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Note Four-Dimensional Polytopes: Alicia Boole Stott’s Algorithm IRENE POLO-BLANCO AND JON GONZALEZ-SANCHEZ Jon Gonzalez...

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

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Four-Dimensional Polytopes: Alicia Boole Stott’s Algorithm IRENE POLO-BLANCO AND JON GONZALEZ-SANCHEZ Jon Gonzalez-Sanchez was partially supported by the Spanish Ministry of Science and Education, grant MTM2008-6680-c02-01, partly with FEDER funds.

etween 1850 and 1852, the Swiss mathematician Ludwig Schla¨fli developed a theory of geometry in n-dimensions. In Theorie der vielfachen Kontinuita¨t ([4]), he defined the n-dimensional sphere, introduced the concept of four-dimensional polytopes, which he called polychemes, and proved that there are exactly six regular polytopes in four dimensions but only three in dimensions higher than four. Unfortunately, his work was not accepted for publication, and only fragments were published some years later. The entire manuscript did not appear until 1901. Thus, mathematicians in the second half of the century were unaware of Schla¨fli’s discoveries. Between 1880 and 1900 the six regular polytopes were independently rediscovered by, among many others, Stringham in 1880 [5], Gosset in 1900 [3] and Boole Stott in 1900 [1]. The work of Stringham and Gosset is well known today, but that of Boole Stott, an amateur mathematician, has remained almost unnoticed. In this note we present her original algorithmic approach, together with her original drawings and models. We emphasize that, as a woman born in the mid-nineteenth century, Boole Stott never received any formal mathematical training. Her discoveries came from an extraordinary capacity to visualize the fourth dimension. Rigorous mathematical proofs can therefore not be expected in her work, but instead we find a watershed of surprising and original ideas. Alicia Boole was born near Cork (Ireland) in 1860, the third daughter of the famous logician George Boole. He died when Alicia was four years old, and her mother became an

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innovative educator. The amateur mathematician Howard Hinton, a frequent guest in their home, was deeply interested in the fourth dimension. He taught the children to visualize four-dimensional shapes with small cubes; this may (or may not) have inspired Alicia’s later research. Whatever the inspiriation, Alicia Boole Stott (she married in 1890) rediscovered the six four-dimensional polytopes by computing their three-dimensional sections. In 1895 she was introduced to the Dutch geometer P. H. Schoute. They collaborated for more than 20 years, combining Schoute’s analytical methods with her unusual visualization ability; in 1914, after Schoute’s death (1913), the University of Groningen awarded Boole Stott an honorary doctorate. After that, she was isolated from the mathematical community until about 1930, when her nephew, G. I. Taylor, introduced her to H. S. M. Coxeter. Despite the nearly 50 year difference in their ages, Boole Stott and Coxeter collaborated productively until her death in 1940. (For more details, see [2].) To clarify her approach to four-dimensional polytopes, we first apply Boole Scott’s method to the five regular polyhedra (Figure 1). Constructing the parallel two-dimensional sections of any polyhedron (i.e., the sections parallel to one of its faces) is quite elementary. To compute, for example, the sections of the cube, we intersect the plane containing a given face of the cube with the cube itself. This intersection is, of course, the face of the cube; that is, the parallel section is a square. Translating the plane towards the center of the cube, we see that all parallel sections are isometric squares. Similarly, parallel sections of the tetrahedron are decreasing triangles, triangles and hexagons for the octahedron, pentagons and decagons for the dodecahedron and triangles, hexagons and dodecagons for the icosahedron. Diagonal sections of a regular polyhedron P are sections H \ P, where H is a plane perpendicular to the segment OV joining the center of the polyhedron with a vertex. We can visualize a regular solid by unfolding it to a planar net. Roughly speaking, this means ‘‘cutting’’ certain edges of the polyhedron and mapping it to a two-dimensional space. The well-known net for the cube is shown in Figure 2. Note that to recover the three-dimensional cube from the unfolded version, one must identify certain edges. This allows us to describe the parallel sections of the cube in a very easy way. Namely, one parallel section could be one of the squares in Figure 2, for example, the middle square (call it MS). In order to obtain the other sections (which will be parallel to the square MS after folding the net) one just needs to move the four edges of MS in the unfolded cube parallel towards the remaining squares. In each case, one obtains a square isometric to the square MS (after necessary identification of end points of the edges). 2010 Springer Science+Business Media, LLC, Volume 32, Number 3, 2010

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Figure 1. The five regular polyhedra: The tetrahedron, the cube, the octahedron, the dodecahedron and the icosahedron.

Boole Stott’s Sections of Polytopes In ‘‘On certain series of sections of the regular fourdimensional hypersolids’’ [1], Boole Stott describes an original method for obtaining the three-dimensional sections of the regular polytopes. A polytope in 4-dimensional space is a subset of the fourdimensional Euclidean space bounded by polyhedra such that every face of each polyhedron is also a face of exactly one other polyhedron. For any polytope, we define a flag (P, L, V, C) to be the figure consisting of a vertex P, an edge L containing P, a face V containing L, and a cell C containing V. The polytope is said to be regular if there is an isometry mapping any flag to any other, in such a way that

Figure 2. An unfolded cube.

ðrðPÞ; rðLÞ; rðV Þ; rðCÞÞ ¼ ðP 0 ; L0 ; V 0 ; C 0 Þ: The number of vertices, edges, faces and cells of the six regular polytopes are listed in Figure 3. Boole Stott begins with an intuitive uniqueness proof, roughly as follows. Let P be a regular polytope whose cells are cubes. Let V be one of the vertices of P, and consider the diagonal section of P by an affine subspace K, close enough to V so that K intersects all the edges coming from V. The corresponding section must be a regular polyhedron bounded by equilateral triangles, i.e., the tetrahedron (bounded by 4 triangles), the octahedron (bounded by 8 triangles) or the icosahedron (bounded by 20 triangles). Therefore the polytope can only have 4, 8, or 20 cubes

Figure 3. Polytopes in four dimensions.

meeting at each vertex. Considering the possible angles in 4 dimensions, Boole Stott shows that P must have 4 cubes at a vertex (8 and 20 are too many), which gives the 8-cell (also called a hypercube). She finds the remaining five polytopes in a similar manner.

AUTHORS

......................................................................................................................................................... IRENE POLO-BLANCO received her Ph.D.

JON GONZALEZ-SANCHEZ received his Ph.D.

in 2007 at the University of Groningen, the Netherlands. Her current position is with the group of Didactics of Mathematics at the University of Cantabria. Her research area is algebraic geometry, and didactics of mathematics. As a hobby, she sings in the choir ‘‘Camerata Coral de la Universidad Cantabria’’ http://camerata.unican.es/.

from the University of the Basque Country. After a two year postdoctorate at the University of Groningen, he obtained a ‘‘Juan de la Cierva’’ research fellowship at the University of Cantabria, where he is currently located. He works in group theory and, more recently, in effective algebraic geometry. He enjoys travelling and doing sports. The authors of this paper are married (to each other) and have a four year old son.

Departamenton de Matema´ticas Estadı´stica y Computacioń Universidad de Cantabria Avda. de los Castros s/n E-39005 Santander, Cantabria Spain e-mail: [email protected]

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THE MATHEMATICAL INTELLIGENCER

Departamento de Matema´ticas Estadı´stica y Computacioń Universidad de Cantabria Avda. de los Castros s/n E-39005 Santander, Cantabria Spain e-mail: [email protected]

Next, she studies three-dimensional parallel sections of these polytopes. Let H be an affine three-dimensional subspace perpendicular to the line OC, where O is the center of a given polytope P, and C the center of one of its cells. The parallel section is H \ P. Although she treats only parallel sections of polytopes in [1], Boole Stott also made models of diagonal sections K \ P. Here K is an affine three-dimensional subspace perpendicular to the segment OV, where again O is the center of P and V is one of its vertices. Boole Stott uses the unfolding of a four-dimensional body in a three-dimensional space, analogous to our discussion above. This unfolding operation can be described as ‘‘cutting’’ some of the two-dimensional edges between the three-dimensional faces and mapping the polytope to the third dimension. For example, the unfolded hypercube is the famous tesseract (Figure 4). Note that some twodimensional faces (i.e., squares) must be identified to

Figure 4. Unfolded hypercube.

Figure 5. Four octahedra of the 24-cell [1]. 2010 Springer Science+Business Media, LLC, Volume 32, Number 3, 2010

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recover the hypercube (this identification, of course, is only possible in four dimensions). We have formalized in the following algorithm Boole Stott’s method for computing these three-dimensional parallel sections. A similar algorithm can be described for the case of diagonal sections.

Figure 6. Section H2 \ P of the 24-cell [1].

Figure 7. Section H3 \ P of the 24-cell [1]. 4


Sketch of the Algorithm Let P be a four-dimensional regular polytope. • Step 1: Unfold the polytope P into the three-dimensional space.

Figure 11. Drawings and models of parallel sections of the 600-cell. (Courtesy of the University Museum of Groningen.)

Figure 8. Five tetrahedra of the 16-cell [1].

Figure 9. Second, third and fourth sections of the 16-cell [1].

Figure 10. Sections of the 120-cell. (Courtesy of the University Museum of Groningen.)

• Step 2: Let C be the graph whose nodes are the vertices and the midpoints of the edges of the unfolded P. Two nodes are connected if one is the midpoint of an edge and the other a vertex contained in that edge.

• Step 3: Fix a cell C of the polytope P on the unfolded figure. • Step 4: The first three-dimensional sections S1 of P will be C. (Note that the cell C, in the folded polytope P, is contained in a three-dimensional subspace H1. The section S1 is therefore H1 \ P = C.) 2010 Springer Science+Business Media, LLC, Volume 32, Number 3, 2010

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• Step 5: Let V 2 be the elements of C at distance 1 of C; these are just the midpoints of the edges of P \C meeting C. (Note that in this case, V 2 , in the folded polytope P, is contained in a hyperplane H2 parallel to H1. These points will be the vertices of a polyhedron S2 that will be the second section of P.) One can easily compute the faces of S2: for any cell D of the unfolded polytope that intersects V 2 , the polygon, segment or point given by the convex hull of D \ V 2 will be a face, edge or vertex of S2, respectively. The natural folding of P gives the identification of every face of the polyhedron S2. • Step 6: Let V 3 be the elements of C at distance 1 of S2 that are not contained in S1. (Note that V 3 is contained, in the folded polytope, in a hyperplane H3 parallel to C. S3 ¼ P \ H3 is the third section of P.) We compute the faces of S3 as above. The natural folding of P gives the identification of every face of the polyhedron S3. • Step 7: Repeat step 6 until V i ¼ ; . For a non-trivial example, let P be the 24-cell. (This polytope is the only one without an analogue in three dimensions). Its cells are octahedra, with 6 meeting at each vertex. Step 1 of the algorithm gives the unfolding of P. In Boole Stott’s representation (see Figure 5) only 4 octahedra are drawn. Note that the figure is again an unfolding. The two A0 should be identified and similarly, for the vertices AE and AC. Fix an octahedron cell of P (step 3). Let H1 be the threedimensional space containing the octahedron ABCDEF. The first section H1 \ P is clearly the octahedron ABCDEF itself (step 4). Let H2 be the space parallel to H1 and passing through the point a (the mid-point between A and AC). The second section H2 \ P is a three-dimensional solid whose faces are either parallel to the faces of the octahedron ABCDEF or to the rectangle BCEF. In Figure 6 two of these faces are shaded. Since the drawing of the octahedra meeting at A is not complete (3 octahedra are missing), we only see part of the final section. The remaining part can be deduced by symmetry (step 5).

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Following step 6, let H3 be the space parallel to H1 and passing through the vertex AC. The section H3 \ P contains a rectangle ABACAEAF parallel to the rectangle BCEF and a triangle AEACCE parallel to the face ACE (the shaded faces of Figure 7). By symmetry, the fourth section passing through a1 (the mid-point between AC and A0 ) is isomorphic to the second section (step 7). Again by symmetry, the last section through A0 is an octahedron (step 8). Exercise: Use the algorithm to compute the threedimensional sections of the 5-cell and the 8-cell. We conclude with Boole Stott’s drawings for the parallel sections of the 120-cell, and her cardboard models for the diagonal sections of the 600-cell, which you can see on display at the University of Groningen (Figures 8–11).

ACKNOWLEDGMENTS

The first author would like to thank Jan van Maanen for introducing her to Boole Scott’s beautiful world. We would also like to thank Marjorie Senechal for her enormous help in improving our text.

REFERENCES

[1] A. Boole Stott, On certain series of sections of the regular fourdimensional hypersolids, Verhandelingen der Koninklijke Akademie van Wetenschappen te Amsterdam 1900; 7 nr. 3:1–21. [2] I. Polo-Blanco, Alicia Boole Stott, A geometer in higher dimension, Historia Mathematica 2008; 35:123–139. [3] T. Gosset, On the regular and semi-regular figures in space of n dimensions. Messenger of Mathematics 1900; 20:43–48. [4] L. Schla¨fli, Theorie der vielfachen Kontinuita¨t, Denkschriften der Schweizerischen naturforschenden Gesellschaft 1901; 38:1–237. [5] W. I. Stringham, Regular figures in n-dimensional space. American Journal of Mathematics 1880; 3:1–14.

Viewpoint

Deus ex Machina and the Aesthetics of Proof ALAN J. CAIN

The Viewpoint column offers mathematicians the opportunity to write about any issue of interest to the international mathematical community. Disagreement and controversy are welcome. The views and opinions expressed here, however, are exclusively those of the author, and neither the publisher nor the editors-inchief endorses or accepts responsibility for them. Viewpoint should be submitted to either editor-in-chief.

nexpectedness and inevitability, two of the aesthetic qualities G. H. Hardy identified as being properties of beautiful proofs [5, §18], together seem paradoxical: how can something be seen as both unexpected and inevitable? Two possibilities are that the inevitability only becomes apparent in hindsight, or that the strategy of the proof is unexpected but, once chosen, proceeds inevitably. In this essay, I suggest a different solution: I argue that the literary concept of deus ex machina can be used to clarify the notion of inevitability in proof and reconcile it with unexpectedness. Deus ex machina (literally, ‘‘god from the machine,’’ henceforth abbreviated to deus) refers to a type of plot device used to resolve a seemingly intractable situation. The term is derived from ancient Greek drama, where such a resolution might be effected by a god intervening, with the actor playing the god being lowered onto the stage by a crane (the ‘‘machine’’). It has come to mean any event in a story that resolves a situation but which does not fit with the internal framework of the plot [7]. Aristotle is the earliest extant author to complain of the unsatisfactory nature of such a resolution [Poetics, 1454a33–b8], and this disdain has continued to the present day. Aristotle gives the example of how Medea, at the end of Euripides’s play of the same name, is rescued from Jason’s vengeance by being carried off to Athens in the chariot of the god Helios [Medea, l.1314]; until this point, the play is free from divine intervention.

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Narrative and Proof Parallels between proof and narrative have been explored by Thomas, who argues that ‘‘[l]ogical consequence is the gripping analogue in mathematics of narrative consequence in fiction; all physical causes, personal intentions, and logical consequences in stories are mapped to implication in mathematics’’ [14, p.45]. My intention is to focus here on the notion of deus in narrative, and argue for a parallel notion of deus in proof: inevitability, in the Hardian sense, can then be thought of as avoidance of deus. In a narrative, the reader is presented with a place, a time, and some characters, and is told how the characters interact with each other and the world they inhabit. The reader gradually builds up a mental conception of the world and the characters’ motives and personalities. A narrative, at its most basic, can be a bare enumeration of events, perhaps disconnected, that conveys only superficial information about the world of the narrative: telling what happened without giving the reader any inkling of why. A better, fuller narrative allows the reader to build up a coherent mental picture and understand the events and the characters’ actions. 2010 Springer Science+Business Media, LLC, Volume 32, Number 3, 2010

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In a proof, the reader is presented with some mathematical objects, and, through the reasoning in the proof, gradually builds up a mental conception of how these objects behave. Part of this mental conception is embodied in the theorem that is proven. A proof, at its most basic, can be a bare listing of statements, each following logically from earlier ones, that lead from the hypotheses to the conclusion. However, a better proof can leave the reader, not simply with knowledge of the theorem’s truth, but with a deeper understanding of it and the objects it concerns. Let me explain this further. In reading a narrative or a proof, a reader has some mental conception of the world of the narrative or the objects with which the proof is concerned. This conception includes formal knowledge and (for want of a better word) intuition. In narrative, formal knowledge consists of facts that are established within the world of the narrative. These may include facts about characters’ past actions, skills, and relationships. In proof, it consists of definitions and proven properties of the objects. Intuition, in narrative, consists of less certain impressions, for instance regarding the motivation and psychology of the characters and expectations for how the plot will proceed. In a proof, intuition consists of an impression of how the objects concerned behave and interact. In both cases, intuition is informed by formal knowledge. Indeed, the reader constructs an intuition of the characters of a narrative or the objects of a proof from those pieces of formal knowledge supplied by the narrative or proof. As one follows a proof or narrative, one acquires new formal knowledge. In a proof, readers will check each new step in the proof to see whether it follows from their existing stock of formal knowledge, and, if so, add it to their formal knowledge. In a narrative, the checking is less important, or at least less active, although presumably a reader would notice if a narrative contained contradictory statements. In both narratives and proofs, the acquisition of new formal knowledge causes readers to modify their intuition.

AUTHOR

.........................................................................

In a narrative, a deus is unsatisfying for two reasons. The first is that any future attempt to build tension is undercut if the author establishes that a difficulty can be resolved by a deus. The second reason—more important for the purposes of this essay—is that the deus does not fit with the internal structure of the story. There is no reason internal to the story why the deus should intervene at that moment. There is only an external explanation: the author wants to extricate the hero. For this reason, readers cannot incorporate the deus and its consequences into the intuitive component of their mental conception, or at least can do so only with difficulty. In the context of proofs, this second reason has a parallel. In a proof, a deus takes the form of an unexplained construction or a calculation of elements or a definition of a function that simply ‘‘happens to work.’’ Like a deus in narratives, such a manoeuvre serves an external purpose, namely the teleological one of proving the theorem at hand. Such features in a proof do not fit with the structure or setting of the proof. Readers cannot see why this construction or this calculation or that definition is being carried out; they cannot perceive a reason for it that is internal to the proof. In short, they have more difficulty in modifying their intuitive conceptions to include the deus. They can follow the proof to its conclusion, checking each step against their formal knowledge of the objects concerned, but the deus is a cataract that their intuition cannot easily navigate. I argue that taking ‘‘inevitability’’ in proof to mean ‘‘avoidance of deus’’ allows one to understand how ‘‘inevitability’’ and ‘‘unexpectedness’’ can both occur in a beautiful proof: for like a pleasing narrative, a beautiful proof can contain unexpectedness provided it fits its structure and setting.

A Case Study: Three Proofs of Morley’s Theorem To illustrate the notion of deus in proof, I shall compare three different proofs of Morley’s trisector theorem, which says that the adjacent trisectors of the angles of any triangle meet at the vertices of an equilateral triangle; see Figure 1. [There are many different proofs of this theorem; see the

ALAN J. CAIN received his Ph.D. from the University of St Andrews in 2005, where he remained for a further three years as a postdoctoral researcher. He then sought the sunnier climes of Portugal, first in Lisbon and then in Porto, where he is presently a research fellow. He generally spends his time acquiring books, from which he occasionally takes a break to do mathematics. Centro de Matema´tica Universidade do Porto 4169-007 Porto Portugal e-mail: [email protected]

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Deus ex Machina


Figure 1. Morley’s trisector theorem.

bibliography in [12].] I shall not gives the proofs in full, I shall merely highlight the salient points. 1. Conway’s proof [3] starts with three angles a, b, c with a + b + c = p/3. It specifies the angles and side lengths of seven triangles, one being equilateral, and then shows that they can be fitted together to form the triangles ABZ, ACY, BCX, AYZ, BXZ, CXY, and XYZ in Figure 1. For example, one triangle (which will become AYZ when the pieces are assembled) is specified to have angles a, c + p/3, b + p/3, with the side between the latter two angles being equal to the side of the equilateral triangle. Thus the proof constructs the triangle ABC. 2. The proof given by Dijkstra [4, p.182–3] again starts from three angles a, b, c. It starts from the equilateral triangle XYZ and constructs the triangles AYZ, BXZ, and CXY, with the angles of each being specified. For example, AYZ is specified to have angles a, c + p/3, b + p/3. It then uses the sine rule to show that a ¼ \CAY ¼ \YAZ ¼ \BAZ; and similarly for the angles at B and C. 3. The proof by Bankoff [1] starts from the triangle ABC and its trisectors, and, by making use of trigonometric identities and the sine rule, first of all calculates the lengths of AY and AZ in terms of the radius of the circumcircle and the angles of ABC, then calculates the angles \AYZ and \AZY : Symmetrical arguments give the angles \BXZ; \BZX; \CXY and \CYX; from which it follows that each angle of XYZ must be p/3. Conway’s proof (1) is the shortest of the three. It is simple and has the merit of avoiding use of trigonometric identities or the sine rule, but the specification of the seven triangles is a deus. The values for what turn out to be the angles of the seven smaller triangles of Figure 1 simply happen to work. Certainly, one sees some of the relationship between the angles, such as the fact that \AYZ is dependent on c. But one does not see why it is dependent only on c and not on a or b or the side length of the triangle. Dijkstra’s proof (2) also involves a deus, albeit a milder one: only the specification for three triangles is produced out of a hat. The use of the sine law then gives the reader some intuitive feeling of how the result follows from the relationships between the sides and angles of the triangle. Bankoff’s (3) is the longest proof, but its approach is unsurprising. It uses the kind of trigonometric arguments one expects in this situation, including several applications of the sine rule. Although this argument is rather more involved than either of the other two, requiring the use of trigonometric identities and several applications of the sine rule, it contains no deus. In particular, the reader can follow the reasoning intuitively. Both Conway’s and Dijkstra’s proofs work in reverse: they start from the equilateral triangle XYZ and show that for any angles a, b, c, a triangle with angles 3a, 3b, 3c can be constructed whose adjacent angle trisectors meet at X, Y, and Z. Bankoff’s, in contrast, starts from the triangle ABC and deduces that XYZ is equilateral. It gives an idea of the relationships holding between the angle trisectors and how

these force XYZ to be equilateral. For example, one can see why \AYZ is dependent only on c: because AY and AZ are also dependent on a, b, and the radius of the circumcircle, but these dependencies cancel each other out. Comparison of these examples also shows the independence of economy (according to Hardy, a third aesthetic quality of beautiful proofs) from the absence of deus. For Conway’s proof, with the strongest deus, is the most economical. The only tools it uses are the most elementary geometric facts; the most ‘‘advanced’’ being the ‘‘angle– side–angle’’ similarity argument. Dijkstra’s has a milder deus but uses a more advanced tool, viz., the sine rule. Bankoff’s avoids deus but requires a still bigger toolkit: the sine rule and various trigonometic identities.

Differences between Narrative and Proof One difference between deus in narrative and deus in proof should be emphasised. Chronology, in the sense of the order in which a reader is informed of events, plays an important roˆle in narrative. Whether a reader views a particular event as a deus is closely linked with chronology. If the readers have been supplied with an explanation for an event (in the sense of knowledge of causes for this event or at least the potentiality of this event), they will not view the event as deus. This holds true even if the explanation is not recognised as such until the reader encounters the event. (The mysterious figure who has been following the hero for days is revealed to be an ally and comes to his aid.) However, an event followed by a post hoc explanation is unlikely to be appreciated. (The hero is rescued by an ally of whom the reader had not been hitherto informed, but who is now said to have been following him for days.) The readers may be able to incorporate the post hoc explanation into their intuition, but the author is unlikely to be able to salvage the ability to create tension in the remainder of the narrative. In proof, chronology plays a lesser roˆle, at least insofar as the evaluation of deus is concerned. If, after a construction or calculation or definition, the proof retrospectively shows why this procedure was necessary, readers will be able to incorporate it into their intuition. A very early example of this is Euclid’s proof of the infinity of prime numbers [Elements IX.20]. Given a collection of primes p1, …, pn, one forms the number N = p1 … pn + 1. Only retrospectively does the reason for this definition for N become clear: because division of N by any pi leaves remainder 1, so that some prime not among the pi must divide N. This retrospective explanation for the choice of N ensures that this move is not seen as a deus; indeed Hardy selects Euclid’s proof as one of his two examples of beautiful proofs [5, §12]. In contrast, in Conway’s proof of Morley’s theorem, for instance, the specification of angles is not justified retrospectively.

Inevitability and Unexpectedness In light of these discussions, the Hardian aesthetic concept of inevitability in proof can be seen as avoidance of deus. Hence, unexpectedness can be reconciled with inevitability, for avoiding deus does not entail avoiding unexpectedness. 2010 Springer Science+Business Media, LLC, Volume 32, Number 3, 2010

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An event in a story can be unexpected yet not a deus if, once the reader has encountered it, it is seen to fit properly with the structure of the tale. The same holds in proof: Netz [10, p.256– 7] noted an example in Archimedes’s Sphere and Cylinder. Archimedes announces his intention of proving decidedly three-dimensional results: that the surface area of a sphere is four times that of its great circle, and that the volume of a sphere is two-thirds of a cylinder that exactly encloses it. Yet he starts by proving two-dimensional results: a series of theorems on circles and polygons. This is followed by results on pyramids and cones; three-dimensional results, true, but apparently irrelevant. Then, suddenly, he imagines rotating polygons about an axis: all the two-dimensional results suddenly acquire three-dimensional analogues, and their applicability to the situation at hand suddenly makes sense to the reader; the desired results follow in short order. This is unexpected, but not a deus, for it fits with the internal structure of the proof and illuminates the preliminary results. Another example is Zagier’s one-sentence proof of Fermat’s theorem that every prime p : 1 (mod 4) is a sum of two squares [17]. It has an unexpected starting point, but thenceforth the proof proceeds inexorably (and rapidly) to its conclusion.

his wanderings since the fall of Troy, reaches the point when he and his men are trapped by the Cyclops Polyphemus, but then jumps ahead to leaving the island, and blandly assures his listeners that his cunning allowed their escape. We would have reason to believe him: he successfully uses stratagems and ruses at many points throughout the Odyssey. Yet, even if we were to accept the truth (within the world of the tale) of what he says, such a turn of events would be a deus, for we could not modify our intuition to take us from the situation of their being in Polyphemus’s power to their leaving the island. Our intuition would be that they are in an inescapable predicament. To suddenly jump ahead to after their escape would be just as aesthetically unsatisfying as, for instance, Polyphemus spontaneously deciding to release them. Unable to modify our intuition, we would be left with unresolved questions: How could they overcome Polyphemus’s great strength? How could they evade the other Cyclopes? The real story, by contrast, tells how Odysseus and his men prepare their plan; how Odysseus creates an opportunity to use it by giving Polyphemus wine; and how he sets up their escape past the other Cyclopes. Each step here allows us to modify gradually our intuition of Odysseus and of Polyphemus.

Other Examples

Connections with Teaching and Exposition

Einstein discussed what he perceived as an ugly and an elegant proof of one direction of Menelaus’s theorem. [Given a triangle ABC and a line dividing the lines AB, BC, and CA into A0 B and A0 C, B 0 C and B 0 A, C 0 A and C 0 B, respectively, A0 C B 0 A C 0 B = A0 B B 0 C C 0 B.] Although the first proof is somewhat simpler, it is not satisfying. For it uses an auxiliary line that has nothing to do with the content of the proposition to be proved, and the proof favors, for no reason, the vertex A, although the proposition is symmetrical in relation to A, B, and C. The second proof, however, is symmetrical, and can be read off directly from the figure. [8, p.38] In the terms used in this paper, the auxiliary line and the favouring of the vertex A form a deus, for nothing either before or afterwards compels these constructions. Some mathematicans have a certain distaste for the Haken–Appel proof of the four-colour theorem, which depends upon a computer-assisted argument (see, for example, [6, p.92–3]). The computer-assisted part of the proof is a deus, albeit of a slightly different kind from the examples discussed above. For in the earlier examples, the deus still allows a reader to formally check the proof, whereas the appeal to a lemma proved by a computer does not allow this checking. Even leaving aside such issues of validity or surveyability (see, for instance, [16]), the computer-assisted part of the proof seems to be a deus, for it represents a step that readers cannot easily incorporate into their intuition. Certainly one can see the strategy of that part of the proof: that all of the unavoidable configurations are reducible. But the proof for this is too long for readers to follow so that they can gradually modify their intuition; the reader is essentially forced to jump ahead in the proof and simply accept the correctness of intermediate steps. Imagine a hypothetical narrative parallel: a variation of book IX of the Odyssey wherein Odysseus tells Alcinous of 10


Aside from aesthetics, various authors have argued that in expositional work, one should avoid manoeuvres that are akin to deus as I use the term: Chow says that ‘‘every step should be motivated and clear’’ [2, p.1] and follows Newman in saying that proofs should be ‘‘natural’’ in ‘‘not having any ad hoc constructions or brilliancies’’ [11, p.59, italics in original]. Tucker explicitly recommends that, when teaching the calculus, one should not use ‘‘deus-exmachina auxiliary functions’’ [15, pp.239–240]. This expositional advice, if followed strictly, would seem to rule out the use of surprise, whereas I have drawn a distinction between deus (in my sense) and unexpectedness in proofs. Certainly, it seems pedagogically safer to avoid both and, following Chow’s advice, motivate every step. Additionally, a proof is probably easier to memorize if it avoids deus, for each deus, not being compelled by the overall structure of the proof, would have to be explicitly remembered. However, complete avoidance of surprise might reduce the appeal of the exposited mathematics.

Aesthetics of Proof Proofs, like narratives, can be aesthetically unsatisfying in ways other than using deus. Just as a narrative text can use inelegant language, clumsy exposition, or bad pacing, a proof can use poor notation, unclear explanation, or unsatisfactory division into lemmata. Each of these factors would decrease the satisfaction of a narrative or a proof that nevertheless avoided deus. Rota [13, p.181] suggests that ‘‘mathematical beauty’’ is a term mathematicians use to avoid describing a piece of mathematics as enlightening. One does not need to accept fully his assertion to see that the identification of beauty with enlightenment is compatible with the arguments above. As readers follow a proof, they modify their intuition. They will

find the proof enlightening if, by the end of the proof, their intuition includes what the theorem describes. If the readers’ intuition does not include it, the proof is unenlightening. Since any deus presents the readers with a difficulty in modifying their intuitions, a proof that involves a deus is less likely to be perceived as enlightening.

[7] Deus ex machina. The Literary Encyclopedia. 28 June 2004. [http://www.litencyc.com/php/stopics.php?rec=true&UID=1436, accessed 17 July 2009.]. [8] A. S. Luchins and E. H. Luchins. The Einstein-Wertheimer correspondence on geometric proofs and mathematical puzzles. Mathematical Intelligencer 12(2):35–43, 1990. [9] P. Mancosu, K. F. Jørgensen, and S. A. Pedersen, editors.

ACKNOWLEDGEMENTS

I would like to thank Yumi Murayama for reading and commenting on an earlier draft of this essay.

Visualization, Explanation and Reasoning Styles in Mathematics, volume 327 of Synthese Library, Dordrecht, 2005. Springer. [10] R. Netz. The aesthetics of mathematics. In Mancosu et al. [9], pp. 251–293. [11] D. J. Newman. Analytic Number Theory, volume 177 of Graduate

REFERENCES

[1] L. Bankoff. A simple proof of the Morley theorem. Mathematics Magazine 35(4):223–224, September 1962. [2] T. Y. Chow. A beginner’s guide to forcing. In T. Y. Chow and D. C. Isaksen, editors, Communicating Mathematics, volume 479 of Contemporary Mathematics, pp. 25–40. American Mathematical Society, 2009. [3] J. H. Conway. Posted to the geometry.puzzles newsgroup, 24th November 1997. [4] E. W. Dijkstra. A collection of beautiful proofs. In Selected Writings on Computing: A Personal Perspective, Texts and Monographs in Computer Science, pp. 174–183. Springer-Verlag, New York, 1982. [5] G. H. Hardy. A Mathematician’s Apology. Cambridge University Press, Cambridge, 1940. [6] J. P. King. The Art of Mathematics. Dover, Mineola, 2006.

Texts in Mathematics. Springer-Verlag, New York, 1998. [12] C. O. Oakley and J. C. Baker. The Morley trisector theorem. American Mathematical Monthly 85(9):737–745, 1978. [13] G. C. Rota. The phenomenology of mathematical beauty. Synthese 111(2):171–182, May 1997. [14] R. S. D. Thomas. Mathematics and narrative. Mathematical Intelligencer 24(3):43–46, 2002. [15] T. W. Tucker. Rethinking rigor in calculus: the role of the mean value theorem. American Mathematical Monthly 104(3):231–240, 1997. [16] T. Tymoczko. The four-color problem and its philosophical significance. Journal of Philosophy 76(2):57–83, 2 1979. [17] D. Zagier. A one-sentence proof that every prime p : 1 (mod 4) is a sum of two squares. American Mathematical Monthly 97(2):144, 1990.

2010 Springer Science+Business Media, LLC, Volume 32, Number 3, 2010

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Three-Dimensional Fractals CHRISTOPH BANDT, MAI THE DUY

ractals such as the Sierpin´ski gasket and the Koch curve (Figure 3) have become standard examples in mathematical textbooks, and there are many related two-dimensional structures [4, 13]. In dimension 3, only two examples are well-known: the Menger sponge and the fractal tetrahedron, shown on many web sites: see for instance http://www.mathpaint.blogspot.com (click on ‘‘April 2008’’). A.G. Bell, known for the development of the telephone, found the fractal tetrahedron several years before Sierpin´ski, in flying experiments with kites (see Figure 2). In 1903, Bell wrote that since 1899 ‘‘I have been continuously at work upon experiments relating to kites. Why, I do not know, excepting perhaps because of the intimate connection of the subject with the flying-machine problem. We are all of us interested in aerial locomotion; and I am sure that no one who has observed with attention

F

Figure 1. A new fractal construction.

12

THE MATHEMATICAL INTELLIGENCER 2010 Springer Science+Business Media, LLC

AND

MATHIAS MESING

the flight of birds can doubt for one moment the possibility of aerial flight by bodies specifically heavier than the air.’’ Bell disproved the belief that an ‘‘air-ship’’ would not be possible because the weight-to-surface ratio must increase for larger machines. His fractal kite, which he considered a ‘‘milestone of progress’’ in the field, could be increased ‘‘indefinitely’’ without changing the ratio of surface and weight [5]. Some years later, Bell’s kite could lift a man, but too late: the Wright brothers’ first flight took place in December 1903. Since then, fractals have found many other applications, including the structure of neurons and DNA, clouds, rough surfaces, soil, and other porous materials. Though all applications concern subsets of three-dimensional space, the number of simple three-dimensional geometric models has not increased significantly. Beside the tetrahedron and the cube, the other three regular polyhedra have been studied.

Figure 2. The fractal tetrahedron, presented as a kite by Bell’s team in 1903. Reprinted from [5].

A3

A2

A4

A1

A21 A14

Figure 3. Left: the Sierpin´ski gasket; right: the Koch curve.

Figure 4. In self-similar sets, pieces are similar to the whole set. If the pieces are too small, they do not touch each other and they form a Cantor set (left). If the pieces are too large, their overlaps can obscure the self-similarity (right). Only for special arrangements is a nice geometric structure obtained.

(See Figure 4 for the octahedron, and [9, 11, 12, 15, 17] for icosahedron and dodecahedron.) Whereas computer scientists used these fractals as a testbed for ray-tracing algorithms rendering complicated 3D scenes, mathematicians apparently were driven by the idea that there should be a small number of fractals with a particular regular and simple structure. After introducing the relevant mathematical concepts, we will address this idea, suggest a definition of

‘‘simple structure’’, and present some new examples, such as the fractal shown in Figure 1.

Geometric Self-Similarity We consider the simplest class of fractals. A set A in Rn is called self-similar if it is the union of sets A1, ... Am, which are geometrically similar to A. These pieces must then contain similar subpieces, and the subpieces must contain still

AUTHORS

......................................................................................................................................................... CHRISTOPH BANDT is a professor of

MAI THE DUY received his MSc from Hanoi

mathematics at the University of Greifswald, Germany. Beside fractals, he is interested in random phenomena and biomathematics with real data.

National University and is a PhD student in Greifswald, supported by the Ministry of Education and Training of Vietnam.

Institute of Mathematics Arndt University, 17487 Greifswald Germany e-mail: [email protected]

Institute of Mathematics Arndt University, 17487 Greifswald Germany


13

smaller similar copies, and so on. This is the idea of fractal structure, according to Mandelbrot who coined the term ‘‘fractal’’ and convinced the world of its relevance [13]. Fractal structure is also apparent when the similarity is not strict, for instance when affine or conformal mappings act between the set and its pieces, as in the Julia sets recently discussed in this journal [16]. Here we focus on strict self-similarity. If the pieces Ai are disjoint, as on the left side of Figure 4, then A is a Cantor set, which is not very interesting from the geometric point of view. The pieces Ai should be allowed to intersect. But their overlap should be small, because otherwise, as on the right side of Figure 4, the self-similar structure can hardly be recognized. Special constructions are needed to guarantee small overlap. P.A.P. Moran [14] defined a self-similar construction by a bounded open set U and geometrically similar subsets Ui , U, i = 1, ..., m, which are disjoint. In each Ui, disjoint similar copies Uij are chosen, and so on. Moran takes F0 ¼ U ; the closure of U, as first approximation, and [ [ F1 ¼ Ui ; F2 ¼ Uij etc., so that F0 F1 F2 . . . i

i;j

The intersection of the decreasing sequence of compact sets Fi is the resulting fractal set A. In 1946, Moran determined the Hausdorff dimension d of A and proved that the d-dimensional Hausdorff measure of A is positive, which provides a natural concept of volume and geometric probability on the set A. For simplicity, we assume that there are m pieces on each level, and that the similarity ratios of the Ui with respect to U are all equal to some positive real number r. In this case the dimension can be explicitly calculated [4, 8, 10, 13] m rd ¼ 1

or

d¼

log m : log r

For the examples below, d indicates to what extent the fractal fills three-dimensional space. Moran’s construction is illustrated in Figure 3. Let U be a triangle. Cut a hole H into U, such that the set U nH consists of a union of similar copies of U. In Figure 3, the hole is a closed triangle. In the Menger sponge, the hole is a threedimensional cross, and in the fractal tetrahedron it is a regular octahedron.

The Topology of the Fractal Octahedron In the fractal octahedron, the holes are tetrahedra. This example has an interesting topological structure. Figure 5 shows that all 8 tetrahedral holes are carved from outside

......................................................................... MATHIAS MESING completed his PhD in

2008 and is now teaching high school in Greifswald. Institute of Mathematics Arndt University, 17487 Greifswald Germany 14

Figure 5. The fractal octahedron – a deflated balloon.


into the basic octahedron U, and all faces of the remaining m = 6 octahedra Ui can be reached completely from outside. By self-similarity, on each level k of the construction, all surface points of the 6k octahedra of the residue set can be reached by an arc from outside U that does not contain other points of the residue set. In the limit, each point of the fractal can be reached by such an arc. This means that each of the 8 faces of the fractal octahedron is the image of a triangle under a continuous map – a three-dimensional version of the Koch curve (Figure 3). However, unlike in two dimensions, many pyramidal spikes will touch each other along an edge so that this ‘‘surface’’ has double points and is not homeomorphic to a triangle. Since the diameter of the remaining octahedra on level n converges to zero for n??, no interior points are left between the fractal faces. The fractal octahedron has eight faces, exactly as the ordinary octahedron, but no interior – similar to a deflated balloon. Any two neighboring fractal faces have as their intersection an ordinary Euclidean triangle! Its three vertices are the two endpoints of their common edge and the center of the original octahedron. Every square, given by 4 vertices of the basic octahedron U that do not enclose a face, belongs completely to A. (To prove all this, use the fact that the edges of U, and hence of all smaller octahedra, are contained in A. Some of these edges form a grid that in the limit becomes a dense subset of the square.) Thus if we look at an octahedral ‘‘face’’ from the other side, we see an ordinary pyramid, and if we remove the boundary, smaller pyramids will appear below. Altogether they represent a small part of the ‘‘face’’ however, since triangles have dimension 2, and the dimen6 sion of the fractal ‘‘face’’ is log log 2 2:58:

Similarity Mappings The method of carving holes H into appropriate open sets U is not very powerful because of our lack of imagination. But in 1981, Hutchinson introduced similarity mappings fi with Ai = fi(A). A mapping f from Rn into itself is a

similarity mapping with factor r [ 0 if jf ðxÞ f ðyÞj ¼ r jx yj

for all points x; y;

that is, all distances |x - y| are contracted by the ratio r. Given f1, ..., fm with factors ri \ 1, the corresponding compact self-similar set A was defined by the equation A ¼ f1 ðAÞ [ . . . [ fm ðAÞ: It is not difficult to show that this equation always has a unique nonempty solution, even in the case of affine or conformal contractive mappings [10]. Letting Ai ¼ fi ðAÞ ¼ fi1 fi2 . . . fin ðAÞ; we see that this algebraic approach addresses small pieces by words i = (i1i2 ... in) from the alphabet {1, …, m} (see Figure 3). This approach made fractals accessible to computers. In a time when the first personal computers with graphics facilities became available, Hutchinson’s concept became extremely popular, and various algorithms for computer visualization were developed. See Barnsley [4] who calls f1, ..., fm an iterated function system (IFS).

Neighbor Maps A basic set U is not needed in the mapping construction of A. It turns out, however, that its Hausdorff dimension and measure can only be calculated, and complicated overlaps such as that in Figure 4 can only be avoided if there is an open set U such that the fi(U) are disjoint subsets of U. An algebraic criterion by Bandt and Graf [1] says that U exists if and only if mappings of the form hij ¼ fi1 fj ; with i ¼ ði1 i2 . . .in Þ; j ¼ ðj1 j2 . . .jn0 Þ and n; n0 2 N; cannot converge to the identity map. It is sufficient to consider those i, j for which the pieces Ai and Aj intersect each other, and have ‘‘almost the same size’’. Since ri = r was assumed here, the last condition means n = n0 so that Ai and Aj have exactly the same size, and the map hij is an

Figure 6. Four possible neighbor positions for the Koch curve. If the bold curve represents A4 in Figure 3, it has only the upper left neighbor, whereas A2 has also the lower right neighbor. A14 and A21 have both upper neighbors. Of course, for each piece of a curve only one neighbor on each end is possible.

isometry. That is, h(x) = Mx + v, with an orthogonal matrix M and a vector v. The map h = hij, called a neighbor map, maps A to a neighbor set h(A), which has the same position relative to A as Aj has to Ai, up to similarity. In other words, fi(A) = Ai and fi(h(A)) = Aj. The set of neighboring positions of A for the Koch curve is shown in Figure 6. The previous criterion says that such neighboring positions must not come arbitrarily close to the position of A.

Finite Type There is one case where this criterion can be checked algorithmically, with the aid of a computer if needed. When the set of neighbor maps is finite, we need only check whether hij 6¼ id; or, equivalently, whether two pieces Ai, Aj coincide. All examples here and almost all examples in the literature have a finite number of neighbor maps. In this case A, and the IFS f1, ..., fm, is said to be of finite type. The neighbor maps can be generated recursively, starting with f-1 i fj for i; j ¼ 1; . . .; m; i 6¼ j: The recursion is given -1 by fik-1fj‘ = f-1 k (fi fj)f‘. It turns out that for neighbor maps – with Ai \ Aj 6¼ ; – only isometries with |v| \ c need to be considered, where c is a small constant, for example, 2 times the diameter of A. The algorithm will stop after finite time when there are finitely many neighbor maps. In this case, the IFS f1, ..., fm, or the generated self-similar set A, is of finite type. The figures that follow were checked with this algorithm [3].

Choice of Maps and Symmetry Our principle was to look for finite type fractals. The number of neighbor types is a measure of the complexity of the geometry [3], and we are searching for the simplest examples possible. This restricts the choice of mappings since composition of rotations in R3 is usually noncommutative. In a series of articles written around 1999, Conway, Radin, and Sadun discussed related questions. Among other things, they found that when two rotations g, h around the origin in R3 with different axes fulfil any algebraic relation such as g2hg-1h3g = id, then special conditions must be fulfilled that are often connected with the Platonic solids [7]. Since in the case of finite type at least two neighbor maps must be equal, hij ¼ hi0 j0 , the rotational parts of the fi must fulfil such a relation. Thus it seemed reasonable to look for mappings associated with regular polyhedra. Actually, symmetry significantly decreases the number of neighbor types (defined as in the following). If A does not possess symmetries, the geometric position of A and h(A) is determined by the map h. However, if s, t are symmetries of A, that is, isometries of Rn with s(A) = t(A) = A, then g = sht and h determine the same geometric neighbor position, up to symmetry of the pair of sets. In other words, s-1(A) = A and s-1(g(A)) = h(A). Thus a neighbor type of the IFS f1, ..., fm has to be defined as an equivalence class with respect to the relation g*h if there are symmetries s, t with g = sht. The Koch curve has two types, as seen in Figure 6, and Sierpin´ski’s gasket as well 2010 Springer Science+Business Media, LLC, Volume 32, Number 3, 2010

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as the tetrahedron has only one type (translation along an edge of the triangle) when reflections are included as symmetries. Menger’s sponge has three types of neighbors – those with a common face, a common edge, or only a common point. In the following we take symmetric sets, saving a lot of computation in checking the finite type condition.

Turning Plane Fractals into Space We now explain the pictures, which were produced with the excellent public domain software chaoscope [6]. We start with Sierpin´ski’s triangle with vertices ci, arranged symmetrically around 0 in the x1, x2-plane: fi(x) = rx + (1 - r)ci for i = 1, 2, 3, where r ¼ 12 : Now we compose each fi with a 90 rotation around the axis [0, ci]. Explicitly, let c1 = (1, 0, 0), and let t denote the 120 rotation in the x1, x2-plane with t(c1) = c2. Then we take f1 ðx1 ; x2 ; x3 Þ ¼ ðrx1 þ 1 r; rx3 ; rx2 Þ and f2 ¼ tf1 t 1 ; f3 ¼ t 1 f1 t: For r ¼ 12 we would get a Cantor set, but for r ¼ 23 the images of the triangle Dc1, c2, c3 under the fi will intersect in a vertical line segment L through 0. In Figure 7, the triangle and its image under f1 are depicted. Since each altitude in Dc1, c2, c3 is mapped into itself by the corresponding fi, the factor r ¼ 23 comes from the fact that altitudes in an isosceles triangle intersect in a ratio 1:2. The three pieces Ai of the resulting fractal will meet in a Cantor set on L. As open set U we can take the double pyramid with vertices c1, c2, c3, c02 , and c03 . Altogether, there are 420 neighbor maps. When neighboring pieces meeting in a single point are neglected, and the symmetry reduction is performed with the full symmetry group of A, there remain only two neighbor types. Neighbor pieces can meet Ai along a long or a short edge of the corresponding set Ui. In both cases, the angle between the pieces is log 3 unique. The dimension of this fractal is log 3=2 2:71: The construction can be applied to all n-gons with n C 3.

Modification of the Fractal Tetrahedron Next, consider the fractal tetrahedron, Figure 2. As a rotation axis for the fi, take the altitude from the vertex ci (the fixed-point of fi) onto the opposite face of the tetrahedron. Rotation around 120 would be a symmetry of A. To obtain something new, combine the fi with a rotation around 60, or, equivalently, around 180. It turns out that the images of the basic tetrahedron will meet along an edge when we increase the factor to r ¼ 35 : The result is shown in Figure 8. To see how we obtain r, consider the basic tetrahedron T as subset of the unit cube, with vertices c3 = (0, 0, 0), c4 = (1, 1, 0), c1 = (1, 0, 1), and c2 = (0, 1, 1). The diagram in Figure 8 shows the projection of T onto the x1, x2plane. As in Figure 7, let c02 and c03 denote the images of c2, c3 under f1 that are outside T. We have to choose r so that both c02 and c03 are images of vertices cj under two other mappings fi, which is indicated by connecting them to the vertices cj. By symmetry, c02 = (t, t, t) and c03 = (1 - t, 1 2 t, t) for some t. The equation jc02 c03 j2 ¼ jcp02 ffiffi pffiffiffi ffi c1 j gives 3 4 0 t ¼ 5 and jc2 c1 j ¼ 5 2: Since jc2 c1 j ¼ 2; the factor of f1 is 35 : For the open set U we can take the convex polyhedron with vertices ci and c0i , that is, the convex hull of the union of the four image tetrahedra fi(T). (Incidentally, they enclose an inner tetrahedron of the same size.) The Ai intersect in Cantor sets, and there are only two neighbor types that share a long or short ‘‘edge’’, but there are many other types of neighbors that meet at a single point. The Ai intersect the outer faces of U in Koch curves (to prove this, calculate the four mappings fi fj which map a face of U into itself). The log 4 dimension log 5=3 2:71 is almost the same as for Figure 7, 20 and also for the Menger sponge, log log 3 2:73: This method also applies to the cube, which is the self-similar set with respect to 8 homotheties with factor 12 and centres in the vertices. If the mappings are combined with 60 rotations around the space diagonals, and the factor is 58 ; Figure 9 is obtained, where the neighboring pieces touch in a single point, similar to the fractal tetrahedron.

C2

C‘ 2

C3 ‘ C3

C 1

Figure 7. A three-dimensional modification of Sierpin´ski’s triangle. The diagram shows the triangle given by the fixed points ci of the mappings fi, and the image triangle under f1. 16


C4

C2 C‘2 C‘3 C3

C1

Figure 8. The modified fractal tetrahedron touches the faces of U with Koch curves. U is obtained from a tetrahedron T by adding small pyramids on the faces of T. In the diagram, U is depicted as a subset of the unit cube, viewed from above. The vertices c1, ..., c4 of T are vertices of the cube; the other upper vertices of the cube were moved inside to the position of c02 and c03 .

Figure 9. Two views of the modified cube, generated by homotheties combined with 180 rotation around the cube’s diagonal.

Figure 10. Two views of the reverse of Figure 7. 2010 Springer Science+Business Media, LLC, Volume 32, Number 3, 2010

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The Reverse Fractal For A generated from the mappings fi, define the reverse fractal of A by the mappings - fi (cf. [2], Section 10). It is obvious that the neighbor maps hij = f-1 i fj coincide for both families of mappings. Thus when one of the fractals is of finite type or fulfils the open set condition, so does the other. When A is centrally symmetric, as in Figures 9 and 5, it coincides with its reverse. In other cases, the appearance can be quite different although the dimension and the number of types remains unchanged. Figure 1 is the reverse of Figure 8. The reverse of Figure 7 is shown in Figure 10. In both cases, the geometry becomes more complicated, and more realistic as a model for natural phenomena. It is also possible to apply the minus sign only to some of the fi, or to some of the three coordinates, but we will stop here and leave it to you to create more examples.

in tetrahedral kites, see http://www.cit.gu.edu.au/*anthony/ kites/tetra. [6] Chaoscope software developed by N. Desprez, http://www. chaoscope.org/. [7] J. H. Conway, C. Radin, and L. Sadun, Relations in SO(3) Supported by Geodetic Angles, Discrete Comput. Geom. 23 (2000), 453–463. [8] K.J. Falconer, Fractal Geometry. Mathematical Foundations and Applications, Wiley 1990. [9] T. A. DeFanti and J. C. Hart, Efficient antialiased rendering of 3-D linear fractals, ACM SIGGRAPH Computer Graphics 25 (3) (1991), 91–100. [10] J.E. Hutchinson, Fractals and self-similarity, Indiana Univ. Math. J. 30 (1981), 713–747. [11] H. Jones and A. Campa, Fractals Based on Regular Polygons and Polyhedra, N.M. Patrikalakis (ed.) Scientific Vizualization of Physical Phenomena, Springer, New York, 1991, 299-314.

[1] C. Bandt and S. Graf, Self-similar sets 7. A characterization of

ski Polyhedra, Pi Mu [12] A. Kunnen and S. Schlicker, Regular Sierpin Epsilon J. 10 No. 8 (1998), 607–619.

self-similar fractals with positive Hausdorff measure, Proc. Amer.

[13] B.B. Mandelbrot, The Fractal Geometry of Nature, Freeman, San

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Math. Soc. 114 (1992), 995–1001. [2] C. Bandt and N.V. Hung, Fractal n-gons and their Mandelbrot sets, Nonlinearity 21 (2008), 2653–2670. [3] C. Bandt and M. Mesing, Fractals of finite type, Banach Center Publications 84 (2009), 131–148.

Francisco, 1982. [14] P.A.P. Moran, Additive functions of intervals and Hausdorff measure, Proc. Camb. Phil. Soc. 42 (1946), 15–23. [15] A. Norton, Generation and rendering of geometric fractals in 3-D, ACM SIGGRAPH Computer Graphics 16(3) (1982),

[4] M.F. Barnsley, Fractals Everywhere, 2nd ed., Academic Press, 1993.

61–67. [16] K.A. Roth, Julia sets that are full of holes, Math. Intelligencer 30,

[5] A.G. Bell, Tetrahedral Principle in Kite Structure, National

No. 4 (2008), 51–56. [17] W. Sternemann, Neue Fraktale aus platonischen Ko¨rpern,

Geographic Magazine Vol. XIV, No. 6, 1903. Available at http:// www.fang-den-wind.de/bell_eng.htm . For recent developments

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Spektrum der Wissenschaft 11 (2000), 116–118.

Mathematically Bent

Colin Adams, Editor

Looking Backward 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]

will now continue my most amazing tale. I expect you have so far found it somewhat hard to believe, but I attest it all to be completely true. As I have previously related, in the year of our Lord 1898, on June the 25th, I was put into a hypnotic state by the esteemed animal magnetist, Dr. Pillsbury, in order to overcome my insomnia and to allow me to obtain a full night’s sleep. After inducing me into a somnambulant state, he left me sleeping on my bed in a specially built chamber embedded deep in the foundation of my house, so as to shield me from the noise and bustle of the city of Boston. Later that evening, the doctor departed permanently for New Orleans, but not before leaving me references for other doctors and instructing my servant on how to wake me in the morning. However, as I slept that night, an unfortunate accident with an oil lamp caused the house above me to burn to the ground. My poor servant Bartholomew perished in the conflagration. It was not until 112 years later, in the year of our Lord 2010, that workmen repairing a sewage line for the house that had replaced mine discovered the chamber in which I lay. After realizing that I was not dead, they called the owner of the house, Dr. Leete, who undertook to revive me. I awoke, physically unharmed, but weak and suffering a great degree of disorientation. Since that time, I have regained my strength, recovered my wits, and learned much about the marvelous new world that exists today. No longer do lines of tall smokestacks belch noxious fumes into the open air. Verily, the inhabitants of this epoch have overcome the dangers of over-industrialization and make no decision that might have negative repercussions for the surrounding environment. Consequently, they live in a verdant lush garden of a world, beautifully cultivated and carefully stewarded for the sake of future generations. To provide sustenance to the populace, fish and livestock are raised in humane conditions. And well-fertilized land produces substantial quantities of grain, fruit and vegetables. A cornucopia of delicious edibles weighs down every table. And no longer do orphaned children fend for themselves on the streets. If any citizens are incapable of caring for themselves and have no one to care for them, then the state takes charge. Homelessness does not exist in this new world. Everyone is guaranteed the right to an abode and the nourishment necessary for life to flourish.

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If someone becomes sick or incapacitated in any way, the costs of hospitalization and doctor’s bills are borne by the government. All citizens are provided with equal and high-quality health care, as one would expect from an enlightened society. Moreover, all adults have the right to fulfilling careers. The most menial jobs are performed by automatons, which are constructed for that purpose. Citizens attend school for 12 years, after which they come to a decision about what profession to pursue. All professions are paid comparably, and one chooses what to pursue based on desire, not on the potential monetary reward. Indeed, it seems that the citizens of this time have truly succeeded in creating what can only be called a utopian existence for all. And much of this have I already related. But now, I shall explain to you what I have learned of mathematics in this new era. As you may remember, I myself once dreamed of becoming a mathematician. However, it was my father’s wish that I forego mathematics to become a lawyer. For a lawyer had the potential to make substantial sums of money, which would have allowed me to support a family in the appropriate manner. As I was betrothed to my beloved Edith, this career choice made eminent sense. But finding myself in this new world, I was very curious to learn how mathematics was perceived, and to perhaps determine if this could be a field that I might now pursue. In my new circumstance, the familial pressures to which I had once yielded had disappeared, leaving such decisions entirely up to me. One evening, as we sat smoking thoughtfully after an excellent meal, I asked Dr. Leete about mathematics in the modern era. ‘‘Ah, it is funny you should ask,’’ he responded. ‘‘For you see, in this year of 2010, the Mathematics World Exposition is taking place right here in the city of Boston. This event occurs once every 10 years, and a city considers itself very lucky to be chosen the venue. The Mathematics World Exposition is a celebration of all things mathematical, and it will give you a sense of how important mathematics is now perceived to be. Lucy and I will take you to view it tomorrow afternoon.’’ Lucy was Dr. Leete’s niece, and the spitting image of my now long dead Edith. Although my heart still ached with the loss of my beloved, there was so much that reminded me of her in Lucy that I could not help but have feelings for the girl. The next day, I accompanied the good doctor and his niece on the moving walkway that transported us to the Exposition Halls. Lucy was dressed in a becoming green dress that, although quite different in style from the dresses to which I was accustomed, was still modest in appearance and suggestive of her virtue. As we rode along, the doctor nodded to passing acquaintances as he explained to me the current state of mathematics. ‘‘Today, all citizens are brought up to be aware of the importance of mathematics. At a very young age, children become proficient in counting, addition, subtraction, multiplication and division. All teaching at this level is in the form of games. So students never memorize as much as play to learn mathematics. Some of the greatest minds spend their

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time trying to come up with new games for the children to play that will teach them the mathematics.’’ I thought to myself how arduous the memorization of all the various mathematical facts had been for me. The primary method used by schoolmasters to reinforce memorization was through judiciously timed ruler raps on the knuckles. No one would have described the process as agreeable. ‘‘By age six, students are proficient with fractions, decimals, percents and the like. They are even capable of difficult computations involving logarithms to the base 10. All students can easily manipulate the most complicated of slide rules, which today have 20 or more moving parts.’’ ‘‘In addition to the teaching of mathematics that occurs in school, there are also performances by troupes of actors who travel from city to city, enacting the greatest moments of mathematics. Just last week, I went to see a reenactment of the death of Galois. It was very moving, but perhaps not appropriate for the younger children due to its violent climax.’’ ‘‘And I went to see the discovery of hyperbolic geometry,’’ added Lucy enthusiastically. ‘‘It was so funny to hear of all the mathematicians who tried to prove Euclid’s parallel postulate. Such silly people.’’ She wrinkled her nose in an endearing manner. ‘‘And there were many children there. They loved it, clapping quite loudly at the end. The actor who played Lobachevsky had a funny thick Russian accent. He made us all laugh.’’ Dr. Leete smiled at his niece and then continued. ‘‘For Halloween, two of the most popular costumes are Carl Friedrich Gauss and Isaac Newton. Last year, my great grandson Henrik went as Leonardo Fibonacci. ‘‘By the age of eight, the children have mastered algebra and trigonometry. Speaking of which…’’ He pointed to a billboard towering above us. Upon it was a picture of the graph of sine of x with various happy cartoon children playing upon the curve. Beneath the picture appeared the words, ‘‘Sine of x, a wonderful tool for your enjoyment.’’ ‘‘Many are the ways in which we sing the praises of mathematics,’’ he said. ‘‘By 10, children have mastered the calculus. You may wonder why calculus is considered important enough that we insist everyone learn it. But even those who ultimately choose a trade to which it is not relevant benefit from the rigors of its study.’’ ‘‘But for those who don’t use it in their professions, don’t they forget it quickly?’’ I asked. ‘‘To maintain an ongoing interest in mathematics amongst the general public,’’ replied Dr. Leete, ‘‘the government produces quiz shows that are disseminated over the telephone. Average citizens compete in an attempt to achieve renown. Thousands of others listen in over their telephones as a lucky participant tries to answer difficult questions. The most popular of these shows is called, ‘‘Who Wants to Use Their Knowledge of Mathematics to Achieve Distinction?’’ Lucy grabbed my hand enthusiastically. ‘‘Oh, Julian, you must meet Wendell Carmody. He will be at the Exposition. He is the most successful of the contestants ever. His knowledge of mathematics is truly encyclopedic.’’ Dr. Leete laughed. ‘‘He was in fact a patient of mine, and I have promised Lucy an introduction. In fact, he is perhaps more famous for winning the Nobel Prize in Mathematics.’’

I had heard of Alfred Nobel’s passing in 1896, and his will, which endowed prizes in various fields. However, by 1898, the prizes had yet to be distributed. ‘‘I thought Nobel did not endow a prize in mathematics,’’ I said, ‘‘only in the sciences, literature and peace.’’ ‘‘You are correct,’’ replied Dr Leete. ‘‘Mathematics was added to the list of categories in 1910.’’ ‘‘I will look forward to meeting this Carmody,’’ I said. Lucy continued to hold my hand, and I certainly made no protest. Dr. Leete went on. ‘‘By age 12, students have learned the methods of differential equations, so as to understand those formulas that govern the movement of electrons on wires. Many of the inventions you see around you, including the automatons, are powered by electricity. They could not exist, were it not for the calculus.’’ I could see several of the automatons, hulking metal hydraulically-powered creatures riveted together out of iron, riding the people movers as they went about their simple errands. ‘‘At the age of 14, students must choose whether or not they intend to continue with mathematics. Those who desire to learn more move on to probability and advanced calculus, wherein they are taught the techniques of rigorous argumentation and the roles Greek letters play within them. ‘‘At the age of 16, students decide whether to pursue applications of mathematics or the pure mathematics, wherein the subject is studied for its aesthetic beauty rather than for its utilitarian benefits. In fact, in Boston alone there are so many students interested in continuing their mathematical education that no lecture hall could possibly be large enough to contain the audience. ‘‘But the miracle of the telephone has allowed thousands of students to hear lectures by world famous mathematicians. In fact, Carmody is giving a lecture tonight on fluxions. We can listen to it if you so desire.’’ ‘‘Yes,’’ I replied. ‘‘I would be very interested.’’ ‘‘Me, too,’’ added Lucy. ‘‘We will plan on it then. In fact, if we were not free to hear the live lecture, it would not matter. For the government records the lectures on gramophone cylinders, copies of which are then distributed worldwide.’’ ‘‘That is quite incredible,’’ I said. ‘‘So all students get to hear the very greatest of mathematicians.’’ ‘‘Yes,’’ replied Dr. Leete. ‘‘By these advanced means, the need for individual teachers has dropped precipitously. Now the local teacher need only record on the blackboard the equations described by the expert over the phone.’’ At this point, the moving walkway deposited us at the entrance to the World Mathematics Exposition. I looked up at the giant metal Greek letter p, the legs of which formed the gate to the park. ‘‘Where do we pay?,’’ I asked, as we walked toward the gate. Both the good doctor and his niece laughed out loud. ‘‘Silly,’’ said Lucy, ‘‘you never have to pay to learn mathematics.’’ We joined the convivial throng funneling into the Exposition park and soon found ourselves walking with many others down a lane surrounded by shiny metal sculptures of

various quadric surfaces. As we approached the main avenue, we could see floats passing by and hear the music of marching bands. ‘‘Is there always a parade?’’ I asked. Lucy smiled, pleased to be able to explain it to me. ‘‘You see, the main street is in fact in the shape of a circle, and the marchers continue around it all day. We call it the infinite parade, as a circle has no beginning or end.’’ As we approached, I could see the float for ex going by. Directly behind it appeared a slightly tired marching band playing a strange, somewhat dissonant tune. ‘‘Those notes correspond to the digits of p,’’ said Dr. Leete. He pointed to a nearby building. ‘‘Over there, computational wizards continue to calculate new digits of p, so as to stay ahead of the band. The song never ends.’’ ‘‘Don’t worry,’’ said Lucy, sensing my concern. ‘‘The bands take turns. They get plenty of time off.’’ ‘‘Some of the greatest musicians of the day have devoted themselves to the creation of songs about mathematics,’’ said Dr. Leete. ‘‘Perhaps the most popular song is one about the quadratic formula. It is so catchy, I sometimes cannot get it out of my head.’’ ‘‘I could sing it for you later,’’ said Lucy, squeezing my hand. ‘‘I am sure I would enjoy that very much,’’ I replied. We followed a small tunnel under the infinite parade route and found ourselves at the entrance to a large wooden rollercoaster. ‘‘What is the mathematical significance of this? ‘‘ I asked. Lucy giggled. ‘‘Do you not see that the track passes over and under itself,’’ she said, ‘‘sometimes passing through the openings in the supporting timbers. Since the track eventually ends where it started, the entire rollercoaster is in fact a knot. I believe that even in your era, people understood the mathematical significance of knots.’’ Indeed, in my own time, I had known of a theory of the atom based on knotted vortices in the ether. But I also knew that it had been discredited when the MichelsonMorley experiment demonstrated there was no ether. There had also been a somewhat less scientific theory that we were all knotted three dimensional cross-sections of fourdimensional creatures. But I hesitated to display my own ignorance on the subject before the lovely Lucy. So I did not venture to seek further explanation. ‘‘Come,’’ said Lucy, pulling me forward, ‘‘We must all go for a ride.’’ ‘‘You won’t get me on that contraption,’’ chortled Dr. Leete. ‘‘And I warn you, Julian, you may regret it.’’ I had never been fond of carnival rides, but the opportunity to be alone with Lucy overrode any reluctance on my part. As we were seated next to each other, our legs momentarily touched, sending a charge of what electricity must feel like up my spine. As the ride began, the car in which we were seated ascended a long tilted track high into the air. As the ground fell away below us, Lucy grabbed onto my arm and hung on tightly. We reached the highest point and then plunged downward at a terrific speed. She screamed, burying her head in my shoulder. As we shot around the twisted 2010 Springer Science+Business Media, LLC, Volume 32, Number 3, 2010

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track, the entanglement enhanced the sense of disorientation to such a degree that one became confused as to which direction was up, down, left or right. By the end, my stomach was as knotted as the track itself. Dr. Leete laughed aloud when he spied the expression on my face. ‘‘Truly, Julian,’’ he said, ‘‘I tried to warn you. Now, come, we will seek out Wendell Carmody.’’ The Nobel Pavilion was in the Greek style with large pillars framing the ornate entrance. As we walked into the marble hall, I could see a row of busts of the previous winners lining the walls. I was stunned to recognize that one of the winners was Buskin, a student I had known when I was at Harvard. No one would have suspected that he might eventually win any kind of prize in mathematics. He had always been a particularly lazy student, rarely attending class, choosing instead to spend his time frequenting a variety of Cambridge pubs. But the surprise I experienced was tenfold greater when I spied a bust of another winner, a woman who had won the prize in 1928. It was none other than my dear lost Edith. ‘‘But, how can this be?’’ I exclaimed. ‘‘Oh, that is my great great grandaunt Edith Wilson,’’ replied Lucy. She received the Nobel Prize in mathematics for her work on neutral groups.’’ ‘‘She became a mathematician?’’ I said more to myself than anyone else. ‘‘An excellent mathematician,’’ interjected Dr. Leete. ‘‘She spent all her time working on mathematics. Never married.’’ ‘‘Indeed,’’ I said, a strange mix of emotions rising in my breast. ‘‘Come,’’ said the doctor. ‘‘There is the dais upon which we will find Carmody.’’ I could sense Lucy’s excitement grow as we approached. A large red banner hung above the dais, proclaiming all of the degrees and awards that had been received by Carmody. Dr. Leete ushered us forward. ‘‘Dr. Carmody, so good to see you.’’ ‘‘Ah, Dr. Leete,’’ he responded, rising to greet us. ‘‘I am still grateful to you for curing my nasal drip.’’ ‘‘It was nothing,’’ responded the doctor. ‘‘Let me present to you my niece, Lucy, and a visitor, Julian.’’ Ignoring me completely, he focused all his attention on Lucy. Taking her hand, he lifted it to his lips and kissed it. ‘‘Doctor, why have you kept your niece hidden from me?’’ he asked without looking away from her. She giggled. ‘‘He has been too busy to escort me here sooner. For I have begged him since the opening of the Exposition to bring me to meet you.’’ A sly smile flitted across his face. ‘‘I am honored by your interest in me,’’ he said. ‘‘Do you have mathematical inclinations?’’ ‘‘Oh, yes,’’ replied Lucy. ‘‘Mathematics is the lifeblood of the sciences.’’ ‘‘Indeed it is.’’ ‘‘And for what did you receive your Nobel Prize?’’ I interrupted. Carmody turned to look at me. ‘‘I would think you would know,’’ he replied. 22


Dr. Leete jumped in. ‘‘Ah, Julian is not from this country. So his schooling has focused on other subjects.’’ Dr. Leete and I had agreed not to divulge my true history until I had a bit more time to adjust. ‘‘I see,’’ said Carmody as he eyed me carefully. ‘‘Well, perhaps I can enlighten you. I solved the Goldbach Conjecture, the greatest open problem in all of mathematics. It states that every even integer greater than 2 is the sum of two primes.’’ ‘‘Yes, I am aware of the conjecture,’’ I replied, ‘‘just not of your solution.’’ ‘‘Your country must be very far away indeed,’’ replied Carmody. ‘‘Ah, yes it is,’’said Dr. Leete uncomfortably. ‘‘Well, look at the time. Perhaps we should be going.’’ ‘‘Must we, Uncle?’’ asked Lucy. ‘‘We just arrived. I so wanted to get to know Dr. Carmody better.’’ ‘‘I think our guest might be getting tired,’’ replied the doctor nodding to me. ‘‘He is still adjusted to a different time zone, having only recently arrived by hot air balloon.’’ ‘‘Don’t leave on my account,’’ I said. ‘‘Well,’’ said Lucy to Carmody, ‘‘For the sake of dear Julian, we should go. But I do hope we get the chance to see you again. We look forward to your lecture tonight, which we will listen to over the telephone.’’ ‘‘I will do my best to make it worth your while,’’ replied Carmody. ‘‘And I would very much enjoy seeing you again.’’ He bowed ceremoniously as we turned to go. After dinner at Dr. Leete’s house, the doctor hooked the telephone up to the gramophone speaker, which immediately emitted a static sound. The three of us made ourselves comfortable, and at the appointed time the static was replaced by an announcer, who gave a flowery introduction to Carmody that included a bloated list of his various accomplishments. Then Carmody began to speak in a pedantic manner. He continued to wax on about fluxions and fluents for the next hour. At various points, his lecture was interrupted by the clapping of what must have been the live audience present in the auditorium. Carmody did his best to sound sophisticated and erudite, using the largest words he could muster to explain the simplest ideas. Lucy sat listening raptly. At the end of his lecture, Carmody summed up and then added, ‘‘I would like to dedicate this lecture to Dr. Leete and his niece Lucy, whose interests in mathematics are an inspiration to us all.’’ Lucy turned bright red and had great difficulty hiding her pleasure. Afterward, the doctor and I walked Lucy home and then returned to the doctor’s parlor to smoke. After discussing various aspects of the lecture, I asked Dr. Leete where I might find Carmody’s proof of the Goldbach Conjecture. ‘‘Why, it appeared in every major magazine,’’ he replied. He fished through a stack of copies of the Gentlemen’s Home Quarterly and handed me a copy. On the cover was Carmody’s supercilious expression. ‘‘Can I borrow this? ‘‘ I asked. ‘‘Certainly,’’ he replied. I snuffed my cigar in the ashtray. ‘‘I am a bit tired.’’ I said. ‘‘I think I shall retire for the evening.’’

The doctor bid me good night, and I ascended the stairs to my room. I immediately settled at the desk and began pouring over Carmody’s proof of the Goldbach Conjecture. Although somewhat technical, I was able to understand the gist of it. It was built upon several results of Riemann that had already existed in 1898. I spent many hours going over it, but eventually, exhaustion overcame me, and, unable to keep my eyes open any longer, I climbed into bed and fell into a deep sleep. At some point, I began to dream. Carmody was tapping on Lucy’s door. ‘‘Let me in,’’ he cooed. Clothed only in her dressing gown, she rose from her bed to unlock the door. I called to her. ‘‘Lucy, Lucy, do not open the door.’’ But she was oblivious to my entreaties. Suddenly, a voice interjected. ‘‘Wake up, sir. It is morning.’’ I opened my eyes to the dim light of an oil lamp, and the face of my long dead servant Bartholomew leaning over me. I leapt from the bed. ‘‘What is this,’’ I cried. ‘‘You died over 100 years ago.’’ He looked at me strangely, and then said, ‘‘That must have been quite a dream, sir.’’ I looked about myself and realized that this was the secret chamber under my house. ‘‘What day is it?’’ I asked, with great trepidation. ‘‘Why, it is Tuesday, sir, June the 26th. Dr. Pillsbury hypnotized you so that you might sleep, and then instructed me to wake you.’’ ‘‘And what of the fire?’’ I asked. ‘‘I know not of a fire,’’ he responded. My mind was in turmoil. Could it be that all that had happened to me had been a dream? That there was no Dr. Leete, that there was no Lucy, that there was no Carmody. ‘‘Sir?’’ asked Bartholomew. ‘‘Um, yes, Bartholomew, I will be all right,’’ I said. ‘‘Just give me a bit of time.’’ ‘‘Sir, Miss Edith is expected within the hour. She said you were to go with her on a carriage ride today.’’ ‘‘Edith, you say. I see.’’ I didn’t know what to think. It appeared that I had lost the lovely Lucy, so gay and so spirited. And yet, at the same time, I had gained my dear Edith back. It was overwhelming. ‘‘I will be up in a bit,’’ I said to Bartholomew. Very good, sir,’’ he replied as he took his leave. I put on my robe and then sat down at the desk. Could it be that the entire world of 2010 had simply been an incredibly intricate creation of my imagination? Could everything that I had experienced have been my mind’s interpretation of the future?

On the other hand, I considered, what if what I was experiencing now was the dream, and 2010 the reality? It certainly had seemed at least as real as this room did now. How was I to know what was dream and what was reality? But then it occurred to me that in 2010, I had read over Carmody’s proof of the Goldbach Conjecture. If I retained the memory of it, then 2010 must have been real. I grabbed a sheaf of paper that lay on the desk, and wrote feverishly for half an hour. When I had finished, there before me on the pages was the proof. It appeared to be correct. I saw no logical contradictions, and the arguments building on Riemann’s work appeared sound. So this meant that indeed I had been in the year 2010, and my experience now was in fact the dream. So all I need do was wake up. And yet everything around me continued to appear completely substantial. I then realized that another possibility existed. Perhaps, the mind being the intricate instrument that it is, my subconscious had come up with the proof itself in the process of creating the dream of 2010. I oscillated between believing either of these two possibilities. My brain seemed to be spinning like a top. I cupped my head in my hands. But finally, I calmed myself. For whichever was the reality, in either case, I was in possession of the most important mathematical discovery of the century. At that moment I determined that as long as I remained in this time period, perhaps for the rest of my life, I would become a mathematician, my father’s opinion notwithstanding. Even he could not voice disapproval when I announced my result. My very first published theorem would be the greatest theorem of the age, a proof of Goldbach’s Conjecture. And perhaps I would win the Nobel Prize in mathematics, if indeed such a prize were to be endowed. And if this other world, this world of 2010 was not just a dream, but in fact would exist in such a form over 100 years hence, I would have the satisfaction of knowing that I had robbed Carmody of his greatest theorem, and perhaps in the process prevented him from winning a Nobel Prize. And, most importantly, perhaps I would have prevented him from besmirching the lovely Lucy. I rose from the desk, and threw on some clothes, contemplating my reunion with Edith. Was she really interested in mathematics? I would ask her at once. And would we become a mathematical couple, two like-minded individuals, united through our love of all that is mathematical? I tucked the papers I had written into the desk drawer for safekeeping. Only time would answer these questions. I rushed up the stairs to meet my future.


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The Last Boat from Lisbon: Conversations with Peter D. Lax ISTVAŃ HARGITTAI eter Lax (born in Budapest, 1926) is Professor Emeritus of New York University (NYU). He started his high-school studies in the Minta Gimna´zium in Budapest. When he was 15 years old, he immigrated with his family to the United States to escape the persecution of Jews. The Lax family left Europe on December 5, 1941, on a boat from Lisbon. They were in the open sea when Japan attacked the United States at Pearl Harbor on December 7. The next day, the United States officially became a belligerent party in World War II, hence the title of this interview. Peter Lax completed his secondary education at Stuyvesant High School in Manhattan. He received his Bachelor’s degree in 1947 and his PhD degree in 1949, both from NYU. He has been with NYU since 1951. In the period 1972–1980, he was Director of its Courant Institute of Mathematical Sciences. In 1945–1946 and in 1950–1951, he worked at the Los Alamos National Laboratory. Professor Lax is a Member of the National Academy of Sciences of the USA and of the American Academy of Arts and Sciences; Foreign Associate of the French Academy of Sciences; Foreign Member of the Russian (formerly Soviet) Academy of Sciences; the Hungarian Academy of Sciences; and the Academia Sinica, Beijing. His honors and awards include the Award in Applied Mathematics and Numerical Sciences of the National Academy of Sciences of the USA; the U.S. National Medal of Science (1986); the Wolf Prize (Israel, 1987); and the Abel Prize in 2005 from the Norwegian Academy of Science and Letters ‘‘for his groundbreaking contributions to the theory and application of partial differential equations and to the computation of their solutions.’’

P

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His first wife, the late Anneli (neé Cahn), was also a mathematics professor at NYU. Their son Johnny was a graduate student in history at Columbia University when he was killed in an automobile accident caused by a drunk driver. Their son Jimmy is a physician in New York City. Lax’s second wife, Lori, is a musician and the daughter of Richard Courant. We recorded three conversations, one in Budapest in November 2005 and the other two in New York in May and June 2007. They were combined, and what follows is an edited and shortened version. Let’s start with your family background. My father, Henry Lax (originally, Lax Henrik, 1894–1990) and my mother, Kla´ra Kornfeld (1895–1973) met in medical school. My mother might have been one of the first women admitted to medical school in Hungary. They started medical school just as WWI was starting and when they graduated they were living in a whole different country. Difficult times followed. First the Commune came and then the White Terror. My parents did not participate in the Commune, but some of their friends did and had to flee to the Soviet Union. My parents corresponded with them, but after a while there were no responses, and after a while, they got messages that for God’s sakes, don’t write to us, and then came the news about arrests, and then about persecutions or that people had been killed or disappeared. I know about the story that in 1920 my father traveled to Szeged by train on one occasion; the next day, the same train was stopped, the Jewish passengers were taken out and shot to death. My father went to work in the Jewish Hospital in Budapest and became Chief of Medicine.

Peter Lax with his parents in London in 1947 (courtesy of Peter Lax).

Your parents decided to emigrate in 1941 and that means that they had foresight, which few people had in Hungary. It was difficult to leave, especially for my father because he was a very successful physician in Budapest, but my mother insisted. We were traveling through Germany by train, and when we reached the Swiss border, the German guard checked our papers and then he said, ‘‘just a moment,’’ and was going to say something. The air froze for us for a moment. But to our relief, all he asked was whether we had any ration coupons left for meat and butter that we received when we entered the Reich. My father gave him the coupons. The irony was that had we not had our papers in order, they would have had us killed, but as it turned out, they asked us for a favor. I would like to ask you about your schooling in Budapest. For high-school, I went to the famous Minta Gimna´zium, which was a very good school; Theodore von Ka´rmań and

AUTHOR

.......................................................................... ´ N HARGITTAI is a physical chemist ISTVA

and head of the George A. Olah PhD School at the Budapest University of Technology and Economics. When he was eleven years old he received a book about coal as a prize in a mathematical competition, and it turned him to chemistry. He and his fellow scientist wife, Magdolna, have coauthored and edited about a dozen books on symmetry, the latest being Visual Symmetry (World Scientific 2009) and Symmetry through the Eyes of a Chemist (3rd edition, Springer 2009). Budapest University of Technology and Economics, Post Office Box 91, H-1521 Budapest, Hungary e-mail: [email protected]

Edward Teller were among its graduates. My interest in mathematics manifested itself strongly. My parents arranged for me to have additional instructions from Ro´zsa Pe´ter1 and advice from Deńes Ko¨nig.2 When we were leaving Hungary, they each sent a letter to John von Neumann. Miss Pe´ter was wonderful; she taught at the Jewish Gimna´zium, and after the war she became a university professor. It was at her suggestion that I participated in the Eo¨tvo¨s competition, which was organized for high-school graduates. I could not officially participate because I was not a high-school graduate. I did well. How well? Very well. As these letters attest, I outperformed even the winners. You must have learned everything in mathematics in Hungary that you needed for the American high-school. And much more. In New York, how did your parents decide to which high-school you should go? Someone advised them, and I went to Stuyvesant High School. Many famous scientists and mathematicians went there. For example, Jack Schwartz and Paul Cohen, who solved the continuum hypothesis. But you were not taking mathematics. I did not, but I was a member of the mathematics team in the citywide competition among high-schools. Stuyvesant won in the year when I was on its team. There were five members and I and two of my teammates later became members of the National Academy of Sciences. How would you compare Stuyvesant and Minta? The comparison is not between these two particular schools but between the atmosphere in the Hungarian gimna´zium and the American high-school. In the gimna´zium I was a very good student, but I was petrified of my teachers. They were very kind to me, but I was still afraid of them. In America, the teachers were friends. However, many excellent people came out of the Hungarian high-school. Perhaps it was efficient? [after a long silence] Possibly. Otto Neugebauer, one of Courant’s students, said and only half in joke that the education in America, which goes back to John Dewey, is pragmatism. Its principle is that the school should prepare you for life. Neugebauer thought that the European schools did that much better than the American schools. In the European school, you recognized who your enemy was— the teachers. That could explain why European schools were so efficient. You had to fight for your life. Then you had to choose college. Von Neumann was extremely nice to me. He was not a warm person; most of the time he was thinking. Immediately upon our arrival, in 1942, after he had received the letters I have mentioned, he called on us here in New York and talked to me. I have his letter he wrote to my father about my education. The advice he gave was wrong. He thought I should go to Columbia University. My father also consulted with Ga´bor Szeg} o, whom he knew very well.

1

Ro´zsa Peter (1905–1977) was a mathematician who never found proper employment until after World War II. One of her books, Playing with Infinity, has been a worldwide success. 2 Deńes Ko¨nig was a mathematician, a pioneer of the theory of graphs. He committed suicide during the Nazi terror in Budapest.


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Szeg} o said that Richard Courant was very good with young people. That was the best possible advice. When did you graduate? I entered Stuyvesant in February 1942 and graduated from it in February 1943. I then entered NYU immediately. I had only three semesters at NYU and spent the summer of 1943 at Stanford. When I turned 18, I was drafted before I could have graduated. I only asked the board for a four-week extension so that I could complete the semester, which they granted. At Stanford, I took several courses and a reading course with an excellent mathematician, Victor Uspensky, an ex-Russian. He was Vinogradov’s teacher, the czar of mathematics in Russia. With Uspensky, we read a fairly advanced book on number theory based on lectures of Dirichlet. Szeg} o was the one who created the department at Stanford. When he came in 1938 there were only two good mathematicians there and one of them just retired, so he built up the department. Mrs. Szeg} o and my mother were first cousins and I lived with the Szeg} os. I learned a lot of mathematics during breakfast and dinner. What happened after you were drafted? First I went to basic training in Florida; it was the usual basic training; and I did well in it. I learned how to shoot a rifle, the machine gun, and so on. When I finished the training, I was sent to Texas A&M and attended an Army Specialized Training Program. They trained people in engineering, in languages, and in other programs. I got a semester’s worth of training in engineering, which was quite good. How were you selected for the program? All recruits, when we entered the Army, took a very detailed intelligence test, and it must have been the result of that test. When did you arrive in Los Alamos? In June 1945. You did not even have a Bachelor’s degree. I only had four semesters at NYU. What did you know about the Los Alamos project when you arrived? Nothing. Once we arrived there, they explained to us what they were doing. We were about 30 people. When you learned about the project, were you shocked? I was. They told us that they were building a bomb out of plutonium, an element that did not even exist in the Universe, but they were manufacturing it in Hanford. Did you realize at that point the role of the Hungarian scientists in initiating the program? I did not. Did you consider yourself a Hungarian at that point? Yes, I did. [after some hesitation] No, I considered myself American. What was your state of mind? I got married in 1948. Anneli, my wife, was interested in languages, and wanted to learn Hungarian for fun, but I said no. I did not want a Hungarian family; I wanted an American family. I don’t think I ever forgave the Hungarians for what they did, although I found that out only later, after the war, about all the horrors of 1944–1945. How did you find out?

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Peter and Anneli Lax in 1953 (courtesy of Peter Lax).

From people and from books; there was a book in particular, Zsido´sors, Jewish Fate, that told what happened in detail. How often do you visit Hungary? Every other year. I still have three cousins there and I have scientific contacts. So you must know that Hungary has never truly faced its past. I know. For that reason I feel more comfortable in Germany. Don’t feel too comfortable though, because facing the past was not done too deeply in German society, according to my experience. I had a student from Germany; he got his degree with me and we became very good friends and still are. His parents were anti-Nazi. His father is still obsessed with what happened; he is giving lectures about the past. Then, I have friends who came from Germany, most of all Ju¨rgen Moser who had spent the war in Germany; he was drafted during the last year of the war and he and his comrades were sent out to fight tanks with rifles. Most of his classmates were killed in battle. His parents were anti-Nazi. He is a wonderful person and so is Stefan Hildebrandt. Courant was a Professor of Mathematics in Go¨ttingen before he… …before he was kicked out of Germany. He came to the United States in 1934 when he was 46 years old, so when I met him in 1942, he was 54. In spite of the age difference, you could work together. Very much so. Did you ever talk about his Go¨ttingen life? Was he bitter? When the war was over, he went back to Germany as soon as he could. He wanted to see what he could do to help. He knew people who were strongly anti-Nazi and he was also looking for young people. He also helped to put people in position who were talented and free of Nazi taint. He had no bitterness at all. My father was the opposite. He went back

to Budapest after the war only to fetch his mother and visit his sister. He never stepped out of his hotel. He never forgave the Hungarians. After his sister died, he never went back again. You don’t have such resentment. I don’t. The present generation is not to be blamed. Because Hungary never truly faced its past, it is being reproduced in members of the new generations. Before the political changes, there was covert anti-Semitism; since the political changes, overt anti-Semitism has become tolerated by some strong political forces, and thus, encouraged. I know you’re right, and I do feel uncomfortable. Let’s return to Los Alamos. What did you do there? Did they use you as a computer? No, they used me as a mathematician. I did a criticality study of an ellipsoidal assembly of explosives. That was a nontrivial study. For me the shape of the plutonium bomb was especially interesting because for the implosion, they constructed a shape that could be described as a truncated icosahedron. Yes, that was for the explosion lenses. The plutonium core itself was spherical. They used two different explosives, which detonated with different speeds, so as to produce as spherical an implosion as possible. Von Neumann had contributed to that. Did you talk with him at that time? A little bit, and I talked with him more when I went back to Los Alamos after I got my PhD in 1949. That was when I became interested in differential equations in solving fluid dynamical problems. Please, tell me more about your first wife. She was a mathematician and she was a PhD pupil of Courant. We were graduate students together. Later, she taught undergraduate courses at NYU. She also edited a series of 40 volumes of mathematical books for high-school students. After her death in 1999 they renamed the series for her. We wrote a calculus book together.3 It was Courant who proposed you initially for membership in the National Academy of Sciences of the USA. Are mathematicians included among scientists? There are two sections entirely devoted to mathematicians, the Section of Applied Mathematics and the Section of Computer Science. I was elected in the Section of Applied Mathematics. You had great mentors, like Ro´zsa Pe´ter, von Neumann, Kurt Friedrichs, Courant, and others. Have you had great pupils? Yes, I have. Two of them are now members of the National Academy, which is, of course, not the only measure. I had 55 PhD students altogether and at least 15, maybe 20, among them have become active scientists; quite a few of them outstanding. You served as director of the Courant Institute, and before that you directed the Computer Center. I have read your statements that you never sought administrative positions but did your best when you had to have one. You

3

Peter Lax in front of his blackboard at the Courant Institute, 2007 (photo by I. Hargittai).

hired good mathematicians, for example. Is there anything else that might be of interest to mention from your experience in these positions? There was a very peculiar incident at the Courant Institute, which occurred in 1970, at the height of the student unrest. The war in Viet Nam was used as an excuse, but the rioters had a much broader agenda. In the spring of 1970, such a group, led by a rogue professor, occupied the Courant Institute and demanded that the administration put up $100,000 bail for the Black Panthers, a revolutionary group. If the university refused, they threatened to destroy the computer at the Courant Institute, worth $3.5 million. At that time I was Director of the Computing Center. The University didn’t give in; after two days the occupiers left; a group of us from the Computing Center were in the lobby of our building. I smelled smoke, so we ran up to the computer room, and found a burning fuse, leading to bottles of flammable liquids tied to the computer. Two of my younger colleagues put out the fuse, and the rest of us removed the bottles. So the computer was saved. Afterwards, Anneli asked how I could be so crazy to run toward a burning bomb. I told her that I was so angry that I didn’t think. There are a few prizes about which people used to say, ‘‘that is the Nobel Prize in Mathematics.’’ Now with the establishment of the Abel Prize, it seems to be truly the supreme recognition in mathematics. Another important award is the Wolf Prize, which you have, and the Fields Medal (below the age of forty), which you don’t. There is certain arbitrariness about prizes. There is a tendency to give the Fields Medal for solving an important problem. Maybe I was better known for posing problems. La´szlo´ Fejes To´th explained to me that there are problem posers, and he was one, and there are problem solvers among mathematicians, and Endre Szemere´di is one of them. He is brilliant. As for myself, I have worked in a number of areas that may have been too applied for the Fields Medal. I’ve worked on numerical solution of partial differential equations, which is a wonderful subject. I’ve worked on scattering theory, and other applications.

Peter Lax, Samuel Burstein, and Anneli Lax, Calculus with Applications and Computing, Springer, 1976.


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I’ve heard about the story of Erd} os introducing you to Albert Einstein as a talented young Hungarian mathematician, and Einstein asked Erd} os, ‘‘Why Hungarian?’’ What did Erd} os answer? I’m sure Erd} os was speechless. My impression about Teller and to a certain extent about Wigner was that they were nationalistic about being Hungarian. Teller came to Budapest after the political changes and started his speech ‘‘Ve´reim, Magyarok!’’ (My blood brother Hungarians!) You would have never said that. When I was elected to the Hungarian Academy of Sciences as a foreign member, I started my inaugural lecture by addressing the audience with ‘‘Kedves Nyelvta´rsaim!’’4 Still, does Hungarian have a connotation of being outstanding in science? There is a culture of science and mathematics and of the way talented young people are picked out and mentored. There was a similar thing in the Soviet Union and it continues in Russia. The mentoring early on was particularly successful in Hungary. Von Neumann thought that the Eo¨tvo¨s competition was very good in picking out good people. Szeg} o and Po´lya initiated a similar system of competitions in California, but it was discontinued… …when they retired. There is now the Olympiad. You have this piece about speed and size, which reads like poetry. It was about the KdV equation that is difficult to explain in layman’s terms, so I put it into the form of haiku: Speed depends on size Balanced by dispersion Oh, solitary splendor. Do you write poetry? No. My brother—a physicist—is deeply interested in poetry and knows a lot about it. He has translated English poetry into Hungarian. I love poetry, but I don’t have his talent for it. In Los Alamos, you stated later, one had to understand not only how nuclear weapons worked, but also how airplanes fought their way through the atmosphere. I must have meant shock waves. In the implosion, for example, the action of implosion is shock waves. When airplanes fly near the speed of sound, shock waves form. The numerical methods I developed for shock waves have been used for describing supersonic flights. Mathematics has the miraculous quality that equations of the same form may apply to many different problems in many different physical situations. Would you single out one of your scientific problems that you could describe in layman’s terms? Let’s talk about shock waves. It started with von Neumann’s brilliant idea of treating the discontinuity that the shock wave represents as a rapid transition. That was a tremendous simplification. My contribution was that this idea works if you write the equations for fluid flow in conservation form, the conservation of mass, momentum,

4

and energy, then the von Neumann technique will properly treat the shock waves. That has been widely accepted. What is a shock wave? A shock wave is a discontinuity. The equations describing it are nonlinear. The speed of sound depends on the state of the medium, and the signals in the more compressed parts of the medium travel faster. You can describe the direction of the derivative… [The explanation gradually turns into equations on the board…] It was discovered in the middle of the 19th century that the equations of flow have no continuous solutions. Riemann pointed out that the true equations are not differential equations, but integral conservation laws, and they make perfect sense for discontinuous solutions. The difference between pure and applied mathematics has often come up in connection with your work. In first approximation, there is no difference. In second approximation, there are differences between the various branches of pure mathematics. Does it annoy you when people distinguish between the two? It used to annoy me, but people no longer do it so much. Or you got used to it. Or I got used to it. Paul Halmos wrote an article with the title, ‘‘Applied Mathematics Is Bad Mathematics.’’ In the conclusion he softened it and said, ‘‘Applied mathematics is ugly mathematics.’’ The fact is that he knew nothing about it, so whatever he said was irrelevant. But it was strange because he had been an assistant and a great admirer of von Neumann. Von Neumann did a lot of applied mathematics, which should have given Halmos some thought. You have mentioned somewhere that Hungarian mathematics tended to be more abstract or esoteric than… …since the rise of computer science, combinatorics, a Hungarian specialty, has become very practical. Maybe this is a point that pure mathematicians like to make that many branches of pure mathematics have unexpected applications. That was a big boost for Hungarian mathematics. You have made a comment about Erd} os that he tended to be… …Erd} os did some very great things, but the best were when he collaborated with first-rate people: like random graphs that Erd} os and Reńyi did, then probabilistic number theory that he did with Mark Kac. What I found strange was that he was willing to work on anything. It was partly kindness: when people came to him with problems, he was very willing to do it, and partly it was just that he was interested in everything. He was very disappointed that he did not get an appointment at the Institute for Advanced Study, which would have suited his temperament. But people like von Neumann did not like his willingness to work on anything. Von Neumann was also open to problems. He was open to problems, but he had very good taste.

This pun is difficult to translate; it is approximately, ‘‘My Dear Comrades in Language!’’ and in Hungarian ‘‘My Comrades’’ (‘‘Elvta´rsaim’’), sounding political, rhymes with ‘‘Comrades in Language’’ (‘‘Nyelvta´rsaim’’), which sounds merely funny.

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Are you implying that Erd} os did not have very good taste? He did not apply his taste when people came to him with problems. What he accomplished was marvelous and we can’t criticize him. Then there was this unfortunate controversy with Selberg, which had to do with the prime number theorem. This story, too, contributed to his not getting his appointment to Princeton.5 Did you coauthor a paper with him? I never wrote a paper with him, but my first paper in 1944 was on a conjecture of Erd} os. Even before that, a paper in 1943 by Erd} os, which appeared in the Annals, had a footnote, which said, ‘‘This proof is due to Peter Lax.’’ What is your Erd} os number? 1.5. Because of the footnote? Yes. Does anybody else have an Erd} os number 1.5? Maybe not. Let us turn to your meetings with the ‘‘Martians of Science,’’ von Ka´rmań, Szilard, Wigner, von Neumann, and Teller. Start with Szilard.6 I met him a few times. Szilard was a very good friend of my uncle on my mother’s side, Albert Kornfeld, an engineer. Albert Kornfeld won the Eo¨tvo¨s competition in mathematics and finished second in physics. Szilard won the Eo¨tvo¨s competition in physics and was second in mathematics. That’s how they met. When Szilard went to Germany, he strongly advised my uncle to do the same and he did. You may remember the refrigerator that Szilard and Einstein invented. My uncle made the engineering design for that. So Szilard was a good friend of my family. I remember one conversation—it must have been in the late 1950s—when I asked him why such an intelligent man as Wigner is so convinced that war between the United States and the Soviet Union is inevitable. Szilard answered that Wigner is, indeed, very intelligent, but his thinking has a legalistic frame. He cannot imagine—as it is hard to imagine—some kind of a negotiated legal settlement of the issues between these two powers. But in real life, this is not how things develop. Things happen; people just act subconsciously, and the result is not a legal arrangement. I found this very interesting. Another thing I remember him saying is that the trouble with international relations is that there are many countries that make a conscious decision to treat their adversaries—or, generally, other countries with whom they have some conflicts—10% better than they treat them. The reason why this does not work is that each country overestimates by 20% the goodness of its own intentions. I think there is something in this. People are usually very generous in interpreting their own motives. How about Wigner? I met Wigner a few times. One time Courant took me to Princeton to have lunch with Wigner. Wigner had a great respect for Courant because Courant was very nice to him

when they met in Go¨ttingen and Wigner was a young man at the time. After lunch Wigner took me aside and asked me, ‘‘What does Courant want? I’ll be happy to help him.’’ But Courant was very indirect. He would never directly say what he wanted. People had to guess it. In this case I had to tell Wigner that I had no idea. Courant’s approach worked well when he was younger. On this occasion he was perhaps too old for this game to work. Did you meet von Ka´rmań, too? I did. It was in the late 1950s. The airplane company Convair had a contract to build the Atlas intercontinental missile, and Convair formed a very distinguished panel of advisers, headed by von Ka´rmań, including Teller and Courant; I was a low man there. We spent a few days in California. Courant stayed in the same hotel as von Ka´rmań and they had some meals together. They had an interesting relationship dating back to their Go¨ttingen days. When Courant came to Go¨ttingen as a student, von Ka´rmań was already a Privatdozent. Von Ka´rmań was one of the most brilliant people there and Courant was very impressed by him. Von Ka´rmań was not especially impressed by Courant; he was just a student. That relationship persisted. On the other hand, Mrs. Courant was the daughter of Professor Runge, who was one of von Ka´rmań’s teachers. So von Ka´rmań treated Mrs. Courant with the greatest respect. It was very funny to watch it. Courant noticed it too because he managed to bring up the subject of the communist regime in Hungary in 1919, when von Ka´rmań was Assistant Minister of Education. He saw to it that the best people were appointed at the universities. When the Commune was over all those people became unemployable. I’m sure this was not what von Ka´rmań wanted to hear. You met all the five Martians. Who was your favorite? Von Neumann. I knew him best and I was considerably influenced by him. Szilard perhaps had the most fantastic imagination. He could foresee the future and act on it. Very few people foresee the future and those who do, don’t do anything, and he did. Perhaps he was the most remarkable among them. But von Neumann had a mind, which was in its power unlike anybody else’s. There was an interesting movie about von Neumann, one of a series the Mathematical Association produced about mathematicians. In it Hans Bethe is saying, half in jest, that he regarded von Neumann’s brain as a mutation upward from the normal human brain. That is the impression he made on everyone. You owe your life to America. I owe my life to America and all the opportunities that I have had here. On a lighter note, do you count in English or in Hungarian? In Hungarian. And you curse in Hungarian. Sorry about that. I do that out of tact.

5

See The Mathematical Intelligencer, vol. 31, no. 3, summer 2009, pp. 18–23. In one variant of the story of the ‘‘Martians’’ label, Enrico Fermi was wondering about the origin of the smart and extraordinary Hungarian scientists; Leo Szilard suggested that they had come from Mars, but disguised themselves by speaking Hungarian. See, I. Hargittai, The Martians of Science: Five Physicists Who Changed the Twentieth Century (New York: Oxford University Press, 2006; 2008).

6


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The first and last (fourth) pages of John von Neumann’s letter to Henry Lax about the courses Peter might be taking at Columbia University and about Stuyvesant. The letter is in Hungarian and the signature is Neumann Jańos.

English Translation of the Letter from Ro´zsa Pe´ter to John von Neumann from Budapest, November 8, 1941 Dear Professor! Allow me to draw your attention to Pe´ter Lax, a highschool sophomore who is about to emigrate. I have never had such a truly talented pupil. I have lived in permanent anxiety lest this talent get lost in my hands. I would have not trained him for more than another year; I would have passed him on to a mathematician superior to myself. For example, he was the non-official winner of the last two Eo¨tvo¨s competitions. He was the only one who solved the most difficult problem of this year’s Eo¨tvo¨s competition, and he did it in such an elegant way that even those who had set the problem were astonished. It has happened that

he had heard or read about a theorem and then proved it without any previous knowledge, and with the greatest independence. If auxiliary means were needed for the proof, he created them. He has a sense for problems and poses them on his own as well. I would like to see him in good hands out there because I am convinced that he may amount to something.

Is there anything that you would like to add to what we have talked about? [big sigh] As I said, I have had many opportunities. It was during my time at Los Alamos when I realized how important computing was, and that it was very much worth doing. It helped being at a center of mathematics because I could learn about every new development. That was very important. Today it may not be that important where you are geographically because communication goes with the speed of light over the Internet. But, when I started, it certainly was.

The mathematical community grew in my lifetime; it has become very much larger. But the number of really outstanding people did not grow that much. One effect of having a much larger community is that once an idea is grasped, its consequences are worked out much faster. In the past people often spent their lives on one area to work out everything that was worth working out. In time things have become much faster, and this suits me fine. I didn’t like to stick to one subject. I advise that to others, but it’s a question of temperament.

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With sincere greetings,

Ro´zsa Pe´ter

Mathematical Entertainments

Michael Kleber and Ravi Vakil, Editors

Elementary Surprises in Projective Geometry RICHARD EVAN SCHWARTZ* TABACHNIKOV

AND

SERGE

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

*Supported by N.S.F. Research Grant DMS-0072607. Supported by N.S.F. Research Grant DMS-0555803. Many thanks to MPIM-Bonn for its hospitality.

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]

he classical theorems in projective geometry involve constructions based on points and straight lines. A general feature of these theorems is that a surprising coincidence awaits the reader who makes the construction. One example of this is Pappus’s Theorem. One starts with six points, three on one line and three on another. Drawing the additional lines shown in Figure 1, one sees that the three intersection (blue) points also lie on a line. Pappus’s Theorem goes back about 1700 years. In 1639, Blaise Pascal discovered a generalization in which the six (green) points lie on a conic section, as shown on the lefthand side of Figure 2. One recovers Pappus’s Theorem as a kind of limit, as the conic section stretches out and degenerates into a pair of straight lines. Another closely related theorem is Briançon’s Theorem. This time, the six green points are the vertices of a hexagon that is circumscribed about a conic section, as shown on the right-hand side of Figure 2, and the surprise is that the three diagonals intersect in a point. Though Briançon discovered this result about 200 years after Pascal’s Theorem, the two results are, in fact, equivalent for a well-known reason that we will discuss below. In this article, we discuss some apparently new theorems in projective geometry that are similar in spirit to Pascal’s Theorem and Briançon’s Theorem. One can think of all these as statements about lines and points in the ordinary Euclidean plane, but setting the theorems in the projective plane enhances them. The projective plane P can be defined as the space of lines through the origin in R3 : A point in P can be described by homogeneous coordinates ðx : y : zÞ; not all zero, corresponding to the line containing the vector (x, y, z). Of course, the two triples (x : y : z) and (ax : ay : az) describe the same point in P as long as a 6¼ 0: One says that P is the projectivization of R3 : A line in the projective plane is the set of lines through the origin in R3 that lie in a plane. Any linear isomorphism of R3 ; — i.e., multiplication by an invertible 3 9 3 matrix — permutes the lines and planes through the origin, and so induces a mapping of P that carries lines to lines. These maps are called projective transformations. One way to define a (nondegenerate) conic section in P is to say that

T

• The set of points in P of the form (x : y : z) such that z 2 ¼ x 2 þ y 2 6¼ 0 is a conic section. • Any other conic section is the image of the one we just described under a projective transformation. We can identify R2 as the subset of P corresponding to points (x : y : 1) and write R2 P: The ordinary lines in R2 2010 Springer Science+Business Media, LLC, Volume 32, Number 3, 2010

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Figure 1. Pappus’s Theorem.

are subsets of lines in P: The conic sections intersect R2 in either ellipses, hyperbolas or parabolas. One of the beautiful things about projective geometry is that these three kinds of curves are the same from the point of view of the projective plane and its symmetries.

The dual plane P is the set of planes through the origin in R3 : Every such plane is the kernel of a linear function on R3 ; and this linear function is determined by the plane up to a nonzero factor. Hence P is the projectivization of the dual space ðR3 Þ : We can identify R3 with ðR3 Þ using the scalar product and think of P as the space of lines in P: Given a point v in P; the set v\ of linear functions on R3 that vanish at v determine a line in P : The correspondence v 7! v? carries collinear points to concurrent lines; it is called the projective duality. A projective duality takes points of P to lines of P and lines of P to points of P : Of course, the same construction works in the opposite direction, from P to P: Projective duality is an involution: Applied twice, it yields the identity map. Figure 3 illustrates an example of a projective duality based on the unit circle: The red line maps to the red point, the blue line maps to the blue point, and the green point maps to the green line. Projective duality extends to smooth curves: the oneparameter family of the tangent lines to a curve c in P is a one-parameter family of points in P ; the dual curve c*. The curve dual to a conic section is again a conic section.

Figure 2. Pascal’s Theorem and Briançon’s Theorem.

AUTHORS

.........................................................................................................................................................

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RICHARD EVAN SCHWARTZ grew up in Los

SERGE TABACHNIKOV grew up in the Soviet

Angeles and attended college at UCLA and graduate school at Princeton. He is currently a Professor of Mathematics at Brown University. He has published a number of articles and monographs in geometry, topology, and dynamics and was an invited speaker at the 2002 ICM. Rich likes simple problems, and often uses the computer as a tool to investigate them. In his spare time, he draws, exercises, listens to music, and plays with his daughters. A. K. Peters has just published his ‘‘You Can Count on Monsters’’, a children’s picture book about prime numbers.

Union and has a PhD from Moscow State University; he does research in topology, geometry and dynamical systems. Since 1990, he has been teaching at universities in the USA. In 1988–1990, he headed the Mathematics Department of the ‘‘Kvant’’ (Quantum), a Russian magazine on physics and mathematics for advanced high-school students and up. Since 2000, he has been the Director of the Mathematics Advanced Study Semesters (MASS) program at Penn State.

Department of Mathematics Brown University Providence, RI 02912 USA E-mail: [email protected]

Department of Mathematics Pennsylvania State University University Park, PA 16802 USA E-mail: [email protected]


Figure 5. If P is an inscribed octagon, then P * T21212(P). Figure 3. Projective duality.

fp1 pkþ1 ; p2 pkþ2 ; . . . pn ; pkþn g:

Figure 4. The pentagram.

Thus projective duality carries the vertices of a polygon inscribed in a conic to the lines extending the edges of a polygon circumscribed about a conic. Projective duality takes an instance of Pascal’s Theorem to an instance of Briançon’s Theorem, and vice versa: the input of Pascal’s theorem is an inscribed hexagon and the output is three collinear points, while the input of Briançon’s Theorem is a superscribed hexagon and the output is three coincident lines. Like Pascal’s Theorem and Briançon’s Theorem, our results all involve polygons. A polygon P in P is a cyclically ordered collection {p1, …, pn} of points, its vertices. A polygon has sides: The cyclically ordered collection {l1, …, ln} of lines in P; where li ¼ pi piþ1 for all i. Of course, the indices are taken mod n. The dual polygon P* is the polygon in P whose vertices are {l1, …, ln}; the sides of the dual polygon are {p1, …, pn} (considered as lines in P ). The polygon dual to the dual is the original one: (P*)* = P. Let X n and X n denote the sets of n-gons in P and P ; respectively. Given an n-gon P = {p1, …, pn}, we define Tk(P) as

1

That is, the vertices of Tk(P) are the consecutive k-diagonals of P, and Tk is an involution. The map T1 carries a polygon to the dual one. Even when a 6¼ b; the map Tab ¼ Ta Tb carries X n to X n and X n to X n : We have studied the dynamics of the pentagram map T12 in detail in [2, 3, 4, 5, 6], and the configuration theorems we present here are a byproduct of that study. The map is so-called because of the resemblence, in the special case of pentagons, to the famous mystical symbol having the same name. See Figure 4. It is a classical result that, when P is a pentagon, P and T12(P) are projectively equivalent. See [2]. We extend the notation: Tabc ¼ Ta Tb Tc , and so on. To save words, we say that an inscribed polygon is a polygon whose vertices are contained in a conic section. Likewise, we say that a circumscribed polygon is a polygon whose sides are tangent to a conic. Projective duality carries inscribed polygons to circumscribed ones and vice versa. Two polygons, P in P and Q in P , are equivalent if there is a projective transformation P ! P that takes P to Q. In this case, we write P *Q. By projective transformation P ! P , we mean a map that is induced by a linear map R3 ! ðR3 Þ :

T H E O R E M 1 The following are true. • If P is an inscribed 6-gon, then P *T2(P). • If P is an inscribed 7-gon, then P *T212(P). • If P is an inscribed 8-gon, then P *T21212(P). Figure 5 illustrates1 the third of these results. The outer octagon P is inscribed in a conic and the innermost octagon T121212(P) = (T21212(P))* is circumscribed about a conic. You might wonder if our three results are the beginning of an infinite pattern. Alas, it is not true that P and T2121212(P) are equivalent when P is in inscribed 9-gon, and the predicted result fails for larger n as well. However, we do have a similar result for n = 9, 12.

Our JavaTM applet does a much better job illustrating these results. To play with it online, see http://www.math.brown.edu/*res/Java/Special/Main.html.


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T H E O R E M 2 If P is a circumscribed 9-gon, then P * T313(P).

T H E O R E M 3 If P is an inscribed 12-gon, then P * T3434343(P). Even though all conics are projectively equivalent, not all n-gons are. For instance, the space of inscribed n-gons, modulo projective equivalence, is n - 3 dimensional. We mention this because our last collection of results all make weaker statements to the effect that the ‘‘final polygon’’ is circumscribed but not necessarily equivalent or projectively dual to the ‘‘initial polygon.’’

T H E O R E M 4 The following are true. • If P is an inscribed 8-gon, then T3(P) is circumscribed. • If P is an inscribed 10-gon, then T313(P) is circumscribed. • (*) If P is an inscribed 12-gon, then T31313(P) is circumscribed. We have starred the third result because we don’t yet have a proof for it. We discovered these results through computer experimentation. We have been studying the dynamics of the pentagram map T12 on general polygons, and we asked ourselves whether we could expect any special relations when the initial polygon was either inscribed or circumscribed. We initially found the 7-gon result, Case 2 of Theorem 1. Then V. Zakharevich, a participant in the Penn State Research Experience for Undergraduates (REU) program in 2009, found Theorem 2. These two results are closely related to self-dual polygons, as discussed in [1]. For instance, if P is the 7-gon in Case 2 of Theorem 1, then Q = T2(P) is equivalent to the dual 7-gon Q*— and all self-dual 7-gons arise this way. Encouraged by the good luck we had with the results just mentioned, we made a more extensive computer search that turned up the remaining results. Curiously, all our results involve transformations Tw, where w is a palindromic word. (In particular, Tw Tw ¼ Id in all cases.) We think that the list above is exhaustive, in the sense that

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no surprises will be found by applying some combination of diagonal maps to inscribed or superscribed polygons. In particular, we don’t think that surprises like the ones we found exist for N-gons with N [ 12. It is worth pointing out, however, that we can get a few additional ‘‘different looking’’ configuration theorems by cyclically relabelling the vertices. For example, rephrasing the last statement of Theorem 4, we get the following result: A 12-gon P is inscribed if and only if T131313(P) is inscribed. Then, a certain cyclic relabelling of the vertices leads to the following variant: If P is an inscribed 12-gon, then T535353(P) is also inscribed. How did we prove these results? In several of the cases, we found some nice geometric proofs which we will describe in a longer version of this article. With one exception, we found uninspiring algebraic proofs for the remaining cases. (These algebraic proofs essentially amount to writing everything out in coordinates and computing symbolically.) We hope to find nice proofs for these cases too, but so far this has eluded us. Perhaps you will be inspired to look for them. We also hope that these results point out some of the beauty of the dynamical systems defined by these iterated diagonal maps. Finally, we wonder if the isolated results we have found are part of an infinite pattern. We don’t have an opinion one way or the other whether this is the case, but we think that something interesting must be going on.

REFERENCES

[1] D. Fuchs, S. Tabachnikov, Self-dual polygons and self-dual curves, Funct. Anal. Other Math. 2, 203–220 (2009). [2] R. Schwartz, The pentagram map, Experiment. Math. 1, 71–81 (1992). [3] R. Schwartz, The pentagram map is recurrent, Experiment. Math. 10, 519–528 (2001). [4] R. Schwartz, Discrete monodomy, pentagrams, and the method of condensation, J. Fixed Point Theory Appl. 3, 379–409 (2008). [5] V. Ovsienko, R. Schwartz, S. Tabachnikov, The pentagram map: a discrete integrable system, ArXiv preprint 0810.5605. [6] V. Ovsienko, R. Schwartz, S. Tabachnikov, Quasiperiodic motion for the Pentagram map, Electron. Res. Announc. Amer. Math. Soc. 16, 1–8 (2009).

The Mathematical Tourist

Dirk Huylebrouck, Editor

A Mathematical Trip to Princeton EZRA BROWN

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

he phone rings and a familiar voice speaks. ‘‘Since you’re going to Princeton this summer, you’ll surely be taking a tour of the Princeton campus, and their math department. When you go there, be sure to take note of a sculpture near the Math Department Commons Room. This piece involves three mathematicians: One was the artist, another commissioned the piece, and the third was the one to whom it was dedicated. When you get back, tell me what you ... what? No time? I’m sorry, you make the time.’’ Orders are orders, so you make the time and tour the beautiful Princeton University Campus. There, you find that sculpture, along with an ivory tower and many other sights worth seeing. The Princeton University mathematics department may or may not be in first place in the mythical USA Mathematics Marathon, but it is certainly up there in the lead pack. Two Fine Halls have housed the Princeton math department, both

T

named for Henry Burchard Fine (1858–1928), the first chair of mathematics at Princeton and President of the American Mathematical Society in 1911–1912. The first building was opened in 1930; it was renamed Jones Hall in 1968 when the second and current Fine Hall was built. In the immediate vicinity of the current Fine Hall are one building and two sculptures worth a second look. The building is the Lewis Science Library [4], which opened in 2008. Designed by architect Frank O. Gehry, it is a mathematical marvel in its own right. (Go inside and look around: It’s time well spent.) Viewed from the Fine Hall tower, it resembles a cell complex (Figure 1). The library is one among many architectural gems at Princeton, as an architect friend from high-school and his historian wife demonstrated during a campus tour. The library stands next to Fine Hall on what was formerly the site of an informal volleyball court used by the math faculty and students. Apparently, when Fields Medalist William Thurston was on the faculty, he supplied the net, which he put up in the morning and took down in the evening. Two steel sculptures of mathematical character stand in the vicinity. The older one is Alexander Calder’s 26-foot sculpture ‘‘Five Disks: One Empty;’’ [3] dedicated in 1971, it stands in the Fine Hall courtyard plaza between the mathematics and physics departments. (Maybe it looks like a horse, and maybe it doesn’t–Modern Art is like that.) The newer one is Richard Serra’s ‘‘The Hedgehog and the Fox,’’ [5] a trio of 90-foot long and 15-foot high nested serpentine steel ribbons standing just east of the library. It’s fun to walk through this huge installation and listen to the echoes. Appropriately, when viewed from the Fine Hall tower, ‘‘The Hedgehog and the Fox’’ looks like a triple integral sign (Figure 2).

â

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

Figure 1. The Gehry Library (photo by Sharon Sells). Ó 2010 Springer Science+Business Media, LLC, Volume 32, Number 3, 2010

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Figure 2. The Serra Sculpture (photo by Sharon Sells).

The present Fine Hall is an imposing 12-story tower—the tallest building on the Princeton campus. The upper nine floors seem identical, with two elevator doors, two stairwell doors, a dozen or so offices and a seminar room. Each floor’s office directory is on a wall near the elevators, and mathematical tourists will recognize some famous names. All the doors are closed. Apparently, the present Fine Hall was designed by an architect who did not have the collaborative/social nature of mathematical research in mind. The result was the quintessential Ivory Tower, an appropriate name because, as one recent Ph.D. graduate from the department put it, the corridors are 99 44/100% empty. The sameness of those upper floors led, according to one source, to some mischief. The night before April Fools Day one year, some of the students rewired the elevator buttons so that, for example, pressing the button for the fifth floor might bring the elevator to the eighth floor. The numbers on

AUTHOR

......................................................................... grew up in New Orleans, has degrees from Rice and LSU, and has been at Virginia Tech since 1969, where he is currently Alumni Distinguished Professor of Mathematics. Most of his research has been in number theory and combinatorics, but one of his favorite papers was written with a sociologist. During the summers, he does applied mathematics in the Washington, DC Area. He has received some writing awards from the Mathematical Association of America. He enjoys singing in operas, playing jazz piano, gardening, and kayaking. He occasionally bakes biscuits for his students.

EZRA (BUD) BROWN

Virginia Polytechnic Institute and State University Blacksburg, VA 24061-0123 USA e-mail: [email protected] 36


the office doors had been similarly switched accordingly. The next morning, all the professors came in, rode the elevators to ‘‘their’’ floors, went to ‘‘their’’ offices–but nobody’s keys worked. (Allow yourself a wry chuckle at the bedlam that ensued.) On the matter of who did this, the source was as silent as the corridors of those upper nine floors. More important than the appearance of the building, however, are the results of the mathematical training occurring therein. These results have been astonishing, and they are reflected in the group photographs of each entering class of graduate students from 1966 to the present that line the third-floor corridor. You can watch changes unfold as you go through the years. The first class was all male and they wore suits (that didn’t last long); eventually, women appeared in the pictures. Among the pictures you see many eminent mathematicians, including colleagues from your home institution and other workplaces, three Fields Medalists, numerous Putnam Fellows and International Math Olympiad participants, a MacArthur Fellow, and a coauthor—all looking very youthful. It’s quite an impressive showcase. It is true that four Princeton Ph.D. mathematicians have been Fields Medalists, not three. However, the student days of John Milnor, the first Princeton Ph.D. to receive a Fields Medal, predate the series of pictures. The next thing you see is in the middle of the run of group photographs, namely, the door of Room 310. Unlike practically every other door in the building, 310 is completely covered with signs, figures and pictures in a fascinating crazy quilt, reflecting the interests of the office’s inhabitant, John H. Conway. There’s no official name on the door, but a sign advertising ‘‘Conway: $9.99’’ gives it away. Then you see the mathematical sculpture, entitled ‘‘The Third Constant of Euclidean Geometry,’’ and it is very striking. This polished five-foot high tower, beautifully made of Inner Mongolian black granite, is the work of the eminent sculptor-mathematician Helaman Ferguson (Figures. 3, 4). The piece stands in a prominent spot across from the departmental Commons Room, back-lit by light from a south-facing window. Ferguson’s work honors the memory and the mathematics of the late Princeton Professor Fred Almgren, who worked in geometric measure theory. It was commissioned by Professor Jean Taylor, Almgren’s second wife and his first Ph.D. student, and dedicated in 2000. This sculpture vividly captures Professor Almgren’s startling and comprehensive theorems on generalizations of the isoperimetric inequality and mass-minimizing hypersurfaces, announced in [1] and described in detail in a legendary 1,700-page preprint. This latter work was edited by Jean Taylor and Vladimir Scheffer and published in a single volume in 2000 [2]— truly a labor of love. The constant of the sculpture’s title is called c, defined by pffiffiffi cðn þ 1Þ :¼ ððn þ 1Þ=2Þ!1=n =ððn þ 1Þ pÞðnþ1Þ=n : For 2 B n B 8, c(n) is written out in base 10 and base k, 2 B k B n on each of seven levels of the sculpture. Each level is a regular n-gon, and the sculpture flows beautifully between levels. Appropriately, for the sculpture’s close connection with p, the nearest room is Room 314.

Figure 3. ‘‘The Third Constant’’ and its creator (photos by Georg Glauser).

those weight limits printed on elevators are not just for show. Ferguson calculated that the combined weight of his assistant, himself, the sculpture and the equipment needed to move and install the sculpture was just barely under the capacity of the freight elevator. On an agreed-upon date, the two of them unloaded the piece from Ferguson’s truck and carefully moved sculpture and equipment into the freight elevator. Just then, someone who knew about the move ran around the corner and leaped joyfully into the elevator just as the doors were closing ... thus loading the elevator beyond its capacity. Now, how can you ask the latecomer to leave, when the latecomer is John Conway? Well, you can’t; you press the button and hold your breath. The elevator shuddered a bit and rose uncertainly to the third floor. After that bit of excitement, the rest of the installation was routine. What else of mathematical interest is there to see in Princeton? Merely the Institute for Advanced Study—but that’s another story!

REFERENCES

[1] Frederick J. Almgren Jr, Optimal Isoperimetric Inequalities, Bull. Amer. Math. Soc. (N.S.), 13, #2 (October 1985), 123–126. [2] Frederick J. Almgren Jr, ‘‘Almgren’s Big Regularity Paper: Q-Valued Functions Minimizing Dirichlet’s Integral and the Regularity of Area-Minimizing Rectifiable Currents Up to Codimension 2’’ (Vladimir Scheffer and Jean E. Taylor, eds.), World Scientific Publishing Co. (2000).

Figure 4. ‘‘The Third Constant’’ (detailed view; photo by Georg Glauser).

Finally, Helaman Ferguson tells a story about the sculpture’s installation. Inner Mongolian black granite is heavy, and

[3] http://blogs.princeton.edu/aspire/2009/01/sculpture_at_princeton_ the_putnam_collection.html [4] http://www.princeton.edu/main/news/archive/S20/84/49I22/index. xml?section=featured [5] http://www.lera.com/projects/usnj/serraprinceton.htm

Ó 2010 Springer Science+Business Media, LLC, Volume 32, Number 3, 2010

37

Pareto’s Law MICHAEL HARDY

T

he Pareto of the title is the economist Vilfredo Pareto (1848-1923), but the ‘‘law’’ of the title is not one that he enunciated.

The ‘‘80/20 Law’’ What is nowadays called ‘‘Pareto’s law’’ states that generally 80% of all effects result from 20% of all causes. In particular, it is alleged (see [6] and [1]) that 20% of employees of any business are responsible for 80% of productive output; 20% of the customers bring in 80% of the revenue; 20% of products bring in 80% of the revenue; 20% of all books in a library account for 80% of the library’s circulation; 20% of all people own 80% of all wealth; 20% of all people receive 80% of all income; 20% of all opportunities to make mistakes account for 80% of mistakes; 20% of all employees account for 80% of all absenteeism; in any meeting, 80% of all decisions are made in 20% of the time; 20% of authors of research papers write 80% of all published papers; 80% of the world’s population lives in 20% of the cities; 80% of all use of software features involves 20% of all features; and so on. Pareto’s law has enjoyed some popularity in the management field, and an entire book [4] requiring no knowledge of mathematics has been written about it. The name ‘‘Pareto’s law’’ originated in 1954 with the paper [3] of Joseph M. Juran, who contrasted the contributions of the ‘‘vital few’’ with those of the ‘‘trivial many.’’ The iterated ‘‘80/20 law’’ The first time I recall hearing of Pareto’s 80/20 law was in a talk involving no mathematics by someone with a business background, who stated it in an ‘‘iterated’’ form: 20% of all sales personnel accomplish 80% of all sales, and 20% of the top 20% achieve 80% of that 80%, and so on, so that 0.2n 100% of the sales force makes 0.8n 100% of the sales. Since I knew that the Pareto distribution, with probability density proportional to x 7! x a1 on an interval (x0, ?), has been used for modeling the distribution of incomes, I immediately wondered whether the Pareto distribution is in some sense equivalent to this iterated 80/20 law. I have not seen that

38


question addressed in accounts of the Pareto distribution. It is an easy exercise that with the right value of a, the iterated 80/20 law holds for integer n, but I have not found that in the literature. I have also found nothing like the converse: that an 80/20-type law entails the Pareto distribution. In this paper I will show that if we allow n to be (positive) real-valued, we can demonstrate equivalence, with the qualification that a [ 1. The integer-n version of the iterated 80/20 law has appeared in Wikipedia’s article titled ‘‘Pareto principle’’ since April 2003, because I put it there.

Pareto’s Model of Income Distribution Pareto did not state an ‘‘80/20’’-type law. But we will see that his data from 400,648 British income tax returns, conjoined with his model (1) below, show that about 27.7% of the population got about (100 - 27.7)% = 72.3% of the income, and about 27.7% of that top 27.7% got about 72.3% of of that 72.3%, and generally about 0.277n 100% of the population got about (1 - 0.277)n 100% of the income— for n between 0 and slightly more than 4.5. The data stop there and give no obvious indication of whether the pattern persists beyond that point. It cannot go beyond n = 10, for 400,648 9 0.27710 &1. Pages 299–345 of Pareto’s book [5] are a chapter that includes more than 40 datasets concerning distribution of incomes in a variety of European and North and South American cities and countries. He proposed [5, page 305] that log N ¼ log A a log x;

ð1Þ

where N is the number of persons whose incomes are higher than x, and A and a [ 0 are parameters varying geographically and over time. The British tax-return data found on page 305 appear below (Figure 1). For each value of x, the corresponding value of N is the number of income-tax returns in Britain reporting on ‘‘Schedule D: commerce and professions’’, an income of more than £x for the years 1893–1894. The line in this plot is an ordinary least-squares fit: log N = 19.331 - 1.3379 log x. The value of p for which log p/(1-p) p = 1.3379 is about 0.277. If the model (1) holds, so that one can

and hence x a N 0 ¼ N0 x

for x x0 :

ð2Þ

The Pareto Distribution

Figure 1. Pareto’s data.

interpolate, then we can predict, for example, that the number of persons with income exceeding £1500 is about exp (19.331 - 1.3379 log 1500) & exp 9.547 & 14, 000. Pareto’s model (1) cannot hold for all x [ 0 because N ? ? as x ; 0, so we would have an infinite population, even though for any x [ 0 only finitely many people would have incomes exceeding x. If we substitute for N the finite size N0 of the population, then the solution x0 of the equation (1) for x must be the minimum income. Everyone’s income is at least x0. (Economists take (1) seriously for sufficiently large incomes but not for the lowest incomes; see [2].) Subtracting both sides of the identity log N0 = log A a log x0 from (1), we get log

N x ¼ a log ; N0 x0

AUTHOR

......................................................................... dropped out of the graduate program in mathematics, then went back and got a Ph.D. in statistics. He has taught mathematics and/or statistics here and there over the years, at (for instance) the University of North Carolina, MIT, and the Woods Hole Oceanographic Institution. He has worked on foundations of epistemic probability, but he hasn’t figured out whether this has practical application in statistical inference.

MICHAEL HARDY

School of Mathematics University of Minnesota Minneapolis, MN 55455 USA e-mail: [email protected]

Pareto’s model, stated either in the form (2) or as Pareto initially stated it [5, page 305] in the form (1), is a continuous probability distribution of an unbounded random variable. As such, it can only be an approximation to a distribution of incomes that must be discrete and bounded because the number of members of a subset of the population is a finite integer. If we treat the income X of a randomly chosen person as a continuous random variable whose distribution is given by (2), it follows that X has cumulative probability distribution function ( x a 0 for x x0 ; 1 ð3Þ F ðxÞ ¼ PrðX xÞ ¼ x 0 for x\x0 ; and hence has probability density function 8 a < ax0 for x [ x0 ; f ðxÞ ¼ F 0 ðxÞ ¼ x aþ1 : 0 for x\x0 :

ð4Þ

D EFINITION The Pareto distribution (also called Pareto’s law, because probability distributions are sometimes called ‘‘laws’’) is any of the continuous probability distributions that satisfy (4), or equivalently (3), for some values of the two parameters a [ 0, x0 [ 0. The case a = ? is the degenerate distribution concentrating probability 1 at the single point x0. The parameter a is called the Pareto index. (The parameter A in (1) depends on both the minimum x0 and the population size N0. Consequently we cannot recover the value of x0 from (1) and we cannot think of (1) as giving us a parametrization of the family of Pareto distributions.)

Continuity Pareto used the notation N ¼ the number of persons whose income strictly exceeds x:

ð5Þ

We will also want to consider M ¼ the total income of all whose income strictly exceeds x:

ð6Þ

If this language is construed literally, then N is integervalued and M has jump discontinuities as x varies. Therefore things like dN/dx or dM/dN, or like the density function in (4) do not make sense. In other words, to model income distribution with such continuous probability distributions is to use continuous approximations to discrete variables. (However, the statement that PrðX [ xÞ ¼ N =N0 is exact, with N and x defined as in (1) and X as in (3).) The purpose here will be to demonstrate certain results about the continuous probability distribution characterized 2010 Springer Science+Business Media, LLC, Volume 32, Number 3, 2010

39

by (3) or (4). The most felicitous way to make suitable definitions of N and M precise appears to be the language of measure theory. However, our argument in succeeding sections can be followed without that language by anyone not squeamish about writing things like dM/dN or ðd=dxÞ PrðX [ xÞ in this context, understanding those to be derivatives of continuous approximations to discrete variables. The present section of the article can be skipped: it serves only to connect this informal interpretation to the measure-theoretic one. Thus • Instead of a ‘‘population’’ we have the underlying set X of a measure space. • Instead of the size of a subset A of the population we have its measure m(A), and we assume mðXÞ\1 (the size of the whole population is finite). • Instead of ‘‘income’’ we have a measurable function X : X ! ½0; 1Þ. • Instead of N as defined above we have N ¼ mfx 2 X : XðxÞ [ xg:

ð7Þ

• Instead of the total income of all members of a subset A of the population, we have Z X dm: lðAÞ ¼ A

(One can state this by saying X is a Radon–Nikodym derivative: X = dl/dm. This terminology will not be put to any use, but it may be amusing to contrast it with the Core Lemma below.) • Instead of M as defined above we therefore have Z X dm ð8Þ M¼ X [x

and the probability that X is within any measurable set B R is PrðX 2 BÞ ¼

mfx 2 X : XðxÞ 2 Bg : mðXÞ

Now (3) and (4) make literal sense for all x, as does the operation of differentiating with respect to x. The only measure-theoretic theorem that we need is this: If all values of X on a set A X are in some particular interval, then so is the average (9).

Four Observations and a Lemma The following four observations make no assumptions about the probability distribution of X (such as 80/20 laws or the particular density functions or cumulative distribution functions stated above). • N and M are weakly decreasing functions of x. • N and M are strictly increasing functions of each other (they are defined as not-necessarily invertible functions of x, and it follows from the definition that they can be computed as functions of each other). • The function X assumes (almost) no values in any interval on the x-axis on which N and M remain constant. See Figure 2(a). (‘‘Almost no values’’ means that the set of all members x 2 X for which X(x) has such a value has m-measure 0.) In the income application, nobody’s income is within the interval on which N and M remain constant. • For any change Dx in the value of x, let DN and DM be the corresponding changes in those variables. If Dx [ 0, then DN ; DM 0 . If these latter inequalities are strict, then the average value of X on that part of X where x\X x þ Dx is DM=DN , and then we have

where ‘‘X [ x ’’ means fx 2 X : XðxÞ [ xg .

x\

The average value of X on any measurable set A X is R X dm RA ; ð9Þ A 1 dm

DM x þ Dx: DN

If Dx\0 and DN ; DM [ 0 , then x þ Dx\

DM x: DN

M

N

M1

N1

x x1

(a)

x2

Figure 2. The case in which no one’s income is between x1 and x2. 40


ð10Þ

N N1

(b)

ð11Þ

In the income application, this says that the average income of those whose income is between x and x þ Dx is between x and x þ Dx . The fact that DM=DN [ x in (10) and B x in (11) means that M is a strictly concave function of N. (In the case where X is constant, which in our application means everybody has the same income, the graph of M as a function of N consists of just two isolated points: (0, 0) and (N0, M0), with no curve connecting them, and so the function is vacuously concave.)

R

X dm ¼ ð1 pÞn ; X dm X

XR[ x

where ‘‘X [ x ’’ means fx 2 X : XðxÞ [ xg:

P R O P O S I T I O N 1 If the generalized 80/20 law holds with a given p, then X is distributed according to the Pareto distribution (as characterized by either (2), (3), or (4)) with index a ¼ logp=ð1pÞ p [ 1:

Core Lemma: • Unless x is within a closed interval on which N and M are constant, the corresponding values of M and N satisfy dM ¼ x: dN

ð12Þ

• If M and N remain constant as x increases from x1 to x2, but vary within [x2, x2 + e) and (x1 - e, x1] for arbitrarily small e, then the left- and right-sided derivatives of M with respect to N are respectively x2 and x1 (see Figure 2(b)). (Larry Gray has pointed out that nearly the same proposition appears in Wikipedia’s article titled ‘‘Lorenz Curve’’. No refereed or other source is cited there).

C O R O L L A R Y At all points where M is a differentiable function of N, the equation (12) holds. Both of the assertions in the core lemma are used in establishing the corollary.

P R O O F O F T H E L E M M A When x is not in a closed interval of constancy of N and M, then N is strictly decreasing on some open interval containing x; hence at that point, x is a continuous function of N. Therefore Dx ! 0 as DN ! 0. Since Dx ! 0, the relations (10) and (11) imply that DM=DN , being squeezed between x and x þ Dx, approaches x. For the second assertion, replace the inequalities (10) and (11) with

This excludes those Pareto distributions in which 0 \ a B 1, of which more will be said below. The minimum value x0 may be any positive number. The reader can check that as p increases from 0 to 1/2, a increases from 1 to ?.

P R O O F Suppose the generalized 80/20 law holds, and let 0\b :¼ logp ð1 pÞ ¼ logpn ðð1 pÞn Þ 1: "

"

Since p; 1 p\1

Since p 1=2

Then in the income application we would say that for any r [ [0, 1], the proportion r (= pn) of the population has rb (= (1 - p)n) of the income. To say this precisely, we let the size of the whole N0 ¼ mðXÞ (in the income application, R population) and let M0 ¼ X X dm (in the income application, the total income of the whole population). With N, M, and x defined as in (7) and (8) above, we then have r = N/N0 and rb = M/M0, so that b M N ¼ for all real N 2 ½0; N0 : ð13Þ M0 N0 Therefore M is a differentiable function of N, so the corollary to the core lemma applies:

DM DM x1 x x2 \ x2 Dx DN DN according as Dx [ 0:

x1 Dx\

By the corollary to the core lemma M0 N b1 # dM ¼b : x¼ dN " N0 N0

Generalized 80/20 Implies Pareto Now we consider something stronger than the ‘‘iterated 80/ 20 law’’ that we saw above. As in that discussion, the portion playing the role of the ‘‘vital few’’ need not be 20% of the whole; so that some p [ (0, 1/2] now replaces 0.2, and we might speak of a ‘‘(1 - p)/p law’’ rather than 80/20. But in addition, we now assume for every nonnegative real n (not just for integer n) that the pn 9 100% of the population with the highest income has (1 - p)n 9 100% of the income. More precisely: ‘‘Generalized 80/20 law’’: For any real (including noninteger) n C 0, there is a unique x [ 0 such that PrðX [ xÞ ¼ pn , and for that value we have

From ð13Þ Since b B 1, x decreases as N increases, and when N attains its largest possible value N0, then x attains its smallest possible value x0 = bM0/N0 [ 0. Thus b1 N ; x ¼ x0 N0 and so x 1=ð1bÞ x a N 0 0 ¼ ¼ N0 x x


41

where a is defined by that last equality. This is the Pareto distribution as characterized by (2). Since 0 \ b B 1, we have 1 \ a B ?. Since b = logp(1 - p) and a = 1/(1 - b), we get a = logp/(1-p) p, as claimed.

R1 R1 Z 1 uf ðuÞdu uf ðuÞdu x N ¼ Rx 1 N0 f ðuÞdu M ¼ R1 x x f ðuÞdu x f ðuÞdu Z 1 ¼ N0 uf ðuÞdu: x

Meanings of the parameters: Since x0 = bM0/N0 is the minimum value of the Pareto-distributed random variable and M0/N0 is the average value, we conclude that

b¼

minimum : average

Consequently, the Pareto index is average a¼ : average minimum If income is so distributed that the generalized 80/20 law holds and p is actually 0.2, so that the most affluent 20% of the population have 80% of the income, then b ¼ log0:2 0:8 ¼ log5 ð5=4Þ 0:1386. . . and the Pareto index is a ¼ log4 5 1:160964. . . .

Pareto Implies Generalized 80/20 The argument of the previous section shows that if there is a probability distribution that satisfies the generalized 80/20 law, then it is the Pareto distribution with a [ 1. It stops short of proving that Pareto distributions with a [ 1 do satisfy the generalized 80/20 law.

P R O P O S I T I O N 2 For 1 \ a B ?, the Pareto distribution, as characterized by (3) or (4), satisfies the generalized 80/20 law.

P R O O F If f is the probability density function of X and x0 is the minimum possible value of X, then the average of all values of X exceeding x [ x0—that is, the conditional expected value of X given the event X [ x—is R1 uf ðuÞdu Rx 1 : x f ðuÞdu The measure N ¼ mfx 2 X : XðxÞ [ xg (corresponding in the income application to the size of that part of the population whose income exceeds x) is the measure N0 ¼ mðXÞ of the whole space (in the application, the size of the whole population) times the probablity PrðX [ xÞ (the proportion of the population whose income exceeds x): Z 1 f ðuÞdu: N ¼ N0 x

R The measure M ¼ lfx 2 X : XðxÞ [ xg ¼ X [ x X dm (corresponding in the income application to the total income of those whose income exceeds x) is the average of all such values of X times the size of that part of the space (this corresponds to the average income of those whose income exceeds x times the number N of such people):

42


The proportion lðfx 2 X : XðxÞ [ xgÞ=lðXÞ (in the income application the proportion of the population’s income received by those whose income is more than x) is therefore R1 uf ðuÞdu M : ¼ Rx1 M0 x0 uf ðuÞdu Applied to the Pareto density function (4), this gives: x a1 M 0 ¼ M0 x for x Cx0, provided a [ 1. Conjoining this conclusion with the relation (2), we conclude that for the unique x satisfying PrðX [ xÞ ¼ ðx0 =xÞa , we have R x a1 0 XR[ x X dm ¼ X dm x X (the proportion (x0/x)a of the population has (x0/x)a - 1 of the income). Letting r = (x0/x)a, we have that for any r [ [0, 1], if PrðX [ xÞ ¼ r, then the ratio of integrals above is r(a - 1)/a = rb, where the last equality defines b (the proportion r of the population gets rb of the income). Since 0 \ b B 1 (being equal to 1 precisely when a = ?), there is a unique p [ (0, 1/2] such that pb =1 - p. Thus (pn)b = (1 - p)n. Therefore when x is so chosen that PrðX [ xÞ ¼ pn , then the ratio of integrals above is (1 - p)n (so the proportion pn of the population has (1 - p)n of the income, for each real n C 0).

What If 0 \ a B 1? If aB1, then the Pareto distribution with density f ðxÞ ¼ ax0a =x aþ1 for x [ x0 has infinite expected value: Z 1 xf ðxÞdx ¼ 1: x0

This corresponds to a finite population whose total income is infinite. The income received by those whose income is less than any particular x [ 0 is finite; the divergence to infinity is in the tail of the distribution. ACKNOWLEDGMENTS

I am happy to thank John Baxter, Daniel Velleman, and an anonymous referee for pointing out things that could be said more clearly, Larry Gray and Charlie Geyer for some useful comments, and Ezra Miller for his LaTeXspertise.

REFERENCES

[4] Koch, Richard, The 80/20 Principle: The Secret of Achieving More

[1] Burrell, Q. L., ‘‘The 80-20 rule: Library lore or statistical law?’’ Journal of Documentation 41 (1985), 24–39.

With Less, Doubleday, New York, 1998. [5] Pareto, Vilfredo, Cours d’Ećonomie Politique: Nouvelle e´dition par

[2] The History of Economic Thought Website, http://homepage.

G.-H. Bousquet et G. Busino, Librairie Droz, Geneva, 1964.

newschool.edu/het (click on Alphabetical Index and then on Pareto).

[6] Ultsch, Alfred, ‘‘Proof of Pareto’s 80/20 law and precise limits for

[3] Juran, Joseph M., ‘‘Universals in management planning and

ABC-analysis’’, Technical Report 2002/c, DataBionics Research

controlling’’, Management Review, 43(11) (1954), 748–761.

Group, University of Marburg.


43

Years Ago

David E. Rowe, Editor

A Friendship of Lasting Value: Answers to Quiz from Vol. 31, No. 3 VOLKER R. REMMERT

AND

DAVID E. ROWE

The photo shows Heinrich Behnke (1898–1979, left) and Henri Cartan (1904–2008, right). It was presumably taken in Mu¨nster in the 1930s. (Photo from: Ju¨rgen Elstradt & Norbert Schmitz, Geschichte der Mathematik an der Universita¨t Mu¨nster, Teil I: 1773–1945, Mu¨nster: WWU Mu¨nster, 2009).

n case you missed it, the image above appeared a year ago without the identifying names. Readers were challenged to fill these in and also to write a short essay explaining why the friendship between these two mathematicians was worth remembering. We received two noteworthy replies. A delightfully amusing response came from Ulrich Elsaesser, who offered this brilliant analysis, worthy of a Sherlock Holmes:

I â

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

44


The figure on the right (a face and moustache that cannot be disguised) is unmistakably Henri Cartan. Since the photo was taken in 1931, the average German on the left must be Heinrich Behnke and the place must be Mu¨nster (Westphalia, Germany). So, the paper in their hands is certainly Peter Thullen’s dissertation ,,Zu den Abbildungen durch analytische Funktionen mehrerer komplexer Vera¨nderliche. Die Invarianz des Mittelpunktes von Kreisko¨rpern.’’ Following this impressive string of deductions, Elsaesser turned his clairvoyant powers to the physical surroundings in the picture: From the lighting conditions in this photograph one can deduce that it was taken on Monday, May 25, 1931 at about 2:47 PM on the porch of Behnke’s home, located at Wilhelmstrasse 15, near the University. Coffee and cake are just being served by Elisabeth Hartmann (note the shadow on Behnke’s shoulder). Evidently Behnke is arguing with Cartan because he thinks that several complex variables are somehow superfluous and unaesthetic. Our second contest winner is Sin Hitotumatu, professor emeritus at Kyoto University, who brought some personal insights to bear while successfully cracking this case. With the sincere modesty one expects from an elderly Japanese scholar, he cautiously wrote: ‘‘The gentleman at right looks like Henri CARTAN. The man at left may be Prof. Heinrich BEHNKE.’’ But then he quickly let the cat out of the bag: ‘‘Nearly a half-century ago, I studied analytic functions of several complex variables and stayed a short time at Behnke’s Institute at Mu¨nster. The photograph may show the start of the key theorem of Cartan-Thullen for domains of regularity.’’ Prof. Hitotumatu declined to write an essay for the contest because of his lack of facility with English, but he added the warm remark that the photograph brought back some fond memories of his days in Mu¨nster. Having now some idea of the mathematical context that linked Cartan and Behnke, a word should be added about the political events that helped forge this special friendship. Behnke, a student of Erich Hecke in Hamburg, taught mathematics in Mu¨nster from 1927 until his retirement in 1967. While in Mu¨nster, he founded a very successful school in complex analysis, which began in 1930 with the work of his Ph.D. student, Peter Thullen. One year later, Behnke invited Cartan to Mu¨nster at a time when Franco-German relationships were far from good. Cartan presented a series of talks there in June 1931, and soon afterward he and Thullen published a paper together in the Mathematische Annalen. This marked the beginning of a friendship that lasted through very trying times. Six years later, Behnke visited Cartan in Strasbourg, and in 1938, Cartan made a return trip to Mu¨nster to see Behnke.

Cartan had been teaching at Strasbourg since 1931, but when the Nazis invaded Poland in September 1939, the inhabitants of Strasbourg had to be evacuated. In this harried atmosphere, Cartan left many of his belongings behind in his apartment, including his mathematical manuscripts. The university faculty soon relocated in Clermont-Ferrand, where Cartan taught for a year before he assumed a professorship at the Sorbonne in Paris. By this time, France had fallen to the German armies, which made it impossible for Cartan to return to his apartment in Strasbourg. He later recalled these circumstances and how he eventually retrieved some of his papers: One day, Behnke offered to try and retrieve some mathematical papers I had left there. He actually went to Strasbourg, but to no avail. He tried again and succeeded. He managed to get hold of some documents, which he left with the library of the University of Freiburg. In 1945, some members of the French Forces in Germany happened to find them there and returned them to me [3]. Among these papers were Cartan’s notes from the very first meeting of the Bourbaki group, which took place in July 1935. The war brought much personal tragedy and loss. Henri Cartan’s brother Louis, who taught mathematical physics in Poitiers, was a member of the Resistance and was arrested in September 1942. Afterward, the family heard nothing more from him and feared the worst. Cartan appealed to Behnke for help, but his friend could do nothing. Only in May 1945 did they learn that Louis Cartan had been beheaded by the Nazis in December 1943. Late in 1945, when World War II had ended, Cartan returned to his post at the University of Strasbourg. One year later, in November 1946, he renewed contacts with his German colleagues when he visited the Research Institute in Oberwolfach, located in the Black Forest. He remembered the scene vividly years afterward:

It was very cold; there was snow and ice. I saw Professor Su¨ss (the founder of Oberwolfach) and Frau Su¨ss, and also Heinrich Behnke. I remember they asked me to play the piano… . The old chaˆteau at Oberwolfach doesn’t exist anymore. I visited Oberwolfach several times after that. [3] In 1949, Cartan made his way to Behnke’s Institute in Mu¨nster once again; and he returned frequently in the years that followed. Beyond their common mathematical interests, these two masters bore testimony to an impressive FrancoGerman friendship and cooperation during times when such relationships were something out of the ordinary.

REFERENCES AND FURTHER READING

[1] Henri Cartan, Quelques souvenirs, Mu¨nster/Westfalen, le 9 Octobre 1978. Glu¨ckwunschadresse zur Vollendung des 80. Lebensjahres von Heinrich Behnke. Heidelberg: Springer-Verlag 1978. [2] K. Hulek and T. Peternell, Henri Cartan, ein franzo¨sischer Freund. Jahresbericht der DMV 111, 85–94 (2009). [3] Allyn Jackson, Interview with Henri Cartan. Notices of the American Mathematical Society 46, 782–788 (1999). [4] Volker R. Remmert, Ungleiche Partner in der Mathematik im ,,Dritten Reich‘‘: Heinrich Behnke und Wilhelm Su¨ss. Mathematische Semesterberichte 49, 11–27 (2002).

Volker R. Remmert and David E. Rowe Fachbereich 08, Institut fu¨r Mathematik Johannes Gutenberg University D-55099 Mainz Germany e-mail: [email protected] e-mail: [email protected]


45

A Spherical Pythagorean Theorem PAOLO MARANER

here are probably many inequivalent statements in spherical geometry, somehow reducing to the Pythagorean theorem in the limit of an infinite radius of curvature r. Among these, the Law of Cosines,

T

cosðc=rÞ ¼ cosða=rÞ cosðb=rÞ; for a spherical right triangle with hypotenuse c and legs a and b, is generally presented as the ‘spherical Pythagorean theorem’. Still, it has to be remarked that this formula does not have an immediate meaning in terms of areas of simple geometrical figures, as the Pythagorean theorem does. There is no diagram that can be drawn on the surface of the sphere to illustrate the statement in the spirit of ancient Greek geometry. In this note I reconsider the issue of extending the geometrical Pythagorean theorem to non-Euclidean geometries (with emphasis on the more intuitive spherical geometry).1 In apparent contradiction with the statement that the Pythagorean proposition is equivalent to Euclid’s parallel postulate, I show that such an extension not only exists, but also yields a deeper insight into the classical theorem. The subject matter being familiar, I can dispense with preliminaries and start right in with Euclid’s Elements [1].

The Pythagorean Theorem The most celebrated theorem in mathematics [3] appears as Proposition 47 of Book I of Euclid’s Elements. It says:

1

In right-angled triangles the square on the side opposite to the right angle equals [the sum of] the squares on the sides containing the right angle. The words ‘the square on the side’ refer to the area of the square constructed on the side, which only incidentally corresponds to ‘the square of the side’ in the sense of the second power of the length of the side. This correspondence no longer holds in spherical or hyperbolic geometry, generating not a little confusion about what the generalization of the theorem should be. On the other hand, since in Euclidean geometry the area of every regular polygon is proportional to the second power of the side, the change of preposition makes clear that the original Pythagorean squares can as well be replaced by equilateral triangles, regular pentagons, regular hexagons or any other kind of regular polygon. Equivalently, since the area enclosed by the circle is again proportional to the second power of the diameter/radius, the Pythagorean squares can also be replaced by circles with diameter/radius equal to the sides of the right triangle. The reach of the Pythagorean theorem can be extended even further. In Proposition 31 of Book VI of the Elements, Euclid himself states that we are actually free to replace the squares with arbitrary shapes provided they are similar: In right-angled triangles the figure on the side opposite to the right angle is equal to the similar and similarly described figures on the sides containing the right angle.

There is already a geometrical non-Euclidean generalization of the Pythagorean theorem [5], but it is not entirely satisfactory, because the figure on the hypotenuse is made to depend on the figures on the sides.

46

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

...

Figure 1. Diagrams representing some of the infinitely many equivalent variants of the Euclidean Pythagorean proposition.

We obtain infinitely many equivalent geometrical statements (see Figure 1), all summarized by the Pythagorean formula c2 = a2 + b2, for any right triangle with hypotenuse c and legs a and b. In spherical and hyperbolic geometry there is no concept of similar figures. The areas of regular polygons with equal sides are no longer proportional. Neither is the area of the circle proportional to that of a regular polygon with side equal to its diameter/radius or to that of another circle with radius equal to its diameter. All Pythagorean statements become inequivalent and none of them remains associated with the Pythagorean formula. The question we pose is whether at least one of these geometrical statements remains true when generalized to non-Euclidean geometries. Clearly, any generalization based on similarity is meaningless, but what about the ones linked by symmetry? To answer this question it is first necessary to decide what the generalization of right triangles, regular polygons, and circles is.

AUTHOR

......................................................................... PAOLO MARANER’S research is on differ-

ential geometry in physics. After a doctorate in Parma, he had postdoctoral appointments there, at MIT and in Budapest. Since 2000, he has been a high-school teacher in Bolzano, also teaching mathematics to economists at the university. His side interests include running and swimming, and (as attested by the present article) the history of mathematics. School of Economics and Management Free University of Bozen/Bolzano via Sernesi 1, Bolzano, 39100 Italy e-mail: [email protected]

For regular polygons and circles, the choice is somehow forced by symmetry. Not so for right triangles. The standard and apparently natural choice of identifying the class of plane right triangles with that of spherical right triangles is unsatisfactory in many respects. In Euclidean geometry the role of the right angle is unambiguous, and so is the distinction between hypotenuse and legs. In spherical geometry a triangle can have two or even three right angles—and, correspondingly, two ‘hypotenuses’ and three ‘legs’ or three ‘hypotenuses’ and three ‘legs’. The very statement of the Pythagorean theorem makes little sense. If one persists in treating right triangles, the existence in spherical geometry of equilateral right triangles immediately provides a counterexample to all Pythagorean statements: The three figures constructed on the congruent sides are identical and the area of one of them can not equal the sum of the areas of the other two. On the other hand, a plane right triangle can be characterized in many different ways. Just to mention the most obvious ones: (a) a triangle with a right angle (whence the name); (b) a triangle with an angle equal to a half of the sum of its interior angles; (c) a triangle obtained by bisecting a rectangle (an equiangular quadrilateral, in preparation for non-Euclidean geometries) by means of its diagonal; (d) an inscribed triangle having a diameter as a side. Each characterization potentially provides a different generalization. The point is whether a generalization exists satisfying at least one of the infinitely many Pythagorean statements. To gain insight into this, let us briefly reconsider a few basic aspects of spherical geometry.

Spherical Triangles Spherical geometry can be obtained by replacing Euclid’s fifth postulate with the statement that no parallel to a given straight line can be drawn through a point not lying on it (in order to achieve a consistent system, however, the first and second postulates must also be partially modified). A model for such a geometry is the curved surface of a sphere of arbitrary radius r: Straight lines are identified with great circles. On the sphere we can draw points, segments, angles, triangles, every kind of polygon and circles. Spherical triangles, in particular, come early on stage. They appear as Definition I of Book I of Menelaus’s2 Sphaerica [4]: A spherical triangle is the space included by arcs of great circles on the surface of a sphere. The absence of a strong notion of parallelism on the sphere invalidates a number of important results of Euclidean geometry. Most remarkably, Proposition 32 of Book I of Euclid’s Elements is replaced by:

2

Menelaus of Alexandria (c. 70–140 CE) was the first to use arcs of great circles instead of parallel circles on the sphere. This marked a turning point in the development of spherical geometry. Being mainly interested in astronomical measurements and calculations, Menelaus did not consider theorems about area, like the Pythagorean theorem.


47

In any spherical triangle the sum of the three interior angles is greater than two right angles. Thus, in spherical geometry (a) above is not equivalent to (b). This provides us with a first alternative generalization of plane right triangles to spherical geometry. The difference between the sum of the interior angles and the straight angle

C

A

B

O

e ¼ sum of interior angles p is called the spherical excess of the triangle and is proved to be proportional to the area A of the triangle itself,

Figure 3. Inscribed spherical triangles having a diameter as a side are not right-angled.

A ¼ r2 e: By triangulation these results straightforwardly extend to every polygon: In any n-sided spherical polygon, the sum of the n interior angles is greater than (2n - 4) right angles, and the area of the polygon equals r2 times its spherical excess. In particular, the sum of the four congruent interior angles of a spherical square is greater than four right angles. Hence, these angles are no longer right. The triangulation of a spherical square by means of its diagonal no longer produces two right triangles. The same holds for every equiangular quadrilateral. It follows that (a) is not equivalent to (c). This provides us with a second possible generalization of plane right triangles to spherical geometry. A third possibility comes from the failure of Proposition 20 of Book III of Euclid’s Elements and of its corollaries. In particular:

pþe : 2 To prove the opposite implication, we just double a spherical triangle ABC with \ ABC ¼ pþe 2 and join the two copies along AC with A and C interchanged. Since \ BAC þ \ ACB ¼ pþe 2 we obtain an equiangular quadrilateral. To see that (d) implies (b), denote by e the spherical excess of the triangle ABC in Figure 3. Draw the segment OC dividing ABC into two isosceles triangles AOC and BOC. Denote by e1 the spherical excess of the first one and by e2 that of the second one. Clearly, e = e1 + e2. Since \OCA \OAC, from the first triangle, we obtain \ ABC ¼

2\OCA þ \ AOC ¼ p þ e1 ; and since \OCB \OBC, from the second one we have 2\OCB þ \ BOC ¼ p þ e2 :

In a given spherical circle, all inscribed angles subtending the diameter are greater than a right angle. Inscribed angles subtending the diameter are no longer right. Therefore, in spherical geometry (a) is not equivalent to (d). Quite remarkably, in spherical geometry (b), (c), and (d) are equivalent. To see that (c) implies (b), consider Figure 2. Since equiangular quadrilaterals have opposite sides congruent, ABC and ACD are congruent. Denote by e their spherical excess. Since spherical excess is proportional to the area and the area of the equilateral quadrilateral ABCD equals the sum of the areas of the triangles ABC and ACD, the spherical excess of the equiangular quadrilateral equals 2e. The sum of its interior angles is therefore 2p + 2e. Given the congruence of the four interior angles, we obtain

D

C

A

B

Figure 2. Spherical triangles obtained by dividing an equiangular quadrilateral by means of its diagonal are not rightangled. 48


Adding term by term, recalling that \ ACO þ \ BCO \ ACB and \ AOC þ \ BOC ¼ p, we obtain \ ACB ¼

pþe : 2

Finally, to prove that (b) implies (d), we consider a spherical triangle ABC with \ ACB ¼ pþe 2 . We now choose point O on AB such that \ ACO equal to \ BAC. Thus, CO:AO. At this point, we observe that \ BCO ¼ pþe pþe 2 \ ACO ¼ 2 \ BAC ¼ \CBA. Thus, CO:BO, and the triangle ABC is inscribed in a circle with diameter AB. The transition from Euclidean to spherical geometry seems to preserve the property of ‘having one angle equal to a half of the sum of its interior angles’ and not the property of ‘having a right angle’. This provides us with a promising class of triangles generalizing plane right triangles to non-Euclidean geometries. Let us therefore introduce a suitable terminology: We say that a triangle is properly angled, or, equivalently, that it is a proper triangle, when it has an angle equal to a half of the sum of its interior angles. That angle is called the proper angle of the triangle; the side opposite to it, the hypotenuse; and the sides containing it the legs. The role of the proper angle is unambiguous, and so is the distinction between hypotenuse and legs. In plane geometry the class of proper triangles corresponds to that

of right triangles. In spherical geometry the class of proper triangles shares at least some of the fundamental properties enjoyed by plane right triangles: Any equiangular quadrilateral is divided by means of its diagonal into two proper triangles; an inscribed triangle having as side a diameter is a proper triangle. It is then natural to wonder whether spherical proper triangles enjoy at least one of the infinitely many symmetric variants of the Pythagorean proposition. Recalling the formula expressing the area of a spherical regular polygon of side l, sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cosðl=rÞ cosð2p=nÞ 2 2 1 Angon ¼ 2pr 2nr sin cosðl=rÞ þ 1 and the formula for the area of a spherical circle of radius r Acircle ¼ 2pr2 ð1 cosðr=rÞÞ; we can simply proceed to a direct check of all of them. It is a wonderful surprise to discover that one of them still holds true.

Pythagoras on the Sphere ... To pay homage to ancient Greek geometers, we state the proposition as follows: In properly angled triangles, the circle on the side opposite to the proper angle equals [the sum of] the circles on the sides containing the proper angle. Here, the words ‘the circle on the side’ mean the area of the circle having the side as radius; this time there is no risk of algebraic confusion. The proposition is illustrated by the beautiful diagram of Figure 4. It is also immediate how to give an analytical proof of it. Parametrizing the sphere by standard spherical coordinates h and /, we consider an arbitrary equiangular quadrilateral ABCD centered at the north pole and with diagonal on the great circle through the pole and (1, 0). Its ^ Cð^ ^ pÞ, for some vertices lie at Að^h; 0Þ, Bð^ h; /Þ, h; pÞ, Dð^ h; / ^ Given the equivalence of (b) and (c), ABC is angles ^h and /. an arbitrary proper triangle. By means of the spherical distance formula for generic points PðhP ; /P Þ and Q(hQ, /Q), PQ ¼ r cos1 cos hP cos hQ þ sin hP sin hQ cosð/Q /P Þ ;

we evaluate the lengths of the sides as ^ h þ sin2 ^ h cos /Þ; AB ¼ r cos1 ðcos2 ^ ^ h sin2 ^ h cos /Þ; BC ¼ r cos1 ðcos2 ^ h sin2 ^ hÞ: AC ¼ r cos1 ðcos2 ^ Dividing by r and taking the cosine of the resulting expressions we have ^ h þ sin2 ^ h cos /; cosðAB=rÞ ¼ cos2 ^ ^ h sin2 ^ h cos /; cosðBC=rÞ ¼ cos2 ^ h sin2 ^ h: cosðAC=rÞ ¼ cos2 ^ Adding the first two equalities and comparing the result with the third one, after a very little algebra we obtain 2pr2 ð1 cosðAC=rÞÞ ¼ 2pr2 ð1 cosðAB=rÞÞ þ 2pr2 ð1 cosðBC=rÞÞ: Recalling the formula for the area of the spherical circle in terms of its radius, we recognize the spherical Pythagorean proposition. Clearly, in the limit of a large radius of curvature r, this expression reduces to the Pythagorean formula 2 2 2 AC ¼ AB þ BC :

... and on the Hyperbolic Plane The proposition straightforwardly extends to the less intuitive hyperbolic geometry. This is proved pretty much in the same way. As hyperbolic plane model we consider the quadric x 2 þ y 2 z 2 ¼ r2 embedded in the Minkowskian space R2;1 . By introducing hyperbolic polar coordinates ! x ¼ ðr sinh w cos /; r sinh w sin /; r cosh wÞ; the plane is parametrized by the hyperbolic latitude w, w C 0, and by the longitude /, - p \ / B p. The distance formula for generic points P(wP, /P), Q(wQ, /Q) reads PQ ¼ r cosh1 cosh wP cosh wQ sinh wP sinh wQ cosð/Q /P Þ : As in spherical geometry, proper triangles are obtained by dividing equiangular quadrilaterals by means of their diagonals. Hence, we again consider an arbitrary equiangular quadrilateral ABCD centered at the pole (0, 0), with diagonal along the hyperbolic line through the pole and (1, 0). The ^ Cðw; ^ pÞ, for ^ 0Þ, Bðw; ^ /Þ, ^ pÞ, Dðw; ^ / vertices lie at Aðw; ^ ^ some values w and / . ABC is an arbitrary proper triangle. The lengths of its sides are evaluated as ^ ^ sinh2 w ^ cos /Þ; AB ¼ r cosh1 ðcosh2 w ^ ^ þ sinh2 w ^ cos /Þ; BC ¼ r cosh1 ðcosh2 w ^ þ sinh2 wÞ: ^ AC ¼ r cosh1 ðcosh2 w

Figure 4. The spherical Pythagorean proposition.

Dividing by r, taking the hyperbolic cosine of the three expressions, and recalling the identity sinh2 x ¼ cosh2 x 1, after some algebra we obtain Ó 2010 Springer Science+Business Media, LLC, Volume 32, Number 3, 2010

49

2pr2 ðcoshðAC=rÞ 1Þ ¼ 2pr2 ðcoshðAB=rÞ 1Þ þ 2pr2 ðcoshðBC=rÞ 1Þ: Recalling the formula for the area of an hyperbolic circle of radius r Acircle ¼ 2pr2 ðcoshðr=rÞ 1Þ; we recognize the hyperbolic Pythagorean proposition. The Euclidean Pythagorean formula is again obtained in the limit of a large radius of curvature r.

Epilogue The Pythagorean theorem is generally claimed to be equivalent to Euclid’s fifth postulate. If so, then it can hold only in Euclidean geometry. As we have seen in this paper, this very much depends on how the proposition is understood. If we insist on squares on the sides of right triangles, no doubt the claim is true. Nevertheless, if we take a slightly wider viewpoint by considering all the equivalent variants of the theorem, and classes of triangles that better embody the properties of plane right triangles in nonEuclidean geometry, we come to a statement that equally holds in Euclidean, spherical, and hyperbolic geometry. Since it is true in Euclidean and hyperbolic geometry, this statement belongs to neutral geometry. In principle, it could be included among the first 28 propositions of the Elements and should be capable of a proof in terms of the

50


first four Euclidean postulates. Since it is also true in spherical geometry, the statement should actually follow from an even smaller set of axioms. In any case, it represents a more basic theorem about area than the original Pythagorean theorem (as in Euclidean geometry, spherical and hyperbolic polygons of the same area are related by scissor congruence [2]). In this paper we presented an analytical proof of the spherical and hyperbolic Pythagorean propositions. In the final analysis, this proof follows from the Euclidean Pythagorean proposition itself. It goes without saying that a synthetic proof based on a minimal choice of postulates would be of great interest. REFERENCES

[1] Euclid, The Elements, translated with introduction and commentary by T. L. Heath, Dover, New York, 1956. [2] R. Hartshorne, Geometry: Euclid and Beyond, Springer-Verlag, New York, 2000. [3] E. Maor, The Pythagorean Theorem, Princeton, Princeton and Oxford, 2007. [4] Menelaus, Sphaerica, translated into Latin from the Arabic version by E. Halley, http://www.wilbourhall.org [5] C. Piel, Der Lehrsatz des Pythagoras in der hyperbolischen Geometrie, Arch. Math. Phys. (1914) 22 199–204.

Reviews

Osmo Pekonen, Editor

Logicomix: An Epic Search for Truth by Apostolos Doxiadis, Christos H. Papadimitriou, Alecos Papadatos, Annie di Donna UITGEVERIJ DE VLIEGENDE HOLLANDER (AUGUST 2009), ISBN: 987-90495-0040-5, SOFTCOVER, 345 PAGES, 19.95€ (PUBLISHED IN ENGLISH BY BLOOMSBURY, NEW YORK) REVIEWED BY KRZYSZTOF R. APT

n the Spring 2006 issue of The Mathematical Intelligencer, it was mentioned that ‘‘Logicomix, ‘a work in progress on which progress is being made,’ will be published in 2007.’’ It took the authors a bit more time to complete the project. In fact, Logicomix has just appeared. It is a most remarkable book which deals with serious philosophical matters in the form of comics. This book genre, called graphic novel, became popular thanks to the successful comic Maus by Art Spiegelman that introduced the readers to the horrors of Auschwitz. Occasionally it has led to most interesting and informative books, like Palestine by Joe Sacco. The book in question introduces the reader to the quest of logicians for laying the foundations of mathematics. It is built around a lecture Bertrand Russell delivered at the outset of World War II, in which he discusses his life and opinions, his work on logic, and his encounters with prominent logicians. Also, David Hilbert and Henri Poincare´ briefly appear in the book in the context of the International Congress of Mathematicians held in Paris in 1900. The book consists of six chapters. The first two chapters essentially focus on the early years of Russell, depicting in a lively way his youth and first marriage. In Chapters 3 and 4 the action moves towards logic. Through Russell’s references to the works of George Boole and Bernard Bolzano, and fictitious encounters with Gottlob Frege and Georg Cantor, the reader is eventually introduced to the Russell paradox (that the set of all sets that are not elements of themselves neither is nor is not an element of itself). The narrative proceeds through references to Giuseppe Peano, Hilbert and Frege to Bertrand Russell’s gargantuan toil with Alfred Whitehead on their Principia Mathematica. In Chapter 5 Ludwig Wittgenstein makes his entrance. This part of the book is devoted to the great debate between Russell and Wittgenstein about the existence of an objective reality. Some incursions are made into the life of Wittgenstein, such as his decision to join the Austro-Hungarian army in World War I. In Chapter 6 there appear Kurt Go¨del, presenting his incompleteness theorem, John von Neumann (very briefly,

I 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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only as a commentator on Hilbert’s programme), and Alan Turing as the inventor of the Turing machine. The historical perspective is dramatically brought to life by a depiction of the rise of Nazism and the murder in 1936 of Moritz Schlick, the founder of the Vienna Circle. The book is interwoven in a truly self-referential way with the sometimes very animated discussions of the neatly drawn authors (Christos: I like your flowery shirt in which you reappear on p. 265) about how to best present the story to the readers. One theme that recurs in the book is the seemingly close affinity of logicians to madness. (On a cartoon on p. 281, one of the authors says: Ils sont fous, ces logiciens!1) The drawings are truly excellent, with a remarkable attention to detail. Thanks to them one experiences the strange sensation of watching a movie inside one’s head, almost hearing the voices of the main characters. (A question to the authors: having in mind an interest of one of them in the question ‘P = NP’, shouldn’t the text on the mug of Doxiadis on p. 229 be ‘P = NP’ instead of ‘P =SP’?) The book makes wonderful reading and intertwines a discussion of serious matters with subtle jokes and detours through Athens. But a nontrivial question arises: How useful might the book be to a reader who would like to understand something about the logical foundations of mathematics? I

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see a natural place for it as introductory reading for first-year mathematics or computer science students. The book does not provide any technical details, yet it gives the reader sufficient clues to understand what kept logicians busy in the critical period from Frege to Go¨del. The historical context is admittedly very sketchy, with brief references to the World Wars and to Nazism. Twenty-five pages of notes form a useful compendium on the work of the foremost logicians (starting with Aristotle) and on relevant concepts and notions (like that of a proof). Hopefully, an interested student could then continue with the more technical but still informal book Engines of Logic: Mathematicians and the Origin of the Computer by Martin Davis, in tandem with a routine course on mathematical logic. This is a review of the Dutch translation of the book, available through http://www.volkskrant.nl/webwinkel that, remarkably, appeared before the English version. The Dutch translation reads very smoothly. Unfortunately, the publisher planted three errors on the back cover. In particular ‘his mission’ (zijn missie) became a ‘vinegar mission’ (azijn missie), and Barry Mazur became Bazzy Mazur. Centrum Voor Wiskunde en Informatica (CWI) Science Park 123, 1098 XG Amsterdam, The Netherlands e-mail: [email protected]

Die Vermessung der Welt by Daniel Kehlmann BERLIN: ROWOHLT VERLAG, 2005; 302 PP., 9.95 EUR; ISBN 978-3-49924100-0

Measuring the World (translated by Carol Brown Janeway) LONDON: QUERCUS FICTION, 2007; 7.99 POUNDS; ISBN 10: 1-84724-114X; ISBN 13: 978-1-84724-114-6 REVIEWED BY ULF PERSSON

ehlmann is a young German writer of fiction who has had great critical and commercial success. Die Vermessung der Welt has already been issued in several editions and has been translated into a large number of languages. However, the enthusiastic response seems not to be shared by mathematicians, to judge from its reception by mathematical reviewers who, with few exceptions, have been highly critical of Kehlmann’s presentation of Gauss. Some even have thought the description borders on libel. The book is plainly a novel and the characters whose names are borrowed are long since dead, so it would be absurd to entertain legal action; nevertheless some interesting and important issues are raised by such criticism, to which I will return at the end. As the title indicates, the novel is about measuring the world and the supposed folly of it. Folly because the world cannot be reduced to fragments and quantitatively described and explained; there is more to it than what can be measured. What is called for is a holistic approach, such as that suggested by Goethe, who in his Farbenlehre opposed Newton’s spectral theory and instead advocated a more sensuous approach to color. Indeed most people instinctively sympathize with Goethe, finding Newton’s approach limited and barren without realizing that this is exactly its point. Goethe does appear marginally in the novel in his character as a sage authorized to bestow blessings on ventures and projects. Yet the author does not delve further into Goethe’s scientific credentials and visions and thus resists making him a spokesman for his own views and a central character. Instead, to dramatize the issue of measuring the world, Kehlmann combines the biographies of two towering German geniuses and scientific heroes—the explorer Alexander von Humboldt and the mathematician Carl Friedrich Gauss. On its face, though contemporaries, they are unlikely bedfellows. They had little to do with each other, although of course by virtue of the small scientific community at the time, and their respective positions in it, they could hardly have been ignorant of one another. What did they have in common? Did they ever meet? The obvious answer to the first question is measurement, with the explorer von Humboldt representing the literally down-to-earth measurements contrasted with the far more theoretical and abstract notions

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that preoccupied Gauss. Whereas the explorer was a busybody, traveling all over the world, Gauss stayed in his study in Go¨ttingen. As to the more anecdotal question of a personal meeting, such a thing of course is harder to disprove than confirm. Leaving the issue of an actual meeting aside as irrelevant, it is natural to open the novel with such an imagined meeting. So we are thrown in media res, with Gauss reluctantly roused from his sleep to travel to Berlin, where he has been invited by von Humboldt to be feted. Gauss is grumpy indeed and although he entertains no tender feelings for his domestic situation, finding his wife a silly goose and his clever son Eugen a stupid bore, he deeply resents leaving his abode and submitting himself to the inconveniences of travel. So we are treated to an imaginatively conceived day, in which Gauss meets an enthusiastic von Humboldt, the photographic documentation of the historic event being (conveniently for the author) bumbled. Furthermore we witness how the former is dragged to a festive dinner, where he meets his future collaborator Weber and is entranced by Weber’s young wife. Meanwhile Gauss’s son haplessly finds himself at a subversive meeting and is arrested; it takes the authority of von Humboldt to get him released, a release made on the condition of his permanent exile to America. The book ends with his subsequent departure. This skeleton of a plot is then padded with extended flashbacks in which segments of the biographies of the explorer and the mathematician are intertwined more or less chronologically. Alexander von Humboldt is presented as a one-dimensional character who supposedly does not recognize anything that cannot be reduced to measurements. Measurements and numbers do away with anxiety and disorder and make up the very essence of knowledge, according to his view of life. His energy and persistence are awesome and obsessively single-minded when he penetrates the wilderness of Amazonia, where he survives one hair-rising adventure after the other. He counts and measures everything he can get his hands on but, as there is no time to lift his gaze, he sees nothing beyond that, to the consternation of his faithful but exasperated companion, the French botanist Bopland, who plays his Sancho Panza. At the end of von Humboldt’s life the heroic explorations of his youth are made a travesty as he travels around Siberia, officially invited by the Czar, encumbered both by the large escort supplied by the latter as well as by interminable social obligations befitting a man of such renown. This would have seemed tragic had he not been presented as a caricature in the beginning; instead, it strikes the reader as merely farcical. To a mathematical readership, however, Kehlmann’s treatment of Gauss is of more interest, so I will concentrate on it, even if doing so skews the review of the book as a whole. It is not often that our mathematical heroes are recognized by the literary world. Not even Gauss, who at least in Germany must be generally known outside the mathematical world (if for no other reason than that he once appeared on a German bank note) has ever before attracted such attention. Just as with von Humboldt, Kehlmann, having done his homework, draws on documented facts. He retells the well-known story about the feat of the child Gauss adding consecutive integers, with the Ó 2010 Springer Science+Business Media, LLC, Volume 32, Number 3, 2010

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twist that in his novel this is an apocryphal story. He refers to Gauss looking at the distribution of primes, although his account is somewhat garbled. He also notes Gauss’s hesitation about whether to devote himself to philology rather than mathematics, with the discovery of the construction of the regular polygon with seventeen sides tipping the balance. And he refers to quadratic reciprocity and his Disquisitiones Arithmeticae as his ticket to fame. Gauss’s extramathematical activities, such as acting as an astronomer, a land surveyor, and in his old age connecting with Weber and measuring magnetism, are duly noted with the regret that such a superior mind should be saddled with such routine assignments. This is of course well known, at least to mathematicians; what is of real interest are the imaginative interpolations. An author is of course free to make them up as he sees fit, nevertheless the quality of such imaginings will depend upon their relevance. Here the mathematical reader may be in a position to judge. I would guess that most would consider the statement that Gauss counted prime numbers when he was nervous to be something of a cliche´. That Gauss should have interrupted his nuptial night by jotting down his idea of least squares lest he forgets it does not square, as this is not the kind of idea that once grasped will ever threaten to evaporate. Most interesting are the intimations of Gauss’s understanding of non-Euclidean geometry, which he famously kept to himself rather than provoking the ire of his contemporaries. Those insights are, Kehlmann suggests, inspired by his activity as a surveyor. It is often claimed by those who should know better that Gauss actually tried to compute the angular sums of triangles, using light beams to decide the physical geometry of the universe experimentally. Kehlmann refers to this but seems to confuse the curvature of the earth with that of its ambient space. Now unless Gauss had the foresight to anticipate Riemann, the idea of being able experimentally to determine the length of the absolute unit (to which Kehlmann refers in a particularly confused sense, admittedly refracted through the supposed limited intelligence of Humboldt) in a homogenous hyperbolic space is preposterous; Gauss clearly would have understood that. Such a short unit would have had dramatic celestial consequences, such as a marked parallax. In fact the determination of the radius of the earth, assumed perfectly spherical, on the basis of angular excess for triangles a couple of kilometers in dimensions (the kind encountered in surveying work) is still not physically feasible. Can we learn anything new about Gauss in this book that we have not gleaned from his many biographies? Facts are one thing, but a fictional account, imaginatively going beyond what can be documented, could probe deeper. Yes, unlike his portrait of Humboldt, the author has attempted to supply Gauss with an inner life. It might not be an inner life that we mathematicians would find realistic. On the general human level Gauss is presented as dismissive toward his children, indifferent toward his first wife, and downright contemptuous of his second, his true love being a Russian prostitute with whom he consorted in his youth. Furthermore he is shown as vain and resentful. A more relevant discussion

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is of course the nature of Gauss’ work and what it means to be a mathematician of such an order. This goes beyond the merely personal and anecdotal, and regardless of the work being fiction or not, getting it right is important, because even when you make up things, those things have to be true in a certain deeper sense. Gauss is also presented as one who measures the world, in fact he is given to the reflection that only through measurement is the world brought into existence, but he is shown to be superior to von Humboldt, because his insights are deeper and his visions grander and his thinking so much quicker. While, as we have already noted, von Humboldt is a restless soul roaming around the world, Gauss stays put knowing that the secrets of the universe can be divined by a powerful intellect alone. In fact, his only regret is being born too early: in the future many of the annoying vicissitudes of life that mar his quotidian existence will have been removed. This exalted view of the power of the mathematical mind, at least in its manifestation through Gauss, might gratify our mathematical vanities, but what exactly is the nature of those insights and visions? Quick thinking is, after all, but a superficial part of a mathematician’s repertoire and many of us do not even possess it. Great men should of course be released from our constricting idealizations of them, torn down from their pedestals, and shown to be (in)human. To complain about this would be absurd. Yet, this does not excuse us from asking whether Kehlmann has any basis for his speculations. As he is not writing a biography he needs none, yet his ambition is nevertheless to present well-rounded pictures of characters, with the historical facts he has brought in as boundary conditions. If his fictional reconstructions are believable, given those conditions, this will subtly influence our estimation of the real historical persons, for better or for worse. As I have noted earlier in this journal a propos Arild Stubhaug’s biography of Abel (The Mathematical Intelligencer, vol. 32, no. 1, pp 68–71); the domains of fiction and biography have a large overlap, and the genres serve similar purposes. The first question is whether this is fair to the historical characters. Of course it is not. On the other hand, there is a price to be paid for achieving fame that survives long after your death: namely, that of being turned into a fictional character. The second question is moral and artistic. Would the book have been as interesting had it concerned two wholly made-up characters? We all find it more interesting to read about what is true and actually happened, rather than what is merely invented. Hence there is a long tradition of authors writing fiction as if it were true, often by inserting real facts to give a sense of veracity. If it is a very good story, the reader will excuse the pretense, if not he or she will invariably feel a bit cheated. Is Kehlmann cheating? There are of course many biographies of Gauss to which readers can turn for a more authoritative view of the great mathematician, and then they can let their own imaginations make the desired interpolations (it seems that Kehlmann has done just that). In recent years the large oeuvre of von Humboldt has been reissued, and a reader unwilling to plow through it could be referred to a recent

anthology [1] to get a taste of the explorer. Otherwise the sympathetic treatment presented in [2] may serve if not as an antidote, at least as a complement to the one provided in the book under review. Finally the reader whose curiosity is whetted by Kehlmann as a writer of fiction may turn to his latest book [3] of nine loosely connected stories exploiting the fictional possibilities provided by modern gadgets such as the cell phone. Kehlmann also explores the relationship between the real and virtual, playing with the idea that they mesh into one another, not unlike the ‘‘two’’ sides of a Mo¨bius strip, so an author could literally step into the fictional world he creates.

REFERENCES

[1] Alexander von Humboldt, Das große Lesebuch. (edited by Oliver Lubrich) Fischer Taschenbuchverlag, 2009. [2] The Art of Travel, Alain de Botton, Penguin, 2002. [3] Ruhm, ein Roman in neun Geschichten, Daniel Kehlmann, Rowohlt, 2009. Department of Mathematics Chalmers University of Technology Go¨teborg Sweden e-mail: [email protected]


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Euler’s Gem—The Polyhedron Formula and the Birth of Topology by David S. Richeson PRINCETON AND OXFORD: PRINCETON UNIVERSITY PRESS, 2008, 317 PP. US $ 27.95, ISBN-13: 978-0-691-12677-7, ISBN-10: 0-691-12677-1 REVIEWED BY JEANINE DAEMS

‘‘

hey all missed it.’’ Richeson’s book begins with a strong and clear motivation for one of his key points on the nature and the historical development of mathematics. ‘‘It’’ is ‘‘Euler’s Gem,’’ Euler’s polyhedron formula, one of the most beautiful formulas of mathematics (in fact, the author informs us, a survey of mathematicians pi found its beauty to be second only to e + 1 = 0, also Euler’s). ‘‘They’’ refers to all of Euler’s predecessors who, though active in the field of geometry, failed to come across this elegant and, to our eyes, even obvious relationship. Euler’s polyhedron formula is elegant and simple: In a polyhedron, the number of vertices (V), edges (E) and faces (F) always satisfy the equality V – E + F = 2. For example, a cube contains 8 vertices, 12 edges and 6 faces, and indeed, 8 – 12 + 6 = 2. But if this formula is so simple, why did no one think of it earlier, especially when, as Richeson explains, people had been fascinated by polyhedra for millennia? The ancient Greeks, for example, were already able to prove that there are exactly five regular polyhedra. Polyhedra are very familiar mathematical objects: They are three-dimensional objects constructed from polygon faces, such as the cube, pyramids, the soccer-ball-shaped truncated icosahedron, and so on. However, there is no historical consensus about the precise definition of a polyhedron. The Greeks and Euler, for example, implicitly assumed that polyhedra are convex, whereas modern definitions do not. And is a polyhedron solid, or is it hollow? Richeson uses Euler’s polyhedron formula as a guiding line on his enthusiastic tour of the wonderful world of geometry and topology. The first part of the book deals with the history of the polyhedron formula, starting with a biographical chapter on Euler. Then Richeson discusses the five regular polyhedra, Pythagoras and Plato, Euclid’s ‘‘Elements,’’ Kepler’s polyhedral universe, and of course Euler’s discovery of his polyhedron formula. And he explains why Euler’s treatment was new: Until then, the theory of polyhedra had dealt with metric properties of polyhedra like measuring angles, finding lengths of sides and areas of faces, and so on. Euler, however, tried to classify polyhedra by counting their features. He was the first one to recognize that ‘‘edge’’ is a useful concept, and he realized it was the vertices, edges and faces he had to count. However, Euler’s proof of his formula did overlook some subtleties and is not completely rigorous by modern standards.

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Then there is an interesting chapter on Descartes (1596– 1650). In 1860 some long lost notes of Descartes surfaced in which he stated a theorem that looks a lot like Euler’s polyhedron formula: P = 2F + 2V - 4, where P is the number of planar angles in a polyhedron, V the number of vertices and F the number of faces. Since the number of planar angles in a polyhedron is twice the number of edges, Euler’s formula follows easily (if one knows the concept of an edge, and it was Euler who introduced that). So, whether Descartes did or did not prediscover Euler’s formula is debatable, but Richeson decides it is not unreasonable to continue ascribing it to Euler. Legendre (1752–1833) gave a proof of Euler’s formula that is correct by our standards, using a projection of the polyhedron on a sphere. A little later it was noticed that Legendre’s proof even worked for a bigger class of polyhedra than the convex ones: The so-called star-convex polyhedra. After this historical exposition, Richeson proceeds by discussing some aspects of more modern mathematics that all have something to do with the polyhedron formula. This part of the book contains some elements of graph theory, the four-color theorem, the discussion of which kinds of polyhedra are exceptions to Euler’s formula and generalizations of the formula that arose from this, and eventually the rise of topology. Does Euler’s formula also apply to objects other than polyhedra? Yes. For example, it applies to partitions of the sphere, something Legendre already used in his proof. Cayley noticed that when Euler’s formula is applied to graphs, the edges need not be straight. Richeson uses such ideas to illustrate the transition from a geometric to a topological way of thinking about shapes. He explains very clearly that in geometry it is crucial that the objects are rigid, but sometimes these rigid features of geometric objects obscure the underlying structures. Richeson’s introduction to topology is very nice. He explains what surfaces are, describes objects like the Mo¨bius strip, the Klein bottle and the projective plane, discusses when objects are topologically the same, states a theorem that relates Euler’s formula to surfaces, gives an introduction to knot theory, differential equations, the hairy ball theorem, the Poincare´ conjecture… The book treats too many subjects to mention all of them. They are all related to Euler’s polyhedron formula in some sense, and together they give a very good overview of the field of topology and its history. But that is not all Richeson achieves with this book: He also shows what it is that mathematicians do. He shows that mathematics is created by people and that it changes over time. Usually, theorems were not stated originally in their current formulation. Richeson’s book is definitely not a mathematical textbook, and it is not just a historical story either. He wants to show what he enjoys about the topology he works on as a research mathematician. As he writes in the preface: ‘‘It is my experience that the general public has little idea what mathematics is and certainly has no conception what a research mathematician studies. They are shocked to discover that new mathematics is [sic] still being created.’’ And he tells us why he was attracted to topology: ‘‘The loose and flexible topological

view of the world felt very comfortable. Geometry seemed straight-laced and conservative in comparison. If geometry is dressed in a suit coat, topology dons jeans and a T-shirt.’’ His playful attitude to mathematics is clearly expressed in the book: There is an abundance of examples, and there are even templates for building your own platonic solids, as well as the Mo¨bius band, the Klein bottle and the projective plane. As he mentions in the preface, Richeson wrote his book for both a general audience and for mathematicians. I think he succeeded. Many insights and theorems he explains are difficult and quite deep. He skips the formal details but does not leave out the mathematical reasoning. And he keeps a good balance between the mathematical arguments and intuitive insights. His explanations are appealing. An example is: ‘‘Even more bizarre, could it [the universe] be nonorientable? Is it possible for a right-handed astronaut to fly away from earth, and return left-handed?’’ The focus of the book lies on the big picture, and for the interested reader there is a list of recommended reads, as well as a long list of references containing many primary sources for the historical part. The fact that ‘‘Euler’s gem’’ has no formal prerequisites does not make it an easy book. As Richeson writes in the preface: ‘‘Do not be misled, though—some of the ideas are quite sophisticated, abstract and challenging to visualize. …

Reading mathematics is not like reading a novel.’’ Which is true. But Richeson believes the audience for this book is self-selecting: ‘‘Anyone who wants to read it should be able to read it.’’ I liked Richeson’s style of writing. He is enthusiastic and humorous. It was a pleasure reading this book, and I recommend it to everyone who is not afraid of mathematical arguments and has ever wondered what this field of ‘‘rubbersheet geometry’’ is about. You will not be disappointed.

OPEN ACCESS

This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.

Mathematical Institute University of Leiden P.O. Box 9512 2300 RA Leiden The Netherlands e-mail: [email protected]


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Modular Forms: A Classical and Computational Introduction by Lloyd J. P. Kilford LONDON: IMPERIAL COLLEGE PRESS, 2008, 236 PP., US$60 / £33, ISBN 13 978-1-84816-213-6, ISBN 10 1-84816-213-8 REVIEWED BY ROBERT JONES

he topic of this book, as its subtitle indicates, is both old and new. The great classical names of mathematics, Fermat, Gauss, Galois, Hilbert, and many others, have contributed to it. The author’s own bailiwick, computational number theory, is a topic of active current research. The book is based on notes for several courses, undergraduate and graduate, given at the Mathematical Institute of the University of Oxford from 2004 to 2006. After a brief historical overview in the first chapter, ranging from the 18th century to the Langlands Program in the 21st century, the book provides references to several books for recommended background reading. These include Koblitz (1993) and Diamond and Shurman (2005). Those who are just beginning to walk along the road of mathematics would be well served by the two-volume introduction by Tom Apostol. One valuable aspect of this book is its topicality. This feature is significant to students and specialists alike. But before taking the plunge into accounts of contemporary research, there is a special problem with which the author must come to terms: this topic suffers from an embarrassment of riches. Mathematics is famous for the interrelatedness of its subdisciplines. The topic of this book is, perhaps arguably, other parts of mathematics more related to than most subdisciplines. For example, modular forms and number theory are closely related to hyperbolic geometry in the plane and especially to hyperbolic 3-space. This has been documented in Elstrodt, Grunewald, and Mennicke (1998). Ratcliffe (1994) and Matsuzaki and Taniguchi (1998) offer introductions to hyperbolic n-dimensional space. The Poincare´ upper half-plane is the book’s point of contact with hyperbolic geometry. One could also imagine the discussion of 3-space leading on to an account of the sensational work of Grigori Perelman, and on again from there to, say, Thurston’s proof of the eight-fold classification of geometries, in Thurston (1997) Chapter Three. The latter steps would be clearly topic-inflation that any book must avoid to remain within publishable bounds. Here one sees a choice of inclusion at work. The latter two topics are certainly related to the subject of the book, but they would be a step too far. Hyperbolic geometry is so closely related to that subject that it cannot be excluded entirely. To do so would mean trimming away the very substance to be discussed.

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The Poincare´ upper half-plane is one of the standard models of hyperbolic geometry. Another is the Poincare´ disk model, and yet another is the Beltrami-Klein disk model. All of these would appropriately appear in a book on hyperbolic geometry, such as Anderson (1999) or Stahl (1993). Kilford draws the dividing line by including only the first model of hyperbolic geometry. Although Kilford situates the book in number theory, Stein (2007) sees modular forms as a branch, or subdiscipline, of complex analysis. Where does all of this rich interconnectedness come from? Has something gone awry? The author explains, ‘‘The modular group…is like an octopus, with tentacles reaching out into many branches of pure mathematics…’’. The phrase, ‘‘the modular group,’’ is used here, rather than say, ‘‘Fuchsian group,’’ or ‘‘modular form,’’ but the underlying explanation of their interconnectedness is similar, and these areas of mathematics are indeed closely connected. Fuchs, a student of Lobachevski; examined the group-theoretic implications of Lobachevskian, or hyperbolic, geometry (also discovered independently by Bolyai). Katok (1992) treats Fuchsian groups. The term ‘‘Fuchsian’’ was coined by Poincare´. An introductory account of Fuchsian groups appears in Chapter 14 of Toth (1998). Apostol (1990) gives an account of modular groups in volume 2. Beardon (1995) provides an account of the closely related Mo¨bius groups. Chapter Two defines modular forms and introduces congruence subgroups. Chapter Three discusses dimensions of spaces of modular forms and how to find a basis for a space of modular forms. Chapter Four discusses Hecke operators and eigenforms. Chapter Five discusses applications of modular forms to elliptic curves and to various classical conjectures and problems. One such problem is Fermat’s Last Theorem (FLT). Gauss was famously dismissive of the FLT, saying that he could easily find similar mathematical problems that were unsolved, but had little general significance (Singh 1997). On the other hand, he generously hailed the progress on the FLT by Sophie Germain. Now that Andrew Wiles has solved the FLT, what other problem could replace it? Arguably, the best candidate may be Goldbach’s conjecture that every even integer greater than 2 is the sum of two primes. But what could ever replace the fiendishly fascinating myth-making power of the wonderful proof that did not quite fit into the margin of the piece of paper? This is the stuff of legend. Here, the FLT will probably never be exceeded. As a small consolation for this loss, Kilford offers us a pertinent quip from an anonymous source: ‘‘this is a one line proof, if you start sufficiently far to the left.’’ In Chapter Six, Kilford discusses Galois representations and Katz modular forms. In Chapter Seven, he enters into his own home territory: computing with modular forms. To limit somewhat his consideration of the plethora of software packages for number theory and modular forms, he sets up two criteria for discussing a particular package: it should be optimized for number theory calculations, and it should contain an extensive modular forms library. These criteria limit much of the discussion to Pari, Magma, Sage, Maple, Mathematica, and MATLAB, although other

packages are not entirely excluded from consideration. Sage, authored by William Stein, allows calls, from Sage, to functions in more than 50 other packages. There are few misprints, and the standard of scholarship is high. But there is a tantalizing peccadillo, an unfinished sentence on page 19, ‘‘We will investigate some of the properties of the Bernoulli numbers in.’’ A plea to the author: Please do not correct this misprint; give us an account of the matter in another book. This fascinating, contemporaneous, and even now unfolding story of current research in a historically brilliant part of mathematics is told with riveting attention to detail. This means, of course, that the book will quickly date. I recommend that you jog across the campus to your nearest bookstore now and get a copy. The reviewer was a student in a course on dynamic systems, in which this book was collateral reading, offered in the Summer Semester of 2009 by Professor Fritz Grunewald at the University of Du¨sseldorf. Many thanks to him for his help in writing this review.

Diamond, Fred, and Jerry Michael Shurman, A First Course in Modular Forms, Graduate Texts in Mathematics, vol. 228, Springer-Verlag, New York, 2005. Elstrodt, Ju¨rgen, Fritz Grunewald, and Jens Mennicke, Groups Acting on Hyperbolic Space, Harmonic Analysis and Number Theory, Springer-Verlag, Berlin, Heidelberg, 1998. Katok, Svetlana, Fuchsian Groups. University of Chicago Press, Chicago, 1992. Koblitz, Neal, Introduction to Elliptic Curves and Modular Forms, 2nd edition, Springer-Verlag, New York, 1993. Matsuzaki, Katsuhiko, and Masahiko Taniguchi, Hyperbolic Manifolds and Kleinian Groups, Clarendon Press, Oxford, 1998. Ratcliffe, John G., Foundations of Hyperbolic Manifolds, SpringerVerlag, New York, Berlin, 1994. Singh, Simon, Fermat’s Last Theorem, The Story of a Riddle that Confounded the World’s Greatest Minds for 358 Years, Fourth Estate Limited, London, 1997. Stahl, Saul, The Poincare´ Half-Plane, A Gateway to Modern Geometry, Jones and Bartlett, Boston, London, 1993. Stein, William, Modular Forms, A Computational Approach, Graduate Studies in Mathematics, vol. 79, American Mathematical Society, Providence, Rhode Island, 2007. Thurston, William P., Three-Dimensional Geometry and Topology, vol.

REFERENCES

Anderson, James W., Hyperbolic Geometry, Springer-Verlag, London, 1999.

1, Princeton University Press, Princeton, New Jersey, 1997. Toth, Gabor, Glimpses of Algebra and Geometry, Springer-Verlag, New York, 1998.

Apostol, Tom M., Introduction to Analytic Number Theory, SpringerVerlag, New York, 1976. Apostol, Tom M., Modular Functions and Dirichlet Series in Number Theory, Springer-Verlag, New York, 1976, 2nd edition, 1990. Beardon, Alan F., The Geometry of Discrete Groups, Springer-Verlag, New York, 1995.

Rurweg 3 D-41844 Wegberg Germany e-mail: [email protected]


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Representation and Productive Ambiguity in Mathematics and the Sciences by Emily Grosholz OXFORD UNIVERSITY PRESS, 2007, 313 PP., US $ 85.00, ISBN 9780199299737 REVIEWED BY MARY LENG

ccording to the logical positivist picture of empirical science that was predominant in the 1930s, scientific theories should be identified with sets of sentences consisting of scientific laws and their deductive consequences. An ideal of reduction held sway, according to which the analysis of the concepts of our scientific theories should show those concepts to be ultimately reducible to a small number of basic building blocks. This ‘syntactic’ view of science, developed for example in Rudolf Carnap’s The Logical Structure of the World (Carnap’s ‘Aufbau’, 1928), took its inspiration from the reductive, foundational projects in mathematics, and particularly from logicism, which sought to reduce the rich and various theories and objects of mathematics to logical constructs built from a small number of ultimately logical concepts. Despite its pleasing neatness, the syntactic view of empirical scientific theories is no longer held to be tenable by most philosophers of science. As Emily Grosholz puts it, in her Representation and Productive Ambiguity in Mathematics and the Sciences, it came to be realised that the objects of science ‘‘must be represented in order to be studied’’, with ‘representation’ being ‘‘a much broader notion than formalization’’. Indeed, ‘‘formalization suits inference, which is indifferent (up to a point) to the things it treats’’ (p. 20). And whereas it ‘‘…makes sense to formalize inference, it doesn’t make sense to formalize a molecule.’’ (p. 63). Paying closer attention to the content of our scientific theories, to their subject matter rather than simply their formal structure, a ‘semantic’ approach to scientific theories has arisen as the now-dominant alternative to the syntactic view. At its narrowest, the semantic approach views scientific theories as collections of their models (in the logician’s sense), thus moving the emphasis away from particular languages and linguistic formulations of our theories. But more broadly, semantic approaches to science pay attention to the activity of scientists in building theory ‘models’ in a sense more closely analogous to the notion of a ‘scale model’, a means of representation which aims to identify and illuminate important features of actual situations to provide tractable tools for the theoretical examination of those situations. A tempting view is that, whereas the syntactic view works well for the mathematical theories that inspired its original formulation, the distinctive nature of empirical scientific theories, whose objects are tangible and physical

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rather than formal and theoretical, means that a syntactic approach is insufficient there. Although it is perhaps plausible to think that there is nothing more to mathematical objects than can be characterized by sentences within a formal theory, such sentences can be at best imperfect, incomplete characterisations of the multilayered objects of the physical world. One of Grosholz’s central claims is that this distinction between mathematical and physical objects is mistaken: the objects of our mathematical theories are at least as multifaceted and complex as those of our physical theories. For Grosholz, as with molecules, so with mathematics: ‘‘it doesn’t make sense to formalize a circle, the number 3, the number pi, or the sine wave.’’ (p. 63). Mathematical objects, like physical objects, must be approached from a variety of angles and from varying depths in order to be rendered intelligible. The syntactic view is therefore as mistaken for mathematics as it is for empirical science. This is not to say that the semantic approach to scientific theories, taken by itself, is correct. According to Grosholz, although the semantic view has part of the story right about mathematical and scientific theories, one cannot give a complete picture of the nature of theorizing unless one also turns one’s attention to pragmatics, and in particular to the ways scientists use their theoretical tools for the purpose of problem-solving. Grosholz’s interest, in particular, is in the syntax, semantics, and pragmatics of notation as a mode of representation of the objects of science. Starting with case studies from chemistry, she shows how one’s choice of notation (such as, for example, Berzelian formulas for molecules) plays a positive and productive role in theorizing, rendering the microscopic intelligible and providing techniques for the solution of problems. Moving on to mathematics, an analogous case is made for mathematical notation. The Arabic numerals, for example, make important aspects of number theory tractable in a way that would be impossible if one stuck, for example, to stroke notation as one’s mode of representation of the numbers. Grosholz’s careful case studies from mathematics and empirical science (particularly chemistry) illustrate a number of themes relating to the pragmatic dimension of our modes of representation. She is interested, for example, in how a single mode of representation can function sometimes as symbol (representing without resembling), sometimes as icon (representing with resemblance), in problem-solving contexts. This dual role is an example of the ambiguity of Grosholz’s title. A productive ambiguity, since reading a single mode of representation in various ways can be the key to mediating between different aspects of a problem context. In chemistry, for example, the attention of theorists—and experimenters—must move between various levels of reality, encompassing microscopic molecules and macroscopic reactions. To facilitate these moves, ‘‘a certain linguistic item (symbol or icon) may stand for either or both’’ (p. 89). In mathematics, productive ambiguity can be seen in, for example, the multiple possible readings of diagrams, such as those provided in Newton’s Principia, whose components can—and must—be read in two incompatible ways. If one reads Newton’s diagrams as consisting of finite line segments and areas, Euclidean theorems can be applied for

problem-solving purposes. But reading them as including infinitesimal quantities allows these solutions, arrived at via Euclidean geometry, to be applied to the issues of motion and force for which the diagrams have been designed. Far from being an error of nonrigorous 17th century analysis, Grosholz argues that this kind of ‘productive ambiguity’ is essential to mathematics even today, in order to render intelligible the infinitary subject matter of mathematics to finite minds capable of working only with finitary tools. In recent years, philosophers of mathematics have increasingly come to see the value of looking beyond the narrow confines of logic and set theory in order to understand mathematics as it is actually practised. Grosholz’s book, with its careful case studies from mathematics and empirical science, is a welcome intervention in this movement, showing as it does how our understanding of mathematics can inform, and be informed by, our

understanding of empirical science in a way that goes beyond the positivists’ narrow focus on predicate logic and formal syntax. Yet, in bringing together mathematics and empirical science, Grosholz’s impressive study also remains in the tradition of the logical positivists, who sought to model the empirical sciences on mathematics. Grosholz is, after all, a self-confessed philosophical grandchild of Carnap et al., respecting that tradition even while she replaces its syntactic view, together with the semantic view of Carnap’s philosophical children, with a more nuanced pragmatic account of mathematics and science. Department of Philosophy University of Liverpool 7 Abercromby Square, Liverpool L69 7WY UK e-mail: [email protected]


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Sophie’s Diary by Dora Musielak BLOOMINGTON, INDIANA, AUTHORHOUSE, 2008. SOFTCOVER, 244 PP., US $11.50, ISBN: 1-4184-0812-3 REVIEWED BY DAVID PENGELLEY

an a fictional teenage diary of the mathematician Sophie Germain have dramatic and captivating appeal to audiences ranging from curious teenagers to professional mathematicians? The answer lies in the delicate balance between what we do and don’t know about her real life, along with the extraordinary historical and mathematical circumstances that coalesced with her stranger-than-fiction initiative, perseverance, and mathematical talent, to make her the first woman we know to achieve important original mathematical research. In order to understand the full stage setting, let us first review what we do know about Sophie Germain, including recent new discoveries about her work, and about Fermat’s Last Theorem, that make her mathematical and personal story both compelling and tantalizing. What we now know leaves both abundant and timely opportunity for the creation of the book under review, a fictional diary from Sophie Germain at age 13 to 17, along with a short nonfiction appendix by the author, Dora Musielak. The real Sophie Germain was born on April 1, 1776, and she lived with her parents and sisters in the center of Paris throughout the upheavals of the French Revolution. Even if kept largely indoors, she must as a teenager have heard, and perhaps seen, some of its most dramatic and violent events. Moreover, her father, Ambroise-François Germain, a silk merchant, was an elected member of the third estate to the Constituent Assembly convened in 1789, when the fictional diary begins [1]. He thus brought home daily intimate knowledge of events in the streets, the courts, etc.; how this was actually shared, feared, and coped with by the real Sophie Germain and her family we do not know. Much of what we know of Germain’s life comes from the biographical obituary [5] published by her friend and fellow mathematician Guglielmo Libri shortly after her death in 1831. He wrote that at age thirteen, Sophie Germain, partly as sustained diversion from her fears of the Revolution beginning outside her door, studied first Montucla’s Histoire des mathe´matiques, where she read of the death of Archimedes by the sword of a Roman soldier during the fall of Syracuse, because he could not be distracted from his mathematical meditations. It seems that Sophie herself followed Archimedes, becoming utterly absorbed in learning mathematics, studying without any teacher from a then common mathematical work by E´tienne Bezout that she found in her father’s library. Her family at first endeavored to thwart her in a taste so unusual and socially unacceptable for her age and sex. According to Libri, Germain rose at night to work by the glimmer of a lamp, wrapped in covers, in cold that often

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froze the ink in its well, even after her family had removed the fire, clothes, and candles from her room to force her back to bed. It is thus that she gave evidence of a passion they thereafter had the wisdom not to oppose. Libri writes that one often heard of the happiness with which Germain rejoiced when, after long effort, she could persuade herself that she understood the language of analysis in Bezout. Libri continues that after Bezout, Germain studied Cousin’s differential calculus, and was absorbed in it during the Reign of Terror (1793–1794). Dora Musielak’s diary ends on April 1, 1793, Germain’s seventeenth birthday. This is a perfect ending point, since it is from roughly 1794 onwards that we have some records of Germain interacting with the public world. And it was then, Libri explains, that Germain did something so opportunistic, so rashly remarkable, so farreaching in its consequences, that it would lack believability if it were mere fiction. Germain, then eighteen years old, first somehow obtained the lesson books of various professors at the newly founded Ećole Polytechnique. She particularly focused on those of Joseph-Louis Lagrange on analysis. The Ećole, a direct outgrowth of the French Revolution, did not admit women, so Germain had no access to this splendid new institution and its faculty. However, the Ećole did have the novel feature, heralding a modern university, that its professors were both teachers and active researchers. Indeed, its professors included some of the best scientists and mathematicians in the world. Libri writes that professors had the custom, at the end of their lecture courses, of inviting their students to present them with written observations. Sophie Germain, assuming the name of an actual student at the Ećole Polytechnique, one Antoine-August LeBlanc, submitted her observations to Lagrange, who praised them, and learning the true name of the imposter, actually went to her to attest his astonishment in the most flattering terms. Can we even imagine such events occurring today in fact rather than fiction? Perhaps the most astounding aspect is that Germain appears to have educated herself to at least the undergraduate level, capable of submitting written work to Lagrange, one of the foremost researchers in the world, work that was sufficiently notable to make him seek out the author. Unlike other female mathematicians before her, such as Hypatia, Maria Agnesi, and E´milie du Chaˆtelet, who had either professional mentors or formal education, Sophie Germain appears to have climbed to university level unaided and entirely on her own initiative. Germain’s appearance on the Parisian mathematical scene, Libri continues, drew other scholars into conversation with her, and she became a passionate student of number theory with the appearance of Adrien-Marie Legendre’s Theórie des Nombres in 1798. Both Lagrange and Legendre became important personal mentors to Germain, even though she could never attend formal courses of study. After Carl Friedrich Gauss’s Disquisitiones Artithmeticae appeared in 1801, Germain took the additional audacious step in 1804 of writing to him, again as LeBlanc (who in the meantime had died), enclosing some research of her own on number theory, particularly on Fermat’s Last Theorem. Gauss entered into serious mathematical correspondence with ‘‘Monsieur LeBlanc’’, whom he considered to have

‘‘completely mastered’’ his Disquisitiones [4]. In 1807 the true identity of LeBlanc was revealed to Gauss when Germain intervened with a French general to ensure Gauss’s personal safety in Braunschweig during Napoleon’s Jena campaign. Gauss’s response to this surprise metamorphosis of his correspondent was extraordinarily complimentary and encouraging to Germain as a mathematician, and quite in contrast to the attitude of many 19th century scientists and mathematicians about women’s abilities. But how can I describe my astonishment and admiration on seeing my esteemed correspondent Monsieur LeBlanc metamorphosed into this celebrated person, yielding a copy so brilliant it is hard to believe? The taste for the abstract sciences in general and, above all, for the mysteries of numbers, is very rare: this is not surprising, since the charms of this sublime science in all their beauty reveal themselves only to those who have the courage to fathom them. But when a woman, because of her sex, our customs and prejudices, encounters infinitely more obstacles than men, in familiarizing herself with their knotty problems, yet overcomes these fetters and penetrates that which is most hidden, she doubtless has the most noble courage, extraordinary talent, and superior genius. Nothing could prove to me in a more flattering and less equivocal way that the attractions of that science, which have added so much joy to my life, are not chimerical, than the favor with which you have honored it. The scientific notes with which your letters are so richly filled have given me a thousand pleasures. I have studied them with attention and I admire the ease with which you penetrate all branches of arithmetic, and the wisdom with which you generalize and perfect [1, p. 25]. The subsequent arcs of Sophie Germain’s two main mathematical research trajectories, her interactions with other researchers, and with the professional institutions that forced her, as a woman, to remain at or beyond their periphery, are complex. Germain’s development of a mathematical theory explaining the vibration of elastic membranes is told by Lawrence Bucciarelli and Nancy Dworsky in their mathematical biography [1]. And Germain’s efforts to prove Fermat’s Last Theorem, including recent large discoveries in her manuscripts, are told by Andrea Del Centina [2] and Reinhard Laubenbacher and David Pengelley [3, 4]. In brief, the German physicist Ernst Chladni created a sensation in Paris in 1808 with his demonstrations of the intricate vibrational patterns of thin plates, and at the instigation of Napoleon, the Academy of Sciences set a special prize competition to find a mathematical explanation. Germain pursued a theory of vibrations of elastic membranes, and based on her partially correct submissions, the Academy twice extended the competition, finally awarding her the prize in 1816 while still criticizing her solution as incomplete, and did not publish her work [1]. The whole experience was definitely bittersweet for Germain. The Academy then immediately established a new prize, for a proof establishing Pierre de Fermat’s 17th century claim that for each fixed p [ 2, there are no positive natural number solutions to the equation xp + yp = zp. Of course this claim, known as Fermat’s Last Theorem, became one of the greatest unsolved problems in mathematics until its

confirmation by Andrew Wiles at the end of the 20th century. While Sophie Germain never submitted a solution to this new Academy prize competition and never published on Fermat’s Last Theorem, we have long known that she worked on it, from a single footnote in Legendre’s own 1823 memoir published on the topic [3]. Once Fermat had proven his claim for exponent 4, it could be fully confirmed just by substantiating it for odd prime exponents. But when Germain began her work, this had been accomplished only for exponent 3. Legendre’s own publication proved Fermat’s Last Theorem for exponent 5, but he also credited Sophie Germain with the first general result applicable to arbitrary exponents, and this has come to be known as Sophie Germain’s Theorem [3]. It states that for an odd prime exponent p in the Fermat equation, if there exists an auxiliary prime h satisfying two particular congruence conditions on the p-th power residues modulo h, then any solution to the Fermat equation would have to have one of x, y, z divisible by p2. Legendre also credited Germain with verifying the existence of such a h for all p \ 100. This theorem played an important role in work on Fermat’s Last Theorem over the next two centuries. It has long been thought that this one theorem represented Sophie Germain’s entire contribution to the Fermat problem, but very recent study of her surviving manuscripts and letters has demonstrated that, on the contrary, this theorem was merely a small piece of a much larger body of work. Germain pursued nothing less than an ambitious, fullfledged plan of attack on Fermat’s Last Theorem in it entirety, with extensive theoretical techniques, side results, and supporting algorithms. What we have called Sophie Germain’s Theorem was only a small part of her big program, a piece that could be encapsulated and applied separately as an independent theorem, as done in print by Legendre. The much larger scope of her manuscripts was lost, but has now been rediscovered and detailed in [2, 4]. The recent resolution of the Fermat problem, and the discovery of Sophie Germain’s much enlarged accomplishments on the problem, create a captivatingly timely context for Dora Musielak’s book. Sophie’s Diary is delightful to read. Each section leaves one anticipating the next, wondering what will happen, whether it be the fictional Sophie’s next mathematical adventure, or her recounting of and views on an episode in the saga of the French Revolution unfolding outside her door. The writing style is that of a truly curious, sensitive, and articulate young person, and the blur between fact and fiction is excellent, seductively leaving one believing that the fictional Sophie’s writing is the real one’s life. The scope is huge, including four years of the fictional Sophie’s mathematical self-education amidst the events of the French Revolution, ranging over the 1793 riots, the storming of the Bastille, the creation of the Constituent Assembly, the assault on Versailles, the nationalization of church property, the nobility’s loss of titles, the subjugation of clergy to the state, and finally the attempted escape, imprisonment, trial, and execution of King Louis XVI. But it also addresses many broader social and political issues of the day, such as Sophie’s family’s views on her education, and Ó 2010 Springer Science+Business Media, LLC, Volume 32, Number 3, 2010

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her own essentially feminist views on how the rights of women were not being addressed by the Revolution. Despite the motto of Liberte´, E´galite´, Fraternite´, Sophie laments the exclusion of her own aspirations as a woman to attend one of the newly founded institutions of higher education, the Ećole Polytechnique. Sophie’s Diary contains much speculation and questioning on Sophie’s part of both non-mathematical and mathematical natures. The fictional Sophie comes across as someone who constantly challenges both herself and orthodoxy, true to what we know of the real Sophie Germain. In the mathematical realm, the fictional Sophie begins at age 13 with ancient topics such as pi and irrationality, and progresses to challenge herself with problems that teach her about algebra, complex numbers and Euler’s identity, calculus, various infinite series of Euler, analysis, differential equations, Goldbach’s conjecture and quadratic forms of primes in number theory, and Pascal’s triangle and combinatorics. Along the way she teaches herself Latin in order to read books by Euler like Introductio and Institutiones, and Newton’s Principia. The reader is taken on a delightful tour of much mathematics from ancient times right up through the 18th century. Finally, as the diary concludes on Sophie’s 17th birthday, she pronounces herself ready to embark on life as a mathematician, and considers how she intends to engage those at the Academy of Sciences. This final touch is a perfect segue into the life of the real Sophie Germain, who at age 18 really did succeed in obtaining serious attention from Lagrange. The author’s historical appendix focuses on Sophie Germain’s biography in the context of Fermat’s Last Theorem. And she writes, Sophie’s Diary was inspired by Sophie Germain. I wanted to honor Germain and make her known to generations of girls (and others as well), to promote her achievements. Knowing so little about her childhood, I wanted to present a perspective as to how the teenage Sophie must have learned mathematics on her own. Writing Sophie’s Diary became my way of bringing Sophie to life.

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Dora Musielak has admirably achieved this goal. There are some small English and typographical errors and mathematical inaccuracies in the diary, but these would easily be remedied in a new printing. Somewhat more serious are a few historical mathematical misstatements in the appendix and on the back cover, and some confusion of wording regarding the two conditions in the hypotheses of Sophie Germain’s Theorem in relation to Case 1 and Case 2 of Fermat’s Last Theorem. These problems, while disappointing, can also easily be corrected. Altogether Sophie’s Diary is a charming, captivating book to read. It should delight mathematicians, and inspire young people, especially young women, about mathematics.

REFERENCES

[1] Louis Bucciarelli and Nancy Dworsky, Sophie Germain: an essay in the history of the theory of elasticity, D. Reidel, Boston, 1980. [2] Andrea Del Centina, Unpublished manuscripts of Sophie Germain and a revaluation of her work on Fermat’s Last Theorem, Archive for History of Exact Sciences 62 (2008), 349–392. [3] Reinhard C. Laubenbacher and David Pengelley, Mathematical expeditions: chronicles by the explorers, Springer, New York, 1999. [4] Reinhard Laubenbacher and David Pengelley, ‘‘Voici ce que j’ai trouve´:’’ Sophie Germain’s grand plan to prove Fermat’s Last Theorem, Historia Mathematica, to appear; and at http://www. math.nmsu.edu/*davidp. [5] Guglielmo (Guillaume) Libri, Notice sur Mlle Sophie Germain, in Sophie Germain (ed. A-J Lherbette), Conside´rations geńe´rales sur l’e´tat des sciences et des letters, aux diffe´rentes e´poques de leur culture, Imprimerie de Lachevardie`re, Paris, 1833, pp. 11–16, reprinted from Journal des De´bats,18 May, 1832. Department of Mathematical Sciences New Mexico State University Las Cruces, NM 88003 USA e-mail: [email protected]

Mathematicians of the World, Unite! by Guillermo P. Curbera WELLESLEY, MASSACHUSETTS: A.K. PETERS, LTD, 2009, HARDCOVER, XVIII + 326 PP., US$59.00/£42.50, ISBN 978-1 56881-330-1 REVIEWED BY GERALD L. ALEXANDERSON

he author, a mathematician at the University of Seville, organized a comprehensive exhibition of historical materials on the occasion of the 2006 International Congress of Mathematicians (ICM), held in Madrid. His extensive efforts are evident in this book with its carefully researched text and extraordinary collection of 400 illustrations, many of which have not appeared elsewhere, at least in recent times. The text is a joy to read, and the lavish layout is a delight to the eye. Even the cover is a hint of good things to come: a wraparound group picture of the 1954 Congress in Amsterdam on the actual cover and a similar group picture of the 1950 Cambridge Congress on the dust jacket. Usually when these group pictures are reproduced in a book, the images are so small it is impossible to make out much of anything. But these covers are large and the pictures are skillfully reproduced, with detail so clear that one can easily pick out faces of old friends. The literature on the International Congresses of Mathematicians is not extensive—until now, only two books were available (in addition to the proceedings volumes issued after each congress), one published prior to the Berkeley Congress in 1986 [1], the other a history of the International Mathematical Union, which now organizes the congresses [2]. The author covers these international meetings as events, with lots of color pictures, showing not only mathematicians and meeting venues, but also ephemera: posters, invitations and tickets, logos, postage stamps, scenes of social events and excursions—and even the sheet music of a song by Tom Lehrer. In a Foreword, Lennart Carleson describes attending his first Congress, appropriately that of 1962 in Stockholm. ‘‘The congress was a great experience. I was amazed to encounter the richness of our field and how unimportant my own specialty was considered by many people. I made friends from different parts of the world and these contacts have lasted through the years. I saw icons of mathematics whose names I knew from theorems and listened to their lectures. I remember in particular Jacques Hadamard. The organizers of the congress had with great effort managed to get a visa for him for a few days, in spite of the risk for the security of the country to let an 85-year-old communist in. This was my first contact with the problem of how politics interferes with mathematics. Much more on this subject can be found in this book. At the congress I listened to lectures by not only Hadamard but also H. Cartan, K. Go¨del, J. Leray,… and S. S. Chern, to just mention a few. I also

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remember the excitement of the Fields Medals—who would win?—and the discussions afterwards.’’ Carleson captures the pleasure of attending one’s first congress, as well as subsequent ones. If one is not standing on the shoulders of giants, one is perhaps able, for a short time, to rub shoulders with them. Mainly we stand aside in awe—seeing Pontrjagin chatting with someone on a street corner, seeing groups of Fields Medalists past and recent together, catching a glimpse of Sierpin´ski or Bombieri or… and hearing some great talks. This is not a book about mathematics; it is primarily about the community of mathematicians. To be sure, by reading titles of plenary sessions, descriptions of major mathematical announcements, and reports on the work of Fields Medalists, the reader can get some picture of the mathematics of the time, but for details, one really needs to consult the published proceedings. Still, Curbera catches the spirit of the congresses beautifully. Mathematicians, like other people, can be difficult, remote, and unwelcoming individuals, but at these meetings they often appear to be genuinely happy to see each other. Of course, as Lehto makes clear in Reference [2], some of the politics of mathematics can be ugly: difficulties after World War I affecting the organization of postwar congresses in Strasbourg, Toronto, and Bologna, and political conflicts during the Cold War delaying the Warsaw Congress in 1982. In Vancouver and Helsinki there were questions about whether Soviet mathematicians would be allowed to attend and how the delegates were being chosen. Curbera does not dwell on these problems; he maintains a light touch, as the book’s proletarian title suggests. In the captions, in particular, one occasionally finds an unexpected remark, as when he says that the design of the postage stamp showing Jean Bernoulli, issued by Switzerland on the occasion of the 1994 Congress in Zu¨rich, ‘‘could have been prettier.’’ The narrative proceeds largely chronologically, from the international ‘‘congress’’ convened in Chicago on the occasion of the World’s Columbian Exposition in 1893 (planned for 1892 to commemorate Columbus’s discovery of America, but postponed because of construction delays), at which Felix Klein spoke. Papers by others not in attendance, including Hermite, Hilbert, Minkowski, Pincherle, and Pringsheim, were presented but not read by the authors. The first ‘‘official’’ congress was held in Zu¨rich in 1897, with participants from 26 countries (108 countries were represented in Madrid). Rivalries, if not outright political conflict, were evident in Zu¨rich: thanks to the organizers’ strong ties to Go¨ttingen, there was no representation from the University of Berlin. Perhaps the most famous congress was held three years later in Paris at the time of another World’s Fair, the 1900 Exposition Universelle. The main event was Hilbert’s speech, in which he announced the first ten problems from his famous list of 23, setting the mathematical agenda for the 20th century. Following successful congresses in Heidelberg (1904), Rome (1908), and Cambridge (England) (1912), held with little drama, there was a break in the series during World War I. The first Congress after the war was held in Strasbourg (1920), a provocative choice. France had, under the provisions of the Treaty of Versailles, again taken control of 2009 Springer Science+Business Media, LLC, Volume 32, Number 3, 2010

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Alsace, which had been lost to Germany in the FrancoPrussian War of 1871. Invitations to Strasbourg were not issued to mathematicians from the Central Powers (Germany, the AustroHungarian Empire, Bulgaria, and Turkey), and the Congress name was changed from International Congress of Mathematicians to International Congress of Mathematics. There’s a difference. The exclusion of German mathematicians and others continued into the Toronto Congress of 1924 with details too complex to go into here. The Toronto Congress did, however, contribute one innovation of note: the first group picture of the delegates was taken. In Bologna, four years later, the president of the honorary committee was Benito Mussolini! But more important, the Congress took back its former name. As Oswald Veblen explained at the Amsterdam Congress (1954), ‘‘The series of International Congresses are very loosely held together. They are not congresses of mathematics, that highly organized body of knowledge, but of mathematicians, those rather chaotic individuals who create and conserve it.’’ The first Fields medals were awarded at the Oslo congress in 1936. Attendance was disappointing because of the Great Depression and the ominous political events in Europe. In Oslo the next site was announced—New York or a city nearby—but that congress would not be held until 1950. Throughout this chronological account, the author intersperses chapters (‘‘Interludes’’) on: ‘‘Images of the ICM,’’ logos and memorabilia; ‘‘Awards of the ICM’’ (the Fields Medal, the Nevanlinna Prize, and the Gauss Prize); ‘‘Buildings of the ICM,’’ with pictures of the meeting sites from the Richelieu Amphitheatre in Paris (the size of a large classroom) to huge convention halls seating thousands; and ‘‘The Social Life at a Congress.’’ This last aspect of congresses is important and had been from the beginning—congresses are, after all, meetings of mathematicians (in early congresses activities were planned for the ‘‘ladies,’’ while their husbands took care of Congress business). The social events take various forms and just as God in his wisdom, it has been remarked, placed significant bodies of water next to large cities, someone sees to it that ICM participants have cruise options: Zu¨rich-See (3 times!), Oslo Fjord, the Strait of Georgia, the Gulf of Finland, and San Francisco Bay, to name a few. Of course there have been exceptions. In Rome participants had only the fountains at Hadrian’s Villa; in Bologna 400 participants made a long journey to Ravenna to see the Adriatic. This book includes a picture of Hadamard on the sand removing his shoes to go wading, still wearing his hat. On the boring train ride back to Bologna, Hadamard tossed out a challenging problem to those in his compartment. They spent the trip quietly working on it, while Hadamard napped. Nonaquatic events for the Berkeley congress in 1986 included a rodeo and a Western barbecue. The 18-day rail trip from Toronto to Vancouver and back in 1924 still holds the record for excursion length. The author notes wryly that after this trip, John Charles Fields, who had organized the Congress (and for whom the medals are named) suffered a sudden decline in his health. Small wonder! And there has been music. The Cambridge Congress in 1950 set records in that department: a concert by the Busch 66


String Quartet, with recitals by Helen Traubel, the reigning Wagnerian soprano of the day, and the folk singer Richard Dyer-Bennett. (Can it be coincidence that Dyer-Bennett’s brother was a mathematician?) In Beijing there were performances of three Chinese operas; this book provides short versions of the plots, thus demonstrating that Verdi and Wagner did not exhaust the supply of silly opera scenarios. If you enjoy reading about mathematicians, their foibles as well as their passion for their subject, this book has much to offer. In addition to the photographs of meeting sites, you will find lists of participating countries; portraits of many of the plenary speakers, Fields Medalists, and presidents of congresses; group pictures of the participants; pictures of statuary in host cities (Lobachevsky and Chebychev in Moscow, Lie in Oslo); displays of calculating equipment at the Cambridge Cavendish Laboratory, in Zu¨rich, at Harvard (with Grace Hopper), and, in Amsterdam, mechanical calculators trying to keep up with a human calculating prodigy; pictures of the medals—Fields, Nevanlinna, and Gauss, in full color (both obverse and reverse); and title pages of pioneering journals, among others. Some of the pictures are surprising. One shows the Premier of France, Paul Painleve´—a mathematician who had given a plenary lecture on differential equations at the Heidelberg congress—standing with a tall, young man who turns out to be Charles Lindbergh! A picture of a wreath, laid at the base of the wellknown Abel monument in Oslo by the German delegation, shows a prominently placed swastika! The lists of talks at the congresses allow us to track the ebb and flow of subdisciplines. Mathematical physics and mechanics were strongly representated in early congresses, less so later. Today connections with physics are highlighted again with Fields Medals for Witten, Jones, Drinfeld, and Kontsevich. The first female plenary speaker was Emmy Noether (Zu¨rich, 1932). The next, Karen Uhlenbeck, spoke 58 years later (Kyoto, 1990). In 1994 (Zu¨rich) there were two women, Ingrid Daubechies and Marina Ratner, but in Berlin (1998), only one (Dusa McDuff). Progress is not necessarily monotonic. As for trends in meeting sites, the first six congresses, and almost all of them since, have been held in Western Europe. But there have been four in North America (Toronto, Cambridge, Vancouver, Berkeley) and recently two in Asia (Kyoto, Beijing), with Hyderabad scheduled for 2010.

REFERENCES

[1] Donald J. Albers, et al., International Mathematical Congresses: An Illustrated History, 1893-1986, Springer, New York, 1986. [2] Olli Lehto, Mathematicians without Borders: A History of the International Mathematical Union, Springer, New York, 1988.

Department of Mathematics and Computer Science Santa Clara University Santa Clara, CA 95053-0290 USA e-mail: [email protected]

A Person of Interest: A Novel by Susan Choi NEW YORK: VIKING PRESS, 2008, 357 PP., US$24.95, ISBN 978-0-67001846-8

Fermat’s Room (La Habitacioń de Fermat) directed by Luis Piedrahita and Rodrigo Openã BARCELONA, NOTRO FILMS, 2007, DVD, US$19.98, ASIN B0026T

No One You Know by Michelle Richmond NEW YORK: BANTAM BOOKS, 2008, 331 PP., US$15.00, ISBN 978-0-38534014-4

Pythagoras’ Revenge: A Mathematical Mystery by Arturo Sangalli PRINCETON, NEW JERSEY: PRINCETON UNIVERSITY PRESS, 2009, 183 PP., US$24.95, ISBN 978-0-691-04955-7

Pythagorean Crimes by Tefcros Michaelides LAS VEGAS, NEVADA: PARMENIDES PUBLISHING, 2008, 272 PP., US$14.95, ISBN 0-312-29252-X

The Girl Who Played with Fire by Stieg Larsson NEW YORK: ALFRED A. KNOPF, 2009, 503 PP., US$25.95, ISBN 978-0-30726998-0 REVIEWED BY MARY W. GRAY

nspired perhaps by the success of Numb3rs [1], a U.S. TV series that features a mathematician solving crimes for the FBI, recently a number of authors have decided to give their protagonists the profession of mathematics. Numb3rs makes occasional missteps even with a number of mathematicians as advisers, but occasionally imbeds some mathematics in compelling drama. However, it seems that many mathematical mysteries have very little mathematics, and some have very little mystery as well. The

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mathematics that does appear often consists of vague references to the key character’s work on the Riemann Hypothesis or the Goldbach Conjecture (but see The Parrot’s Theorem [2] and Uncle Petros and the Goldbach Conjecture [3] where the conjecture is dealt with in more depth). In post-Wiles settings, these seem to have replaced Fermat’s Last Theorem as the mathematics topic of choice. Infecting many of the books is not so much an absence of technical expertise, but rather a complete lack of feeling for what mathematics, its practitioners, and sometimes even storytelling are all about. One characteristic shared by most mathematicians in mystery fiction is strangeness, or, as the author of 351 Books of Irma Arcuri [4] puts it, ‘‘mathematicians are murky.’’ An episode of Numb3rs even suggested an association between a ‘‘math gene’’ and schizophrenia. Reviewer Alex Kasman [5] conjectures that whereas some authors who write about mathematics and mathematicians do it from love of the subject or to make use of some specific result, others choose to use mathematics as the profession most likely to make a character’s weirdness believable. Professor Lee, the ‘‘person’’ in Susan Choi’s A Person of Interest, is forged from the real life Unabomber mathematician Ted Kaczynski, the accused Los Alamos scientist Wen Hoo Lee, and Steven Hatfill, the scientist first suspected in the anthrax scare, with maybe a touch of John Nash blended in. Lee becomes a leading suspect in the bombing-death of his young colleague Hendley, initially largely because he fails to join public expressions of remorse. Nearing retirement, reclusive, not producing much research, not particularly liked by students, and jealous of the other man’s success, he seems to be the perfect suspect, having isolated himself from any possible network of support; his self-loathing grows and he unravels as suspicion focuses on him. Certainly the Unabomber demonstrated in greater degree some of these same characteristics, but on the other hand Hatfill, harassed by government agencies for some six years with multiple searches of his home, his computer, and his trash, and named a ‘‘person of interest’’ by then U.S. Attorney General John Ashcroft, presented quite a different profile; he fought back. What his motive for distributing anthrax could have been was never satisfactorily explained. In fact the U.S. government settled his suit against it for $5.82 million dollars two months before the attention of investigators turned to another government scientist, Bruce Ivins. Ivins’ motive was conjectured to have been to seek attention for anthrax research and vaccine development to enhance his own prestige. This suspect also was far from reclusive, being active in his church and community, entertaining with juggling and guitar-playing until psychological deterioration led to his suicide. So to be a suspect one need not be a mathematician lacking in people skills, but it helps. Lee does not get the vindication that Hatfill obtained, although eventually he seems to put his life back together to some degree. As the story develops, Lee receives a letter from an old nemesis that suggests what might be behind his torment. Overcoming years of passivity, he sets out to confront the letter writer, his extraordinary journey culminating in a confrontation in a setting resembling the isolated site where the Unabomber was eventually captured. The target 2010 Springer Science+Business Media, LLC, Volume 32, Number 3, 2010

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of Lee’s search, also a mathematician, is the REALLY strange one, very clearly modeled on the Unabomber. Choi’s centering on the aggressive pursuit of Lee by the FBI is broken up by flashbacks to a past when Lee was more interesting if no more likeable. It’s hard to tell who the good guys are here, if there are any, but those of the younger generation who eventually appear are at least more promising. Choi’s father was a mathematics professor (who reportedly went to graduate school with Kaczynski) so perhaps that, together with the all-too-prevalent notion that mathematicians lack the ability to relate to others or sometimes to the world, led her to give that profession to her protagonist, but there is really nothing about mathematics in the book except for passing references, including a somewhat obscure reference to group theory in the denouement. On the whole, the climax is unlikely and unsatisfying, an unfortunate conclusion to a clever portrayal of a man in torment, with an afterword at once both weird and dull. Choi seems to run out of ideas in a rush to finish. What she does capture well is the condition of not very productive nor engaged faculty coming to realize that neither are they successful even by the standards of a second-rate institution nor do their long-held selfimages conform to reality. She leaves unexplored the significance of the fact that the victim of the persecution is foreignborn, although clearly it is a factor in both the public’s and Lee’s own perceptions and actions. One anachronism: it seems unlikely that the secretarial staff of a contemporary math department would be generally unfamiliar with the use of the Internet (or would be called secretarial staff). The personas of strange mathematicians in other mysteries seem even more unlikely than those of the professors in Choi’s tale. In Irene Dische’s Sad Strains of a Gay Waltz [6], Waller, a terminally ill recluse mathematician, fills his life by adopting a young boy and his pianist mother, leading to a bizarre triangle. At the opening of the book, Waller is conversing with long-dead Einstein in a chapel, whereas at the end he consults Einstein for advice. The original € (A foreign feeling) would German title Ein Fremdes Gef ubl be more descriptive of this tale. Waller is said to work on solitons, but his work is described in such a way as to trivialize it. The eponymous mathematician in Tigor [7] runs out on a conference in Trieste, supposedly because his work in the geometry of snowflakes is threatened by chaos theory. Tigor apparently studied the geometry of snowflakes by collecting them. Given this description of his work, it is a blessing that the narrative quickly moves on from his mathematics. Kasman [5] has accused author Jungk of slandering mathematics, to which the author responded: ‘‘as you may have guessed, math has always been my anathema…in school and after…but i thought i could only understand mathematics better by making my hero belong to the very world i had no keys to…no wonder my math seemed odd to you…’’ Tigor says ‘‘My science is founded on astoundingly flimsy assumptions, it puts its trust in groping forward movements, and stumbles over every line of its conclusions,’’ concluding with ‘‘Stay away from mathematics!’’ However, there is a nice bit when he meets up with Cantor’s grandson and another when he indulges in a discussion of the Goldbach Conjec-

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ture. After trying his hand as a stagehand in Paris, he finds his life is more interesting as a teacher of high-school mathematics in the Soviet republic of Armenia and ultimately as a pilgrim to Mount Ararat. In neither this nor Dische’s story do we hear much of the mathematics of the protagonists, although we are assured of their wide acclaim in the profession. In Amos Oz’s Touch the Water, Touch the Wind [8], a Holocaust survivor, said to have been on the verge of a great discovery, undertakes via levitation a surrealistic odyssey involving everything from growing a tail to breaking his isolation in order to take over the finances of a kibbutz and rescue it from ruin, somehow proving something important about infinity in the process, while all along understanding the profound connection between mathematics and music. In another work Oz has said that he is devoted to imagining the other as a moral imperative. Just how the fertile imagination of this tale relates to the protagonist’s life as a mathematician—if it does—isn’t clear. In Life after Genius [9], the burned-out prodigy Mead retreats to his father’s profession of funeral director after undergoing adventures at the Institute for Advanced Study and fleeing from misplaced praise for his rumored resolution of the Riemann Hypothesis. Although the blurb for the book describes it as a combination academic thriller and coming-of-age story, the only mystery is why Mead left his promising career, and it is not clear that he ever ‘‘comes of age.’’ A warning for the squeamish: there is more than most want to know about the family business. The only characteristic consistently exhibited by the failed prodigy, which is thought by many to be typical of mathematics, is perhaps the tendency to overanalyze everything. The time in which the story is set is not specified although a description of the Cray I places it in the early 1970s. In The Book of Getting Even [10], Gabriel, a budding astronomer with a picture of Gauss on his wall, engages in a series of bizarre behaviors in pursuit of the 17-gon, infinitesimal calculus and other forays into mathematics and physics. He describes mathematics as ‘‘calculability, sweet detachment from the corporeal universe.’’ Although there is little of mathematics, it is a sensitive story of three young people. The author’s reason for including mathematics here may be to indicate an elegant smartness in his characters. Unfortunately their bad choices lead to sadness and defeat. As might be expected from the title, Orpheus Lost [11] brings together mathematics and music, not to mention a substantial infusion of espionage. While playing Gluck’s Che faro senza Euridice in a Boston subway, the musician, who comes to be known as Orpheus, is encountered by Leela, whose research involves the mathematics of music. But Orpheus soon turns into Euridice as he undergoes rendition to Iraq and torture. Mathematics is not central to this tale of terrorism, but this is still another example of authors who are not mathematicians cavalierly remarking upon the lack of social skills and disconnect from reality claimed to be characteristic of mathematicians. Leela’s former dissertation advisor is given to remarks such as ‘‘We understand numbers, not people.’’ The advisor also feels compelled to remark about the scarcity of women mathematicians of

Leela’s caliber. All in all, Leela and her Orpheus/Euridice and the other characters who emerge from their past are interestingly characterized, but not so their mathematics. Also in the genre of ‘‘Maybe not all mathematicians are strange, but it helps if you are’’ is a recent film, Fermat’s Room (La Habitaci on de Fermat). The improbable but absorbing plot keeps the viewer in suspense throughout. It centers not around Fermat’s Last Theorem, as one might think from the title, given the public fascination with the result, but the Goldbach Conjecture (again!). Four people apparently receive invitations to a mathematical evening. What their motivation might be for trekking far out to a remote lake, across which they must row to a mysterious structure, is not explained. Perhaps it is not required, for after all if they are interested in mathematics they must be at least a bit strange. Or perhaps more positively, mathematicians are all imagined to be imbued with a sense of adventure. Name tags are provided to identify them as Hilbert, Galois, Pascal, and Oliva. The first reaction of mathematician viewers is likely to be ‘‘Why Oliva?’’ Why not a woman mathematician (and why the sexist technique of using only a first name)? Non-mathematicians may be more likely to assume that Oliva Sabuco was a mathematician or perhaps that there are no women mathematicians. In fact, Oliva Sabuco was a 16th century Spaniard credited with being the first to understand how the brain controls the body (but that doesn’t explain why the first name). Why the four characters are chosen does eventually become clear. The ostensible host locks the four in a room with the message that they will be presented with a series of mathematical problems. If each is not solved within one minute, the room will shrink, ultimately crushing the four of them. Things are, of course, not as they seem. The putative Galois has not actually proved the Goldbach Conjecture (as the mathematicians in the audience would know) although it has been widely rumored that he has. Moreover, the host who disappears is not the imaginary Fermat, but the characters in the film are all connected—although one is merely connected with the machinery that is shrinking the room. The math problems are not actually mathematical problems, but rather are puzzles or riddles such as the one concerning the transportation of a wolf, a sheep, and a cabbage across a river. One reviewer [12] has remarked: ‘‘Mathematics is scary all by itself even without the mysterious parties, angry strangers, and a freaky shrunken room.’’ But for those for whom mathematics is NOT scary, it is a wonderful film. In Michelle Richmond’s No One You Know the mathematician Lila is lost before the story begins—just as she too was about to prove the Goldbach Conjecture. The mystery revolves around how and why she died, explored by her sister Ellie, who is as obsessed with coffee as Lila was with mathematics, recalling the famous Erd} os claim that a mathematician is a device for turning coffee into theorems. Ellie remarks that ‘‘like the kitchens of famous restaurants, the bowels of mining shafts, and the most prestigious mathematics departments, the coffee industry was [at the start of her career] a man’s world.’’ Has the character of any of these changed today? One of Ellie’s techniques for feeling close to her sister is visiting the graves of mathematicians—Pascal at Saint Etienne-du-Mont in Paris, Gauss at the Albanifriedhof in

G} ottingen, Leibniz in Hannover (apparently she missed Germain in Pe`re Lachaise). Relating that her mathematical prodigy sister experienced the world through her intellect whereas as a coffee buyer she experiences the world through her senses, Ellie says that ‘‘writing about mathematics is a way of tackling my demons’’ but asserts that as a coffee buyer she wanted the ‘‘flavor’’ of math to be part of the story. This she has achieved remarkably well. Much is made of the claim that mathematicians are held to a higher standard of proof than anyone else—that in mathematics one must be absolute, a view quite different from that of Tigor, whereas in science there is always some doubt. It is Ellie’s goal to resolve the doubts about the fate of her sister. Once again we hear not only of the Goldbach Conjecture, but of Fibonacci, Hypatia, Agnesi, and Germain, including the story of Agnesi’s proving theorems while sleepwalking. Lila’s genius is said to ‘‘lay in her fierce imagination, her ability to envision things that she had not yet been taught,’’ a characterization often applied, at least implicitly, to promising young students. On the first page of Lila’s notebook is Hardy’s: ‘‘A mathematical proof should resemble a simple and clear-cut constellation, not a scattered cluster in the Milky Way.’’ Richmond’s exposition follows this prescription. In contrast to most authors’ description of mathematicians, Richmond claims that mathematicians as a group are extremely interconnected, constantly sharing information. But whether or not it helps to be crazy or withdrawn in order to be a mathematician, most people probably believe that it helps to be obsessed; in this case it requires obsession to solve the mystery. It’s a pageturner, the characters are exceptionally well developed, and the story is compellingly related; it is likely to be as attractive to coffee lovers as to mathematicians. Gifted [13] is based on the real life story of a young woman who entered Oxford to study mathematics at the age of thirteen, but who left early without fulfilling her promise. So also with the young woman in Gifted, who rebels against the strict control of her Asian-Anglo father and abandons mathematics. Here and in Life after Genius, No One You Know, and Sad Strains of a Gay Waltz there is another familiar stereotype—that mathematics is the domain of the young. As in The Book of Getting Even, amicable numbers make an appearance, here accompanied by an introduction to Mersenne primes and the prime number theorem. The unhappy fate of mathematical prodigies is clearly a recurring theme in literature and sometimes in real life. Mathematicians seem to do better with mathematical mysteries than do others and better than they themselves do when venturing far from their discipline. Perhaps they should stick with their own specialty if The Book of Murder [14], the disappointing second book of Guillermo Martıńez, the mathematician author of The Oxford Murders [15], is any example. On the other hand, The Book of Murder was well received by many reviewers, with the U.K.’s Independent on Sunday claiming that the mathematician author’s ‘‘creativity and the mastery of logic necessary to that stern discipline are both evident in this brilliant crime thriller.’’ Creativity is apparent, but logic is woefully missing, as are both mathematics and effective storytelling. Not often does a mathematical mystery make the bestseller lists as has The Girl Who Played with Fire. The central 2010 Springer Science+Business Media, LLC, Volume 32, Number 3, 2010

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character is the brilliant, but disturbed, eponymous young hacker of The Girl with the Dragon Tattoo [16], the first of Stieg Larsson’s posthumously published trilogy. Lisbeth Salander has now discovered mathematics, compulsively reading an imaginary book, Dimensions of Mathematics, by the equally imaginary Harvard don L.C. Parnault. In Kasman’s categorization of authors’ employment of mathematics as a character’s profession, one would have to classify Larson’s use both as a means to demonstrate extraordinary intelligence and to make bizarre behavior seem more believable. Lisbeth has an epiphany that she has discovered Fermat’s actual proof. She realizes that the mathematics used by Wiles could not have been known by Fermat, but the author spoils the illusion a bit by having her dismiss Wiles’ proof as having been found by ten years’ work with ‘‘the world’s most advanced computer programme’’! But never mind—if not much of a mathematician, Larsson was a terrific writer, a genius at creating unforgettable characters and keeping the reader enthralled in their escapades, even if he relies a bit too much on coincidence. One might wonder whether Lisbeth’s revelation that she has discovered the proof is like Hardy sending the legendary postcard to Bohr asserting that he had solved the Riemann Hypothesis to serve as insurance that he would survive a turbulent voyage. But after all, there is another book to come. Sadly Larsson died just after completing the trilogy; it would be wonderful if we could expect many more adventures of Salander and the investigator journalist Mikael Blomkvist with whom she is teamed. In their encounters in The Girl with the Dragon Tattoo the ‘‘bad guys’’ were corrupt and criminal businessmen whose violence was gruesome but limited. In the second book not only have Lisbeth’s intelligence and bizarre behavior escalated but so has the viciousness of the crimes, here the trafficking of women from Eastern Europe to Sweden accompanied by extensive torture and homicide. The story shifts from the prologue featuring unspeakable cruelty to a young girl to Lisbeth’s idyllic retreat in Grenada. Why Grenada? At the time the prime minister of the country was a statistician, but he doesn’t appear in the story. Rather Lisbeth is engrossed in Parnault’s book, advancing through Archimedes, Newton, Martin Gardner, and a dozen other classical mathematicians ‘‘with unmitigated pleasure.’’ That Gardner is found in this company is perhaps explained by Salander’s fascination and skill with puzzles, of which she considers Fermat’s Last Theorem to be just an example, one she can tackle with confidence. She describes math as ‘‘actually a logical puzzle with endless variations.’’ Her version of the certainty expected in mathematics, not unlike that of Lila in No One You Know, is ‘‘The mathematician must be able to stand on a podium and say the words ‘This is so because …’’’ Although she can live more than comfortably wherever she chooses as a result of complex electronic financial manipulations, she is soon back in Sweden, enmeshed in murders and mayhem. Blomkvist, whom she rescued in the previous volume, undertakes an odyssey to return the favor. In spite of her lack of what one would consider normal emotional development and social skills, at the distance of cyberspace they edge back together. Aside from the FLT 70


obsession and a brief mention of perfect numbers, little is seen of mathematics except for elementary equations that introduce each section of the book. Perhaps the third volume will reveal other mathematical exploits of the heroine and be even better than the first two, which are so artfully crafted with a cast of fascinatingly-evoked characters. One of the periodic revivals of the controversy about whether Pythagoras was a real person may have inspired two mysteries that invoke his image: Pythagoras’ Revenge and Pythagorean Crimes. The mathematical settings of both appear, at least in part, to be motivated by a love for the subject. However, the first of these contains little mathematics in spite of the misleading subtitle: A Mathematical Mystery. It poses the questions: What if there exists a longlost scroll written by Pythagoras, and where might it be? Jule, an American mathematician commissioned by a neoPythagorean cult, and an Oxford historian intrigued by what he has seen in an Arabic text being hawked by a representative of the Franciscan order, seek the rumored scroll. The tale seems modeled on Dan Brown’s success with The Da Vinci Code, but author Arturo Sangalli, a mathematician by trade, lacks the vivid imagination and flare for excitement and engagement that attracted the Code’s fanatic fans and detractors. He sees fit to define prime numbers and explain who Euclid was while writing vaguely about contrasting views of mathematics and reality, but the only ‘‘mathematics’’ discussed in detail is the famous ‘‘15’’ permutation puzzle, used as an entrance exam for Jule’s employment in the search. References to randomness, chaos, and string theory are thrown in from time to time apparently to appear contemporary (the setting seems to be at the end of the 20th century). There is the suggestion that modern mathematics and the Pythagorean belief that ‘‘All is number’’ clash, but how or why is not made clear. Nor why the scroll, if found, would change the world or what would constitute Pythagoras’ revenge is not even hinted at. Certainly such an artifact would be extremely valuable—think of the Archimedes Palimpsest—but hardly earthshaking. Another search is also underway; the neo-Pythagorean cult known as The Beacon is actually seeking the living reincarnation of the Master! There is a large cast of characters in the story, ranging from the pair mentioned above to the reincarnated Pythagoras or maybe Anti-Pythagoras or maybe both, to the young woman who opens and closes the story. A biologist who studied salmon reproduction, she arrives at Oxford seeking help for the Royal Ontario Museum of Science: ‘‘We are counting on Pythagoras to sell the museum to the general public!’’ So too with Sangalli and his book! Saving the best for last, we arrive at Pythagorean Crimes. For sheer enjoyment it tops the list for those who like their mysteries lively and fast-moving, sprinkled with portrayals of mathematics and mathematicians, real and imaginary, with whom they might like to identify and whose presence relieves the guilt that might otherwise be engendered by spending time on reading mysteries. The question asked is: Can the solution to a mathematics problem inspire a passion so intense and perilous as to drive someone to murder? The strong dose of 20th-century Greek history in this book may be off-putting to some, but why not learn something along with the fun, especially since a particularly

informative map of 1913 Europe is included? The source of the first of the Pythagorean crimes is the largely discredited story of the execution of Hippasus to prevent the anticipated destruction of the number system—and thus the world of the Pythagoreans—by what we now know as the discovery of irrationality. Michael Igerinos, the narrator, at the time a student at G} ottingen, finds himself at the 1900 Paris International Congress of Mathematicians where tension builds as Hilbert is about to speak. In the audience are his rival Peano, Hadamard, de la Valleé Poussin, Minkowski, Jordan, Russell, and Frege, as well as a fellow Greek, Stefanos Kandartzis, currently studying in Paris. The author-mathematician Tefcros Michaelides is especially skilled at putting Hilbert’s lecture in the context of Poincare´’s lecture at the 1896 ICM and the debate on the limits of knowledge surrounding Emil du BoisReymond’s motto ‘‘ignoramus et ignorabimus’’ (we do not know and we will not know) and creating for the reader the politics of mathematics and of the broader society. The role of the motto in the future of mathematical research could be said to reflect the original Pythagorean crime, and its significance figures prominently in this story. The author, by attention to details both mathematical and personal, conveys well what he calls the ‘‘magical experience’’ of Hilbert’s lecture. But beyond that, the mathematics ranges from Omar Khayam’s work with cubics to the Tartaglia-Cardano dispute, the ill treatment by Cauchy of Abel and Galois, and Newton’s ingratitude to Halley and Barrow. Because the book is so filled with mathematics from Gauss and Galois to G} odel, one might wonder whether the goal is to teach mathematics in an enjoyable setting or to entertain mathematicians with a mystery story filled with familiar characters. Whether or not the first is possible, the second is admirably achieved and anyone not very accomplished as a polymath is certain to learn from the extensive glossary if nothing else. The bond that the two Greek students established at the ICM and after evenings spent with the artists of Montmartre, including Picasso (then known as Pablo Ruiz) is reestablished years later back in Greece where Stefanos, after having obtained a Ph.D. in Paris, secured only a position in a provincial high-school, largely because of Greek politics. Having taken over the prosperous family business, Michael has still maintained an amateur’s interest in mathematics; they talk of mathematics over a weekly session of chess. The mystery opens with the discovery in 1929 of Stefanos’ body and ends with the solution of the mystery several years later. Not all of the Paris adventures are strictly historically accurate—the date of Picasso’s appearance in Paris is off a few months (revealed in the postscript) and absent Twitter, actual accounts of the ICM do not provide the

on-the-scene reporting that could substantiate some of the action described. But what fun to imagine Hilbert at Moulin Rouge or Picasso discussing tiling problems with the two young Greeks! Mathematicians will wish they were there— both at the ICM to see and hear first-hand Hilbert’s sensational presentation and to join in the bustling social scene. Without spoiling the story, it can be said that G} odel’s theorem plays a central role in the solution and leads to the book’s title, for Stefanos’ murder too was a crime committed to suppress knowledge. That the context might be as unbelievable as the original Pythagorean crime doesn’t really matter.

REFERENCES

[1] N. Falacci and C. Heuton, Numb3rs, produced by Scott Free Productions, CBS Television, 2004. [2] D. Guedj, The Parrot’s Theorem, New York: Thomas Dunne, 2001. [3] A. Doxiadis, Uncle Petros and the Goldbach Conjecture, New York: Bloomsbury, 2000. [4] D. Bajo, The 351 Books of Irma Arcuri, New York: Penguin Books, 2008. [5] A. Kasman, http://kasmana.people.cofc.edu/MATHFICT/mfview. php?callnumber=mf449. [6] I. Dische, Sad Strains of a Gay Waltz, New York: Metropolitan Books, 1993. [7] P.S. Jungk, Tigor, New York: Handsel Books, 2003. [8] A. Oz, Touch the Water, Touch the Wind, San Diego: Harvest Books, 1974. [9] M.A. Jacoby, Life after Genius, New York: Grand Central Publishing, 2008. [10] B. Taylor, The Book of Getting Even, Hanover: Zoland Books, 2008. [11] J.T. Hospital, Orpheus Lost: A Novel, New York: W.W. Norton & Co., 2007. [12] S. Weinberg, ‘‘Review of Fermat’s Room,’’ http://www. cinematical.com/2008/04/30/tribeca-review-fermat-room/ . [13] N. Lalwani, Gifted, New York: Random House, 2007. [14] G. Martıńez, The Book of Murder, London: Abacus Books, 2008. [15] G. Martıńez, The Oxford Murders, London: Abacus Books, 2006. [16] S. Larsson, The Girl with the Dragon Tattoo, New York: Vintage Books, 2009. Mathematics and Statistics American University 4400 Massachusetts Avenue NW Washington, DC 20016-8050 USA e-mail: [email protected]


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Stamp Corner

Robin Wilson

Recent Mathematical Stamps: 2005, Chaos and Fractals ROBIN WILSON n November 2005, as part of a series in science and technology, the Macau Post Office issued a set of stamps featuring chaos and fractals. The topics featured in the series are: Hilbert’s space-filling curve, a binary fractal tree, the Sierpinski triangle with fractal dimension log 3/log 2, Michael Barnsley’s ‘chaos game’, the von Koch curve with fractal dimension log 4/log 3, and the Cantor set. There was also a souvenir sheet featuring the recurrence 2 relation zn+1 = zn + c, a stamp depicting the Julia set, and a picture of the Mandelbrot set. A special first-day cancellation postmark depicted a version of the Hilbert curve.

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ä

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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