued fractions. Therefore, it looks like Wallis’s question on the existence of other arithmetic formulas for p, similar to his infinite product, fell on ground carefully prepared by Brouncker.

make the series 1 3 7 24 1; ; ; ; ; ... : 2 5 12 41

Epilogue ð57Þ

Approximations 1 and 1/2 are too crude. Approximation 3/ 5 is used in Eastern music. Approximation 7/12 is the best. It divides the musical scale into 12 semitones, and 7 such semitones correspond to the fifth. If the interval between two notes is a ratio of small integers, these two notes are called consonant. Otherwise, they are called dissonant.3 This happens again due to the restricted abilities of human beings. Computers would have another opinion on this matter. There are seven intervals which are commonly considered to be consonant (they had already appeared in Descartes’s table; see [1, p.13]). The most important among them are 3/2 (the perfect fifth) and 5/4 (the major third), since the binary logarithms of other consonant intervals are linear combinations of 1, log2 3/2 and log2 5/4 with the coefficients in {0, 1, -1}. Hence, the error of the 3

I give another citation from the astrological site: ‘‘Jupiter, the King of the Gods, is the ruler of Sagittarius. In Astrology, Jupiter is a planet of plenty. It is tolerant and expansive. The first of the social planets, Jupiter seeks insight through knowledge. Some of this planet’s keywords include morality, gratitude, hope, honor and the law. Jupiter is a planet of broader purpose, reach and possibility’’. Things changed for Brouncker when in 1658 Cromwell died, and Brouncker started to move gradually from the protection of Saturn to that of Jupiter . Already, in 1660, he was elected as a member of Parliament. In 1662 Brouncker was promoted by King Charles II to an important position. In 1663, the Royal Society of London was created, and Lord Brouncker was nominated as its first president. See [2] for other details of Brouncker’s remarkable career. However, things changed for him in mathematics, too. He left forever a very fruitful and important area that he had discovered. Later,

Euler developed an original theory of sound classification by ‘‘degree of pleasure’’ in his monograph [9].

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Euler developed this area into the theory of special functions. Stieltjes introduced his theory of moments. One may observe that the proof of Brouncker’s formula is the starting point of Stieltjes’s theory. Chebychev discovered orthogonal polynomials. Brouncker’s polynomials fPn gn>1 turned out to be orthogonal with respect to the weight 1 1 þ it 4 dl ¼ 3 C dt; 8p 4 interestingly related to the Gamma function. These polynomials are placed at the very center of the family of Wilson polynomials, which include all the classical orthogonal polynomials. More can be found in my book [20]. I have a question for the reader. Which protection do you prefer, Saturn’s or Jupiter’s ? Please think before giving an answer. Brouncker enjoyed them both, but which was better? Where is the yacht constructed by Brouncker for King Charles II? Nobody knows, but his formula and two Great Theorems are in front of you.

[13] L. Euler. ‘‘De usu novi algorithmi in Problemate Pelliano solvendo’’, Novi Commentarii academiae scientiarum Petropolitane., 11, (1767), pp. 29–66; reprinted in Opera Omnia, Ser. 1, Vol. 3, pp. 73-111; E323. [14] L. Euler. ‘‘Integral Calculus’’, Vol. I (Impeofis Academiae Imperialis Scientiarum, St. Petersburg, 1768). Russian translation: Moscow, GITTL, 1956; E342. [15] L. Euler. ‘‘Vollstandige Anleitung zur Algebra’’ (Leipzig, 1770); E387–E388. [16] A. Van Helden. ‘‘Huygens’s Ring, Cassini’s Division and Saturn’s Children’’ (Smithsonian Institution Libraries, Washington D. C., 2006), http://www.sil.si.edu/silpublications/dibner-library-lectures/ 2004-VanHelden/2004_VanHelden.pdf. [17] J. Havil. ‘‘Gamma. Exploring Euler’s Constant’’ (Princeton University Press, Princeton, 2003). [18] C. Huygens. ‘‘De circuli magnitudine inventa. Accedunt eiusdem problematum quorundam illustrium constructiones’’ (J. and D. Elzevier, Leiden, 1654). [19] S. Khrushchev. ‘‘A recovery of Brouncker’s proof for the quadrature continued fraction’’, Publicacions Matematiques 50 (2006), pp. 3–42.

REFERENCES

[1] W. Brouncker. ‘‘Animadversions upon the Musick-Compendium of Descartes’’ (London, 1653).

[20] S. Khrushchev. ‘‘Orthogonal Polynomials and Continued Fractions: From Euler’s point of view’’ (Cambridge University Press, Cambridge, 2008).

[2] J. J. O’Connor and E. F. Robertson. ‘‘William Brouncker’’, (The

[21] H. Koch. ‘‘Number Theory. Algebraic Numbers and Functions’’

MacTutor History of Mathematics Online Archive, 2002). http://

(AMS, Providence, 2000). [22] F. D. Kramar. Integration Methods of John Wallis, in: Historico-

www.history.mcs.st-and.ac.uk [3] M. Jesseph Douglas. ‘‘Squaring the Circle: The War between Hobbes and Wallis’’ (University of Chicago, Chicago, 2000). [4] E. G. Dunne and M. McConnell. Pianos and continued fractions, Mathematics Magazine 72:2 (1999), pp. 104–115. [5] J. Dutka. ‘‘Wallis’s product, Brouncker’s continued fraction, and Leibniz’s series’’, Archive for History of Exact Sciences 26:2 (1982), pp. 115–126.

mathematical Research 14, pp. 11–100, in Russian (FizMatGiz, Moscow, 1961). [23] J. L. Lagrange ‘‘Solution d‘un proble´me d’arithme´tique’’, Miscellanea Taurinensia, 4 (1766–1769); = Oeuvres I pp. 671–731 Paris, Gauthier-Villars, MDCCCLXVII. [24] J. L. Lagrange. ‘‘Sur la solution des proble´mes inde´termine´s du second degre´’’, Me´moires de l’Acade´mie royale des sciences et

[6] H. M. Edwards. ‘‘Fermat’s Last Theorem: A Generic Introduction

belles-lettres (de Berlin), anne´e 1767; Oeuvres II, pp 377–538,

to Algebraic Number Theory’’ (Springer, New York, 1977). [7] L. Euler. ‘‘Euler’s letter to Goldbach on August 10, 1730’’, OO723

Paris, Gauthier-Villars, MDCCCLXVIII.

(in the Euler Archive, http://www.math.dartmouth.edu/*euler/). [8] L. Euler. ‘‘De solutione problematum diophanteorum per numeros integros’’, Commentarii academiae scientiarum Imperials Petropolitanae VI (1738), 175–188 (presented on May 29, 1733); reprinted in Opera Omnia, Ser. 1, Vol 2, pp. 6–17; E029 [9] L. Euler. ‘‘Tentamen novae theoriae musicae ex certissismis harmoniae principiis dilucide expositae’’ (Petropoli, Academiae Scientiarum, 1739); reprinted in Opera Omnia, Ser. 3, Vol. 1, pp. 197–427; E033. [10] L. Euler. ‘‘De fractionibus continuus, dissertatio’’, Commentarii

American Mathematical Society 49:2 (2002), pp. 182–192. [26] E. Maor. ‘‘e: The Story of a Number.’’ (Princeton University Press, Princeton, 1994). [27] A. S. Posamentier and I. Lehmann. ‘‘A Biography of the World’s Most Mysterious Number’’ (Prometheus Books, New York, 2004). [28] J. H. Rahn. ‘‘An introduction to algebra, translated out of the High Dutch into English by Thomas Brancker, M.A. Much altered and augmented by D.P.’’ (Moses Pitt, London, 1668).

Academiae Scientiarum Imperials Petropolitane IX(1744) for 1737,

[29] G. H. Hardy, Seshu Aiyar, B. M. Wilson (eds.). ‘‘Collected Papers

98–137 (presented on February 7, 1737); reprinted in Opera Omnia,

of Srinivasa Ramanujan’’ (Chelsea Publishing Co./American

Ser. 1, Vol. 14, pp. 187–216; E071; translated into English: Mathematical Systems Theory (1985) 4:18. [11] L. Euler. ‘‘Introductio in analysin infinitorum’’ (Apud Marcum – Michaelem Bousquet & Socios, Lausanne, 1748); E101. [12] L. Euler. ‘‘De fractionibus continuus, observationes’’, Commentarii Academiae Scientiarum Imperials Petropolitane XI (1750b) for 1739, pp. 32–81 (presented on January 22, 1739); reprinted in Opera Omnia, Ser. I, Vol. 14, pp. 291–349; E123.

30

[25] H. W. Lenstra Jr. ‘‘Solving the Pell equation’’, Notices of the

THE MATHEMATICAL INTELLIGENCER

Mathematical Society, 2000). [30] Sir Reginald R. F. D., Palgrave, K.C.B.. ‘‘Oliver Cromwell H. H., the Lord Protector and the Royalist Insurrection Against His Government of March, 1655’’ (Sampson, Low, Marston and Co. Ltd., London, 1903). [31] J. A. Stedall ‘‘Catching proteus: the collaborations of Wallis and Brouncker I. Squaring the circle’’, Notes and Records of the Royal Socitey of London 54:3 (2000), pp. 293–316.

[32] J. A. Stedall ‘‘Catching proteus: the collaborations of Wallis and

[35] J. Wallis. ‘‘Arithmetica Infinitorum’’ (Typis Leon: Lichfield Acade-

Brouncker II. Number Problems’’, Notes Rec. R. Soc. London 54:3 (2000), pp. 317–331.

mia Typographi, Icnpenfis Tho. Robinson, London, 1656). [36] E. E. Whitford. ‘‘The Pell Equation’’ (College of the City of New York,

[33] J. A. Stedall. (English translation) The Arithmetics of Infinitesimals:

New York, 1912), available online in the University of Michigan

John Wallis, 1656’’ (Springer-Verlag, New York, 2004).

Historical Math Collection (http://www.hti.umich.edu/u/umhistmath/).

[34] I. Vardi. ‘‘Archimedes’ Cattle Problem’’, The American Mathematical Monthly 105:4 (1998), pp. 305–319.

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Mathematical Entertainments Michael Kleber and Ravi Vakil, Editors

Tilings* FEDERICO ARDILA , AND RICHARD P. STANLEYà This column is a place for those bits of contagious

The region must be covered entirely without any overlap. It is allowed to shift and rotate the seven pieces in any way, but each piece must be used exactly once One could start by observing that some of the pieces fit nicely in certain parts of the region. However, the solution can really only be found through trial and error.

mathematics that travel from person to person in the 4

community, because they are so elegant, surprising, or appealing that one has an urge to pass them on.

5

6

3 2

Contributions are most welcome. 1 7

C

onsider the following puzzle. The goal is to cover the region For that reason, even though this is an amusing puzzle, it is not very intriguing mathematically. This is, in any case, an example of a tiling problem. A tiling problem asks us to cover a given region using a given set of tiles completely and without any overlap. Such a covering is called a tiling. Of course, we will focus our attention on specific regions and tiles that give rise to interesting mathematical problems. Given a region and a set of tiles, there are many different questions we can ask. Some of the questions that we will address are the following:

using the following seven tiles.

1

2

4

3

6

7

5

• • • • • • • • •

Is there a tiling? How many tilings are there? About how many tilings are there? Is a tiling easy to find? Is it easy to prove that a tiling does not exist? Is it easy to convince someone that a tiling does not exist? What does a ‘‘typical’’ tiling look like? Are there relations among the different tilings? Is it possible to find a tiling with special properties, such as symmetry?

â

Please send all submissions to the Mathematical Entertainments Editor, Ravi Vakil, Stanford University, Department of Mathematics, Bldg. 380, Stanford, CA 94305-2125, USA e-mail: [email protected]

Is There a Tiling? From looking at the set of tiles and the region we wish to cover, it is not always clear whether such a task is even

* This paper is based on the second author’s Clay Public Lecture at the IAS/Park City Mathematics Institute in July, 2004 Supported by the Clay Mathematics Institute à Partially supported by NSF grant #DMS-9988459, and by the Clay Mathematics Institute as a Senior Scholar at the IAS/Park City Mathematics Institute

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THE MATHEMATICAL INTELLIGENCER 2010 SPRINGER SCIENCE+BUSINESS MEDIA, LLC

possible. The puzzle at the beginning of this article is such a situation. Let us consider a similar puzzle where the tiles, Solomon Golomb’s polyominoes, are more interesting mathematically. This puzzle can be solved in at least two ways. One solution is shown above. A different solution is obtained if we rotate the shaded block by 180. In fact, after spending some time trying to find a tiling, one discovers that these (and their rotations and reflections) are the only two possible solutions. One could also ask whether it is possible to tile two 6 9 5 rectangles using each pentomino exactly once. One way of doing it is shown below. There is only one other such tiling, obtained by rearranging two of the pentominoes; it is a nice puzzle for the reader to find those two tiles.

A pentomino is a collection of five unit squares arranged with coincident sides. Pentominoes can be flipped or rotated freely. The figure shows the 12 different pentominoes. Since their total area is 60, we can ask, for example: Is it possible to tile a 3 9 20 rectangle using each one of them exactly once?

Knowing that, one can guess that there are several tilings of a 6 9 10 rectangle using the 12 pentominoes. However, one might not predict just how many there are. An exhaustive computer search has found that there are 2,339 such tilings.

AUTHORS

......................................................................................................................................................... FEDERICO ARDILA was born and grew up in Bogota´, Colombia. He received his Ph.D. from MIT under the supervision of Richard Stanley. He is an assistant professor at San Francisco State University and an adjunct professor at the Universidad de Los Andes in Bogota´. He studies objects in algebra, geometry, topology and phylogenetics by understanding their underlying combinatorial structure. He leads the SFSU–Colombia Combinatorics Initiative, a research and learning collaboration between students in the United States and Colombia. When he is not at work, you might find him on the fu´tbol field, treasure hunting in little record stores, learning a new percussion instrument, or exploring the incredible San Francisco Bay Area.

RICHARD STANLEY is currently the Norman

Levinson Professor of Applied Mathematics at MIT. His main research interest is combinatorics and its connection with such other areas of mathematics as algebraic topology, commutative algebra and representation theory. He is the author of three books, including Enumerative Combinatorics, Volumes 1 and 2, and over 150 research papers. He has supervised over 50 Ph.D. students and maintains on his web page a dynamic list of exercises related to Catalan numbers, including over 175 combinatorial interpretations. Department of Mathematics Massachusetts Institute of Technology Cambridge, MA 02139 USA e-mail: [email protected]

Department of Mathematics San Francisco State University San Francisco, CA 94132 USA e-mail: [email protected] 2010 Springer Science+Business Media, LLC, Volume 32, Number 4, 2010

33

These questions make nice puzzles, but are not the kind of interesting mathematical problem that we are looking for. To illustrate what we mean by this, let us consider a problem that is superficially somewhat similar, but that is much more amenable to mathematical reasoning. Suppose we remove two opposite corners of an 8 9 8 chessboard, and we ask: Is it possible to tile the resulting figure with 31 dominoes?

Our chessboard would not be a chessboard if its cells were not colored dark and white alternatingly. As it turns out, this coloring is crucial in answering the question at hand. Notice that, regardless of where it is placed, a domino will cover one dark and one white square of the board. Therefore, 31 dominoes will cover 31 dark squares and 31 white squares. However, the board has 32 dark squares and 30 white squares in all, so a tiling does not exist. This is an example of a coloring argument; such arguments are very common in showing that certain tilings are impossible.

cells with a domino; they are also white and dark, respectively. Continue in this way, until the path reaches the second hole of the chessboard. Fortunately, this second hole is white, so there is no gap between the last domino placed and this hole. We can, therefore, skip this hole and continue covering the path with successive dominoes. When the path returns to the first hole, there is again no gap between the last domino placed and the hole. Therefore, the board is entirely tiled with dominoes. We now illustrate this procedure.

What happens if we remove two dark squares and two white squares? If we remove the four squares closest to a corner of the board, a tiling with dominoes obviously exists. On the other hand, in the example below, a domino tiling does not exist, since there is no way for a domino to cover the upper left square

.

A natural variation of this problem is to now remove one dark square and one white square from the chessboard, as shaded above. Now the resulting board has the same number of dark squares and white squares; is it possible to tile it with dominoes? Let us show that the answer is yes, regardless of which dark square and which white square we remove. Consider any closed path that covers all the cells of the chessboard, like the following one.

Now start traversing the path, beginning with the point immediately after the dark hole of the chessboard. Cover the first and second cell of the path with a domino; they are white and dark, respectively. Then cover the third and fourth

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This question is clearly more subtle than the previous one. The problem of describing which subsets of the chessboard can be tiled by dominoes leads to some very nice mathematics. We will say more about this topic in the Section ‘‘Demonstrating That a Tiling Does Not Exist’’ below. Let us now consider a more difficult example of a coloring argument, to show that a 10 9 10 board cannot be tiled with 1 9 4 rectangles.

Giving the board a chessboard coloring gives us no information about the existence of a tiling. Instead, let us use four colors, as shown above. Any 1 9 4 tile that we place on this board will cover an even number (possibly zero) of squares of each color

As we saw earlier, there are 2,339 ways (up to symmetry) to tile a 6 9 10 rectangle using each one of the 12 pentominoes exactly once. It is perhaps interesting that this number is so large, but the exact answer is not so interesting, especially since it was found by a computer search. The first significant result on tiling enumeration was obtained independently in 1961 by Fisher and Temperley [7] and by Kasteleyn [12]. They found that the number of tilings of a 2m 9 2n rectangle with 2mn dominoes is equal to m Y n Y jp kp mn 2 2 þ cos cos : 4 2m þ 1 2n þ 1 j¼1 k¼1

. Therefore, if we had a tiling of the board, the total number of squares of each color would be even. But there are 25 squares of each color, so a tiling is impossible. With these examples in mind, we can invent many similar situations where a certain coloring of the board makes a tiling impossible. Let us now discuss a tiling problem that cannot be solved using such a coloring argument. Consider the region T(n) consisting of a triangular array of n(n + 1)/2 unit regular hexagons.

T(1) T(2)

T(3)

T(4)

Call T(2) a tribone. We wish to know the values of n for which T(n) can be tiled by tribones. For example, T(9) can be tiled as follows.

Here P denotes product, and p denotes 180, so the number above is given by 4mn times a product of sums of two squares of cosines, such as cos

2p ¼ cos 72 ¼ 0:3090169938. . . : 5

This is a remarkable formula! The numbers we are multiplying are not integers; in most cases, they are not even rational numbers. When we multiply these numbers we miraculously obtain an integer, and this integer is exactly the number of domino tilings of the 2m 9 2n rectangle. For example, for m = 2 and n = 3, we get: 46 ðcos2 36 þ cos2 25:71. . . Þ ðcos2 36 þ cos2 51:43. . . Þ ðcos2 36 þ cos2 77:14. . . Þ ðcos2 72 þ cos2 25:71. . . Þ ðcos2 72 þ cos2 51:43. . . Þ ðcos2 72 þ cos2 77:14. . . Þ ¼ 46 ð1:4662. . .Þð1:0432. . .Þð0:7040. . .Þ ð0:9072. . .Þð0:4842. . .Þð0:1450. . .Þ ¼ 281:

Since each tribone covers 3 hexagons, n(n + 1)/2 must be a multiple of 3 for T(n) to be tileable. However, this does not explain why regions such as T(3) and T(5) cannot be tiled. Conway and Lagarias [3, 21] showed that the triangular array T(n) can be tiled by tribones if and only if n = 12k, 12k + 2, 12k + 9 or 12k + 11 for some k 0: The smallest values of n for which T(n) can be tiled are 0, 2, 9, 11, 12, 14, 21, 23, 24, 26, 33 and 35. Their proof uses a certain nonabelian group that detects information about the tiling that no coloring can detect, while coloring arguments can always be rephrased in terms of abelian groups. In fact, it is possible to prove that no coloring argument can establish the result of Conway and Lagarias [16].

Counting Tilings, Exactly Once we know that a certain tiling problem can be solved, we can go further and ask: How many solutions are there?

Skeptical readers with a lot of time to spare are invited to find all domino tilings of a 4 9 6 rectangle and check that there are, indeed, exactly 281 of them. Let us say a couple of words about the proofs of this result. Kasteleyn expressed the answer in terms of a certain Pfaffian, and reduced its computation to the evaluation of a related determinant. Fisher and Temperley gave a different proof using the transfer matrix method, a technique often used in statistical mechanics and enumerative combinatorics. There is a different family of regions for which the number of domino tilings is surprisingly simple. The Aztec diamond AZ(n) is obtained by stacking successive centered rows of length 2, 4, ..., 2n, 2n, . . ., 4, 2, as shown.

AZ(1)

AZ(2)

AZ(3)

AZ(7)

The Aztec diamond AZ(2) of order 2 has the following 8 tilings:

2010 Springer Science+Business Media, LLC, Volume 32, Number 4, 2010

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pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃ ½N T ¼ 42 ¼ 1:189207115. . . For the 2n 9 2n square, the exact formula for the number of tilings is somewhat unsatisfactory, because it does not give us any indication of how large this number is. Fortunately, as Kasteleyn, Fisher and Temperley observed, one can use their formula to show that the number of domino tilings of a 2n 9 2n square is approximately C4n^2, where

Elkies, Kuperberg, Larsen and Propp [6] showed that the number of domino tilings of AZ(n) is 2n(n+1)/2. The following table shows the number of tilings of AZ(n) for the first few values of n 1

2

3

4

5

6

2

8

64

1024

32768

2097152

Since 2(n+1)(n+2)/2/2n(n+1)/2 = 2n+1, one could try to associate 2n+1 domino tilings of the Aztec diamond of order n + 1 to each domino tiling of the Aztec diamond of order n, so that each tiling of order n + 1 occurs exactly once. This is one of the four original proofs found in [6]; there are now around 12 proofs of this result. None of these proofs is quite as simple as the answer 2n(n+1)/2 might suggest.

Counting Tilings, Approximately Sometimes we are interested in estimating the number of tilings of a certain region. In some cases, we will want to do this, because we are not able to find an exact formula. In other cases, somewhat paradoxically, we might prefer an approximate formula over an exact formula. A good example is the number of domino tilings of a rectangle. We have an exact formula for this number, but this formula does not give us any indication of how large this number is. For instance, since Aztec diamonds are ‘‘skewed’’ squares, we might wonder: How do the number of domino tilings of an Aztec diamond and a square of about the same size compare? After experimenting a bit with these shapes, one notices that placing a domino on the boundary of an Aztec diamond almost always forces the position of several other dominoes. This almost never happens in the square. This might lead us to guess that the square should have more tilings than the Aztec diamond. To try to make this idea precise, let us make a definition. If a region with N squares has T tilings, we will say that it pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ has ½N T degrees of freedom per square. The motivation, loosely speaking, is the following: If each square could decide independently how it would like to be covered, and pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ it had ½N T possibilities to choose from, then the total number of choices would be T. The Aztec diamond AZ(n) consists of N = 2n(n + 1) squares, and it has T = 2n(n+1)/2 tilings. Therefore, the number of degrees of freedom per square in AZ(n) is: 36

THE MATHEMATICAL INTELLIGENCER

C ¼ e G=p ¼ 1:338515152. . .: Here G denotes the Catalan constant, which is defined as follows: 1 1 1 G ¼ 1 2 þ 2 2 þ 3 5 7 ¼ 0:9159655941. . .: Thus, our intuition was correct. The square board is ‘‘easier’’ to tile than the Aztec diamond, in the sense that it has approximately 1.3385. . . degrees of freedom per square, while the Aztec diamond has 1.1892. . ..

Demonstrating That a Tiling Does Not Exist As we saw in the Section entitled ‘‘Is There a Tiling?’’, there are many tiling problems where a tiling exists, but finding it is a difficult task. However, once we have found it, it is very easy to demonstrate its existence to someone: We can simply show them the tiling! Can we say something similar in the case where a tiling does not exist? As we also saw in the Section entitled ‘‘Is There a Tiling?’’, it can be difficult to show that a tiling does not exist. Is it true, however, that if a tiling does not exist, then there is an easy way of demonstrating that to someone? In a precise sense, the answer to this question is almost certainly no in general, even for tilings of regions using 1 9 3 rectangles [1]. Surprisingly, though, the answer is yes for domino tilings! Before stating the result in its full generality, let us illustrate it with an example. Consider the following region consisting of 16 dark squares and 16 white squares. (The shaded cell is a hole in the region.)

One can use a case-by-case analysis to become convinced that this region cannot be tiled with dominoes. Knowing this, can we find an easier, faster way to convince someone that this is the case? One way of doing it is the following. Consider the 6 dark squares marked with a •. They are adjacent to a total of 5 white squares, which are marked with an *. We would need 6 different tiles to cover the 6 marked dark squares, and each one of these tiles would have to cover one of the 5 marked white squares. This makes a tiling impossible.

* *

* *

*

Philip Hall [10] showed that in any region that cannot be tiled with dominoes, one can find such a demonstration of impossibility. More precisely, one can find k cells of one color which have fewer than k neighbors. Therefore, to demonstrate to someone that tiling the region is impossible, we can simply show them those k cells and their neighbors! Hall’s statement is more general than this and is commonly known as the marriage theorem. The name comes from thinking of the dark cells as men and the white cells as women. These men and women are not very adventurous: They are only willing to marry one of their neighbors. We are the matchmakers; we are trying to find an arrangement in which everyone can be happily married. The marriage theorem tells us exactly when such an arrangement exists.

Tiling Rectangles with Rectangles One of the most natural tiling situations is that of tiling a rectangle with smaller rectangles. We now present three beautiful results of this form. The first question we wish to explore is: When can an m 9 n rectangle be tiled with a 9 b rectangles (in any orientation)? Let us start this discussion with some motivating examples. Can a 7 9 10 rectangle be tiled with 2 9 3 rectangles? This is clearly impossible, because each 2 9 3 rectangle contains 6 squares, while the number of squares in a 7 9 10 rectangle is 70, which is not a multiple of 6. For a tiling to be possible, the number of cells of the large rectangle must be divisible by the number of cells of the small rectangle. Is this condition enough? Let us try to tile a 17 9 28 rectangle with 4 9 7 rectangles. The argument of the previous paragraph does not apply here; it only tells us that the number of tiles needed is 17. Let us try to cover the left-most column first.

Our first attempt failed. After covering the first 4 cells of the column with the first tile, the following 7 cells with the second tile, and the following 4 cells with the third tile, there is no room for a fourth tile to cover the remaining two cells. In fact, if we manage to cover the 17 cells of the first column with 4 9 7 tiles, we will have written 17 as a sum of 4 s and 7 s. But it is easy to check that this cannot be done, so a tiling does not exist. We have found a second reason for a tiling not to exist: It may be impossible to cover the first row or column, because either m or n cannot be written as a sum of a s and b s. Is it then possible to tile a 10 9 15 rectangle using 1 9 6 rectangles? In fact, 150 is a multiple of 6, and both 10 and 15 can be written as a sum of 1 s and 6 s. However, this tiling problem is still impossible! The full answer to our question was given by de Bruijn and by Klarner [4, 13]. They proved that an m 9 n rectangle can be tiled with a 9 b rectangles if and only if: • mn is divisible by ab, • the first row and column can be covered; i.e., both m and n can be written as sums of a s and b s, and • either m or n is divisible by a, and either m or n is divisible by b. Since neither 10 nor 15 is divisible by 6, the 10 9 15 rectangle cannot be tiled with 1 9 6 rectangles. There are now many proofs of de Bruijn and Klarner’s theorem. A particularly elegant one uses properties of the complex roots of unity [4, 13]. For an interesting variant with fourteen (!) proofs, see [20]. The second problem we to discuss is the following. pﬃﬃwish ﬃ Let x [ 0, such as x ¼ 2. Can a square be tiled with finitely many rectangles similar to a 1 9 x rectangle (in any orientation)? In other words, can a square be tiled with finitely many rectangles, all of the form a 9 ax (where a may vary)? For example, for x = 2/3, some of the tiles we can use are the following.

1.5

2 1

4

2π

3 6

3π

They have the same shape, but different sizes. In this case, however, we only need one size, because we can tile a 2 9 2 square with six 1 9 2/3 rectangles. 1

1 x = 2/3 2/3 2/3

?

For reasons which will become clear later, we point out that x = 2/3 satisfies the equation 3x - 2 = 0. Notice also that a similar construction will work for any positive rational number x = p/q. 2010 Springer Science+Business Media, LLC, Volume 32, Number 4, 2010

37

Let us try to construct a tiling of a square with similar rectangles of at least two different sizes. There is a tiling approximately given by the picture below. The rectangles are similar because 0.7236. . ./1 = 0.2/0.2764. . ... 1

.7236...

.2764... 1/5

How did we find this configuration? Suppose that we want to form a square by putting five copies of a rectangle in a row, and then stacking on top of them a larger rectangle of the same shape on its side, as shown. Assume that we know the square has side length 1, but we do not know the dimensions of the rectangles. Let the dimensions of the large rectangle be 1 9 x. Then the height of each small rectangle is equal to 1 - x. Since the small rectangles are similar to the large one, their width is x(1 - x). Sitting together in the tiling, their total width is 5x(1 - x), which should be equal to 1. Therefore, the picture above is a solution to our problem if x satisfies the equation 5x(1 - x) = 1, which we rewrite as 5x2 - 5x + 1 = 0. One value of x that satisfies this equation is pﬃﬃﬃ 5þ 5 ¼ 0:7236067977. . .; x¼ 10 giving rise to the tiling illustrated above. But recall that any quadratic polynomial has two zeros; the other one is pﬃﬃﬃ 5 5 x¼ ¼ 0:2763932023. . .; 10 and it gives rise to a different tiling that also satisfies the conditions of the problem. It may be unexpected that our tiling problem has a solution for these two somewhat complicated values of x. In fact, the situation can get much more intricate. Let us find a tiling using 3 similar rectangles of different sizes. 1

x = .5698...

.4302... .2451...

.7549...

Say that the largest rectangle has dimensions 1 9 x. Imitating the previous argument, we find that x satisfies the equation x 3 x 2 þ 2x 1 ¼ 0:

38

THE MATHEMATICAL INTELLIGENCER

One value of x that satisfies this equation is x ¼ 0:5698402910. . . : For this value of x, the tiling problem can be solved as above. The polynomial above has degree three, so it has two are pﬃﬃﬃﬃﬃﬃﬃother zeros. They p ﬃﬃﬃﬃﬃﬃﬃ approximately 0:215 þ 1:307 1 and 0:215 1:307 1. These two complex numbers do not give us real solutions to the tiling problem. In the general situation, Freiling and Rinne [8] and Laczkovich and Szekeres [14] independently gave the following amazing answer to this problem. A square can be tiled with finitely many rectangles similar to a 1 9 x rectangle if and only if: • x is a zero of a polynomial with integer coefficients, and • for the of least degree satisfied by x, any zero ppolynomial ﬃﬃﬃﬃﬃﬃﬃ a þ b 1 satisfies a [ 0. It is very surprising that these complex zeros, that seem completely unrelated to the tiling problem, actually play a fundamental role in it. In the example above, a solution for a 1 9 0.5698. . . rectangle is only possible because 0.215. . . is a positive number. Let us further illustrate this result with some examples. pﬃﬃﬃ The value x ¼ 2 does satisfy a polynomial equation with integer coefficients, namely, pxﬃﬃ2ﬃ - 2 = 0. However, the other root of the equation is 2\0 . Thus, a square cannot pﬃﬃﬃ be tiled with finitely many rectangles similar to a 1 2 rectangle. pﬃﬃﬃ On the other hand, x ¼ 2 þ 17 12 satisfies the quadratic 2 408x + 1 = 0, whose other root is equation 144x pﬃﬃﬃ 2 þ 17 12 ¼ 0:002453 [ 0. Therefore, a squarepcan ﬃﬃﬃ be tiled with finitely many rectangles similar to a 1 ð 2 þ 17 12Þ rectangle. How would we actually do it? pﬃﬃﬃ xp33ﬃﬃp-ﬃﬃ p 2 ﬃﬃﬃﬃﬃﬃ = ﬃ 0. The Similarly, x ¼ 3 2 satisfies the equation p3 ﬃﬃ 2 3 2 1 : Since other two roots of this equation are p3 ﬃﬃ 2 2 22 \0, a square cannot be tiled with finitely many rectpﬃﬃﬃ angles similar to a 1 3 2 rectangle. pﬃﬃﬃ Finally, let r/s be a rational number, and let x ¼ rs þ 3 2. One can check that this is still a zero of a cubic polynomial, whose other two zeros are: p ﬃﬃﬃ p ﬃﬃﬃpﬃﬃﬃ 3 3 2 2 3 pﬃﬃﬃﬃﬃﬃﬃ r 1: 2 2 s It follows that a square canpbe ﬃﬃﬃ tiled with finitely many rectangles similar to a 1 ðrs þ 3 2Þ rectangle if and only if p ﬃﬃﬃ 3 2 r : [ 2 s As a nice puzzle, the pﬃﬃﬃ reader can pick his or her favorite fraction larger than 3p2ﬃﬃ=2 ﬃ , and tile a square with rectangles similar to a 1 ðrs þ 3 2Þ rectangle. For other tiling problems, including interesting algebraic arguments, see [18]. The third problem we wish to discuss is motivated by the following remarkable tiling of a rectangle into nine squares, all of which have different sizes. (We will soon see what the sizes of the squares and the rectangle are.) Such tilings are now known as perfect tilings.

b

c

a e

d

f h

g i

To find perfect tilings of rectangles, we can use the approach of the previous problem. We start by proposing a tentative layout of the squares, such as the pattern shown, without knowing what sizes they have. We denote the side length of each square by a variable. For each horizontal line inside the rectangle, we write the following equation: The total length of the squares sitting on the line is equal to the total length of the squares hanging from the line. For example, we have the ‘‘horizontal equations’’ a + d = g + h and b = d + e. Similarly, we get a ‘‘vertical equation’’ for each vertical line inside the rectangle, such as a = b + d or d + h = e + f. Finally, we write the equations that say that the top and bottom sides of the rectangle are equal, and the left and right sides of the rectangle are equal. In this case, they are a + b + c = g + i and a + g = c + f + i. It then remains to hope that the resulting system of linear equations has a solution and, furthermore, is one where the values of the variables are positive and distinct. For the layout proposed above, the system has a unique solution up to scaling: (a, b, c, d, e, f, g, h, i) = (15, 8, 9, 7, 1, 10, 18, 4, 14). The large rectangle has dimensions 32 9 33. Amazingly, the resulting system of linear equations always has a unique solution up to scaling, for any proposed layout of squares. (Unfortunately, the resulting ‘‘side lengths’’ are usually not positive and distinct.) In 1936, Brooks, Smith, Stone and Tutte [2] gave a beautiful explanation of this result. They constructed a directed graph whose vertices are the horizontal lines found in the rectangle. There is one edge for each small square, which goes from its top horizontal line to its bottom horizontal line. The diagram below shows the resulting graph for our perfect tiling of the 32 9 33 rectangle.

8

We can think of this graph as an electrical network of unit resistors, where the current flowing through each wire is equal to the length of the corresponding square in the tiling. The ‘‘horizontal equations’’ for the side lengths of the squares are equivalent to the equations for conservation of current in this network, and the ‘‘vertical equations’’ are equivalent to Ohm’s law. Knowing this, our statement is essentially equivalent to Kirchhoff’s theorem: The flow in each wire is determined uniquely, once we know the potential difference between some two vertices (i.e., up to scaling). Brooks, Smith, Stone and Tutte were especially interested in studying perfect tilings of squares. This also has a nice interpretation in terms of the network. To find tilings of squares, we would need an additional linear equation stating that the vertical and horizontal side lengths of the rectangle are equal. In the language of the electrical network, this is equivalent to saying that the network has total resistance 1. While this correspondence between tilings and networks is very nice conceptually, it does not necessarily make it easy to construct perfect tilings of squares, or even rectangles. In fact, after developing this theory, Stone spent some time trying to prove that a perfect tiling of a square was impossible. Roland Sprague finally constructed one in 1939, tiling a square of side length 4205 with 55 squares. Since then, much effort and computer hours have been spent trying to find better constructions. Duijvestijn and his computer [5] showed that the smallest possible number of squares in a perfect tiling of a square is 21; the only such tiling is shown below. 27

35 50 8

15 9 7 29

25

17 11 19 2 6 24

18

16

4 33

37

42

What Does a Typical Tiling Look Like? Suppose that we draw each possible solution to a tiling problem on a sheet of paper, put these sheets of paper in a bag, and pick one of them at random. Can we predict what we will see?

9

15 1 7

10 4

18 14

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39

The random domino tiling of a 12 9 12 square shown, with horizonal dominoes shaded darkly and vertical dominoes shaded lightly, exhibits no obvious structure. Compare this with a random tiling of the Aztec diamond of order 50. Here, there are two shades of horizontal dominoes and two shades of vertical dominoes, assigned according to a certain rule not relevant here. These pictures were created by Jim Propp’s Tilings Research Group. .

Relations Among Tilings When we study the set of all tilings of a region, it is often useful to be able to ‘‘navigate’’ this set in a nice way. Suppose we have one solution to a tiling problem, and we want to find another one. Instead of starting over, it is probably easier to find a second solution by making small changes to the first one. We could then try to obtain a third solution from the second one, then a fourth solution, and so on. In the case of domino tilings, there is a very easy way to do this. A flip in a domino tiling consists of reversing the orientation of two dominoes forming a 2 9 2 square.

This may seem like a trivial transformation to get from one tiling to another. However, it is surprisingly powerful. Consider the two following tilings of a region.

Although they look very different from each other, one can, in fact, reach one from the other by successively flipping 2 9 2 blocks.

This very nice picture suggests that something interesting can be said about random tilings. The tiling is clearly very regular at the corners, and gets more chaotic as we move away from the edges. There is a well defined region of regularity, and we can predict its shape. Jockusch, Propp and Shor [11] showed that for very large n, and for ‘‘most’’ domino tilings of the Aztec diamond AZ(n), the region of regularity ‘‘approaches’’ the outside of a circle tangent to the four limiting sides. Sophisticated probability theory is needed to make the terms ‘‘most’’ and ‘‘approaches’’ precise, but the intuitive meaning should be clear.

Thurston [21] showed that this is a general phenomenon. For any region R with no holes, any domino tiling of R can be reached from any other by a sequence of flips. This domino flipping theorem has numerous applications in the study of domino tilings. We point out that the theorem can be false for regions with holes, as shown by the two tilings of a 3 9 3 square with a hole in the middle. Here, there are no 2 9 2 blocks and, hence, no flips at all. There is a version, due to Propp [17], of the domino flipping theorem for regions with holes, but we will not discuss it here.

Confronting Infinity This result is known as the Arctic Circle theorem. The tangent circle is the Arctic Circle; the tiling is ‘‘frozen’’ outside of it. Many similar phenomena have since been observed and (in some cases) proved for other tiling problems.

40

THE MATHEMATICAL INTELLIGENCER

We now discuss some tiling questions that involve arbitrarily large regions or arbitrarily small tiles. The first question is motivated by the following identity: 1 1 1 þ þ þ ¼ 1: 12 23 34

Consider infinitely many rectangular tiles of dimensions 1 12 ; 12 13 ; 13 14 ; . . .: These tiles get smaller and smaller, and the above equation shows that their total area is exactly equal to 1. Can we tile a unit square using each one of these tiles exactly once? good

1/2

1/3

1/4

1/2

1

1/5

1/3

...

1/6 1/4

1/5

*

*

* bad

It is easy to see why it is impossible to tile the whole plane with the bad collection shown above. Once we lay down a tile, the square(s) marked with an asterisk cannot be covered by any other tile. However, we can still ask: How large of a square region can we cover with a tiling? After a few tries, we will find that it is possible to cover a 4 9 4 square.

1

1 This seems to be quite a difficult problem. An initial attempt shows how to fit the first five pieces nicely. However, it is difficult to imagine how we can fit all of the pieces into the square without leaving any gaps. 1 1/2

1/3

1/3 1/4 1/2

1/5 1/6

1/5 1/4

To this day, no one has been able to find a tiling or prove that it does not exist. Paulhus [16] has come very close; he found a way to fit all these rectangles into a square of side length 1.000000001. Of course, Paulhus’s packing is not a tiling as we have defined the term, since there is leftover area. Let us now discuss a seemingly simple problem that makes it necessary to consider indeterminately large regions. Recall that a polyomino is a collection of unit squares arranged with coincident sides. Let us call a collection of polyominoes ‘‘good’’ if it is possible to tile the whole plane using the collection as tiles, and ‘‘bad’’ otherwise. A good and a bad collection of polyominoes are shown below.

It is impossible, however, to cover a 5 9 5 square. Any attempt to cover the central cell of the square with a tile will force one of the asterisks of that tile to land inside the square as well. In general, the question of whether a given collection of polyominoes can cover a given square is a tremendously difficult one. A deep result from mathematical logic states that there does not exist an algorithm to decide the answer to this question.1 An unexpected consequence of this deep fact is the following. Consider all the bad collections of polyominoes that have a total of n unit cells. Let L(n) be the side length of the largest square that can be covered with one of them. The bad collection of our example, which has a total of 22 unit squares, shows that Lð22Þ 4: One might expect L(22) to be reasonably small. Given a bad collection of tiles with a total of 22 squares, imagine that we start laying down tiles to fit together nicely and cover as large a square as possible. Since the collection is bad, at some point we will inevitably form a hole that we cannot cover. It seems plausible to assume that this will happen fairly soon, since our tiles are quite small. Surprisingly, however, the numbers L(n) are incredibly large! If f(n) is any function that can be computed on a computer, even with infinite memory, then L(n) [ f(n) for all large enough n. Notice that

1

A related question is the following: Given a polyomino P, does there exist a rectangle that can be tiled using copies of P? Despite many statements to the contrary in the literature, it is not known whether there exists an algorithm to decide this.

2010 Springer Science+Business Media, LLC, Volume 32, Number 4, 2010

41

computers can compute functions that grow very quickly, such as n

f ðnÞ ¼ nn ; f ðnÞ ¼ nn ; or n

f ðnÞ ¼ nn

:::

ða tower of lengthnÞ; . . .:

In fact, all of these functions are tiny in comparison with certain other computable functions. In turn, every computable function is tiny in comparison with L(n). We can give a more concrete consequence of this result. There exists a collection of polyominoes with a modest number of unit squares2, probably no more than 100, with the following property: It is impossible to tile the whole plane with this collection; however, it is possible to completely cover Australia3 with a tiling. A very important type of problem is concerned with tilings of infinite (unbounded) regions, in particular, tilings of the entire plane. This is too vast a subject to touch on here. For further information, we refer the reader to the 700-page book [9] by Gru¨nbaum and Shephard devoted primarily to this subject.

[5] A. Duijvestijn. Simple perfect squared square of lowest order. J. Combin. Theory Ser. B 25 (1978), 240–243. The unique perfect tiling of a square using the minimum possible number of squares, 21, is exhibited. [6] N. Elkies, G. Kuperberg, M. Larsen and J. Propp. Alternating sign matrices and domino tilings I, II. J. Algebraic Combin 1 (1992), 111–132, 219–234. It is shown that the Aztec diamond of order n has 2n(n+1)/2 domino tilings. Four proofs are given, exploiting the connections of this object with alternating-sign matrices, monotone triangles, and the representation theory of GL(n). The relation with Lieb’s square-ice model is also explained. [7] M. Fisher and H. Temperley. Dimer problem in statistical mechanics—an exact result. Philosophical Magazine 6 (1961), 1061–1063. A formula for the number of domino tilings of a rectangle is given in the language of statistical mechanics. [8] C. Freiling and D. Rinne. Tiling a square with similar rectangles. Math. Res. Lett 1 (1994), 547-558. The authors show that a square can be tiled with rectangles similar to the 1 9 u rectangle if and only if u is a zero of a polynomial with integer coefficients, all of whose zeros have positive

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3

cussed include perfect tilings of rectangles and aperiodic tilings of

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2

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extremely close to a perfect circle inscribed in the diamond.

The author studies the problem of tiling an n-dimensional box of

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integer dimensions A1 9 . . . 9 An with bricks of integer dimen-

dimer arrangements on a quadratic lattice. Physica 27 (1961),

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1209-1225. Kasteleyn proves exact and asymptotic formulas for the number

multiple of the brick if it can be tiled in the trivial way. It is shown

of domino tilings of a rectangle, with edges or with periodic

that, if a1|a2, a2|a3, . . ., an-1|an, then the brick can only tile boxes that are multiples of it. The converse is also shown to be true.

boundary conditions. He then discusses the relationship between

Say ‘‘unit squares’’ have a side length of 1 cm. which is very large and very flat

42

In a random tiling of a large Aztec diamond, the central region is

THE MATHEMATICAL INTELLIGENCER

this problem and the Ising model of statistical mechanics.

[13] D. Klarner. Packing a rectangle with congruent n-ominoes. J.

It is shown that the set of orientations of a graph that have the

Combin. Theory 7 (1969), 107-115. Klarner investigates the problem of tiling a rectangle using an odd

same flow-differences around all circuits can be given the structure of a distributive lattice. This generalizes similar constructions

number of copies of a single polyomino. He also characterizes the

for alternating-sign matrices and matchings.

rectangles that can be tiled with copies of an a 9 b rectangle, and

[18] S. Stein and S. Szabo´. Algebra and tiling. Homomorphisms in the

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[14] M. Laczkovich and G. Szekeres. Tilings of the square with similar

Washington, DC, 1994.

rectangles. Discrete Comput. Geom 13 (1995), 569-572.

This book discusses the solution of several tiling problems using

The authors show that a square can be tiled with rectangles

tools from modern algebra. Two sample problems are the fol-

similar to the 1 9 u rectangle if and only if u is a zero of a polynomial with integer coefficients, all of whose zeros have positive

lowing: A square cannot be tiled with 30–60–90 triangles, and a square of odd integer area cannot be tiled with triangles of unit

real part. [15] I. Pak. Tile invariants: New horizons. Theoret. Comput. Sci. 303

area. [19] W. Thurston. Conway’s tiling groups. Amer. Math. Monthly 97

(2003), 303-331.

(1990), 757–773.

Given a finite set of tiles T, the group of invariants G(T) consists of

The author presents a technique of Conway’s for studying tiling

the linear relations that must hold between the number of tiles of

problems. Sometimes it is possible to label the edges of the tiles

each type in tilings of the same region. This paper surveys what is

with elements of a group, so that a region can be tiled if and only if

known about G(T). These invariants are shown to be much stronger than classical coloring arguments.

the product (in order) of the labels on its boundary is the identity element. The idea of a height function that lifts tilings to a three-

[16] M. Paulhus. An algorithm for packing squares. J. Combin. Theory

dimensional picture is also presented. These techniques are

Ser. A 82 (1998), 147–157. Paulhus presents an algorithm for packing an infinite set of

applied to tilings with dominoes, lozenges, and tribones. [20] S. Wagon. Fourteen proofs of a result about tiling a rectangle.

increasingly small rectangles with total area A into a rectangle of

Amer. Math. Monthly 94 (1987), 601–617.

area very slightly larger than A. He applies his algorithm to three

Wagon gives 14 proofs of the following theorem: If a rectangle

known problems of this sort, obtaining extremely tight packings.

can be tiled by rectangles, each of which has at least one integral

[17] J. Propp. Lattice structure for orientations of graphs, preprint, 1994, arXiv: math/0209005.

side, then the tiled rectangle has at least one integral side.

2010 Springer Science+Business Media, LLC, Volume 32, Number 4, 2010

43

A World Record in Atlantic City and the Length of the Shooter’s Hand at Craps S. N. ETHIER

AND

FRED M. HOPPE*

t was widely reported in the media that, on 23 May 2009, at the Borgata Hotel Casino & Spa in Atlantic City, Patricia DeMauro (spelled Demauro in some accounts), playing craps for only the second time, rolled the dice for four hours and 18 minutes, finally sevening out at the 154th roll. Initial estimates of the probability of a run at least this long (assuming fair dice and independent rolls) ranged from one chance in 3.5 billion [5] to one chance in 1.56 trillion [10]. Subsequent computations agreed on one chance in 5.6 (or 5.59) billion [2, 6, 9]. This established a new world record. The old record was held by the late Stanley Fujitake (118 rolls, 28 May 1989, California Hotel and Casino, Las Vegas) [1]. One might ask how reliable these numbers (118 and 154) are. In Mr. Fujitake’s case, casino personnel replayed the surveillance videotape to confirm the number of rolls and the duration of time (three hours and six minutes). We imagine that the same happened in Ms. DeMauro’s case. There is also a report that Mr. Fujitake’s record was broken earlier by a gentleman known only as The Captain (148 rolls, July 2005, Atlantic City) [8, Part 4]. However, this incident is not well documented (specifically, the exact date and casino name were not revealed) and was unknown to

I

*Supported by NSERC.

44

THE MATHEMATICAL INTELLIGENCER Ó 2010 Springer Science+Business Media, LLC

Borgata officials. In fact, a statistical argument has been offered [4, p. 480] suggesting that the story is apocryphal. Our aim in this article is not simply to derive a more accurate probability, but to show that this apparently prosaic problem involves some interesting mathematics, including Markov chains, matrix theory, generating functions, and Galois theory.

Background Craps is played by rolling a pair of dice repeatedly. For most bets, only the sum of the numbers appearing on the two dice matters, and this sum has distribution pj :¼

6 j j 7j ; 36

j ¼ 2; 3; . . .; 12:

ð1Þ

The basic bet at craps is the pass-line bet, which is defined as follows. The first roll is the come-out roll. If 7 or 11 appear (a natural), the bettor wins. If 2, 3, or 12 appears (a craps number), the bettor loses. If a number belonging to P :¼ f4; 5; 6; 8; 9; 10g appears, that number becomes the point. The dice continue to be rolled until the point is repeated (or made), in which

case the bettor wins, or 7 appears, in which case the bettor loses. The latter event is called a seven out. The first roll following a decision is a new come-out roll, beginning the process again. A shooter is permitted to roll the dice until he or she sevens out. The sequence of rolls by the shooter is called the shooter’s hand. Notice that the shooter’s hand can contain winning 7s and losing decisions prior to the seven out. The length of the shooter’s hand (i.e., the number of rolls) is a random variable we will denote by L. Our concern here is with tðnÞ :¼ PðL nÞ;

n 1;

ð2Þ

the tail of the distribution of L. For example, t(154) is the probability of achieving a hand at least as long as that of Ms. DeMauro. As can be easily verified from (3), (6), or (9) below, t(154)&0.178 882 426 9 10-9; to state it in the way preferred by the media, this amounts to one chance in 5.59 billion, approximately. The 1 in 3.5 billion figure came from a simulation that was not long enough. The 1 in 1.56 trillion figure came from (1 - p7)154, which is the right answer to the wrong question.

Two Methods We know of two methods for evaluating the tail probabilities (2). The first is by recursion. As pointed out in [3], t (1) = t (2) = 1 and ! X X pj tðn 1Þ þ pj ð1 pj p7 Þn2 tðnÞ ¼ 1 j2P

þ

X j2P

pj

rediscovered several times since, is based on a Markov chain. The state space is S :¼ fco; p4-10; p5-9; p6-8; 7og f1; 2; 3; 4; 5g;

ð4Þ

whose five states represent the events that the shooter is coming out, has established the point 4 or 10, has established the point 5 or 9, has established the point 6 or 8, and has sevened out. The one-step transition matrix, which can be inferred from (1), is 0 1 12 6 8 10 0 B 3 27 0 0 6 C C 1 B B 0 26 0 6 C ð5Þ P :¼ B 4 C: 36 @ 5 0 0 25 6 A 0 0 0 0 36 The probability of sevening out in n - 1 rolls or fewer is then just the probability that absorption in state 7o occurs by the (n - 1)th step of the Markov chain, starting in state co. A marginal simplification results by considering the 4 by 4 principal submatrix Q of (5) corresponding to the transient states. Thus, we have tðnÞ ¼ 1 ðP n1 Þ1;5 ¼

4 X ðQn1 Þ1;j :

ð6Þ

j¼1

Clearly, (3) is not a closed-form expression, and we do not regard (6) as being in closed form either. Is there a closedform expression for t(n)?

j2P

n1 X

ð1 pj p7 Þl2 pj tðn lÞ

ð3Þ

l¼2

for each n C 3. Indeed, for the event that the shooter sevens out in no fewer than n rolls to occur, consider the result of the initial come-out roll. If a natural or a craps number occurs, then, beginning with the next roll, the shooter must seven out in no fewer than n - 1 rolls. If a point number occurs, then there are two possibilities. Either the point is still unresolved after n - 2 additional rolls, or it is made at roll l for some l [ {2, 3, . . ., n - 1} and the shooter subsequently sevens out in no fewer than n - l rolls. The second method, first suggested, to the best of our knowledge, by Peter A. Griffin in 1987 (unpublished) and

Positivity of the Eigenvalues We begin by showing that the eigenvalues of Q are positive. The determinant of 0 1 12 36z 6 8 10 C 3 27 36z 0 0 1B B C Q zI ¼ B C A 36 @ 4 0 26 36z 0 5

0

0

25 36z

is unaltered by row operations. From the first row, subtract 6/(27 - 36z) times the second row, 8/(26 - 36z) times the third row, and 10/(25 - 36z) times the fourth row, cancelling the entries 6/36, 8/36, and 10/36 and making the (1,1) entry equal to 1/36 times

AUTHORS

......................................................................................................................................................... S. N. ETHIER is professor of mathematics

at the University of Utah, and he specializes in applied probability. His book, The Doctrine of Chances: Probabilistic Aspects of Gambling, will be published this year by Springer. Department of Mathematics University of Utah Salt Lake City, UT 84112 USA e-mail: [email protected]

is professor of mathematics and statistics, and associate faculty of the Booth School of Engineering Practice, at McMaster University. At times, he has published on branching processes, population genetics, probability bounds, nuclear risk, and lotteries.

FRED M. HOPPE

Department of Mathematics and Statistics McMaster University Hamilton, ON L8S 4K1 Canada e-mail: [email protected] Ó 2010 Springer Science+Business Media, LLC, Volume 32, Number 4, 2010

45

12 36z 3

6 8 10 4 5 : 27 36z 26 36z 25 36z

ð7Þ

The determinant of Q zI, and therefore the characteristic polynomial q(z) of Q, is then just the product of the diagonal entries in the transformed matrix, which is (7) multiplied by (27 - 36z)(26 - 36z)(25 - 36z)/(36)4. Thus, qðzÞ ¼ ½ð12 36zÞð27 36zÞð26 36zÞð25 36zÞ 18ð26 36zÞð25 36zÞ 32ð27 36zÞð25 36zÞ 50ð27 36zÞð26 36zÞ=ð36Þ4 : We find that q(1), q(27/36), q(26/36), q(25/36), and q(0) alternate signs, and therefore the eigenvalues are positive and interlaced between the diagonal entries (ignoring the entry 12/36). More precisely, denoting the eigenvalues by 1 [ e1 [ e2 [ e3 [ e4 [ 0, we have 1 [ e1 [

where rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 5 u 349 þ a eðu; vÞ :¼ þ 8 72 3 sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ v 698 a 3 þ 2136u : 72 3 349 þ a Next we need to find right eigenvectors corresponding to the five eigenvalues of P. Fortunately, these eigenvectors can be expressed in terms of the eigenvalues. Indeed, with rðxÞ defined to be the vector-valued function 0 1 5 þ ð1=5Þx B 175 þ ð581=15Þx ð21=10Þx 2 þ ð1=30Þx 3 C B C B C B 275=2 ð1199=40Þx þ ð8=5Þx 2 ð1=40Þx 3 C B C B C 1 @ A 0 we find that right eigenvectors corresponding to eigenvalues 1, e1, e2, e3, e4 are

27 26 25 [ e2 [ [ e3 [ [ e4 [ 0: 36 36 36

The matrix Q, which has the structure of an arrowhead matrix, is not symmetric, but is positive definite. A nonsymmetric matrix is positive definite if and only if its symmetric part is positive definite. This is easily seen to be the case for Q by applying the same type of row operations to the symmetric part A ¼ 12 ðQ þ QT Þ to show that the eigenvalues of A interlace its diagonal elements (except 12/ 36), and hence are positive.

ð1; 1; 1; 1; 1ÞT ; rð36e1 Þ; rð36e2 Þ; rð36e3 Þ; rð36e4 Þ; respectively. Letting R denote the matrix whose columns are these right eigenvectors and putting L :¼ R1 , the rows of which are left eigenvectors, we know by (6) and the spectral representation that tðnÞ ¼ 1 fR diagð1; e1n1 ; e2n1 ; e3n1 ; e4n1 ÞLg1;5 : After much algebra (and with some help from Mathematica), we obtain

A Closed-Form Expression The eigenvalues of Q are the four roots of the quartic equation q(z) = 0 or 23328z 4 58320z 3 þ 51534z 2 18321z þ 1975 ¼ 0;

ð8Þ

tðnÞ ¼ c1 e1n1 þ c2 e2n1 þ c3 e3n1 þ c4 e4n1 ;

where the coefficients are defined in terms of the eigenvalues and the function f ðw; x; y; zÞ :¼ ð25 þ 36wÞ 4835 5580ðx þ y þ zÞ

while P has an additional eigenvalue, 1, the spectral radius. We can use the quartic formula (or Mathematica) to find these roots. We notice that the complex number a :¼ f

1=3

þ

9829 f1=3

;

þ 6480ðxy þ xz þ yzÞ 7776xyz= ½38880ðw xÞðw yÞðw zÞ as follows: c1 :¼ f ðe1 ; e2 ; e3 ; e4 Þ;

where f :¼ 710369 þ 18i

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1373296647;

appears three times in each root. Fortunately, a is positive, as we see by writing f in polar form, that is, f = reih. We obtain pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 710369 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ : a ¼ 2 9829 cos cos1 3 9829 9829

c2 :¼ f ðe2 ; e3 ; e4 ; e1 Þ; c3 :¼ f ðe3 ; e4 ; e1 ; e2 Þ; c4 :¼ f ðe4 ; e1 ; e2 ; e3 Þ: Of course, (9) is our closed-form expression. Incidentally, the fact that t (1) = t (2) = 1 implies that c1 þ c2 þ c3 þ c4 ¼ 1

The four eigenvalues of Q can be expressed as e1 :¼ eð1; 1Þ; e2 :¼ eð1; 1Þ; e3 :¼ eð1; 1Þ; e4 :¼ eð1; 1Þ;

46

ð9Þ

THE MATHEMATICAL INTELLIGENCER

ð10Þ

and c1 e1 þ c2 e2 þ c3 e3 þ c4 e4 ¼ 1: In a sequence of independent Bernoulli trials, each with success probability p, the number of trials X needed to

achieve the first success has the geometric distribution with parameter p, and PðX nÞ ¼ ð1 pÞn1 ;

n 1:

It follows that the distribution of L is a linear combination of four geometric distributions. It is not a convex combination: (10) holds but, as we will see, c1 [ 0;

c2 \0;

e1 0:862 473 751 659 322 030; e2 0:741 708 271 459 795 977; e3 0:709 206 775 794 379 015;

c3 \0;

e4 0:186 611 201 086 502 979; and the coefficients in (9) are c1 1:211 844 812 464 518 572; c2 0:006 375 542 263 784 777; c3 0:004 042 671 248 651 503;

c4 \0:

In particular, we have the inequality tðnÞ\c1 e1n1 ;

n 1;

ð11Þ

c4 0:201 426 598 952 082 292:

ð12Þ

These numbers will give very accurate results over a wide range of values of n. The result (12) shows that the leading term in (9) may be adequate for large n; it can be shown that

as well as the asymptotic formula tðnÞ c1 e1n1

as n ! 1:

Another way to derive (9) is to begin with the recursive formula (3). The generating function of the tail probabilities (2) is T ðzÞ :¼

1 X

tðnÞz n1 ;

1\c1 e1n1 =tðnÞ\1 þ 10m for m = 3 if n C 19; for m = 6 if n C 59; for m = 9 if n C 104; and for m = 12 if n C 150.

n¼3

and by (3) we have T ðzÞ ¼ 1

X

Crapless Craps

! pj zðz þ T ðzÞÞ

j2P

þ

þ

P0 :¼ f2; 3; 4; 5; 6; 8; 9; 10; 11; 12g

X pj ð1 pj p7 Þz 2 1 ð1 pj p7 Þz j2P X

p2j z 2

j2P

1 ð1 pj p7 Þz

ð1 þ z þ T ðzÞÞ:

Solving for T (z) using (1), we find that T ðzÞ ¼

In crapless craps [7, p. 354], as the name suggests, there are no craps numbers and 7 is the only natural. Therefore, the set of possible point numbers is

but otherwise the rules of craps apply. More precisely, the pass-line bet is won either by rolling 7 on the come-out roll or by rolling a number other than 7 on the come-out roll and repeating that number before 7 appears. With L0 denoting the length of the shooter’s hand, the analogues of (4)–(6) are

z 2 ð20736 33828z þ 16346z 2 1975z 3 Þ ; 23328 58320z þ 51534z 2 18321z 3 þ 1975z 4

S0 :¼ fco; p2-12; p3-11; p4-10; p5-9; p6-8; 7og f1; 2; 3; 4; 5; 6; 7g;

the denominator of which can be written (cf. (8)) as 23328ð1 e1 zÞð1 e2 zÞð1 e3 zÞð1 e4 zÞ:

0

B1 B B B2 1 B B P 0 :¼ B 3 36 B B4 B B @5

A partial-fraction expansion leads to (9), except that f is replaced by f ðw; x; y; zÞ :¼

1975 16346w þ 33828w2 20736w3 : 23328w2 ðw xÞðw yÞðw zÞ

Using Vieta’s formulas, this alternative version of (9) can be shown to be equivalent to the original one; in fact, yet another version uses 1975 16346w þ 33828w2 20736w3 ; f ðw; x; y; zÞ :¼ 2 3w ð6107 34356w þ 58320w2 31104w3 Þ which has the advantage of depending only on w.

Numerical Approximations Rounding to 18 decimal places, the non-unit eigenvalues of P are

6

0

2

4

6

8

10

29 0 0 28 0 0

0 0 27

0 0 0

0 0 0

0 0

0 0

0 0

26 0

0 25

0

0

0

0

0

0

1

C C C C C C C; C 6 C C C 6 A 36 6 6 6

and t0 ðnÞ :¼ PðL0 nÞ ¼ 1 ðP 0n1 Þ1;7 : There is an interesting distinction between this game and regular craps. The non-unit eigenvalues of P 0 are the roots of the sextic equation 0 ¼ 15116544z 6 59206464z 5 þ 93137040z 4 73915740z 3 þ 30008394z 2 5305446z þ 172975;

Ó 2010 Springer Science+Business Media, LLC, Volume 32, Number 4, 2010

47

and the corresponding Galois group is, according to Maple, the symmetric group S6. This means that our sextic is not solvable by radicals. Thus, it appears that there is no closed-form expression for t0(n). Nevertheless, the analogue of (9) holds (with six terms). All non-unit eigenvalues belong to (0, 1) and all coefficients except the leading one are negative. Thus, the analogues of (11) and (12) hold as well. Also, the distribution of L0 is a linear combination of six geometric distributions. These results are left as exercises for the interested reader. Finally, t0(154) & 0.296 360 068 9 10-10, which is to say that a hand of length 154 or more is only about one-sixth as likely as at regular craps (one chance in 33.7 billion, approximately).

[2] Bialik, C. Crunching the numbers on a craps record. The

ACKNOWLEDGMENTS

[6] Peterson, B. A new record in craps. Chance News 49 (2009).

Numbers Guy, Wall Street Journal blog. 28 May 2009. http:// blogs.wsj.com/numbersguy/crunching-the-numbers-on-a-crapsrecord-703/ [3] Ethier, S. N. A Bayesian analysis of the shooter’s hand at craps. In: S. N. Ethier and W. R. Eadington (eds.) Optimal Play: Mathematical Studies of Games and Gambling, pp. 311–322. Institute for the Study of Gambling and Commercial Gaming, University of Nevada, Reno, 2007. [4] Grosjean, J. Exhibit CAA. Beyond Counting: Exploiting Casino Games from Blackjack to Video Poker. South Side Advantage Press, Las Vegas, 2009. [5] Paik, E. Denville woman recalls setting the craps record in AC. Newark Star-Ledger, 27 May 2009. http://www.nj.com/news/ local/index.ssf/2009/05/pat_demauro_remembers_only_one.html

We thank Roger Horn for pointing out the interlacing property of the eigenvalues of Q. We also thank a referee for suggesting the alternative approach via the generating function T(z).

http://www.causeweb.org/wiki/chance//index.php/Chance_ News_49 [7] Scarne, J. and Rawson, C. Scarne on Dice. The Military Service Publishing Co., Harrisburg, PA, 1945. [8] Scoblete, F. The Virgin Kiss and Other Adventures. Research Services Unlimited, Daphne, AL, 2007.

REFERENCES

[1] Akane, K. The man with the golden arm, Parts I and II. Around Hawaii, 1 May 2008 and 1 June 2008. http://www.aroundhawaii. com/lifestyle/travel/2008-05-the-man-with-the-golden-arm-part-i. html and http://www.aroundhawaii.com/lifestyle/travel/2008-06the-man-with-the-golden-arm-part-ii.html

48

THE MATHEMATICAL INTELLIGENCER

[9] Shackleford, M. Ask the Wizard! No. 81. 1 June 2009. http://wizardofodds.com/askthewizard/askcolumns/askthewizard 81.html [10] Suddath, C. Holy craps! How a gambling grandma broke the record. Time, 29 May 2009. http://www.time.com/time/nation/ article/0,8599,1901663,00.html

How to Win Without Overtly Cheating: The Inverse Simpson Paradox ORA E. PERCUS

AND

JEROME K. PERCUS

nyone contemplating a statistical analysis is warned, at an early stage of the game, ‘‘But don’t combine the statistics of monkey wrenches and watermelons,’’ or the equivalent. Failure to heed this instruction – at a more sophisticated level to be sure – gives rise frequently to Simpson’s Paradox: if choice A is ‘‘better on average’’ than choice B in each of two differing circumstances, it may nevertheless happen that merging the two sets of data produces the opposite conclusion. We are going to look at this familiar pitfall, and then analyze the less familiar danger that it may occur ‘‘accidentally on purpose.’’

A

Simpson Consider the following specially constructed example for the sake of illustration: Two workers A and B are evaluated on performance in one easy task (#1) and one hard task (#2). Worker A: 20 tries, mean success rate 0.8 Task#1 Worker B: 80 tries, mean success rate 0.6 Task#2

Worker A: 80 tries, mean success rate 0.4 Worker B: 20 tries, mean success rate 0.2

Here 0.8 [ 0.6, 0.4 [ 0.2, showing A’s superiority on both tasks. But if we made the mistake of considering the total number of successes out of 100 tries for each worker,

we would see 16 + 32 \ 48 + 4, and B might seem preferable, just because B has de-emphasized the hard task. Or, more generally, if we tally successes (S) for A and B in tasks #1 and #2, and find Worker A has SA1 successes out of NA1 tries on the first, SA2 successes out of NA2 tries on the second task, with similar notation for Worker B, and if we set SA = SA1 + SA2, NA = NA1 + NA2, etc., it may very well SA SB SA1 SB1 SA2 SB2 happen that \ even though [ ; [ : NA NB NA1 NB1 NA2 NB2 This phenomenon is well known and well documented [3, 5–9] – but hope springs eternal. Only recently [1] a drug manufacturer, whose current potential blockbuster drug (Xinlay) failed to better a placebo in two clinical trials with uncorrelated protocols, proposed to a regulatory agency to pool the two sequences. If that criterion had been used, their drug would have appeared to outperform the placebo, allowing them to move forward. The regulatory agency panel was not unaware of the possibility of paradox, and denied the reinterpretation of the data.

Inverse Simpson The Simpson Paradox is data driven. It may, or may not, hold in a given situation; that is, data sets which indicate a similar statistical conclusion, when combined, may or may not point to the opposite conclusion. In fact, if the component data sets are sufficiently similar (say, as an extreme, identical), then pooling them surely will not reverse the conclusion. We will speak of the Inverse Simpson Paradox in case we start with a comparison between two large data sets – say, Ó 2010 Springer Science+Business Media, LLC, Volume 32, Number 4, 2010

49

successes and failures with drug A, and similarly with drug B – and ask if deceptive conclusions can result from decomposing it into two comparisons. Instead of pooling or ‘‘aggregating’’ two experiments, we ask whether it is possible to concoct a decomposition or ‘‘de-aggregation’’ into two experiments and give apparently opposite conclusions. The end result will then be the same old Simpson Paradox, only it arose by imposing a decomposition not a merging. There is a variety of purposes one may have in mind: a) Most directly and legitimately, it may be realized that data from two sources were combined for simplicity, and so a unique natural decomposition is called for, which may be instructive even if it does reverse the conclusion. This appears to be the case in the oft-quoted Berkeley sex discrimination controversy [5] discussed below. b) Least directly and least legitimately – but perhaps an effective strategy in litigation – one can ask for that decomposition that maximally reverses the conclusion, and then artfully invent excuses for lumping the data to give those subsets. c) Putting a different spin on b), one can ask for that decomposition that maximally comes jointly to either conclusion, and use this as an investigative tool to recognize hidden properties of significant subsets of data. We see at once that artificially inducing Inverse Simpson is indeed generally possible. If we break down our given pool of data into two artificial tasks and arrange that Task #1 contains no failures for A and Task #2 contains no successes for B, then to be sure Worker A will be dramatically superior to Worker B. Anyone scanning the results would sound the alarm. But this is only a suspiciously extreme version of the strategy. A clever manipulator can make results look more reasonable. To put it in context, let us consider the well-known Berkeley sex discrimination case [5]. We exhibit the phenomenon with simplified numbers for clarity. See Table 1. The aggregate figures show 41 males admitted out of 100 applicants, but only 29 females out of 100 applicants. The appearance of discrimination against females is strong, but

Table 1. Simplified Berkeley Admission Data Dept. 1

Dept. 2

Male Applicants

30

70

Males Admitted

6

35

Female Applicants

70

30

Females Admitted

14

15

Total Male Admissions/Applicants 41/100 = .41 Total Female Admissions/Applicants 29/100 = .29

dubious. The success rate for either sex in Dept. 1 was .2; in Dept. 2, .5. Women had been applying to the tougher department. Then combining the two departments created a statistical artifact. We are entitled to wonder whether a sneaky administrator could cover up a case of out-and-out discrimination, by taking what was in reality one big pool and making assignments of applicants to one or another department so as to make the imbalance seem attributable to this kind of artifact. Let us see what this would involve in a general situation. We are given NA and PA = SA/NA, NB and PB = SB/NB, for which, without loss of generality, PA [ PB. The cover-up is to consist in compartmentalizing the A-pool as NA1 = aNA, NA2 = (1 - a)NA, and the B-pool as NB1 = bNB, NB2 = (1 - b) NB; the successes in the various subsets will be denoted SA1 = PA1NA1, SA2 = PA2NA2, SB1 = PB1NB1, SB2 = PB2NB2. Denote by a and b, respectively, the proportions of the A and B total data to be allocated to compartment # 1. The question then is whether they can be chosen so that PA1 ¼ PB1 k PA2 ¼ PB2 l

ð2:1Þ

indicating no advantage to A or B in either case. No problem! Since SA1 = akNA, SA2 = (1 - a)lNA, SB1 = bkNB, SB2 = (1 - b)lNB, the condition is PA ¼ ak þ ð1 aÞl PB ¼ bk þ ð1 bÞl

ð2:2Þ

Thus PA and PB are both averages of k and l, which therefore must lie outside the interval (PB, PA) as in the

AUTHORS

......................................................................................................................................................... ORA E. PERCUS received an M.Sc. in Mathematics at Hebrew University, Jerusalem, and a Ph.D. in Mathematical Statistics from Columbia University in 1965. She has been active in several areas of mathematics, including probability, statistics, and combinatorics.

Courant Institute of Mathematical Sciences New York University New York, NY 10012 USA e-mail: [email protected] 50

THE MATHEMATICAL INTELLIGENCER

JEROME K. PERCUS received a B.S. in Elec-

trical Engineering, an M.A. in Mathematics, and a Ph.D. in Physics, in 1954, from Columbia University. He has worked in numerous areas of applied mathematics, primarily in chemical physics, mathematical biology, and medical statistics. Courant Institute of Mathematical Sciences New York University New York, NY 10012 USA

PB

we want the two pairs of trials to reverse the initial assertion at a common level of confidence

PA

0

1

ðPB1 PA1 Þ=r1 ¼ C 0 ¼ ðPB2 PA2 Þ=r2

Figure 1. Placement of Averaging Parameters k and l.

figure. In fact, if they are chosen to lie outside this interval then the desired decomposition specified by a and b is uniquely determined. Explicitly, l PA ; lk PA k ; 1a¼ lk a¼

l PB lk PB k 1b¼ lk

ð2:3Þ

The deceptive administrator would be prudent to invent a decomposition in which k is roughly in the middle of the (0, PB) interval, and l roughly in the middle of (PA,1), in order to allay suspicion. In the (simplified) Berkeley example, where PA = 0.41, PB = 0.29, a = 0.3, b = 0.7, we see that the observed k = .2, l = .5 do satisfy this criterion. With (2.3), we find that a suitable decomposition removes the apparent bias against females: no assertion of discrimination can then be made. But we also saw that a very simply constructed decomposition can lead to an apparent success rate much higher for A than for B. What is wrong with that construction, aside from looking awfully suspicious? Nothing, but to see whether the extreme behavior is rightly suspect we should attend to the statistical significance of the new assertions, a point that was emphasized by the FDA panel cited above. This is the subject of the following discussion, see also Zidek [11].

Statistical Significance of the Inverse Simpson Paradox Statistical significance is customarily quantified [3] by attaching a confidence level to the assertion made. In particular, quite generally if our initial data is characterized by success numbers SA, SB, and total numbers NA, NB with NA + NB = N, and if we define the success rates PA = SA/NA, PB = SB/NB, then the confidence level with which we can assert that the process underlying the observations had probabilities pA and pB satisfying pA C pB is given, in the large sample limit, by Z x 1 2 1=2 e y dy /ðN CAB Þ; where /ðxÞ ¼ pﬃﬃﬃﬃﬃﬃ 2p 1 CAB ¼ ðPA PB Þ=rAB [ 0 ð3:1Þ P ð1 P Þ P ð1 P Þ A A B B þ r2A þ r2B : r2AB ¼ NA =N NB =N Our objective is to supply a decomposition into two hypothetical trials (SA1, NA1, SB1, NB1) and (SA2, NA2, SB2, NB2) which reverse the original conclusion at a common level of confidence. Hence, if

r2i

PAi ð1 PAi Þ PBi ð1 PBi Þ þ ; ¼ NAi =N NBi =N

with C [ 0. To start, we need to find the restrictions on C 0 under which the required (PA1, PA2, PB1, PB2) satisfying (3.2) can be found. The solution is direct but algebraically cumbersome, and is presented in detail in [2]. Using the notation if 0 x 1; then x 1 x;

b¼

Ci0 ¼ ðPBi PAi Þ=ri [ 0; i ¼ 1; 2 where PAi ¼ SAi =NAi ; PBi ¼ SBi =NBi

ð3:3Þ

0

ð3:2Þ

ð3:4Þ

the conclusion is that if a C b, then PA bPA aPB b aPB aPB bPA aPB bPA : ; ; ; C 0 min A arB arB brA br ð3:5Þ Since we require C 0 C 0, this implies that a=b PA =PB 1;

a PB =PA 1: b=

ð3:6Þ

The expression is a bit involved and, even worse, contains the unknown parameters pAi, pBi implicitly. But it can be simplified by reducing its right-hand side and thereby strengthening the requirement on C 0 a bit. This is also carried out in [2], resulting in:

THEOREM If a C b then PA þ PB 1 : C 0 2ðccÞ1=2 ððPB =PA Þ2 1ÞðPA PB ÞðPB =PA Þ= " 2 # PA PB 1 PB PA PA þ PB 1 : C 0 2ðccÞ1=2 ððPA =PB Þ2 1ÞðPA PB ÞðPA =PB Þ= " 2 # PA PB 1 PB PA where c ¼ NA =N

ð3:7Þ

are sufficient to carry out the apparent reversal of ranking of A and B. Let us take a simple example that has been previously quoted [4, 8]. We will paraphrase it and use rounded-off data. Hospitals A and B specialize in treating a certain deadly disease. NA = 1000 patients are treated at A and NB = 1000 at B. Of these, SA = 900 recover, whereas SB = 800 recover, so that PA = .9, PB = .8 and Hospital A is apparently the place to go. In fact, one computes CAB = .05, so that this conclusion is supported at the .05 9 (2000)1/2 = 2.24 standard deviation level. Detailed investigation shows that matters are not so simple. Some patients enter in otherwise good shape, others in poor shape. Of the former, NA1 = 900 enter hospital A, and 870 recover; of the latter, NA2 = 100 enter and 30 recover, so PA1 = .967, PA2 = .3. On the other hand, NB1 = 600 enter Hospital B in good shape and SB1 = 590 recover, whereas NB2 = 400, SB2 = 210. Thus, PB1 = .983, PB2 = .55. We see that by not mixing the two classes of patients, Hospital B is superior for each class – at levels C10 = .038 (1.7 standard deviations)

Ó 2010 Springer Science+Business Media, LLC, Volume 32, Number 4, 2010

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Table 2. Simplified Hospital Recovery Data Good Shape

Poor Shape

Admissions to Hospital A

900

100

Recovered in Hospital A

870

30

Admissions to Hospital B

600

400

Recovered in Hospital B

590

210

from different sources (and trying to justify the decision to combine). What we have seen here is that the Inverse Simpson paradox, even in its most ‘‘sophisticated’’ version in which mean differences are weighted by appropriate standard deviations, is nearly universally applicable. This can be an effective tool of analysis, but it is also a dangerous technique for distorting statistical data.

Total Recovered/Admissions in A: 900/1000 = .9 Total Recovered/Admissions in B: 800/1000 = .8 REFERENCES

and C20 = .176 (7.9 standard deviations). Simpson is certainly exemplified. Equally, however, if only the combined data have been recorded, the person controlling the presentation of the evidence may be tempted to engage in Inverse Simpson. The criteria as to which patients entered in good shape, which in poor shape, are inevitably a bit fuzzy, after all. Given the aggregate data, the decomposition into the two classes could be planned with the intention of most convincingly asserting the opposite of the conclusion from the aggregate data. If this had been done according to the prescription of the Theorem, then with the same input data, we would have found a = .935, b = .738 (not far from the a = .9, b = .6 corresponding to the additional data presented in Table 2) and would have concluded with the superiority of Hospital B at a confidence level corresponding to C 0 B .107 or 4.79 standard deviations for each class of patients.

[1] Abboud L. (2005) ‘‘Abbott Seeks to Clear Stalled Drug’’. Wall Street Journal, Sept. 12. [2] ArXiv: 0801.4522. [3] Berger J. O. (1985) Statistical Decision Theory and Bayesian Analysis. Springer-Verlag, New York. [4] Bickel, P. J., Hammel, E. A., and O’Connell, J. W. (1975). ‘‘Sex Bias in Graduate Admissions: Data from Berkeley.’’ Science 187, 398–404. [5] Blyth C.R. (1972) ‘‘On Simpson’s Paradox and Sure-Thing Principle,’’ JASA 67, No. 338, 364-366. [6] Capocci, A. and Calaion, F. (2006). ‘‘Mixing Properties of Growing Networks and Simpson’s Paradox.’’ Phys. Rev. E74, 026122. [7] Moore, D. S. and McCabe, G. P. (1998). Introduction to the Practice of Statistics, 100 – 201. W. Freeman and Co., New York. [8] Moore, T. ‘‘Simpson and Simpson-like Paradox Examples.’’ see http://www.math.grinnell.edu/*mooret/reports/SimpsonExamples. pdf

Concluding Remarks The Simpson paradox, one of the simplest examples of the common misuse of statistics, has received increasing attention, because its consequences can be quite drastic (and sometimes profitable). In the classic Simpson Paradox, the only question is whether or not to combine data

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[9] Saari, D. (2001). Decisions and Elections, Cambridge University Press, Cambridge. [10] Simpson, E. H. (1951). ‘‘The Interpretation of Interaction in Contingency Tables.’’ J. Roy. Stat. Soc. B13, 238–241. [11] Zidek, J. (1984). ‘‘Maximal Simpson-Disaggregation of 2 9 2 Tables.’’ Biometrica 71, No. 1, 187–190.

Confounded Lawrence M. Lesser 3 of 8 poems I submitted to the classic journal were accepted, while 1 of 3 my rival did were, so I won. 2 of 3 poems I sent to the modern journal were accepted, while my rival had 3 of 5, so I won. But overall, my rival had half of hers accepted and I did not, so she won after all. I was confounded! I found that numbers don’t lie, but don’t explain why. Why try comparing if comparison can be reversed with a Peterson roll by underdog wrestling data, rival, or self? When my parts are summed, am I less than some of my parts?

The University of Texas at El Paso El Paso, USA. e-mail: [email protected] Ó 2010 Springer Science+Business Media, LLC, Volume 32, Number 4, 2010

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Understanding Coin-Tossing JAROSLAW STRZALKO, JULIUSZ GRABSKI, ANDRZEJ STEFANSKI, PRZEMYSLAW PERLIKOWSKI AND TOMASZ KAPITANIAK

t is commonly known that a toss of a fair coin is a random event and this statement is fundamental in the classical probability theory [1, 2, 3]. On the other hand, the dynamics of the tossed coin is described by deterministic equations, with no external source of random influence [4, 5, 6], so one can expect predictability in the results. It is possible to construct a mapping of the initial conditions (position, configuration, momentum, and angular momentum at the beginning of the coin motion) to a final observed configuration, that is, the coin terminates with its head (tail) side up or on its edge. The initial conditions which are mapped onto heads create a heads basin of attraction while those mapped onto tails create a tails basin of attraction [4]. The boundary which separates heads and tails basins consists of initial conditions mapped onto the coin standing on the edge. The structure of these boundaries has a significant impact on the problem of the coin-tossing predictability, that is, smooth basin boundaries allow predictability while fractal boundaries can lead to unpredictability [10, 11]. However, the precise structure of the heads-tails basin boundaries for a realistic model of a coin-tossing is unknown. Here, we show that heads-tails basin boundaries are smooth, so the outcome of the coin-tossing is predictable. We have found that an increase in the number of impacts in the period when the coin bounces on the floor makes the basin boundaries more complex, and in the limiting case of an infinite number of impacts the behavior of the coin is chaotic and the basins of heads and tails become intermingled [12, 13, 14, 15, 16]. Our results demonstrate that although the coin-tossing is predictable, it can also approximate the random process and can serve as the foundation for understanding the behavior of physical (mechanical) randomizers [17, 18, 19]. We expect our results to be a new point in the discussion of the nature of random processes [17, 20].

I

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THE MATHEMATICAL INTELLIGENCER 2010 Springer Science+Business Media, LLC

The Coin Model A coin can be modeled as a rigid body, namely a cylinder with a radius r and height h as shown in Figure 1. We consider a nonsymmetrical coin (the so-called cheat coin) for which the center of mass C is located at the distance nC = 0, gC = 0, fC = 0 from the geometrical center B. Any arbitrary position of a rigid body with respect to the fixed reference frame Oxyz can be described by a combination of the position of the origin of the local reference frame x 0 y 0 z 0 and the orientation (angular position) of this frame n, g, f [21, 22]. The local reference frame x 0 y 0 z 0 is rigidly attached to the body and its axes are parallel to the xyz frame; n, g, f is the frame embedded and fixed in the body. For the origin of the local frames it is convenient to choose the geometric center of the body model B. In our studies, we consider the following motion of the coin. We assume that the coin is thrown at the height z0 with the initial conditions U0 ¼ fx0 ; y0 ; z0 ; x_ 0 ; y_0 ; z_0 ; w0 ; m0 ; /0 ; xw0 ; xm0 ; x/0 g, that is, the initial position of the center of mass is {x0, y0, z0}, its initial velocity fx_ 0 ; y_0 ; z_0 g, the coin’s initial orientation {w0, m0, /0}, and the initial angular velocity {xw0, xm0, x/0}. After a free fall when the z coordinate is zero, the coin collides with the parallel base (floor). It is assumed that at the collision, a portion of the coin energy is dissipated, that is, the collision is described by the restitution coefficient v \ 1. The friction at the contact between the coin and the floor is described by the friction coefficient lfr [23]. After the collision, the coin’s center of mass moves to height z1 at which the total mechanical energy of the coin E is equal to its total energy in the moment after the collision E 0 minus the energy dissipated because of air resistance. Next, the coin moves on until it collides with the floor again. The calculations terminate when, after n-th collision, the total mechanical energy of the coin is smaller than the potential energy at the level of the coin’s center of mass (approximately mgr, where g is the gravitational

Figure 1. 3-dimensional model of the coin and its orientation in space.

acceleration), as this condition prevents the coin from flipping over. We also consider rotations of the coin on the floor. Full details of our model are given in [9]. The equations of motion describing the tossed coin are Newton’s equations, with no external source of random influence, that is, fluctuations of air, thermodynamic or quantum fluctuations of the coin. These equations are discontinuous, so in analysing them one cannot apply continuity theorems or direct calculations of Lyapunov exponents. If the outcome of a long sequence of the coin-tossings is to give a random result, it can only be because the initial conditions vary sufficiently from toss to toss. The flow given by the equations of motion maps all possible initial conditions into one of the final configurations. The set of initial conditions which is mapped onto the heads configuration creates a heads basin of attraction b(H) while the set of initial conditions mapped onto the tails configuration creates a tails basin of attraction b(T). The boundary separating the heads and tails basins consists of initial conditions mapped onto the coin-standing-on-edge configuration [24]. For an infinitely thin coin, this set is a set of zero measure and thus with probability 1 the coin ends up either heads or tails. For a nonzero thinness of the coin this measure is not zero, but the probability that the edge configuration is stable is low. Assume that one can set the initial conditions U0 with uncertainty , where is small. If a ball B in the phase space centered at U0 contains only the points which go to one of the final states, the outcome is predictable and repeatable. If in the ball B there are points leading to different final states (denote the set of points leading to heads as b0 (H)

and the set points leading to tails as b0 (T)), then the result of tossing is not predictable. One can calculate the probability of heads (tails) as probðheadsÞ ¼ lðb0 ðH ÞÞ=lðBÞ ðprob ðtailsÞ ¼ lðb0 ðT ÞÞ=lðBÞÞ, where l is a measure of the sets b0 (H), b0 (T) and B. The possibility that heads-tails basin boundaries are fractal [10], riddled [12, 13, 14, 15], or intermingled [12, 16], is worth investigating. Near a given basin boundary, if the initial conditions are given with uncertainty , a fraction f() of the initial conditions give an unpredictable outcome. In the limit ? 0, f() a where a \ 1 for fractal and a = 1 for smooth boundaries. Fractal basins’ boundaries are discontinuous (for example an uncountable sequence of disjoint stripes) or continuous (a snowflake structure) [11]. From the point of view of the predictability of the coin-toss the possibility of intermingled basins is the most interesting. Let us briefly explain the meaning of the term intermingled basins of attraction. The basin b(H) is said to be riddled by the basin b(T) when it satisfies the following conditions: (i) it has a positive Lebesgue measure, (ii) for any point in b(H), a ball in the phase space of arbitrarily small radius has a nonzero fraction of its volume in the basin b(T). The basin b(T) may or may not be riddled by the basin b(H). If the basin b(T) is also riddled by the basin b(H), the basins are said to be intermingled. In this case, in any neighborhood of the initial condition leading to heads there are initial conditions which are mapped to tails, that is, there does not exist an open set of initial conditions which is mapped to one of the final states: an infinitely small inaccuracy in the initial conditions makes the state of the coin tossing unpredictable. In our numerical calculations we consider the following coin data: m = 20 grams, r = 1.25 cm, h = 0.2 cm (former Polish 1 PLN coin made of a light aluminum-based alloy) and nC = 0.1 cm, gC = 0.1 cm, fC = -0.02 cm. We considered the air resistance acting on the coin in both tangential and normal directions and described by the following coefficients kn = 0.8, ks = 0.2 [9]. The friction between the coin and the floor during the impact is described by friction coefficient lfr = 0.2.

Results and Discussion Figure 2 (a-d) shows the basins of attraction of heads and tails calculated for various coin models. The dark regions correspond to heads and the white ones to tails. The case of the coin terminating on the soft floor (restitution coefficient

AUTHORS

......................................................................... ..... ......................................................................... JAROSLAW STRZALKO teaches classi-

JULIUSZ GRABSKI teaches classical and

cal and analytical mechanics and is a coauthor of a number of text books. In his free time he likes to go fishing.

analytical mechanics at all levels and pursues research in these areas. He loves jogging and skiing.

Division of Dynamics Technical University of Lodz Stefanowskiego 1/15, 90-924 Lodz Poland e-mail: [email protected]

Division of Dynamics Technical University of Lodz Stefanowskiego 1/15, 90-924 Lodz Poland e-mail: [email protected] 2010 Springer Science+Business Media, LLC, Volume 32, Number 4, 2010

55

Figure 2. Basins of attraction of heads (black) and tails (white); (a) coin lands on the soft surface, (b-d) coin bounces on the floor, (c,d) enlargements of (b). The following parameters have been used: x0 ¼ y0 ¼ 0; x_0 ¼ y_0 ¼ z_0 ¼ 0; u0 ¼ w0 ¼ 0; J0 ¼ 7p=180 rad, xf0 = 0, xg0 = 40.15 rad/s.

v = 0) is shown in Figure 2(a). The case which allows the bouncing of the coin on the floor surface (v = 0.6) is shown in Figure 2(b). The structure of the basin boundaries for the case without bouncing on the floor is similar to the structure in the Keller model [7]. One can notice that the structure of the basin boundaries is more complex (looks fractal or intermingled) when the coin is allowed to bounce on the floor as can be seen in Figure 2(b). To check the possibility that these basins are fractal (or intermingled), the appropriate enlargements are presented in Figure 2(c,d). It can be seen that apart from the graininess because of the finite number of points, the boundaries are smooth (see Fig. 2d). Under further magnification no new structure can be resolved, that is, no evidence of intermingled or even fractal basin boundaries is visible. The same conclusion has been reached in the studies of simple one- or two-dimensional models [8, 4, 5]. Figure 2 (a-d) is based on the results

obtained from numerically integrated equations of motion. We fixed all initial conditions except two, namely: the position of the coin mass center z0 and the angular velocity xn0. We check that similar structures of the basin boundaries are observed when different initial conditions are allowed to vary. The two-dimensional sections of the phase space presented in Figure 2 (a-d) are a good indication of what happens in the entire phase space. We point out that the same structure of basins of attraction has been observed for the symmetrical coin [9]. This allows us to state our main result: for any initial condition U0 there exists [ 0 such that a ball with radius centered at U0 contains points which belong either to the set b(H) or the set b(T). In other words, if one can settle the initial condition with appropriate accuracy, the outcome of the coin-tossing procedure is predictable and repeatable. Now we try to explain why for particularly small (but not infinitely small) the coin-tossing procedure can approximate a random process. A sequence of coin-tosses will be random if the uncertainty is large in comparison to the width W of the stripes characterizing the basins of attraction, so the condition [[ W is essential for the outcome to be random [4]. It is interesting to note that the uncertainty depends on the mechanism of coin tossing while the quantity W is determined by the parameters of the coin. In the case of the coin bouncing on the floor the structure of the heads and tails basin boundaries becomes complex (Figure 2b). In Figure 3(a-c) we show the calculations of these basins for a different number of impacts n. One can observe the face of the coin which is up after the n-th collision. Figure 3(a-c) shows the results for respectively 0, 3 and 10 collisions. With the increase of the collision numbers it is possible to observe that the complexity of the basin boundaries increases with the number of impacts. With the finite graininess (resolution) of Figure 3(a-c) these basin boundaries look fractal and one can speak about a fractalization-like process which can be observed with an increasing number of impacts. To explain this process, consider the limiting case of an infinite number of impacts. Such a case neglects air resistance and assumes elastic impacts, that is, v = 1, and cannot be realized in a real experiment. Consider the map

......................................................................... ............................................................................... ANDRZEJ STEFANSKI is also a professor

PRZEMYSLAW PERLIKOWSKI is a post-

of mechanics. He works on nonlinear dynamics, particularly the synchronization of chaotic systems. He loves good drinks and plays soccer.

doc at Humboldt University of Berlin (he received his Ph.D. from the Technical University of Lodz, however). He works on dynamical systems, particularly systems with time delay. During the summer holidays he goes trekking and camping in Croatia.

Division of Dynamics Technical University of Lodz Stefanowskiego 1/15, 90-924 Lodz Poland e-mail: [email protected]

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THE MATHEMATICAL INTELLIGENCER

Institute of Mathematics Humboldt University of Berlin Unter der Linden 6, 10099 Berlin Germany e-mail: [email protected]

Figure 3. Basins of attraction indicating the face of the coin which is up after the n-th collision: (a) n = 0, (b) n = 3, (c) n = 10, heads and tails are indicated in black and white, respectively. The same parameters as in Figure 2 have been used.

0.50006. This indicates that in the case of n ? ? this probability tends to 0.5. In a real experiment, such a very large number of impacts cannot be realized because of the dissipation (inelastic impacts and air resistance) so the fractalization-like process has to stop. In our experiments [9] we observed that a typical coin falling from the height of 186 cm bounces on a wooden floor about 8-14 times. The existence of the chaotic process described by the map U introduces a time-sensitive dependence on initial conditions characterized by the positive maximum temporal Lyapunov exponent [25, 26, 27]. This sensitivity is responsible for the ‘‘fractalization’’ shown in Figure 3(a-c) and explains why the coins behave in practice as perfect randomizers. ACKNOWLEDGMENTS

This study has been partially supported by the Polish Department for Scientific Research (DBN) under project No. N N501 0710 33.

REFERENCES

[1] W. Feller: An Introduction to Probability: Theory and Examples,

Figure 4. Basins of attraction of heads (black) and tails (white) in the case when the dissipation of energy is neglected; n = 1000 impacts, (b,c) are enlargements of (a). The same parameters as in Figure 2 have been used.

Wiley, New York, 1957. [2] E.T. Jaynes: Probability Theory: The Logic of Science, Cambridge University Press, Cambridge, 1996. [3] J.E. Kerrich: An Experimental Introduction to the Theory of Probability, J. Jorgensen, Copenhagen, 1946.

U : [0, 2p] ? [0, 2p] which maps the point /n on the edge of the coin, which hits the floor at the nth impact, to the point /n+1 which hits the floor at the (n + 1)st impact. Analysis of the time series of points (/1, /2, …) shows that the dynamics of U is chaotic when the largest Lyapunov exponent is positive (it has been numerically estimated from the time series to be 0.08). In this limiting case (n ??) the basins of heads and tails are intermingled and the outcome of the coin-tossing is unpredictable. Numerically, this can be observed when in the successive enlargements of the heads-tails basin boundaries the new structure is visible, as in Figure 4 (a-c) where the basins of heads and tails are calculated for n = 1000 impacts. The probability (we consider 106 different initial conditions) that a coin side which is up initially will still be up after 15 impacts is equal to 0.50987 and after 1000 impacts to

[4] V.Z. Vulovic and R.E. Prange: Randomness of true coin toss. Physical Review, A33/1: 576 (1986). [5] P. Diaconis, S. Holmes, and R. Montgomery: Dynamical Bias in the Coin Toss, SIAM Rev., 49, 211 (2007). [6] T. Mizuguchi and M. Suwashita: Dynamics of coin tossing, Progress in Theoretical Physics Supplement, 161, 274 (2006). [7] J.B. Keller: The probability of heads, Americam Mathematical Monthly, 93, 191 (1986). [8] Y. Zeng-Yuan and Z. Bin: On the sensitive dynamical system and the transition from the apparently deterministic process to the completely random process, Appl. Math. Mech., 6, 193 (1985). [9] J. Strzalko, J Grabski, A. Stefanski, P. Perlikowski, and T. Kapitaniak: Dynamics of coin tossing is predictable, Phys. Rep., 469, 59 (2008). [10] C. Grebogi, S.W. McDonald, E. Ott, and J.A. Yorke: Metamorphosis of basins boundaries, Phys. Lett., 99A, 415 (1983).

......................................................................... TOMASZ KAPITANIAK is a professor of

mechanics and a head of the Division of Dynamics; his research is concentrated on nonlinear dynamics and chaos theory. He paints in his free time.

[11] B.B. Mandelbrot: Fractal Geometry of Nature, Freeman, San Francisco, 1982. [12] J.C. Alexander, J.A. Yorke, Z. You, and I. Kan: Riddled basins, Int. J. Bifur. Chaos 2, 795 (1992). [13] J.C. Sommerer and E. Ott: Riddled basins, Nature 365, 136 (1993). [14] E. Ott, J.C. Alexander, I. Kan, J.C. Sommerer, and J.A. Yorke: The transition to chaotic attractors with riddled basins, Physica D, 76, 384 (1994).

Division of Dynamics Technical University of Lodz Stefanowskiego 1/15, 90-924 Lodz Poland e-mail: [email protected]

[15] T. Kapitaniak, Yu. Maistrenko, A. Stefanski, and J. Brindley: Bifurcations from locally to globally riddled basins, Phys. Rev. E57, R6253 (1998). [16] T. Kapitaniak: Uncertainty in coupled systems: Locally intermingled basins of attraction. Phys. Rev. E53, 53 (1996)

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[17] H. Poincare´: Calcul de Probabilite´s, George Carre, Paris, 1896.

[23] J.I. Nejmark and N.A. Fufajev: Dynamics of Nonholonomic

[18] E. Hopf: On causality, statistics and probability, Journal of Mathematical Physics, 13, 51 (1934). [19] E. Hopf: U¨ber die Bedeutung der Willku¨rlincken Funktionen fu¨r

Systems, Translations of Mathematical Monographs, (American Mathematical Society, vol. 33, 1972).

die Wahrscheinlichkeitstheorie, Jahresbericht der Deutschen Mathematiker-Vereinigung, 46, 179 (1936). [20] J. Ford: How random is a coin toss, Physics Today, 40, 3 (1983). [21] H. Goldstein: Classical Mechanics, Addison-Wesley, Reading, 1950. [22] J.E. Marsden and T.S. Ratiu: Introduction to Mechanics and Symmetry, Springer, New York, 1994.

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[24] D.B. Murray and S.W. Teare, Probability of a tossed coin landing on edge, Phys. Rev. E 48, 2547 (1993). [25] T. Kapitaniak: Distribution of transient Lyapunov exponents of quasi-periodically forced systems, Prog. Theor. Phys., 93, 831 (1995). [26] T. Kapitaniak: Generating strange nonchaotic attractors, Phys. Rev. E, 47, 1408 (1993). [27] T. Tel: Transient chaos, J Phys A: Math Gen, 22, 691 (1991).

The Mathematical Tourist

Dirk Huylebrouck, Editor

A Walk Through Mathematical Turin SANDRO CAPARRINI Does your hometown have any mathematical tourists attractions such as statues, plaques, graves, the cafe´ where the famous conjecture was made, the desk where the famous initials are scratched, birthplaces, houses, or memorials? Have you encountered a mathematical sight on your travels? If so, we invite you to submit an essay to

Turin’s architecture will be of particular appeal to visitors of mathematical inclination. The plan of the city is remarkably regular, revealing the work of Ancient Rome’s military architects and the later influence of the eighteenth-century French Enlightenment. The streets are usually wide and straight, intersecting at right angles and punctuated with public squares geometrically regular in shape. Most of the historical buildings date back to the Baroque period. However, there is a streak of fine madness running through this apparently tranquil and orderly city. Scattered throughout Turin there are some wildly imaginative, early twentiethcentury Art Nouveau buildings that rival those of Barcelona. One of the weirdest examples of eccentricity in architecture is the Casa Scaccabarozzi, popularly called Fetta di polenta (‘‘Slice of polenta’’) because of its yellow color, designed by the architect Alessandro Antonelli in 1881 (Fig. 1). This fivestorey building stands on a tiny right triangle having one side (along Corso San Maurizio) of 4 m and the hypotenuse (along Via Giulia di Barolo) of 21 m. If you happen to be in Turin, this little gem is worth a visit.

this column. Be sure to include a picture, a description of its mathematical significance, and either a map or directions so that others may follow in your tracks.

ntil recent years, Turin (Torino) had the dubious distinction of being one of the few historic Italian cities rarely visited by tourists. Its traditional image was that of an industrial area, mainly known for the automobile industry and related activities. Yet there are many things that make Turin stand apart from other Italian cities. While most of the places of interest in Italy are historically connected with Roman history or with the Renaissance, Turin flourished during the nineteenth century, when it became an example of economy propelled by science and technology. The rich cultural heritage of the city is reflected in the varieties of its museums. Turin has one of the few automobile museums in the world, and an Egyptian museum displaying what is perhaps the oldest collection of its kind. There is also a spectacular cinema museum showing that, before World War I, this was one of the most important centers of the cinema industry in Europe.

U

â

Please send all submissions to Mathematical Tourist Editor, Dirk Huylebrouck, Aartshertogstraat 42, 8400 Oostende, Belgium e-mail: [email protected]

Figure 1. The Fetta di polenta.

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While to mathematically minded people Turin is usually associated with Joseph-Louis Lagrange (1736–1813), many other important figures in the history of mathematics also spent long periods here. Indeed, the Turin scientific school ranks high in Italy. As early as the second half of the sixteenth century, Giovanni Battista Benedetti (1530–1590), an important forerunner of Galileo in mechanics, was court mathematician to the Duke of Savoy. In the eighteenth century the physicist Giambattista Beccaria (1716–1781), one of the founders of the scientific study of electricity and a correspondent of Benjamin Franklin, was a professor at the university of Turin; Lagrange, the greatest scientist Turin ever produced, was one of his students. Physics and chemistry in Turin are also well represented by the Abbe´ Nollet (1700–1770), who came to Turin in 1739, and by Amedeo Avogadro (1776–1856), known for the Avogadro number. Among nineteenth-century mathematicians who worked in Turin there were Giovanni Plana (1781–1864), Augustin-Louis Cauchy (1789–1857) and Luigi Federico Menabrea (1809–1896). Around the turn of the century, the most productive period for mathematics in Turin, the names of Giuseppe Peano (1858–1932), Mario Pieri (1860– 1913), Corrado Segre (1863–1924), Vito Volterra (1860– 1940) and Cesare Burali-Forti (1861–1931) must be cited. In the twentieth century there were Guido Fubini (1879–1943) and Francesco Tricomi (1897–1978). A lively description of the University of Turin in 1900 was given by J. L. Coolidge [1]. There are many interesting remarks on the intellectual life in fin de sie`cle Turin in the books by H. C. Kennedy [2], E. Marchisotto and J. T. Smith [3] and J. R. Goodstein [4]. Today, some traces of the past mathematical glories of Turin are still visible. Since Turin is not a large city and its centre is best explored on foot, we suggest taking a stroll past its mathematical points of interest. This is best done on weekdays, since some of the places are closed on weekends. The main part of the walk is about 2 km long and takes almost a full day to complete. If you wish, you can of course divide this walk into several separate trips. Motorists

AUTHOR

......................................................................... SANDRO CAPARRINI holds degrees in

Physics and in Mathematics and a Ph.D. in Mathematics from the University of Turin. However, all of his work has been in the history of mathematics. His research is mainly focused on the interaction between mathematics and mechanics from 1750 onward. In 2004 he was awarded the Slade Prize from the British Society for the History of Science. Institute for the History and Philosophy of Science and Technology Victoria College 316 91 Charles St. West Toronto M5S 1K7 Canada E-mail: [email protected]

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Figure 2. The statue of Lagrange.

are warned that most of the city center is closed to traffic during the day. The walk starts at Porta Nuova railway station. As you stand in Piazza Carlo Felice, with the fac¸ade of the station behind you, walk under the arcades along the right side of the square. The first narrow road on the right leads to a small square, the Piazzetta Lagrange. In the centre of the square stands the monument to Lagrange (who else?). The monument was conceived and sculpted by the Piedmontese artist Giovanni Albertoni in 1867. The statue shows Lagrange as a middle-aged man, standing upright, slightly stooping, wearing an old-fashioned waistcoat. He is gazing downward, apparently immersed in profound thoughts. His arms hang down; he has a quill pen in his right hand and a manuscript in his left. There are four books at his feet, perhaps representing the treatises he published late in life. Although not a great work of art, the statue is a simple but effective depiction of a quiet and bookish man (Fig. 2). Leave Piazzetta Lagrange at its upper right-hand (northeast) corner. Walk along Via Lagrange for a couple of blocks. At No. 29, on the so-called piano nobile, (i.e., the storey immediately above the ground floor) Lagrange was born on January 25, 1736 (Fig. 3). A simple commemorative plaque recalls the figure of the illustrious mathematician (Fig. 4). The fac¸ade of the building has been carefully restored, so that it is not difficult to imagine how the place looked in the eighteenth century. Unfortunately, Lagrange’s apartment, now in private hands, has been remodelled since then.

Figure 3. The house where Lagrange was born, showing the commemorating plaque.

From Lagrange’s home, continue walking down Via Lagrange in the same direction until you reach the end of the street, at the intersection with Via Maria Vittoria. Pause for a moment to admire on the right the Baroque Chiesa di San Filippo, where Lagrange was christened, then turn your attention to the seventeenth-century building on your left, the Collegio dei Nobili. Any tour guide will tell you that this is the location of the Egyptian Museum and of the Galleria Sabauda, both requisite stops for tourists. For us, this building is important as the location of the Turin Academy of Science (Fig. 5). The Academy was founded in 1757 by Lagrange together with two friends, the physician and physicist Giovanni Francesco Cigna (1734–1790) and Count Angelo Saluzzo di Monesiglio (1734–1810). A few kindred spirits joined this initial group in the years that followed. At first, this was an informal institution, devoted essentially to discussing the works and readings of its own members, and was thus called Societa` privata (‘‘Private society’’). The members met at the house of the Count of Saluzzo. In 1759 the Society began publishing a scientific journal, originally entitled Miscellanea Philosophico-Mathematica Societatis Privatae Taurinensis. After 14 years, five volumes had been completed; these initiated the long series of Me´langes, Me´moires and Atti that have spread the fame of the earlier Society and later Academy throughout the world of science. Among the contributors to these early volumes were some of the leading scientists of the time: Jean le Rond d’Alembert (1717–1783), Marie Jean

Figure 4. Close-up of Lagrange’s plaque.

Figure 5. The Collegio dei Nobili.

Antoine Nicolas de Caritat Marquis de Condorcet (1743– 1794), Leonhard Euler (1707–1783), Albrecht Haller (1708– 1777), Pierre Simon Laplace (1749–1827), Gaspard Monge (1746–1818) and, of course, Lagrange himself. Shortly after the publication of the Miscellanea, the Society obtained the permission to add the adjective reale (‘‘royal’’) to its name. Finally, the creation of an Academy under royal patronage was suggested to the King; it was formalized on 25 July 1783. Lagrange, then in Berlin, was elected honorary president. Ó 2010 Springer Science+Business Media, LLC, Volume 32, Number 4, 2010

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Figure 6. Plana’s monument in the Palazzo dei Nobili.

Figure 7. The Genocchi bust.

The Academy’s location also deserves attention. The Collegio dei Nobili, a wonderful example of seventeenthcentury Italian baroque architecture, was built between 1679 and 1687 by the great architect Guarino Guarini (1624–1683) as a school for the sons of nobles. (In fact, behind this project was a Jesuit plan to infiltrate the centres of political power in Piedmont.) After the suppression of the Jesuits in 1773, the building passed to the state, and in 1784 King Vittorio Amedeo III designated it as the seat of the new Academy. Now that you are at the Collegio dei Nobili, go to the point at which Via Maria Vittoria meets Piazza San Carlo. Here, a plaque on the wall indicates that ‘‘Giovanni Plana, while living in this building, composed the theory of the movement of the moon between 1807 and 1832.’’ The astronomer and mathematician Giovanni Plana, author of the The´orie du mouvement de la lune (1832) was once considered the most important Italian scientist of his time. The The´orie was an attempt to improve the approximations Laplace had devised for the movements of the moon; it consists of three massive volumes full of incredibly long and complicated formulae. If you want to become better acquainted with Plana, enter the Collegio through its main entrance in Via Accademia delle Scienze, then go all the way to the back of the atrium and turn left along the corridor. After a few meters, you will find on your left a slightly larger than life statue of Plana. The famous astronomer is shown in his old age, sitting in an armchair, a book in his hand, a pensive look on his face (Fig. 6). The statue was made in 1870 by Giovanni Albertoni,

the same sculptor who made the monument to Lagrange. These two statues give you an idea of how mathematicians were viewed at the end of the nineteenth century. Immediately before the statue of Plana there is a bust of Angelo Genocchi (1817–1889), professor of analysis at the University of Turin from 1865 to 1884, now remembered mainly for a polemic with Peano (Fig. 7). In 1883, Peano, still Genocchi’s assistant, was given the task of writing a textbook based on the professor’s lectures. But Peano did not limit himself to merely transcribing what the professor had said. With youthful enthusiasm, he added several pages of endnotes full of important observations and ingenious counterexamples. Unfortunately, these remarks had the collateral effect of undermining many of Genocchi’s proofs. Obviously Genocchi was not happy with the result. Infuriated, he sent a letter to several mathematical journals renouncing authorship of the final text. Today the GenocchiPeano is considered one of the most significant textbooks on analysis ever published. History has not been kind to Genocchi, who, while not on the same level as Peano, was in effect a rigorous mathematician for his time. To visit the Academy it is necessary to request permission a couple of months in advance; send an email to biblioteca@ accademia.csi.it. The Academy is open to visitors from 9 am to 1 pm and from 3 p.m. to 5 p.m., Monday to Friday. The entrance to the Academy is a small door on Via Maria Vittoria 3. While the entire Academy extends over several floors, its core consists of three salons. The main room is called the Sala dei Mappamondi (Fig. 8). Imagine a

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Figure 8. The Sala dei Mappamondi. (photo: Marco Saroldi)

large space having approximately the structure of a church, complete with apses, columns and frescoes. However there is no altar, and the walls are almost completely covered with shelves of old books. This arrangement of the rooms, dating back to the foundation of the Academy, is the work of the architect Mario Ludovico Quarini. It is a remarkable example of architecture in the service of science rather than of religion. Also of interest are the frescoes, painted between 1786 and 1787 by Giovannino Galliari, all depicting scientific subjects. The mathematical tourist should look for the portraits of Pythagoras and Euclid over the two entrances to the second room, each recognizable from appropriate geometric symbols. Leaving the Collegio dei Nobili, follow Via Maria Vittoria as far as the intersection with Via Carlo Alberto, then turn left and go on until you reach number 10. This is the main entrance to Palazzo Campana, a seventeenth-century building which now houses the modern Department of Mathematics of the University of Turin. In October of 2008, the Department was renamed in memory of Giuseppe Peano, who taught there for about 40 years. Like many old buildings, Palazzo Campana hides a few secrets. Under Fascism it became the Casa Littoria, the provincial headquarters of the party; underneath it remain the hiding places in which the city leaders could take refuge in the event of air strikes during the war. Since you are interested in the history of mathematics, you will probably want to visit the Library of the Department, which houses many old books of great interest. Not far from Palazzo Campana, on the right-hand side of Piazza Carlo Alberto, there is also the Biblioteca Nazionale, a real treasure-trove of rare and important texts. A description of the riches of Turin libraries can be found in the catalogue of an exposition held in 1987 [5]. From Piazza Castello, take Via Po, keeping to the left. Turn left at the first intersection with Via Giovanni Virginio. A few steps later, you will arrive at Piazzetta Accademia Militare. Here you will find a line of columns, all that remains after the bombing in the last war of the Turin Military Academy, where Lagrange taught analysis and mechanics from 1755 to 1766 (Fig. 9). A plaque on the wall succinctly recounts the history of the site.

The Military Academy was founded in 1678 primarily as a school for the pages and nobles of the court. There were many changes before it assumed its definitive form in 1815. The rules of 1692 explain how, other than mathematics and design, students would learn how to ride horses, how to joust, how to handle weapons, and how to dance. To become perfect gentlemen, the young men were also encouraged to participate in court festivals. The rules of 1754 indicate that the Academy accepted men and boys between 10 and 30 years of age, divided into three groups. The first consisted of true cadets, the second, of university students who took part in only some of the activities of the Academy, and the third was made up of younger boys. The success of this institution may be judged by the fact that many of the students came from abroad. There is a vivid description of life in the Academy around 1760 in the autobiography of the dramatist Vittorio Alfieri (1749–1803). Military academies were among the best places to learn mathematics during the eighteenth century. In the period when Lagrange taught there, the Military Academy of Turin offered courses in arithmetic, algebra, plane and solid geometry, trigonometry, surveying, mechanics, hydrostatics and the elements of calculus. (Lagrange’s Turin lectures on calculus have been published in [6].) The founders of the renowned E´cole Polytechnique in Paris were inspired by military academies, and this institution, in turn, became the model for West Point. Lagrange was not the only mathematician of importance to teach at the Military Academy of Turin. After him came, among others, Plana, Menabrea, Peano, and Burali-Forti. Any university would be proud of such a faculty of professors. The ruins of the Military Academy take only a few minutes to explore, but the next building on our tour would require hours or days to be fully appreciated. Turning onto Via Po, turn right and walk toward Piazza Castello. On the far side of the piazza, just to the right of Via Palazzo di Citta`, you will find the Church of San Lorenzo, one of Guarini’s masterpieces. We already encountered the Theatine priest Guarini when discussing the Turin Academy of Science. While he is justly considered one of the major architects of the seventeenth century, it would not be a stretch at all to call him a mathematician turned architect. Had he not designed a few innovative buildings, he would be remembered today as the author of

Figure 9. The old Turin Military Academy, circa 1890. Ó 2010 Springer Science+Business Media, LLC, Volume 32, Number 4, 2010

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several excellent texts on pure and applied mathematics. His Euclides adauctus (‘‘Euclid Augmented,’’ 1671), summarizes a good part of the mathematical knowledge of the seventeenth century, while his Coelestis mathematica (‘‘Celestial Mathematics,’’ 1683) is a kind of astronomical encyclopedia. On a slightly different note, his Placita philosophica, physicis rationibus, experientiis, mathematicisque ostensa (‘‘Philosophical Thoughts Demonstrated by Means of Physical Reasoning, Experiments and Mathematics,’’ 1665) is mostly a reflection on scientific methodology. All these works demonstrate a remarkable knowledge of the mathematics of that time. The Euclides adauctus, for example, contains a reference to the then recent descriptive geometry of Desargues. In many respects, Guarini is comparable to his contemporary Christopher Wren, who, like him, was both a great architect and an important scientist. Visiting one of Guarini’s buildings is like entering a giant three-dimensional geometric construction. The plan of San Lorenzo is a curvilinear octagon formed from the intersection of eight circles around a central area (Fig. 10). Looking up, you will see a cupola, parabolic in cross-section, criss-crossed by ribs that, seen from below, form abstract polygons. As interesting as this superimposition of geometrical forms may

Figure 11. Interior of San Lorenzo.

Figure 10. Ground plan of San Lorenzo. 64

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be, it gives only a hint of just how complicated the structure of the church really is. In fact, if you look harder, you will begin to see many unusual things (Fig. 11). You might notice, for example, that the oval windows in the cupola are so large that they compromise its stability. Lowering your gaze to ground level, you will become aware that the columns are too slender to support the enormous weight of the mass overhead. And if this weren’t enough, the arches that appear to hold up the entire weight of the building are perforated with holes just where the keystones should be, that is, exactly where the weight of the arches should bear down on the columns. In essence, most of what you see is fake. The columns and arches do not support anything. The real weightbearing structure is hidden inside the walls; even today, researchers are not sure exactly how it works. This double structure, one visible but false, and the other real but hidden, is typical of Guarini, as are the secret stairways, the hidden frescoes and the complicated system of passages in the space between the internal and external cupolas. Within the Church of San Lorenzo lurk enough eccentricities to delight the readers of historical mysteries like Umberto Eco’s The Foucault Pendulum. As you leave San Lorenzo, look to the left. Above the roofs of the Palazzo Reale you will see the spiral steeple of the Church of the Holy Shroud (1694), another of Guarini’s works. The Church of the Holy Shroud, one of the major masterpieces of Baroque architecture, is mainly known because it is home to the sheet in which, according to tradition, the body of Christ was wrapped after the

crucifixion. The structure of this church was even more complicated than that of San Lorenzo. What you see today, though, is only an empty shell: The inside was completely destroyed by a fire in 1997. From Piazza Castello, retrace your steps back to Via Po. At number 17 you will find the entrance of the old Palazzo dell’Universita`. This is the place where distinguished mathematicians, such as Cauchy, Volterra and Peano, gave their lectures. The Palazzo was built between 1712 and 1720 under the direction of the architect Michaelangelo Garove. King Vittorio Amedeo II decided to modernize university studies in Piedmont, and the new edifice was to be the tangible marker of this reform. Thanks to teachers such as Beccaria and the Abbe´ Nollet, in a few years the University of Turin became an important centre for physics. This tradition continued in the following century: Avogadro and Cauchy taught fisica sublime (‘‘sublime physics’’) in Turin, corresponding more or less with modern theoretical physics. The courtyard of the Palazzo dell’Universita` is another interesting site for mathematical tourists. Starting around 1860, a collection of busts of a number of the most celebrated professors of the University of Turin began to be assembled, following the example of the collections of statues of famous Italians displayed in the arcades of the Uffizi Museum in Florence. Today, the names of these once famous professors are little known, and one feels a twinge of sadness upon contemplating these busts gathering dust. The mathematicians collected here are not among the most important, confirmation that fame can play strange tricks. There are the effigies of Beccaria, Avogadro, Felice Chio` (1813–1871), Tommaso Valperga Caluso (1737–1815), Carlo Ignazio Giulio (1803–1859), Plana and Genocchi. The busts are arranged along the side walls, on two floors. Entrance is free, though the Palazzo dell’Universita` is now home to the Rector’s offices, restricting its hours to between 9 a.m. and 5 p.m. Turn again toward Via Po and follow it in the direction of the river. On the other side of the street, just one block from the Palazzo dell’ Universita`, you will spot the fac¸ade of the Church of the San Franceso da Paola. Construction of the church was begun in 1632 and was completed by the end of the century. It is a fine specimen of Baroque architecture, but for us its real interest lies elsewhere: according to Antonio Maria Vassalli Eandi (1761-1825), an early biographer of Lagrange, this is the birthplace of the calculus of variations [7, p. 50]. Vassalli Eandi writes that in 1755 Lagrange, then only 19 years old, while assisting at Mass in the church, was inspired by the music to create the delta algorithm. He went home immediately to write down the result, which he then sent to Euler on 12th August 1755. Today, anyone seriously interested in mathematics will gladly spend a few minutes in contemplation of the church. Continue along Via Po until you reach the intersection with Via Montebello. Turn left on Via Montebello and look up: There before you stands the stately Mole Antonelliana, designed by Antonelli in 1862. Since the Mole is mentioned in every guidebook and in every tourist brochure about Turin, it hardly needs comment here. Suffice it to say that it is an absolute must for anyone visiting Turin, particularly now that it houses the Cinema Museum. It is an exciting experience to take the elevator with the glass floor all the

Figure 12. The Fibonacci numbers on the Mole.

way up (about 170 m). On a fine day, the view from the top is impressive. However, mathematical tourists will perhaps be more interested in another, more eye-catching feature of the Mole. Displayed on the south side of the dome is the Fibonacci sequence executed in red neon, each number about 2 m high (Fig. 12). This is the work of one of the most distinguished Italian postwar artists, Mario Merz (1925–2003). Merz was fascinated, almost obsessed, by the Fibonacci sequence, in which he saw a ‘‘spiraliform mathematical organisation that differs from the Renaissance perspective and is organic’’ [7, p. 200]. He used the sequence in many of his works, beginning in the late 1960s, most notably on the chimney of the Turku Power Station in Turku, Finland, in 1994 (see [8]) and outside the Zentrum fur Internationale Lichtkunst (International Center for Light Art) in Unna, Germany, in 2001. The story of the Turin installation began in 1984, when Merz presented the Fibonacci sequence as part of an art exhibition held at the Mole. When, in 2000, Merz was requested to contribute to the ‘‘Luci d’Artista,’’ an open-air exhibition of large-scale light installations by Italian artists, held every year in Turin during the winter, he again submitted his old project. The Fibonacci sequence on the Mole has already become a new symbol of the city. The sequence is invisible in daylight; to see it, come here after dusk and approach Ó 2010 Springer Science+Business Media, LLC, Volume 32, Number 4, 2010

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Figure 13. The obelisk commemorating Beccaria.

the Mole from the south. However, you can get a better view of the Fibonacci sequence from the hills surrounding Turin, especially from the Monte dei Cappuccini, on the other side of the River Po. The next stop on our tour is at a rather demanding distance to be reached by walking. Go back to Via Po and take tram number 13 toward Piazza Castello. Stay on the tram until the stop just after Porta Susa railway station, in Piazza Statuto. A few metres away, in the middle of the little gardens in the southern part of the piazza, you will find the monument to Giambattista Beccaria. Aside from the aforementioned experiments on electricity, from 1760 to 1764 Beccaria busied himself with measurements of an arc of meridian in Piedmont. These measurements were important not only for the production of an accurate geographical map, but also to precisely evaluate the flattening of the earth and, consequently, to verify Newton’s theories. Beccaria published his own results in the Gradus Taurinensis (1774), a book that was widely discussed throughout Europe. To have a base for his triangulations, Beccaria measured with extreme precision a distance of about 8 kilometers. The end points of this segment were marked by signs on two slabs of marble. In 1808, two obelisks topped with armillary spheres were raised near these slabs in memory of Beccaria and his measurements (Fig. 13). They were one of the first examples of monuments dedicated to pure science. Oddly enough, to some people who dabble in esotericism, the obelisks are now considered to possess magical significance. 66

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From Beccaria’s obelisk it is only 10 minutes’ walk to the last point of interest. Turn west and go to Via San Donato. At number 31 you will find the Institute of Faa` di Bruno, built by Francesco Faa` di Bruno (1825–1888). Faa` di Bruno was a remarkable man by any standard (Fig. 14). He came from an old Piedmontese family; after receiving his primary education at the Military Academy, he went to the Sorbonne, where he graduated under Cauchy. Then he became in turn a soldier, a cartographer, a composer, a mathematician, an inventor, a social reformer, an architect, a publisher and, finally, a Catholic priest. Somehow he managed to be successful in all of his enterprises. He also created a religious order of nuns that still exists today, the Suore Minime di Nostra Signora del Suffragio. In 1988 the Church formally approved his veneration with his beatification by Pope John Paul II. While there are many books available about Faa` di Bruno, there is no satisfactory overall treatment of his many achievements. The best single source available is a recent collection of essays [9]. Faa` di Bruno’s work as a social reformer is extraordinary. He was particularly concerned with women’s problems: For instance, the welfare of teenaged mothers and of servant women fired when they became old. To help them, Faa` di Bruno created an institution dedicated to providing these women with respectable work and housing. Bit by bit he added other activities: A retirement home for old washerwomen, a school for female teachers, a high school. Faa` di Bruno had a modest apartment here, in which he kept his books and his collection of scientific instruments. When we look at Faa` di Bruno’s variety of interests and duties, we cannot but wonder how he could find the time to do serious work in mathematics. His favorite topic of research was the algebra of invariants according to the

Figure 14. Francesco Faa` di Bruno (1825–1888).

Figure 15. Faa` di Bruno’s apartment.

views of Cayley, Salmon and Sylvester. He wrote one of the best textbooks on the subject, the The´orie des formes binaires (1876), which was also translated into German (1881). Today, Faa` di Bruno is best known in connection with a complicated formula which gives the n-th derivative of composite functions (1857). When Faa` di Bruno died, the nuns did what they could to keep his memory alive, preserving his home and personal effects for future generations. His apartment is now a fascinating little museum, one not ordinarily seen by tourists. Faa` di Bruno’s rooms are almost like a time

Figure 16. The campanile.

capsule, preserved exactly as they were in the 1880s, with his hat and his cane still lying on the table. (It is an unsettling experience to look at a photograph of Faa` di Bruno taken immediately after his death and, at the same time, to stand in front of the very chair where his corpse was laid out.) One entire wall of his living room consists of bookshelves full of classics of mathematics (Fig. 15). Among the other exhibits on display is a fine collection of teaching aids for physics, chemistry and mathematics, and even some rare cameras from the 1860s. The Faa` di Bruno Museum can be visited only by appointment. Call or fax +39-011-489145, or send an e-mail to [email protected]. For more information, see the website www.faadibruno.com. In the middle of the complex of buildings making up the Istituto stands the campanile (i.e., the bell tower), a 75-mhigh edifice with yellow and white walls, designed by Faa` di Bruno himself and erected in 1876 (Fig. 16). The campanile is made up of three different parts: A lower square section, a middle octagonal section, and, at the top, a circular steeple. With its vivid colors and unusual forms, it contrasts strongly with the plain architecture of its surroundings. The architectural style of the campanile is difficult to describe, since it shows several different influences. Some of its design features were probably inspired by the Gothic Revival style that was popular at the time it was built. However, its overall appearance is quite different from other buildings of the same period. At first sight, the campanile appears much too tall and narrow to be stable—the base is a square of only 5 m by 5 m— especially considering that it was built at a time when reinforced concrete had not yet been discovered. It is not surprising that, originally, the project caused many objections. However, the building turned out to be structurally sound. Indeed, it was so sturdy that the only damage it suffered during the bombings of World War II was the loss of the angel decorating the top. The secret of this stability lies in the campanile’s innovative structure. Most notably, the belfry is not located at the top of the campanile, but 35 m up, or a little over halfway to the top. It is composed of 32 cast iron columns. These columns are a particularly unusual feature, since the use of cast iron in construction was almost unheard of when the tower was built. The belfry columns serve to join the upper and lower parts of the tower, making the campanile roughly equivalent to two rigid bodies connected by a spring. Unless an external force is periodic, with a period close to the natural frequency of the system, any dangerous oscillation will be rapidly attenuated. This mechanism for ensuring stability was then completely new, a triumph of applied mathematics. If you are still in the mood for mathematical memorabilia, you might want to take a look at the monument dedicated to the mathematical physicist Galileo Ferraris (1847–1897). Ferraris is mainly known for his work on the technical application of electricity, but he also wrote one of the first treatises on vector calculus (1895). The monument was made in 1903 by the sculptor Luigi Contratti. It is situated at the intersection of Corso Montevecchio and Corso Trieste, quite a long way from our main itinerary. On the pedestal there are two bas-reliefs, showing Ferraris meeting Helmholtz (1891) and Edison (1893). The bronze figure of a Ó 2010 Springer Science+Business Media, LLC, Volume 32, Number 4, 2010

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naked woman on the front side, representing the Electric Science, was considered quite risque´ in its time. We are now at the end of our walk. This guide to historical-mathematical Turin was written in the hope of being as informative as possible. But, for a few hours, why not go off the beaten track? With its many cultural attractions, Turin will amply repay even a casual exploration.

[2] Kennedy, Hubert Collings. 1980. Peano: Life and Works of

ACKNOWLEDGMENTS

[5] Giacardi, Livia and Clara Silvia Roero, eds. 1987. Biblioteca

Giuseppe Peano. Dordrecht: Reidel. [3] Marchisotto, Elena Ann and James T. Smith. 2007. The Legacy of Mario Pieri in Geometry and Arithmetic. Dordrecht: Reidel. [4] Goodstein, Judith R. 2007. The Volterra Chronicles: The Life and Times of an Extraordinary Mathematician, 1860–1940, Providence, RI: American Mathematical Society; London: London Mathematical Society.

I gratefully acknowledge the information about Faa` di Bruno supplied by the Suore Minime di Nostra Signora del Suffragio. For her assistance with the illustrations, my thanks go to Cristina Palermo. I also wish to thank Prof. Livia Giacardi (University of Turin) for her support.

mathematica: documenti per la storia della matematica nelle biblioteche torinesi. Turin: Allemandi. [6] Borgato, Maria Teresa and Luigi Pepe. 1987. Lagrange a Torino e le sue lezioni inedite nelle R. Scuole di Artiglieria. Pp. 3–200 in Bollettino di Storia delle Scienze Matematiche 7. [7] Eccher, Danilo, ed. 1995. Mario Merz. Turin: Hopefulmonster. [8] Gyllenberg, Mats and Karl Sigmund. 2000. The Fibonacci Chim-

REFERENCES

[1] Coolidge, Julian Lowell. 1904. The Opportunities for Mathematical Study in Italy. Pp. 9–17 in Bulletin of the American Mathematical Society 11.

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ney. P. 46 in The Mathematical Intelligencer 22, 4 (December, 2000). [9] Giacardi, Livia, ed. 2004. Francesco Faa` di Bruno: ricerca scientifica, insegnamento e divulgazione. Turin: Deputazione subalpina di storia patria.

Reviews

Osmo Pekonen, Editor

The Man Who Flattened the Earth by Mary Terrall THE UNIVERSITY OF CHICAGO PRESS, 2002, 408 PAGES, ISBN 0-22679360-5 (CLOTH) $48 REVIEWED BY ANDREW J. SIMOSON

short quiz: Name a half-dozen mathematicians of the eighteenth century. To make the quiz a little more difficult, we limit the list to include only those who did most of their work in the eighteenth century. By this rule we disallow both Gottfried Leibniz and Isaac Newton, who belong mostly to the seventeenth century, and mathematicians such as Pierre-Simon Laplace and Joseph Fourier, who belong more to the nineteenth century. Okay, time’s up. Who’s on your list? If your list is like most, it may include one of the Bernoullis, perhaps Johann or Daniel, but not Jacob, as he’s too old. Of course, almost everyone’s list will feature Leonhard Euler. A few might include Alexis Claude de Clairaut, as, among other things, he gave conditions for the equivalence of mixed partial derivatives and for the existence of integrating factors for solving first-order differential equations. One important mathematician of this period might fail to appear on almost anyone’s list. Who is that? Aha! As you are reading this review, you might guess the man who flattened the Earth, the title of a recent book by Mary Terrall. Pierre-Louis Moreau de Maupertuis (1698– 1759) was one of the more colorful and influential mathematicians of his time. As the reader learns in this book, Maupertuis was mentored by Johann Bernoulli at the start of his career and died in the home of Johann Bernoulli II, Johann’s son. In mid-career, Clairaut and Maupertuis were team members on a grand French Academy of Sciences expedition that captivated the imagination of the reading public of their time. And at the end of his career, as president of the Berlin Academy of Sciences, Maupertuis’s chief lieutenant was Euler. Mary Terrall’s overarching thesis is that Maupertuis, as a man of science, was motivated by ‘‘personal honor and ambition.’’ But how does one get inside another’s head? Despite his voluminous writings—Terrall catalogs 58 works in her book’s appendices—she observes that Maupertuis ‘‘was not an introspective person.’’ Therefore, Terrall argues indirectly by summarizing Maupertuis’s scientific arguments, by copiously translating passages into English (each of which is footnoted in the original), and by analyzing his actions.

A Feel like writing a review for The Mathematical Intelligence? You are welcome to submit an unsolicited review of a book of your choice; or, if you would welcome being assigned a book to review, please write us, telling us your expertise and your predilections

â Column Editor: Osmo Pekonen, Agora Centre, 40014 University of Jyva¨skyla¨, Finland e-mail: [email protected]

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As the reader learns, many of Maupertuis’s enterprises had minimal a priori chances for success. For example, his mentor, Johann Bernoulli, was notorious for ‘‘being easily roused to anger’’ and for having a ‘‘vituperative’’ and a ‘‘possessive jealousy’’ with respect to his methods. Why choose to start one’s career with a veritable volcano? Yet after nine months, Bernoulli says their time together was one of ‘‘revealing to [Maupertuis] the deepest parts of my small stock of wisdom, without hiding anything from him.’’ Secondly, in the French Academy of Sciences, Maupertuis championed the idea of geodesic expeditions, of measuring degrees of arc along the Earth’s surface to help determine the shape of the Earth. He then captained a geodesic expedition to Lapland, the region now divided between Finland and Sweden, putting his reputation literally on the line. Voltaire called Maupertuis’s subsequent account of the expedition ‘‘a story and piece of physics more interesting than any novel.’’ Incidentally, Voltaire went on to write a romance, called Microme´gas, in which an extraterrestrial giant stumbles upon Maupertuis and his team in the Baltic who then converse at length about how they know what they know (a new annotated translation appears in [4]).

Figure 1. Engraved portrait of Maupertuis by Jean Daulle´ of Tournie`res, courtesy of the Owen Gingerich Collection. 70

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Finally, Maupertuis, an extremely successful member of the Paris Academy, chose to be president of a fledgling Prussian Academy of Science. Such a career move was a risk to the nth degree. For in the aftermath of this gamble, in the midst of the Seven Years’ War between Prussia and France, Maupertuis was branded a traitor to France. Therefore, Maupertuis did not pursue ‘‘personal honor and ambition’’ out of a sense of entitlement, but out of risk-taking and the spirit of enlightenment, or as Terrall puts it, Maupertuis thrived on being part of a ‘‘circle of like-minded friends, where amusement and science came together in sociable and witty terms,’’ which in turn ‘‘pushed each other to take more decisive [and insightful] steps than they might have taken otherwise.’’ Since it will illustrate the idea of the book’s title, let’s see why Maupertuis championed a French expedition to Lapland, or to a region as far north as was then possible. In the Principia, Isaac Newton proposed a theory of gravitation and concluded that our rotating Earth must be flattened at its poles. He estimated [2] the difference Dr in equatorial radius q and polar radius R, to be 17.1 miles (where Dr = q - R). This calculation expressly contradicted the reigning view on the continent that the Earth was bulging at the poles, a view championed by the father and son team of astronomers Giovanni Domenico Cassini (1625–1712) and Jacques Cassini (1677–1756). Newton gave some data: In 1635, Richard Norwood measured 1 degree of arc along a meridian near London to be 57,300 toises, while the Cassinis at the turn of the century measured 1 degree of arc near Paris to be 57,061 toises. To convert to miles, take 1 toise as 1.949 meters and 1 mile as 1.609 km. Thus, the Norwood measure is 69.39 miles between, let us say, 51° and 52° N, and the Cassini measure is 69.10 miles between, let us say, 48.5° and 49.5° N. Based upon these data points, what is Dr ? The Cassinis and Newton agreed that the difference between q and R was relatively small and that the profile of the Earth was more or less elliptical. Had they chosen to do so, with a lot of work Maupertuis and his associates could have determined that the best radii guesses for q and R using Newton’s two old data points are 4000 and 3872 miles, giving Dr & 128 miles. Such a result means that Newton’s two old measurements are inconsistent. Of course, these measurements were taken years apart using different instruments and were very close in latitude, which was precisely Maupertuis’s argument for launching twin geodesic expeditions: One to the equator and one to Lapland. Technology had improved, and now was the time to settle this 40-year-old dispute. Contrasting Maupertuis’s leadership—of emceeing the measurement of one degree of arc along a meridian (1736–1737) in Lapland—versus the confusion of operations for the expedition sent to Peru (to the region now called Ecuador) lasting nine years (1735–1744), suggests that Maupertuis was a skilled administrator. Furthermore, after his return to Paris, he defended the team’s results decisively. Despite heated objections by Jacques Cassini and his allies about issues of accuracy, Maupertuis got along rather well with Cassini de Thury, the second son of the outraged astronomer. Thury subsequently remeasured Parisian arclength and ultimately vindicated the Lapland

team’s findings, thereby augmenting Maupertuis’s prestige. Of this accomplishment, Voltaire said that ‘‘Maupertuis had flattened the Earth and the Cassinis too.’’ To commemorate this event, Maupertuis commissioned a self-portrait, an engraved version of which is Figure 1. To demonstrate the improvement in the French 1735– 1744 geodesic measurements over Newton’s two old measurements, Maupertuis’s team measured 1 degree of arc between 66° and 67° N as 57,395 toises (69.52 miles), and the equatorial team—one of whose leaders was Charles Marie de La Condamine (1701–1774) who in returning home to Paris from Peru began by going down the Amazon to the Atlantic, collecting cinchona bark and seedling samples, which in turn led to a very effective kind of quinine—measured 1 degree of arc between 0.5° S and 0.5° N as 56,768 toises (68.75 miles) [1]. Using these two new arclength values gives q& 3974.0 and R & 3956.5 miles, for a difference of Dr & 17.5 miles—very close to Newton’s original estimate of 17.1 miles. (Earth’s actual measurements are q & 3964.1, R & 3950.8 and Dr & 13.3 miles.) Terrall’s secondary thesis is ‘‘that Maupertuis made a strategic move by writing in [a] hybrid genre,’’ namely, writing scientific ideas for the reading public, most representatively, Ve´nus physique. As one critic observed, ‘‘Our ladies have abandoned their novels to read it.’’ Other critics decried this unprofessional behavior as one ‘‘seeking fame and reputation, for being fashionable.’’ But such damnation has, over the years, transformed into approbation—and just as Voltaire is ‘‘a poet who writes geometry’’ and La Mettrie (Frederick the Great’s court physician) is ‘‘a doctor who writes about the soul,’’ so Maupertuis is a mathematician who writes about pleasure. That is, in Maupertuis’s words, ‘‘In spite of a thousand obstacles to the union of two hearts and a thousand torments that are bound to follow, pleasure directs the lovers to the goal nature intended.’’ Maupertuis’s personality fitted him well for this venture, for the public enjoyed following the literary exploits of an eccentric yet important savant. For example, Maupertuis ‘‘had a reputation as a libertine man-about-town, equally happy to consort with duchesses and their maids.’’ Of Maupertuis’s life-force experiments: ‘‘He threw salamanders into the fire to show that they burn, and allowed scorpions to bite dogs to test the effect of their venom; he enclosed scorpions with spiders to watch their battles.’’ He traced the genealogies of six-fingered men, conducted breeding experiments with his pet dogs, and maintained a houseful of exotic animals. Maupertuis describes this managerie, ‘‘You would not believe the multiplication of animals of all species I have at my home. When one has lived like this one finds almost as much stimulation from them as from people.’’ Any account of Maupertuis’s accomplishments is sure to include two items: The Lapland expedition, which we have already summarized, and the physical principle of least action compounded by the Ko¨nig affair. Maupertuis considered his work on the principle of least action to be his finest. Intuitively, the principle says ‘‘that nature acts as simply as possible,’’ and formally, that nature acts on matter so as to minimize a product-like combination of its velocity and position. Terrall amplifies these ideas at chapter-

length, giving, among other things, the example of how light in following Snell’s law follows the path of least time rather than least distance. However, Samuel Ko¨nig, a longtime friend of Maupertuis, accused Maupertuis of plagiarizing these ideas from Leibniz. As Terrall points out, ‘‘Following the dispute meant following a complicated trail of print, often mediated by journal articles and letters, as authors and publishers printed a bewildering array of old and new texts.’’ Terrall guides the reader through this labyrinth for 18 pages of spirited give and take. In the midst of the Ko¨nig affair, Maupertuis wrote Lettre sur le progre`s des sciences in 1752. As president of the Berlin Academy, in this open letter he proposes a number of projects for the scientific community to consider. His list is grand and sweeping, not unlike David Hilbert’s 1900 list of 23 problems with which the mathematical community might wrestle into the next century or two. Sprinkled amidst mostly sound proposals are these: Test new medical procedures on criminals, such as removing the kidney as a treatment for kidney stones; use opium to explore the mind; and raise a group of children in isolation from adults to determine the language they would develop. This last item jumps off the page. Was Maupertuis serious? Of course, one of the reasons that the Ko¨nig affair mushroomed beyond anyone’s expectations is that Voltaire entered the debate in defense of Ko¨nig. As Terrall points out, Maupertuis and Voltaire had been long-time friends. Both Maupertuis and Voltaire had had an affair with the same woman, E´milie du Chaˆtelet (1706–1749), whose life work was the translation of Newton’s Principia from Latin into French. Voltaire lived with E´milie for her last 15 years. Maupertuis had tutored both E´milie and Voltaire in mathematics. After Maupertuis’s return from Lapland, Voltaire recommended Maupertuis to Frederick the Great as candidate for leading the Berlin Academy, saying, ‘‘A man like [Maupertuis] would establish in Berlin an academy of science that would outdo the Parisian one.’’ After accepting the presidency of the academy, Maupertuis returned the favor to Voltaire, arranging a private first meeting between Frederick and Voltaire. But, a mutual friend to both Maupertuis and Voltaire predicted, ‘‘Maupertuis and Voltaire are not made to live together in the same room.’’ Terrall explains Voltaire’s ‘‘perverse’’ entrance into the Ko¨nig affair: ‘‘[Voltaire] was motivated by personal animosity, [cloaking his actions as] a self-styled enemy of tyranny and defender of press freedom. He concentrated on ridiculing Maupertuis as a tyrant and a buffoon, and on making fun of the more speculative parts of his works [such as Maupertuis’s Lettre as described above].’’ But to be fair to Voltaire, there’s more. Ko¨nig had been a two-year house guest of E´milie and Voltaire while serving as tutor helping her to understand the Principia. Furthermore, Voltaire had a habit of championing the underdog, even if the underdog advocated ideas contrary to his own. He also thrived on crossing verbal swords with almost anybody. As one example among many, of Jean-Baptiste Rousseau’s poem Ode to Posterity, Voltaire said that ‘‘it was unlikely to reach its destination.’’ Indeed, one of the reasons E´milie invited Voltaire to a long-term stay in the country was to protect Voltaire from himself, that is, from his almost uncontrollable wit. So in the Ó 2010 Springer Science+Business Media, LLC, Volume 32, Number 4, 2010

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midst of the controversy at the Prussian Academy of Science, when ‘‘[Maupertuis’s] more outlandish suggestions had become stock jokes at court’’ [3], Voltaire yielded to defending the underdog once again. To wrap up this Ko¨nig affair, Terrall concludes that ‘‘it did not incapacitate [Maupertuis],’’ as he continued to work on ideas of heredity and the nature of matter until his death. Mary Terrall’s book is the fruit of 20 years of work on showcasing a quasi-forgotten, yet prominent, member of the scientific community of the enlightenment. The man who flattened the Earth, who was called the first French Newtonian, who popularized science for the masses, and whose somewhat endearing nickname was the Flea— Terrall tells a fascinating story backed by interesting detail, careful citation and enlightening insight.

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REFERENCES

[1] Hoare, Michael Rand (2005). The Quest for the True Figure of the Earth. Surrey: Ashgate. [2] Newton, Isaac (1999). The Principia, translated by I.B. Cohen and A. Whitman. Berkeley: University of California Press. [3] Pearson, Roger (2005). Voltaire Almighty New York: Bloomsbury Press. [4] Simoson, Andrew J. (2010). Voltaire’s Riddle: Microme´gas and the Measure of All Things. Washington, DC: The Mathematical Association of America. Professor of Mathematics King College Bristol, TN 37620, USA e-mail: [email protected]

Mathematics and Music (Mathematical World, Vol. 28) by David Wright PROVIDENCE, RI: AMERICAN MATHEMATICAL SOCIETY, 2009, 161 PP., US$35.00, ISBN-10: 0-8218-4873-9; ISBN-13: 978-0-8218-4873-9 REVIEWED BY EHRHARD BEHRENDS

t is a commonplace that there are links between the world of mathematics and the world of music. But in the literature on these connections, the two areas play asymmetric roles. The reader is usually assumed to have some mathematical background: Mathematical terms and theories are used with little explanation. These investigations are hardly accessible to nonspecialists. Mathematics and Music is written in a different spirit. It reviews some basic concepts in both mathematics and music from the very beginning, presuming no background in either of these fields. It’s addressed to students of all fields who are interested in both subjects. The 12 chapters cover a wide variety of mathematical and musical themes. Chapter 1 is devoted to ‘‘basic concepts.’’ Here, the various sets of numbers are introduced (N, Q, etc.), and one also learns, for example, that the integers are well ordered, how to visualize functions by their graphs, and how an equivalence relation is defined. Basics for the musical counterpart include the translation of pitches to notes by the treble and bass clefs, musical intervals (for example, the fifth or the octave), and the use of accidentals. At the end of this chapter, cyclic permutations are introduced to explain how the different modes (Ionian, Dorian, etc.) can be derived from a single scale. Chapter 2 is concerned with ‘‘horizontal structures.’’ How are whole notes, half notes, and so on written, which symbols are used for rests, and how do dots change the length of a note? I never realized before that the length d of a note increases to d(2 - 1/2m) if the note is m-dotted, a fact proved here by geometric series. It is also explained that translation (resp. transposition, resp. retrogression) of patterns corresponds to replacing f(x) by f(x - c) (resp. f(x) + c, resp. -f(x)) for functions f. Let’s turn to Chapter 3: Harmony and Related Numerology. The mathematics starts with the algebraic structure of Z12 . In this setting, a major chord is just the sequence (4, 3, 5) of modular intervals. Similarly, diminished chords and many others are described and correctly translated to musical notation. (That is, one must write E # and not F in the major chord of C #.) Chapter 4 introduces ratios as equivalence classes which one can hear as pitches: The octave, pﬃﬃfor ﬃ example, corresponds to 2 : 1. Clearly, the number 12 2 plays an important role here: This is the ratio associated with a semitone. One also learns how intervals can be converted to cents and vice versa. Logarithms and the exponential function are introduced at the beginning of Chapter 5. The graphs are sketched and

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the basic properties summarized. This is used to transform the description of intervals from ratios to a measurement in semitones (‘‘the interval ratio r is measured in semitones by 12 log2r.’’) In Chapter 6, it is noted that the partition of the octave into 12 semitones is rather arbitrary: For every n 2 N one couldpconsider the n-chromatic scale based on the interval ﬃﬃﬃ ratio n 2. To be able to discuss this scale in more detail, Euler’s /-function is introduced (/(n) is the number of integers k \ n which are relatively prime to n): There are /(n) ways to generate the n-chromatic scale by considering k, 2k, 3k, ... (modulo n). In Chapter 7, the modular arithmetic continues, starting with the properties of N and ending with a number of basic algebraic notions: monoids, groups, homomorphisms, and so on. It is shown how modular arithmetic can be used to generate a 12-tone row in 12-tone music. Algebraic investigations are also central to Chapter 8; they culminate in the proof of the fact that Z is a principal ideal domain. It is noted in passing what prime numbers are and how one can find them with the sieve of Eratosthenes. It is then easy to describe how musical passages are built where patterns of m notes stand against patterns of n notes (with m ^ n ¼ 1). n-chromatic scales are studied in more detail in Chapter 9. For example, the 19-chromatic scale is appropriate if one wants to generate a scale by an interval of ratio 3. Calculus is introduced in Chapter 10. The e-d definition of continuity is given, and piecewise smooth and periodic functions are defined. As the main result, one learns that such functions have a Fourier expansion. With this background, it is easy to explain the importance of formants for the sound of instruments and the human voice. Chapter 11 starts with the old observation that two pitches played simultaneously sound ‘‘harmonious’’ when the ratio of their frequencies is rational with small numerator and denominator. It is shown how a scale constructed by using only a just fifth necessarily has a small imperfection, the comma of Pythagoras. And it is proved that it is impossible to avoid irrational numbers in the n-chromatic scales. Problems concerning tuning are investigated further in Chapter 12. For many centuries, various scales have been proposed: The problem is to have as many justly tuned intervals as possible, and, at the same time, intervals which sound ugly have to be avoided. Advantages and disadvantages of the Pythagorean scale, the mean tone scale and the equal temperament are discussed in some detail in this final chapter. Each chapter is complemented by (mostly mathematical) exercises of various difficulty, and understanding is facilitated by many graphics and musical scores. Wright has packed an ambitious overview into 150 printed pages. He had to make choices, and it is legitimate that he followed his own preferences. And, of course, it is a matter of taste which of the many aspects of the theme ‘‘mathematics and music’’ a reader will consider more fascinating, or less. The author writes that ‘‘the treatise is intended to serve as a text for a freshman college course.’’ This purpose is completely achieved. This book can be an inspiring basis 2010 Springer Science+Business Media, LLC, Volume 32, Number 4, 2010

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for lectures presented to all students. For readers with a mathematical background who are interested in music, the book will be a valuable resource. And even those who have studied music for many years may learn many new facts (formants, just fifths, equal temperament, to mention a few).

• J. Fauvel, R. Floyd, R. Wilson, (eds.): ‘‘Music and Mathematics: From Pythagoras to Fractals,’’ Oxford University Press, Oxford, 2003. • L. Harkleroad: ‘‘The Math behind the Music,’’ Cambridge University Press, Cambridge and the Mathematical Association of America, Washington D.C., 2007.

I also recommend these books on the same subject: • G. Assayag, H. G. Feichtinger, J. F. Rodrigues (eds.): ‘‘Mathematics and Music: A Diderot Mathematical forum, Lisbon, Paris and Vienna, December 3–4, 1999’’ Berlin, Springer, 2002.

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Free University of Berlin Arnimallee 2-6 Berlin, D-14195 Germany e-mail: [email protected]

Mathematicians Fleeing from Nazi Germany: Individual Fates and Global Impact by Reinhard Siegmund-Schultze PRINCETON AND OXFORD: PRINCETON UNIVERSITY PRESS, 2009, 472 PP. US $90.00, ISBN: 978-0-691-12593-0 (CLOTH); US $38.22, ISBN 978-0-691-14041-4 (PAPERBACK) REVIEWED BY G. L. ALEXANDERSON

his is an important book for mathematicians and other scientists, for those in the field of intellectual history, and for general readers interested in our not-too-distant past. I got so caught up in it I could scarcely put it down. It is well written and meticulously researched and documented. For many, there will be connections to personal experience. Anyone who spent the 1950s or even later in the vicinity of Stanford University will have encountered many in its cast of characters: Faculty at Stanford (S. Bergman(n), C. Loewner, G. Po´lya, G. Szeg} o, H. Samelson and M. Schiffer) and visitors (R. Courant, H. Lewy, O. Neugebauer, I. Schoenberg and S. Warshawski, among others). These people shaped the professional and personal lives of many Stanford students. Similar stories could be told of research universities throughout the United States at that time. This book follows by a few years another fascinating work, Mathematicians Under the Nazis, by Sanford Segal (Princeton, 2003), which covered those ‘‘German speaking’’ mathematicians who remained in Germany and the occupied countries during World War II. Siegmund-Schultze’s book covers, in a sense, the complement. Up until now, the literature on the e´migre´s has not been large, due to a variety of reasons: Archives that remained closed to scholars, unwillingness of some to speak on the subject because of political sensitivities, and the possibility that the reminiscences of the e´migre´s were sometimes unreliable due to the passage of time. For those interested in the subject, one of the best sources has been Max Pinl’s series of articles (some written with A. Dick) ‘‘Kollegen in einer dunklen Zeit,’’ that appeared in the Jahresbericht der Deutschen MathematikerVereinigung in the 1960s and 1970s. Siegmund-Schultze points out, however, that Pinl is incomplete and, on occasion, can be misleading, but he has high praise for Constance Reid’s biography of Courant and gives credit for much of his work to Courant’s papers at New York University. The author is extremely conscientious in defining the words he uses. For example, he does not use ‘‘National Socialism’’ because it was neither socialism nor national in character, nor does he use ‘‘Aryan,’’ ‘‘Third Reich,’’ and other such words, because after the war they carried too much additional baggage. He prefers ‘‘Nazis,’’ though sometimes he refers to ‘‘Hitler’s regime.’’ He also makes careful distinctions between those who left Germany or Austria before

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1933 or after, and between those who left ‘‘voluntarily’’ or those who were ‘‘forced.’’ The author decided to concentrate on ‘‘forced emigration’’ after 1933. He also relies most on e´migre´s fleeing racial persecution, though other groups were also leaving central Europe—pacifists, some Catholics and homosexuals. Some who left were able to reach ‘‘safe’’ countries like Sweden, Switzerland and England, though many came directly to the United States. Some were unable to leave or chose to stay, believing the situation would improve. Notable among the latter was F. Hausdorff who stayed, but eventually, with his wife and sister-in-law, committed suicide rather than face the death camps. These personal stories are heart-wrenching. There are many personal stories of death or hardship such as the grim task of travelling across Siberia to end up eventually on the West Coast of the United States (Max Dehn and Kurt Go¨del, for example). We also read here of the difficult questions of just how much the state of mathematics changed as a result of this mass movement of some of the most brilliant mathematicians of the time from one continent to another. The author warns of the post hoc, ergo propter hoc phenomenon: The widespread assumption that mathematics prospered in the United States as never before because of the infusion of all that talent. Though probably true, we have no proof that American mathematics might not have shown remarkable growth in any case. The war itself created jobs, particularly in applied mathematics, and this provided work for many American scientists. At the same time, to say that European mathematics declined in prominence only because of the emigration may be simplistic. These are provocative ideas and will surely be discussed for years to come. The author states that he is quite aware that his will not be the last word on the subject. In particular, he points out that the questions raised about G. D. Birkhoff’s alleged anti-Semitism could only be treated by a much larger biographical study of Birkhoff, well beyond the scope of this volume. He correctly argues, however, that good work is more easily done in a community of scholars: It is best to be able to communicate one’s ideas directly with colleagues rather than relying solely on reading published work. In this way the United States, and other countries, obviously benefited significantly from the emigration. Chapter One covers questions of terminology. Chapter Two is devoted to the extent to which the emigration affected mathematics more than some of the other sciences. The author observes that the United States accepted more e´migre´s by far than other countries, and a disproportionate number of these were mathematicians. One side-effect of the migration was to make English the lingua franca of mathematics, finishing off German as the international language of science. A complication in compiling statistics about the e´migre´s was that some mathematicians (notably E. Artin, K. Friedrichs, E. Kamke and H. Weyl) were not Jewish but were forced to leave their positions because their wives were. And we note, too, that age made a difference: The oldest of those who came to the United States (F. Bernstein, M. Dehn, H. Hamburger, E. Hellinger and A. Rosenthal) failed to get regular appointments. Among those who were eventually successful in locating positions 2010 Springer Science+Business Media, LLC, Volume 32, Number 4, 2010

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appropriate to their ability were those at the Institute for Advanced Study (IAS) in Princeton, or at Stanford, Berkeley or NYU. But with very few exceptions, mainly due to specialized fields of expertise like applied mathematics or history of mathematics, e´migre´s did not get regular positions at the leading departments at Harvard, Princeton, Yale, Chicago, Brown, MIT or Caltech. Chapter Three largely deals with e´migre´s to the United States prior to 1933, mainly motivated by the economic conditions in Europe. Some of these had illustrious careers in America: E. Hille, E. Hopf, T. Rado´, D. Struik, J. von Neumann and A. Wintner. Others from non-German speaking countries also came to the United States at that time: C. Lanczos, I. Sokolnikov, J. Tamarkin, S. Timoshenko, and T. von Ka´rma´n, and, in the critical year 1933, one of the most illustrious additions to the IAS faculty, H. Weyl. A striking table in Chapter Four shows that 90 of the 145 e´migre´s, and 130 of the 234 persecuted (including nonemigrants and those killed) came from only four of the 42 cities covered (from Berlin, 41 faculty members out of 62, and from Go¨ttingen 24 out of 28). The Hitler regime was remarkably effective in clearing out the best and the brightest. In this and the next chapter, we read letters and documents pertaining to those who succeeded in their efforts to emigrate as well as those who waited too long or were just plain unlucky. Among the latter were: O. Blumenthal and A. Tauber, who both died at Theresienstadt; Hausdorff, who was mentioned earlier; and F. Noether, who made the mistake of going to Russia, where he was executed by the Soviets. Chapter Six is devoted to those who emigrated to ‘‘safe’’ European countries, the Middle East, Australia or India. Many of these were eminent mathematicians, but the numbers were comparatively small, and some were also in transit to other destinations. In Chapter Seven the author addresses the attitudes of the e´migre´s following their move to the United States. Curiously enough, though they were grateful for having been saved from almost certain death in Germany, often they still held out hope that they could at some time return to Germany and the colleagues and institutions that had been hospitable to them early in their careers. Since Gauss, the German mathematical community had been extremely strong, with support from outside the universities by the government and publishers such as Springer, for example. Many e´migre´s retained their concern for the health of German science and culture. Germany was, after all, the country of Heine, Schiller and Goethe, Bach and Beethoven. A few mathematicians even returned (notably Eberhard Hopf and, at least temporarily, Carl Ludwig Siegel). Some who could have left Germany did not, for a variety of reasons. A prime example was the Dutch algebraist, B. L. van der Waerden, prompting Courant to write to him in 1945, ‘‘Your friends in America, for example, could not understand why you as a Dutchman chose to stay with the Nazis.’’ This criticism followed van der Waerden through the remainder of his long career. Some who left openly expressed their regret over leaving behind German culture—von Neumann and Feller, for example. Even 76

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Courant found it hard to give up his loyalty to Springer and advised Szeg} o to publish his Orthogonal Polynomials with that eminent publisher. (But Szeg} o did not agree, and it was published instead in 1939 by the American Mathematical Society.) There are many well-known stories here—Neugebauer’s eventually establishing Mathematical Reviews to substitute for the largely unavailable Zentralblatt during the war, and many not so well known The author explores at some length the reactions in the United States to the crisis in Europe (Chapter Eight) and the various committees and organizations set up to expedite the granting of visas and making the necessary arrangements for appointments, even when only temporary. He cites faculty at three institutions who stood out for their efforts: (1) The Institute for Advanced Study; (2) The graduate school at NYU under Courant; and (3) The graduate school at Brown under R. G. D. Richardson. On the other hand, there were those like G. D. Birkhoff who questioned the wisdom of hiring the e´migre´s during the Great Depression when native-born Americans were having such difficulties in getting jobs, a view expressed in Birkhoff’s well-known statement that American mathematicians would be reduced to being ‘‘hewers of wood and drawers of water,’’ on the occasion of the semicentennial of the American Mathematical Society in 1938. Some viewed the remark as clearly anti-Semitic. The American government under Roosevelt could have done more to speed up the process of getting the e´migre´s into the United States, but it was politically difficult because of the Depression and a strong wave of isolationism in the country. SiegmundSchultze, however, makes it clear that ‘‘it is imperative to stress that this kind of anti-Semitism cannot be compared, let alone put on an equal level, with the criminal, institutionally legalized and incited anti-Semitism in Germany after 1933.’’ These observations are supported by a large number of citations of documents and letters and are followed by an assessment of the effect of the immigration on mathematics in the United States (Chapter Ten) and an Epilogue. Much of this is concerned with the question of how well e´migre´s adjusted to American life. With so much attention paid to undergraduates in American universities, the European professors were disappointed in American students who needed background in mathematics that would have been covered in the gymnasium in Europe. Further, in Germany, professors had traditionally held a higher social position than was common in America. These conditions made adjustment difficult. The author quotes L. Coser: ‘‘The intimacy of the coffee house had to give way to the distance and strangeness of the American lifestyle, and so they were for the most part happy but not glu¨cklich.’’ Further, in a quotation from M. R. Davis, we read that, ‘‘Another bar between the foreign professor and his students was the difference in attitude which characterized the European as distinguished from the American professor. The former had developed to a fine art the technique of social distance from his students.’’ C. L. Siegel wrote, ‘‘I no longer have the hope, which led me to America four years ago, of finding a tolerable position abroad…. I can no longer adapt, I am too much of a Prussian.’’ He also wrote to Courant in 1935, ‘‘It

would be meaningless to escape the sadism of Go¨ring’s only to get under the yoke of Mrs. Eisenhart’s notion of morality…. Please do not be offended that I do not like your America.’’ (Luther Eisenhart was dean at Princeton.) These seem to be the exceptions, however. Most adjusted as best they could. Richardson wrote that Loewner, ‘‘the most distinguished mathematician in Kentucky,’’ held a position at the University of Louisville, teaching many sections of trigonometry each term. He eventually moved on to Brown and Stanford. Many had little choice. They had to adjust to whatever positions were open to them. There is no shortage of heroes in these pages, but one, for me, stands out: Charles Loewner, who, before he emigrated, travelled back and forth from Germany to Prague where he had received his Ph.D., because it was easier to get news out of Prague to American friends, reporting on the state of colleagues and asking for assistance in locating positions for them. In a letter here to L. Silverman at Dartmouth, quoted in its entirety, he makes the case for I. Schur, G. Szeg} o and S. Cohn-Vossen. The tone of the letter is consistent with my own impressions of Loewner, who

was one of the kindest people I have ever met and one of the most popular mentors and teachers at Stanford, where he advised the Ph.D. dissertations of 16 students. But he was quiet and self-effacing and never got the credit he deserved for his work for the e´migre´s and his very important mathematical contributions to the solution of the Bieberbach conjecture, for example. Almost as interesting as the text itself are the numerous appendices—lists of those who escaped, those killed, and those persecuted in various ways, along with many letters and documents. All around, this is a rewarding and impressive piece of scholarship, a story that is at once grim but also uplifting, since for many of those who escaped, there was a happy ending. Department of Mathematics and Computer Science Santa Clara University 500 El Camino Real Santa Clara, CA 95053-0290 USA e-mail: [email protected]

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Perfect Rigor by Masha Gessen BOSTON, NEW YORK: HOUGHTON MIFFLIN HARCOURT, 2009, 256 PP., US $26.00, ISBN: 978-0-15-101406-4 REVIEWED BY REUBEN HERSH

his book is a biography of Grigori (Grisha) Perelman, the Russian mathematician who is now famous for proving Thurston’s classification of 3-manifolds. As a corollary, he proved the Poincare´ conjecture—one of the outstanding open problems in mathematics. Thurston had conjectured, and proved in important special cases, that all 3-dimensional manifolds can be classified into combinations of 8 basic types, each of which can be represented geometrically using 3-dimensional nonEuclidean geometry. The simplest of these cases would just be the 3-sphere, which is the subject of Poincare´’s centuryold conjecture. In the course of telling about Perelman, Gessen tells much else that is of great interest. She leads us into the hidden inner life of ‘‘under cover’’ mathematics in the Soviet Union, including ‘‘special schools,’’ ‘‘math circles’’, and ‘‘math clubs’’. There, dedication to truth itself remained possible, for years on end, right under the noses of the Party and the KGB. All this was closely connected with the beneficent influence and inspiration of one man—Andrei Kolmogorov. He was, of course, a great international pioneer and researcher in many different fields of mathematics. But he was also the energizer and inspirer of a whole special Russian system of mathematical education and indoctrination for talented young people. Gessen paints an amazing portrait of him, hitherto quite unknown to me, including his long-time intimate friendship with the great topologist Pavel Sergeevich Alexandrov, and his dedication to an all-round life devoted to beauty and refinement, both cultural and physical. Masha Gessen has never met her subject, Grigori Perelman. Indeed, it seems that for a while now nobody at all has met him—except for his mother, Lyubov, who shares their modest apartment on the outskirts of St. Petersburg (formerly Leningrad). Gessen thinks that her never having met Perelman may have been an advantage in writing the book. She certainly seems to have met and thoroughly interviewed every major friend, acquaintance, and influence in Perelman’s life (except for his mother and his sister). As a result, she has been able to paint a convincing and fascinating psychological portrait of him that makes credible and understandable his refusal of the Fields Medal and the Clay Prize, and even his present total withdrawal, not only from the mathematics community of Russia and of the world, but even from almost all human contact. This life story raises deep, disturbing questions about the stresses and the values of a life entirely devoted to mathematics, especially in the world as it is today. Grisha’s mother Lyubov herself is mathematically gifted. In fact, she declined the offer of a position as a graduate

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student of mathematics in Leningrad in order to give birth to and nurture her son Grigori. When Grigori was 10 years old, she went back to her mentor Professor Natanson, to tell him that her son was mathematically talented. Natanson sent Lyubov and Grisha to Sergey Rukshin, a famous coach of mathematical problem-solving teams, and boss of a math club in St. Petersburg. It seems that Rukshin is more than just a famous math coach; he is the greatest math coach in the world. He has sent many contestants to the International Mathematics Olympiad. Rukshin and Grisha became inseparable companions. Under Rukshin’s coaching, Grisha actually did become one of the best, maybe the very best mathematical problem-solver in the world. First in Rukshin’s math club, and then in national and international competitions, Grisha seems almost never to have found a problem he couldn’t solve. In sessions of the math club, he sat quietly in the back. He was often the last to speak, for his solutions usually were clearly optimal. Nothing left out, nothing unnecessary, nothing open to challenge. While working on the problem, he might rub his leg, hum softly, or toss a ping pong ball back and forth. Not only did he solve the hardest problems, he then explained his solutions in perfectly clear, concise language to anyone who asked. His only difficulty seemed to be how to help anyone who failed to understand his clear explanation. In such a case, he seemed to have no recourse but to simply repeat the same explanation. As a boy, Grisha was reasonably fit physically. At some meetings of Olympiad contestants, he played volley ball with the others. But his mental energy seems to have been totally focused on mathematics, from early childhood until maturity. Another geometer, who was reported to have been Perelman’s friend during his stay in the United States, told Gessen that they often had conversations, and that the conversations were never on any topic except mathematics. Perelman did have one setback. The first time he competed in the all-Russian mathematics Olympiad, he came in second. This was a very severe shock and disappointment. Gessen writes that Grisha decided that he hadn’t worked hard enough in preparation. He resolved never again to allow such a mishap to occur. In fact, it never did. He always came in first, before and after that one ‘‘failure.’’ Like many other male mathematicians of relatively young years, Perelman gave little attention to matters of physical appearance. He always wore the same brown corduroy jacket. He did not waste time or effort about cutting his hair or his fingernails. With food he also preferred simplicity. It seems that while in the United States he rarely ate anything but bread and cheese. He did prefer one particular variety of black bread which he procured, while living and working in New York, at a bakery on the far south side of Brooklyn, at Brooklyn Beach. He would walk there after each day’s work at the Courant Institute in Manhattan. Gessen’s book gives a rather brief treatment of the Poincare´ conjecture itself. Many readers of this journal will know that the strongest attack on it had been made by Richard Hamilton of Columbia University. Hamilton used what he called ‘‘Ricci flow,’’ a nonlinear parabolic partial-differential equation satisfied by a certain geometric quantity associated to a 3-dimensional manifold. The time-evolution of the solution to the equation decribes a smoothing of an arbitrary 3-manifold. The smoothing action eventually would bring an

arbitrary manifold to a form recognizable according to Thurston’s classification. However, before reaching that stage, the evolution could get stuck by encountering a geometrical singularity, one of several possible kinds of singularity. To get past such a singularity, it was necessary to perform what topologists like to call ‘‘surgery’’—that is, a cutting and pasting operation which removes the singularity and renders the evolving manifold again sufficiently regular. Hamilton was unable to show that such surgery was always possible. Perelman succeeded in doing so. Complex, detailed geometrical and analytical reasoning permitted Perelman to provide the necessary surgery instructions to complete Hamilton’s Ricci flow program, thereby proving both the Thurston Classification and the Poincare´ Conjecture. Perelman never submitted his solution for publication in a journal. He posted three announcements on a well-known site intended for such early warnings of new results. He never even announced that he had proved Thurston or Poincare´, merely that he had obtained certain technical results about the Ricci flow. Those who are qualified to read his abstracts would understand their significance. Those who are not so qualified need not attempt to read them. Once the word got around to the ‘‘Ricci-flow community’’ and other interested topologists, they had to decide whether Perelman really had solved those problems. This was not very quick or easy, for his abstracts were concise, even in certain places perhaps a bit obscure. It took a year and a half for several teams of topologists to render the verdict—yes, he did it! During this process, Perelman spent time traveling the U.S., giving talks and answering questions. People found him well-prepared, patient and forthcoming. It was always clear that this work was very likely going to win a Fields Medal and a Clay Prize. As Perelman traveled, giving talks at elite math departments, he received job offers, some very favorable. However, he expressed very little interest in any of them. It is now clear that rather than being excited and flattered by this experience, Grisha was disappointed, repelled, perhaps even disgusted. This was not what he had expected, not what he was looking for. Hamilton did not seek him out, did not express great enthusiasm or gratitude to him. Others who wanted to talk to him about job offers at high salaries for little work did not seem to have even studied or understood his mathematical work. In fact, Grisha was becoming a celebrity, something it seems he had never sought, expected, desired, understood or valued. His celebrity status, even within the

academic community, seemed to outweigh and overbalance the actual content of his mathematical achievement. To Grisha, this was unattractive, unpleasant, even immoral. He practiced mathematics only for its own sake, he believed in mathematics only for its own sake. Mathematics for the sake of fame, money or power were alien to him, perhaps even incomprehensible. Certainly alien, repellent. Unclean. Degenerate. In Russia also there were unpleasant incidents involving money, and horrible surrounding incidents by the Russian press media. Grisha quit his position at the Steklov Institute. There was a kind of embarrassment—I wouldn’t say a scandal—when Shing-Tung Yau, one of the greatest living geometers, seemed to try to squeeze some of the credit for the proof of the Poincare´ conjecture from Grisha toward two of his prote´ge´s—possibly for the sake of political clout in the People’s Republic of China. Then Sylvia Nasar and David Gruber managed to get Grisha to spend time with them in St. Petersburg, and published a somewhat sensational article in The New Yorker. Of course Grisha refused the Fields Medal, refused to attend the International Congress of Mathematicians, and finally refused one million dollars from the Clay Institute. In her book, Masha Gessen reports that Grisha has now broken off from his lifelong friend and mentor Rukshin. He has told people he is looking for something new to do instead of mathematics. He continues to live in their apartment with his mother. Masha Gessen devotes one chapter of her book to the topic of Asperger’s Disorder, a form of autism disproportionately found among mathematicians. She never actually suggests that Grisha Perelman suffers from Asperger’s. Whether he does or not is a medical question. But there is an issue here of good taste and good manners. People may wonder about such things and talk about them privately. Decent consideration for the feelings of the subject of her book would have suggested abstaining from publishing such a chapter. Much more important is the cultural and moral question which this story forces one to ask. Does today’s world have room for a mathematician who practices mathematics for its own sake, and only for its own sake?

1000 Camino Rancheros Santa Fe, NM 87505 USA e-mail: [email protected]; [email protected]

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Mythematics: Solving the Twelve Labors of Hercules by Michael Huber PRINCETON, OXFORD: PRINCETON UNIVERSITY PRESS, 2009, XIX + 183 PP., US $24.95, ISBN: 978-0-691-13575-5 REVIEWED BY JOHN J. WATKINS

ichael Huber is not the first author to have been inspired by the enduring myth of the twelve labors of Hercules. In 1947, Agatha Christie published The Labors of Hercules, a novel in which her brilliant Belgian detective Hercule Poirot decides he will ease his way into retirement by solving precisely twelve final cases: cases selected only with reference to the ‘‘twelve labors of ancient Hercules.’’ In the hands of Dame Christie the foes of the mighty Hercules are wonderfully transformed. The fearsome Nemean lion becomes for Hercule Poirot a small Pekinese dog; the awe inspiring flock of Stymphalian birds becomes two ominous women with long curved noses dressed in cloaks walking by a lake at a European resort; the filth of the Augean stables becomes instead a political scandal at the very highest level of government, a mess which Poirot is called upon to clean up; and appropriately in his final ‘‘labor’’ Poirot is forced, as was Hercules, to deal with an all-too-real Cerberus guarding the gates of Hell. Each chapter in Mythematics: Solving the Twelve Labors of Hercules, by Michael Huber, is also based on one of twelve tasks imposed upon Hercules by Eurystheus. Hercules was born the son of the god Zeus and the mortal woman Alcmena. From infancy, the jealous wife of Zeus, Hera, had but one goal, the destruction of Hercules, and she almost succeeded. Hera was able to eventually drive Hercules mad and he murdered his own three sons. Hercules was thus forced into exile to serve Eurystheus and perform twelve labors. Upon the completion of these labors, he would become immortal. Each chapter of the book follows the same general format and begins with a quote from Apollodorus, the most reliable author of ancient times who wrote about Hercules and his labors, describing the particular task assigned to Hercules. Huber then uses this task as a springboard from which to pose three or four mathematical problems for the reader to attempt. Next, he provides detailed solutions for these problems and also—in passages that are by far the most entertaining sections of the book—elaborates further on the characters and stories from Greek mythology. There is much to be admired in this book. Michael Huber, who teaches mathematics at Muhlenberg College in Pennsylvania, has a real passion for Greek mythology and a creative flair for making connections with a wide range of mathematical topics. This book could be used in many ways. Its most obvious use will be as a source of lively versions of familiar problems that can be used in fresh new

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ways in courses. Or, more ambitiously, I can imagine using this book as the main text in an interdisciplinary course that is co-taught by a mathematician and a classicist where the goal is to introduce students simultaneously to the ancient Greek world and also many of the varied fields of mathematics. This is a course I would truly love to teach. Hercules’ first task is to bring back the skin of the Nemean lion, and he attempts to shoot the lion with an arrow. Huber uses this episode to pose a pair of routine questions: What is the speed at which an arrow strikes the lion at a distance of 200 meters given a launch angle of 20 degrees, and how long does it take the arrow to travel the distance from the bow of Hercules to the invulnerable lion? Huber does ‘‘solve’’ this problem in that he correctly finds the speed at which the arrow leaves Hercules’ bow (about 200 kilometers per hour) and also the time of travel, but he never gets around to saying how fast the arrow is going when it strikes the lion. Of course, the answer is ‘‘about 200 kilometers per hour’’ (here I would invoke conservation of energy, but one could also plug the time of travel into the velocity function to compute this speed). Unfortunately, all Huber says on the matter is ‘‘the speed of the arrow remains constant in flight’’, which of course is utter nonsense. So, while this book is both entertaining and at times inspired, it does need to be used with some care. Hercules’ third labor deals with capturing the Cerynitian deer. Huber turns the first part of this tale into a familiar problem in optimization. The deer, in trying to escape from Hercules, must swim across the Ladon River (which is 250 meters wide) and reach shelter in a forest 1600 meters along the shore on the other side. Of course, the deer runs faster than she swims (8 meters per second versus 5). Where should she land in order to reach the forest as quickly as possible? The artificiality of this particular problem reminded me of a similar problem I came across a few years ago in a new calculus book touting applications to biology and one ‘‘applied’’ problem involved a duck wishing to get from point A to point B as quickly as possible. This mathematically inclined duck could fly at a certain speed over land but could fly faster over water due to an often observed phenomenon whereby water birds fly extremely close to the surface of the water in order to increase efficiency. I was also somewhat bothered in Huber’s version of this problem by his unrealistic assumption that the deer maintained a constant swimming speed of 5 meters per second independent of the angle at which she was swimming relative to the river’s current. Once Hercules captures the deer (presumably by anticipating its landing point) he must carry the deer back to Eurystheus in Mycenae. Huber asks the reader to determine the work needed to carry the deer a distance of 80 kilometers given that the mass of the deer is 125 kilograms. He computes the animal’s weight (a vertical force) and multiplies this force by 80 kilometers (a horizontal distance) to get a completely meaningless answer of 98,000,000 newtonmeters (this is in fact the amount of work it would take to haul this deer to the top of a tower 80,000 meters high!). Huber makes a similar blunder about work later in the book when, having just computed the mass of the earth, he asks, ‘‘How much work does Hercules do in placing the earth on

his massive shoulders?’’ Leaving the reader to answer this question, he quips: ‘‘No wonder Atlas was tired of holding up the earth.’’ But, of course, it takes no work at all for Atlas to hold the earth in one place, even if its mass is 6 9 1024 kilograms. In a good problem that is typical of the sort of modeling problems that Huber favors, Hercules has shot his friend Chiron in the knee with a poison arrow. However, Hercules can administer a protective antidote at 5 minute intervals. Huber models Chiron’s immune system by p(t + 1) = .75p(t) + .1 (that is, his immunity is breaking down continuously by 25% every 5 minutes but also the medication provides an instantaneous boost). At this point, Huber asks the question: How long before Chiron’s immune system falls below the .5 level? (This is where he will constantly be in great pain.) Now, since p(0) = 1 is the starting level for his immunity, the most natural thing for students to do is a few iterations of this function and they quickly discover that p(6) = .5068 and p(7) = .4801. So, just prior to 30 minutes his immunity falls briefly below .5 (before the antidote again brings it back above .5), but after 35 minutes it will remain below the .5 threshold. Note that as long as p(t) [ .4 this function will decrease. But, instead, without ever saying what it is that he solving, Huber says the ‘‘solution’’ is p(t) = c(.75)t + d. Of course, what he intends by this solution is a continuous function that agrees with the original discrete function p(t) modeling Chiron’s immune system for t = 1, 2, 3, . . . . Yet he never explains this strategy nor how the form of this particular continuous function is arrived at. He merely solves for c and d and checks that this function then agrees with the values p(1) and p(2). (Of course, countless other continuous functions also agree with p(1) and p(2) without necessarily agreeing with the other values p(3), p(4), . . . .) He also provides a nice graph purporting to represent Chiron’s immune system protection level, but since this graph exhibits no step-function behavior, it is instead a graph of his continuous approximation p(t) = .6(.75)t + .4. Huber uses the 2,000-year-old The Greek Anthology as a source for several of his problems. Here is a nice combinatorial problem he adapts slightly to suit his needs. Hercules calls for wine and the centaur Pholus poses the following problem. Five centaurs have 45 jars of wine, of which 9 each are full, three-quarters full, half-full, one-quarter full, and empty. The centaurs want to divide the wine and the jars without transferring the wine from jar to jar in such a way that each centaur receives the same amount of wine and the same number of jars, and so that each centaur also receives at least one of each kind of jar and no two of them receive the same number of every kind of jar. Can the wine be so divided? Another problem taken from The Greek Anthology is related to the Labor of the Augean Stables and asks us to find how many herds of cattle Augeas, the king of Elis, had. Hercules the mighty was questioning Augeas, seeking to learn the number of herds, and Augeas replied: ‘‘About the streams of Alpheius, my friend, are half of them; the eighth part pasture around the hill of Cronos, the twelfth part far away by the precinct of Taraxippus; the twentieth part feed in holy Elis, and I left the thirtieth part in Arcadia; but here you see the remaining fifty herds.’’ Huber, as is common,

tends to treat these as simple problems in algebra, but I prefer to use basic ideas about numbers that were certainly well known at the time The Greek Anthology was written. I feel this is more in the spirit in which these problems were intended. We are told that the number in question is divisible by 2, 8, 12, 20, and 30; hence, by 8, 3, and 5. Therefore, the number of herds is either 120, or a multiple of 120. A simple check shows that 120 is not the number (since 60 + 15 + 10 + 6 + 4 + 50 = 120) but that 240 is the number of herds (since 120 + 30 + 20 + 12 + 8 + 50 = 240). Huber also treats another problem from The Greek Anthology as an algebra problem that I suspect may well have originally been an Egyptian problem about unit fractions. In this problem, three Hesperides pour water into a tank at varying rates, and he asks how long it will take the three together to fill the tank. The rates at which the three women are pouring water are, respectively, 12 ; 14 ; and 16 (of a 1 tank per hour) and the final answer is that it takes them 11 of a day (one day = 12 hours) to fill the tank; all of these fractions are unit fractions. Huber takes the story of how Hercules made the river Strymon unnavigable by filling it with rocks and turns it into a nice mathematical problem about the relative degree to which various lattice structures fill space. He chooses to compare three cubic lattices: simple cubic, face-centered cubic, and body-centered cubic. In each case, Hercules stacks equal-sized spherical boulders in the appropriate lattice pattern. Then Huber compares the packing factor in each case. This is extremely well done, and I can just imagine the mighty Hercules sitting in the middle of the river Strymon like a child playing in a sandbox, stacking massive spheres of stone and creating beautiful cubic lattices. Huber can be very inventive, not so much in terms of creating original problems, but by the way in which he can associate standard mathematical problems with the Hercules myth in quite surprising ways. Most ancient sculptures of Hercules show him holding three apples in his left hand. These are the golden apples he was required to fetch from the Hesperides in his eleventh labor. To get the apples, Hercules first relieves Atlas of his burden of holding up the earth and sends Atlas to fetch the apples; then, when Atlas returns with the three apples, Hercules tricks Atlas into once again accepting the responsibility for holding up the earth while Hercules departs with the apples. This provides Huber with a convenient jumping off point for a problem in basic physics: What is the mass of the Earth, given the following three pieces of information: i) the radius of the earth is 6378 kilometers; ii) the acceleration due to gravity is 9.8 meters per second squared (a critical point Huber neglects to make is that this is valid only at the surface of the earth); iii) the universal gravitational constant is 6.67 9 10-11 meters per kilogram per second squared (this quantity, whose value even at this level of precision is still in doubt, was first determined by Henry Cavendish in 1798; unfortunately Huber twice refers to this universal constant as ‘‘earth’s gravitational constant’’). At the end of every fourth chapter, Huber offers readers a reward for our own labors in the form of sudoku puzzles thematically related to a task just completed by Hercules. The first such puzzle follows the Labor of the Erymanthian Ó 2010 Springer Science+Business Media, LLC, Volume 32, Number 4, 2010

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Boar and replaces the numbers 1-9 with the letters A,E,H,I,M,N,R,T,Y (and the solution contains a very pleasant surprise). His second sudoku puzzle follows the eighth labor, which called upon Hercules to bring the four mares of Diomedes to Mycenae. (Agatha Christie turned these savage mares into four high-spirited young women in an English country house.) Huber presents us with a clever variation on a standard sudoku puzzle: a ‘‘knight’s puzzle’’, in which a single number is placed in the center, then sixteen more numbers are symmetrically arranged in groups of four, each group itself forming a knight’s move in chess. The additional condition in this delightful puzzle is that no two squares a knight’s move apart can contain the same number. I must confess that I then skipped ahead to the last sudoku puzzle long before I had completed the twelfth labor. The special wrinkle in this puzzle was appropriately inspired by the three heads of Cerberus, and so uses each of the numbers 1–6 exactly once, but the number 7 three times in each row, column, and 3 9 3 block. The twelfth labor imposed on Hercules was to bring Cerberus from Hades. Huber characterizes Hercules’ descent into the Underworld using a multivariable function f(x, y) = -2x + y2 - 2xy to represent the terrain and asks at various points along the way whether Hercules is ascending or descending and in which direction he should travel to descend as quickly as possible. Then, since Hercules is required to capture Cerberus with his bare hands he decides to strangle the three-headed beast (in another Greek myth, Orpheus chooses a method I much prefer: he lulls him to sleep with music from his lute). Huber models Hercules’ attempted strangulation by assuming there are 6 milliliters

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per second of blood flowing in each of the dog’s three brains to begin with and that, when Hercules is strangling one head, the blood flow is reduced by 7 percent per second. This yields a differential equation db/dt = -.07b. Hercules hangs on until the blood is less than 2 milliliters and then grabs another head to strangle, at which point of course some blood begins to return to the head just released. The dramatic tension Huber creates is whether Hercules can succeed in getting the blood flow in all three brains below the critical threshold of 2 milliliters per second, thus subduing the beast and completing his final task. Well, perhaps not his final task after all. As Huber explains in an appendix about the authors of the Hercules myth, the twelve labors sometimes vary in order and detail, and even the stories may not be the same from author to author. A version describing the deeds of Hercules in The Greek Anthology adds a thirteenth ‘‘labor’’ which neither Agatha Christie nor Michael Huber found suitable to include among their own collections of ‘‘the labors of Hercules’’: First, in Nemea he slew the mighty lion. Secondly, in Lerna he destroyed the many-necked hydra. Thirdly, after this . . . . Twelfthly, he brought to Greece the golden apples. in the thirteenth place he had this terrible labour: In one night he lay with fifty maidens.

Department of Mathematics and Computer Science Colorado College Colorado Springs, CO 80903 USA e-mail: [email protected]

Life After Genius by M. Ann Jacoby NEW YORK/BOSTON: GRAND CENTRAL PUBLISHING, 2008, 400 PP., US $24.99, ISBN: 9780446199711

Monster’s Proof by Richard Lewis NEW YORK: SIMON & SCHUSTER, 2009, 288 PP., US $15.99, ISBN-10: 1416935916, ISBN-13: 9781416935919 REVIEWED BY ALEX KASMAN

brief plot summary of M. Ann Jacoby’s Life After Genius may sound quite familiar: A young man escapes from a small town and the family business but then returns to both when he fails to achieve his dreams of success in the big city. However, there are a few unusual twists in the story of Theodore Mead Fegley. For instance, the Fegley family business is a combination of a furniture store and a funeral home, with many aspects of the latter described in gory detail. It is also unusual that, rather than being a businessman or athlete, the protagonist of this book is a prodigy who will be graduating from a major university at the age of 18 and presenting his research on the Riemann Hypothesis to an audience of prestigious mathematicians. Another unusual feature of this book is that the reader is focused on understanding past events rather than on seeing what will happen in the future. Since the chapters are presented out of chronological order, we know from the start that Mead quit school just days before his graduation and research presentation, but not why he would throw away years of work and tuition in this way. Similarly, for much of the book we know of the death of his cousin, but not how he died nor why Mead feels responsible. Even though there is no murder and no detective, these puzzles give the book the feeling of a mystery novel. There is not much more I can say about these plots and subplots without damaging the effectiveness of this clever literary technique. Instead, we will be concerned here with analyzing what this book says about mathematics and about the nature of genius. The Riemann Hypothesis is currently the most famous open problem in mathematics. With any even, negative integer as input, the zeta function outputs the value zero, and so these input values are ‘‘roots’’ of the function. It was conjectured by the great Bernhard Riemann that, aside from these, all of the roots of the analytic continuation of the zeta function are complex numbers with real part equal to 1/2. Since the zeta function can be written as an infinite product involving the prime integers, a proof of the conjecture would have implications for the distribution of primes. On the other hand, finding even one root that does not have the conjectured property would disprove it. And so, much work has gone into finding the roots of zeta. As Jacoby

A

mentions in the novel, Alan Turing was one person who worked on automating their computation, finding more than a thousand of them. By the late 1960s, the number of computed roots reached into the millions, and by now the number computed is much higher even than that. Needless to say, all of the roots found so far support the conjecture. Aside from this ‘‘statistical evidence,’’ there are some notable theoretical results relating to the Riemann Hypothesis. Hugh Montgomery studied the distribution of zeta roots in the 1970s, and physicist Freeman Dyson pointed out an unexpected connection between Montgomery’s work and distributions of energy levels in mathematical physics. More recently, Michael Berry built upon this connection to physics by showing a more specific relationship between the distribution of zeta roots and the transition to chaotic dynamics. Rather than creating completely fictional mathematical results for Life After Genius (a task which would require great imagination as well as expertise in mathematics), the author ascribes some of these real mathematical results to Mead Fegley. In particular, he is shown traveling to Princeton to use a supercomputer where, supposedly, he is the first person to compute a large number of zeta function roots. Later, by chance, he stumbles upon a physics paper containing familiar formulas and recognizes the connection between the Riemann Hypothesis and chaos. Among the people who are said to be attending the planned presentation of his discoveries are Hugh Montgomery and Michael Berry. Perhaps the inclusion of the names of these people who really contributed to the field is a sort of apology/ acknowledgement from the author. (This would explain the anachronism of including Berry’s name among the experts on the Riemann Hypothesis, since the book takes place around the year 1980 and Berry had not yet published any work on this topic.) The first few times mathematics was discussed in Life After Genius, it was done so smoothly I was certain that the author had advanced mathematical training herself. However, the discussion of infinite series between Mead and his first major advisor at college convinced me otherwise. Other revealing errors include the suggestion that Number Theory (a very large and ancient branch of mathematics) grew out of the Riemann Hypothesis and is nothing other than the attempt to prove it, reference to the ‘‘function plane,’’ and the common mistake of describing the Riemann Hypothesis as ‘‘an equation’’ which needs to be solved. Also, the author refers several times to the periodicals that Mead consults in preparation for his presentation: The Mathematical Intelligencer and the American Mathematical Monthly. I hope I do not offend these fine publications, when I claim that they are not the right sources to utilize in preparing a research talk on the Riemann Hypothesis to an audience of experts on that subject! But why am I being so critical? Overall, Jacoby demonstrates a reasonable understanding of the basic concepts. She recognizes that the key point is to determine the location of the roots of a certain function, and that a single counterexample would disprove the conjecture while a reasoned argument would be required to prove it. Dwelling on the mathematical errors, irrelevant to the plot or literary Ó 2010 Springer Science+Business Media, LLC, Volume 32, Number 4, 2010

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quality of the work, is precisely what Mead Fegley would do! Mead is especially insecure, competitive, confrontational, and cynical. When dealing with a kind offer from others, he is immediately suspicious of their motives. When other people make a mistake in logic or grammar, he is sure to point it out, as if their failure improves his own stature in comparison. The book does implicitly offer some explanation for these behaviors. The other children in his town are often particularly mean to him. His mother is overbearing. He cannot hope to share in the popularity of his charismatic and athletic cousin. Because he has skipped years in school, he is always younger than his classmates and misses out on some socialization. Of course, simply being rude and unpleasant alone are not enough to qualify a person as a ‘‘genius.’’ The title and book jacket certainly suggest that this book is a character study of a genius; this is the word people in Mead’s hometown use to describe him. What does this mean? It is interesting to note that Mead never does anything particularly brilliant in Life After Genius. Starting college at age 15 and graduating at 18 is impressive, but here it appears to be a consequence of the fact that his social failure left him more time for school work. Using a formula given to him by his professor to compute roots of the zeta function by hand or with the aid of a computer demonstrate hard work, but require no insight. Stumbling upon a misshelved physics paper that makes use of a distribution function he recognizes from a different context is pure luck. Is this a deliberate statement on the part of the author, arguing against the idea that a genius is a person whose mind functions in a fundamentally different way, or does she not realize how unimpressive Mead’s achievements seem? Mead does make fun of the concept of ‘‘genius’’ when he teases the naive people in his small town, telling them that geniuses have a body temperature one degree lower than that of ‘‘ordinary people.’’ In fact, aside from being rather anti-social and having a lot of time to devote to his studies, Mead seems essentially to be an ordinary person. If Jacoby does indeed intend to offer Mead as a counterexample to the common view that geniuses are so different from ordinary people as to almost be a different species, then I would certainly be sympathetic. I believe that exaggerated anecdotes about geniuses, and a bias for people with notable ‘‘quirks’’ to be described as geniuses when seemingly ordinary people with equally impressive intellectual qualifications are not, result in a somewhat unrealistic image of what the word really means. However, there is one thing that leads me to suspect that the reader is supposed to view Mead as being a ‘‘true genius.’’ Like John Nash in the film A Beautiful Mind and Catherine in the Pulitzer Prize-winning play Proof, Mead has discussions with people who are not really there and cannot easily distinguish when this is taking place. (He seems to realize that his discussions with Bernhard Riemann are figments of his imagination, but is less sure about similar conversations with classmates, professors, and relatives.) So far as I know, no real famous mathematician has had this problem. (Even John Nash, who truly does suffer 84

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from schizophrenia, did not have delusions of such conversations. This was a clever and effective plot device invented specifically for the film.) I have certainly never heard anyone outside of this book suggest that Albert Einstein, the twentieth century’s canonical genius, had such delusions. Yet, one of the men Mead meets in Princeton tells him that as a young boy delivering newspapers, he saw Einstein having a conversation with someone who was not really there. Since she gives both Mead and Albert Einstein this trait shared by other fictional mathematical geniuses, Jacoby probably does intend readers to put Mead in this category. In the book, there are professors and other students who try to benefit from Mead’s discoveries even though they did not really contribute to the work. Probably, we are supposed to see them as not merely unscrupulous, but also as being unable to have done what Mead did on their own since they are not geniuses like him. If so, her inability to convey Mead’s exceptional mathematical brilliance is a flaw in an otherwise enjoyable and thought-provoking book. Like Life After Genius, Monster’s Proof by Richard Lewis features a mathematical prodigy who is teased and abused by his classmates, as well as detailed discussions of the Riemann Hypothesis. Yet, despite these similarities, the books could hardly be more different. In contrast to the realism and adult themes in Life After Genius, Monster’s Proof is a book for young adults making use of elements of both fantasy and science fiction. The general plot outline here is also likely to sound familiar, this time in the Frankenstein tradition. By the very act of proving a conjecture, young prodigy Darby Ell releases a powerful being of pure mathematics on an unsuspecting world. This entity, who goes by the name of ‘‘Bob’’ and says he comes from Hilbert Space, seems benevolent at first but is soon recognized as being a significant danger. The source of tension here is not what has happened but whether Darby (with help from an angel, a demon, and his grandmother) will be able to save the world from his own creation. Unlike Mead Fegley, Darby is a trusting and kind boy. If Mead’s upbringing is supposed to be responsible for his antisocial tendencies, then perhaps Darby’s more pleasant demeanor is a reasonable consequence of one major difference. Although Darby is also subjected to humiliation by his classmates and also faces competition from unprincipled academic competitors, Darby is only one of a family of geniuses. His grandmother, father, and mother are also all brilliant scientists and mathematicians. (Only his teenage sister is ‘‘normal.’’) Consequently, he would have emotional support that Mead lacked. While working for the United States government’s top secret nuclear weapons program, Darby’s grandmother made a conjecture about an unusual Hilbert Space, which she called the ‘‘thingamabob’’ conjecture because she did not quite understand what it was. Recognizing the potential risks that it entailed, she left it unproved. But, at age 10, Darby Ell is able to prove this conjecture that had stymied teams of top mathematicians at the National Security Agency. This demonstrates more than just academic excellence and hard work. Even though he had some help from ‘‘Bob’’ (who recognized and encouraged the boy’s brilliance

in the hopes that he could prove the conjecture and release him in this universe), Darby Ell seems to better capture what I would mean by ‘‘genius’’ than does Mead Fegley. A much broader range of mathematical topics are touched upon in Monster’s Proof than in Life After Genius. In addition to the Riemann Hypothesis, we see a continued fraction expansion of 4/p, the Mandelbrot set, a Pythagorean cult, operator theory, a classroom discussion of why the product of negative numbers is positive (which the teacher handles very poorly), and many popular mathematical jokes and anecdotes. I believe this reflects not just the greater freedom of the genre, but also the author’s greater familiarity with the subject. Aside from odd descriptions of the Thingamabob Conjecture itself, which can be forgiven as being entertainingly cute even if mathematically nonsensical, all of the mathematics in Monster’s Proof is essentially correct. For a cheerleader who is failing algebra, Darby’s ‘‘ordinary’’ sister gives a surprisingly nice summary of the Riemann Hypothesis: ‘‘The Hypothesis was this incredibly exciting idea that all the zeros of something called the zeta function were on a straight line. Well, excuse me, she thought, the nontrivial zeros.’’ In a subplot, ‘‘Bob’’ and Darby’s father work together on the Riemann Hypothesis. When they eventually disprove it, the father seems to be crushed. This may be based on the common misconception that it would somehow be a horrible thing for mathematics if the hypothesis were false. In any case, ‘‘Bob’s’’ reaction is to say that within mathematics, beauty is truth, and so if they have shown that there is a nontrivial root of the zeta function off of the critical line, then this is a beautiful thing and should be appreciated as such. The more important mathematical subplot in Monster’s Proof is the Thingamabob Conjecture itself, which involves ‘‘the Hilbert Space of all Hilbert Spaces’’. This is a cute idea,

reminiscent of the notion of ‘‘the set of all sets,’’ which seems reasonable until one considers Russell’s Paradox. However, a Hilbert Space is required to have algebraic and topological properties that the set of all Hilbert Spaces is not likely to have. In a fantasy/science fiction story like this, such technical details should not be a problem, and once one ignores this concern (and other similarly cute definitions, such as the operator on the Hilbert Space whose action is defined in analogy to the behavior of sharks), the mathematical work shown in this book is probably more realistic than that in Life After Genius. In particular, the mechanics of working towards a proof used by the characters here, not by rote computation or finding formulas in previously published papers, but by building up a sequence of logical arguments, is what research mathematicians really do. Here we have considered two relatively recent additions to the library of ‘‘mathematical fiction.’’ That the author of Life After Genius lacks the mathematical background to get all of the technical details correctly and misses an opportunity to explore what is ‘‘genius level’’ mathematical research does not detract from the book’s thought-provoking analysis of the affect of the environment on the personality of a young mathematician (and vice versa). Monster’s Proof, which devotes a lot of pages to the romance between a cheerleader and a demon, does not address such sober topics, but it is a fun book and actually does a better job of conveying mathematical ideas to the reader. I recommend both books. Department of Mathematics College of Charleston Robert Scott Small Building, Room 339 Charleston, SC 29424 USA e-mail: [email protected]

Ó 2010 Springer Science+Business Media, LLC, Volume 32, Number 4, 2010

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Stamp Corner

Robin Wilson

Recent Mathematical Stamps: 2005 Avicenna (980–1037) Avicenna, also known as ibn Sinah, was the most celebrated of Persian philosopher-scientists, best known for his treatises on medicine. He contributed to arithmetic and number theory, produced a celebrated Arabic summary of Euclid’s Elements, and applied his mathematical knowledge to problems from physics and astronomy.

Albert Einstein (1879–1955) In 1905, Einstein published his ‘special theory of relativity’, asserting that the basic laws of motion (including Maxwell’s equations) are the same for all observers in uniform motion relative to one another. He thereby extended Newton’s ideas on mechanics to include electromagnetism and Maxwell’s results. A consequence is that mass is a form of energy, and that the energy E and mass m are related by the well-known equation E = mc2, where c is the speed of light.

GAMM 2005 In 2005 the Gesellschaft fu¨r Angewandte Mathematik und Mechanik (Society of Applied Mathematics and Mechanics) organized the 76th International Congress of Applied

Mathematics and Mechanics in Luxembourg. The commemorative stamp illustrates the calculation of the airstream of a turbine in a hydroelectric power station.

Josiah Willard Gibbs (1839–1903) Gibbs was an American physicist and mathematician who spent his working life as professor of mathematical physics at Yale. In mathematics he combined Grassmann’s ideas on exterior algebra with Hamilton’s quaternions, applying his conclusions in vector analysis to areas of mathematical physics. He also contributed to statistical mechanics, helping to provide a mathematical framework for quantum theory.

Edmond Halley (1656–1742) While still an Oxford University student, Halley sailed to St Helena to prepare the first accurate catalogue of the stars in the southern sky. In 1684 he persuaded Isaac Newton to publish his ideas on gravitation in the Principia Mathematica. In 1704 Halley became professor of geometry at Oxford, where he prepared a definitive edition of Apollonius’s Conics. He is primarily remembered for the comet, named after him, whose return he predicted.

William Rowan Hamilton (1805–1865) Hamilton was a child prodigy who mastered several languages at an early age, discovered an error in Laplace’s treatise on celestial mechanics while still a teenager, and became Astronomer Royal of Ireland while an undergraduate. He made several important advances in mechanics, and discovered the noncommutative ‘quaternions’ of the form. a þ bi þ cj þ dk; where i2 ¼ j 2 ¼ k2 ¼ ijk ¼ 1:

Gibbs Einstein

GAMM 2005

Avicenna

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Please send all submissions to the Stamp Corner Editor, Robin Wilson, Faculty of Mathematics, Computing and Technology, The Open University, Milton Keynes, MK7 6AA, England e-mail: [email protected]

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