No title

Letter to the Editors

Letter to the Editor SIOBHAN ROBERTS The Mathematical Intelligencer encourages comments about the material in this issue. Letters to the editor should be sent to either of the editors-in-chief, Chandler Davis or Marjorie Senechal.

o the Editors: As a sequel of sorts to my first mathematical biography, on the geometer Donald Coxeter (King of Infinite Space), I have taken up the Boswellian job of writing a biography of John Horton Conway (Authorized! Or, at least, Semi-authorized!). To that end, I am in search of all manner of Conway stories, anecdotes, lore, rumors, facts, fictions, and defamations which anyone in the mathematical community (and beyond) would be willing to share — for attribution, or not for attribution. Information from all eras is welcome, with a special interest in the period from 1986 onward when Conway landed at Princeton. The biography is to be published by Bloomsbury/ Walker circa 2013. Final deadline for Conway data: March 2012, though ASAP is preferable.

T

School of Mathematics Institute for Advanced Study Princeton, NJ 08540 USA e-mail: [email protected]

Ó 2011 Springer Science+Business Media, LLC, Volume 33, Number 3, 2011

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DOI 10.1007/s00283-011-9236-1

Note

P R O P O S I T I O N 1 If j + l B j, then j + l = j.

Cantor-Bernstein’s Theorem in a Semiring

P R O O F . If j + l B j, then l  @0  j, by axiom (2). This means that, for some k, l  @0 þ k ¼ j. We then have j þ l ¼ l  @0 þ l þ k ¼ l  ð@0 þ 1Þ þ k ¼ l  @0 þ k ¼j

MARCEL CRABBE´ onsider a commutative semiring with sum +, product , the identity 1 for , and the identity 0 for +, which is absorbing for . Suppose moreover that the semiring contains the element @0, and, to the usual axioms for commutative semirings, we add:

C

@0 ¼ @0 þ 1

ð1Þ

if j þ l  j; then l  @0  j

ð2Þ

Here the relation B is defined by: j B l if and only if j + m = l, for some m. This relation is clearly transitive and reflexive. Without being committed to implementation of cardinal numbers in a specific set theory, it is easily seen that all these axioms are satisfied in the usual interpretation, even without assuming the axiom of choice: the sum and the product of cardinal numbers are explained through union of disjoint sets and cartesian product, respectively; 0 is the cardinal of the empty set, 1 is the cardinal of a singleton, and @0 the cardinal of the set of natural numbers. I observe that according to this interpretation, j B l means that there is an injective function of a set of cardinal j into a set of cardinal l. Finally, to see that axiom (2) is verified in this interpretation, let K, M be two disjoint sets of cardinal j and l, respectively. Let also f be an injective mapping from K [ M into K. Then to each x in M and natural number n, we associate the result of applying the (n + 1)-fold iterate of f to x:

axiom ð1Þ:

Cantor-Bernstein’s Theorem states that B is a partial order: if j  l and l  j; then j ¼ l:

P R O O F . If l B j, there exists k such that l + k = j. Then, if j B l, we have l + k B l and, by Proposition 1, l + k = l, i.e., j = l. REMARKS 1. Axioms (1) and (2) are equivalent to the statement: j þ l ¼ j if and only if

l  @0  j

which generalizes the well-known feature of Dedekindinfinite cardinals, namely, jþ1¼j

if and only if

@0  j

2. @0 is the unique element satisfying (3). 3. The axioms allow one to derive directly the properties @0 + @0 = @0 and @0  @0 ¼ @0 . Indeed, as @0 + 1 + 1 = @0, we have ð1 þ 1Þ  @0  @0 and @0 + @0 B @0. Hence, by Proposition 1, @0 + @0 = @0. It follows that @0  @0  @0 : But @0  @0 ¼ ð@0 þ 1Þ  @0 ¼ @0  @0 þ @0 : Therefore, @0  @0  @0 and, by CantorBernstein’s Theorem, @0  @0 ¼ @0 : 4. Remark 1 and the fact that @0  @0 ¼ @0 immediately entail jþl¼j

if and only if

j þ l  @0 ¼ j

From this we get jþj¼j

That is, the function g defined by the recursive equations gðx; 0Þ ¼ f ðxÞ gðx; n þ 1Þ ¼ f ðgðx; nÞÞ is an injective mapping of M  N into K. 80

THE MATHEMATICAL INTELLIGENCER Ó 2011 Springer Science+Business Media, LLC

DOI 10.1007/s00283-011-9242-3

ð3Þ

if and only if

Institut Supe´rieur de Philosophie Universite´ Catholique de Louvain 1348 Louvain-La-Neuve Belgium e-mail: [email protected]

@0  j ¼ j

Note

A Simple2 Proof that f(2) = p6 MICHAEL D. HIRSCHHORN

I shall show that for 0 \ a \ 1,  2 X  1 2 a a4nþ2 1 pffiffiffiffiffiffiffiffiffiffiffiffiffi \ 2 tan a : 2 \2 tan 2 ð2n þ 1Þ 1þ 1a n0

It follows that Z Z

Z Z 1 1 dA\ dA 2 2 2 2 Rb 1  x y Sa 1  x y Z Z 1 \ dA: 1  x 2 y2 Ra

ð1Þ If we let a? 1, we find X

1 p2 ; 2 ¼ 8 n  0 ð2n þ 1Þ

Now, Z Z

1 dA ¼ 2 2 Sa 1  x y

and hence fð2Þ ¼

X 1 X 1 X 1 4 p2 p2  ¼ : ¼ ¼ n2 n  0 22n n  0 ð2n þ 1Þ2 3 8 6 n1

Let 0 \ a \ 1 and



¼

a

Z

0

a

0

X

x 2n y 2n dydx

n0

X

a4nþ2 2: n  0 ð2n þ 1Þ

¼ Also, Z Z

Sa ¼ fðx; yÞ : 0  x  a; 0  y  ag; Ra

Z

 uv uþv ;y ¼ ; u  v  u; 0  u  a ðx; yÞ : x ¼ 1  uv 1 þ uv

1 dA ¼ 2 2 Ra 1  x y

Z 0

a

Z

u

u

j Jj dvdu 1  x2 y2

where

and

J ¼ det

Rb

  uv uþv ;y ¼ ; u  v  u; 0  u  b ¼ ðx; yÞ : x ¼ 1  uv 1 þ uv

¼

ox ou oy ou

ox ov oy ov

! ¼ det

1v2 ð1uvÞ2 1v2 ð1þuvÞ2

2

1u  ð1uvÞ 2

!

1u2 ð1þuvÞ2

2ð1  u2 Þð1  v2 Þ ð1  u2 v2 Þ2

and where a pffiffiffiffiffiffiffiffiffiffiffiffiffi : b¼ 1 þ 1  a2 Then Rb  Sa  Ra as shown in the diagram.

ð1  u2 v2 Þ2  ðu2  v2 Þ2 ð1  u2 v2 Þ2 ð1  u2 þ v 2  u2 v 2 Þð1 þ u2  v 2  u2 v2 Þ ¼ ð1  u2 v 2 Þ2 ð1  u2 Þð1 þ v 2 Þð1 þ u2 Þð1  v2 Þ ¼ : ð1  u2 v2 Þ2

1  x2 y2 ¼

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81

DOI 10.1007/s00283-011-9217-4

It follows that Z Z

1 dA ¼ 1  x2 y2 Ra

Z

a

0

Z

Z

u

2 dvdu 2 Þð1 þ v 2 Þ ð1 þ u u

a

4 tan1 u du 1 þ u2 0 h  2 ia ¼ 2 tan1 u 0  2 ¼ 2 tan1 a : ¼

Apostol [1]. Note that I have managed to cut out the singularity of the integrand at (1,1). ACKNOWLEDGEMENT

I would like to acknowledge the contribution of my colleague J. Steele, who prepared the diagram.

REFERENCES

Similarly, Z Z

 2 1 dA ¼ 2 tan1 b 2 y2 1  x Rb  2 a 1 pffiffiffiffiffiffiffiffiffiffiffiffiffi ; ¼ 2 tan 1 þ 1  a2

and (1) is proved. Remarks. This proof was derived from that of Huylebrouck [2], which, in turn, bears some relationship to that of

82

THE MATHEMATICAL INTELLIGENCER

[1] T. M. Apostol, A proof that Euler missed: evaluating f(2) the easy way, Math. Intelligencer 5 (1983), No. 3, 59–60. [2] D. Huylebrouck, Similarities in irrationality proofs for p; ln2; f(2) and f(3), Amer. Math. Monthly 108 (2001), 222–231. School of Mathematics and Statistics UNSW Sydney, NSW 2052 Australia e-mail: [email protected]

Mathematica with thanks to Lisa Martin Laura Eleanor Holloway

Her voice is, first, a tsunami of Long Island sound: pitch and timbre loud and sharp as roiling debris, the unyielding gift of tongues, a fierce rushing. Translation is a furious shifting of vowels, violent swells hauling r’s out to sea, scattering them to the depths, and then accarezze´vole: the vowels are still shifted, the r’s still scattered, but they have become reverent, a hymn. Who was once priestess is mathematician. This is no Eleusis, no Qabalah, no illumination shrouded behind vague predictions and jumbled quatrains, curtains that won’t be pushed aside. She scrawls the mother tongue across the wall, cosmic ciphers of serpentine integrals and ancient alphabets, cryptic as cuneiform, beautiful as Lascaux. Graphite-black fingers smudge the edges of the cosmos, the twists and bends of space, light, darkness, depth – and everything is more astonishing than could have been imagined.

Washington Crossing PA 18977 USA e-mail: [email protected]

Ó 2011 Springer Science+Business Media, LLC, Volume 33, Number 3, 2011

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DOI 10.1007/s00283-011-9230-7

The Whirling Kites of Isfahan: Geometric Variations on a Theme PETER R. CROMWELL

AND

ELISABETTA BELTRAMI

Introduction

I

n medieval times the city of Isfahan was a major centre of culture, trade and scholarship. It became the capital of Persia in the Safavid era (16–17th centuries) when the creation of Islamic geometric ornament was at its height. Many of the most complex and intricate designs we know adorn her buildings, including multi-level designs in which patterns of different scales are combined to complement and enrich each other. In this article we study five 2-level designs from Isfahan built around a common motif. They illustrate a variety of techniques and the analysis exposes some of the ingenuity and subtle deceptions needed to reconcile incompatible geometries and symmetries, and produce satisfying works of art.

Theme Kites are a characteristic design element in Islamic geometric art. They can be arranged as motifs in their own right or used to provide a structural framework for other elements. Figure 1 shows two patterns created by arranging kites with squares. Part (a) shows a chiral arrangement of four kites chasing around a central square in a finite composition. For want of a name we shall refer to this as the Whirling Kites pattern. We shall also say that the pattern with this orientation is the clockwise variant, and that its mirror-image is counterclockwise. Part (b) shows a repeating pattern that can be extended to fill the plane. It contains the Whirling Kites pattern in both its mirror-image forms. There are three canons of Islamic ornament: calligraphy, arabesque and geometric. All have been applied to the Whirling Kites figure as a secondary form of decoration. In 84

THE MATHEMATICAL INTELLIGENCER  2011 Springer Science+Business Media, LLC

DOI 10.1007/s00283-011-9225-4

Figure 2(a) the compartments are decorated with stylised Kufic calligraphy; the design is taken from a small tiled panel in the al-Hakim Mosque (Masjid Hakim), Isfahan; photograph IRA 1017 in Wade’s collection [17] shows the original. The website [14] is a useful resource on Kufic calligraphy and gives translations of many inscriptions. Figure 2(b) shows an arabesque design carved in relief on a wooden door panel in the Great Mosque of Uqba in Kairouan, Tunisia. Another floral example from the TillaKari Madrasa in Samarkand, Uzbekistan, can be seen in photograph TRA 0732 in [17] or in [15, p. 236]. It is of silver gilded plasterwork and has simple floral trails along the bands. Two much larger and more elaborate floral examples are placed on either side of one of the great iwans in the Imam Mosque (Masjid-i Imam) in Isfahan, formerly known as the Royal Mosque (Masjid-i Shah). They form a mirror-image pair of Whirling Kites motifs and are executed in painted polychrome tiles. Photograph IRA 0225 in [17] and [15, p. 260] show an overall view. We shall see geometric examples later. While it is the finite figure in Figure 1(a) that is the focus of this article, we shall also cite a few examples of the repeating pattern of Figure 1(b). Wade’s archive contains photographs of carved stone reliefs from the Fort at Agra (IND 0404 and IND 0407), and a latticework screen in the Maharajah’s Palace in Jaipur (IND 1019). Examples in wooden door panels and brickwork from the Khan Mosque, Isfahan, can be seen in [1]. As these examples show, the Whirling Kites figure is widespread in the Islamic world. Besides being a motif in its own right, it also provides a useful device to organise a

The Geometry of the Whirling Kites Figure

(a)

(b)

Figure 1. Finite and unbounded Whirling Kites patterns.

(a)

(b)

Figure 2. Examples of simple decoration applied to Whirling Kites patterns. (a) Kufic calligraphy. (b) Floral arabesque.

larger composition and, consequently, a range of styles and techniques have been applied to build complex designs on this simple form. Figure 3 shows the west iwan of the Friday Mosque (Masjid-i Jami), Isfahan. An iwan is an open, high, vaulted porch that provides a large fac¸ade for decoration. This example is of interest as it has five Whirling Kites panels of three different designs: there are two in each of the tall narrow panels that run the full height of the front face either side of the arch, and another on the north side of the inner wall. We shall examine the constructions of these designs plus two others.

A kite is a convex quadrilateral having two pairs of adjacent equal-length sides. We shall assume that a kite is not equilateral so that it has two short sides of length s and two long sides of length t. In all the examples used here, the two angles where sides of different lengths meet are right angles. If h is the acute angle between the two long sides then the obtuse angle between the two short sides is 180 - h. Note h ¼ 2 tan1 ðs=tÞ: The geometry of Whirling Kites patterns is straightforward. Call the four lines making the outer square the frame and the four lines bounding the inner square and radiating from it the rotor. Let x be the length of a side of the frame and y be the length of a side of the small square in the rotor. Then x = t + s and y = t - s. In fact, any pair of x, y, s and t determine the other two. If we ignore the scale, the whole figure is determined by h. Figure 4 shows one way to lay out a Whirling Kites pattern. First take a square ABCD with side length s + t. Mark each side with a point that divides it into segments of lengths s and t so that the long and short segments alternate around the square. In the figure, two such points are marked E and F. Scribe a circular arc centred at E of radius EA, and another centred at F of radius FA. The two arcs intersect at G, and AEGF is the required kite. The Whirling Kites figure is simple to construct, but this property is not sufficient to explain its origin as an ornamental motif. It is possible that mathematical diagrams provided the inspiration. The 10th century Persian mathematician and astronomer Abu’l Wafa wrote On the Geometric Constructions Necessary for the Artisan, which includes references to meetings between geometers and craftsmen at which theoretical constructions were presented and practical applications discussed [13]. In Chapter 10 cut-and-paste arguments are used to construct squares of given area. For example, to construct a square of area 5, place two unit squares so they share an edge and cut the resulting rectangle along a diagonal; two sets of these pieces plus another unit square can be arranged to form a square of area 5—Figure 5(a). Removing the dashed segments produces a template for the periodic pattern in Figure 1(b); the template is repeated by reflection in the sides of the bounding square.

AUTHORS

......................................................................................................................................................... graduated from Warwick, and completed his Ph.D. at Liverpool working in knot theory, is interested in anything geometric, and has written books on polyhedra and knot theory.

graduated from Milan, and studied for her Ph.D. at Pisa and Liverpool, also working in knot theory. She has taught at Trinity College Dublin and Liverpool. She enjoys walking and cooking.

PETER R. CROMWELL

ELISABETTA BELTRAMI

Pure Mathematics Division Mathematical Sciences Building University of Liverpool, Peach Street Liverpool, L69 7ZL England e-mail: [email protected]

Pure Mathematics Division Mathematical Sciences Building University of Liverpool, Peach Street Liverpool, L69 7ZL England

 2011 Springer Science+Business Media, LLC, Volume 33, Number 3, 2011

85

Figure 3. The west iwan of the Friday Mosque, Isfahan. (Photograph reproduced courtesy of Paul Rudkin).

(a)

(b)

Figure 5. Possible sources of inspiration for the Whirling Kites motif. In (a) t : s = 2 : 1 and in (b) t : s = 4 : 3.

Figure 4. Construction of a kite in a square. 86

THE MATHEMATICAL INTELLIGENCER

A similar figure occurs in one of the many proofs of the Pythagorean Theorem. The ancient Chinese text The Arithmetical Classic of the Gnomon and the Circular Paths of Heaven contains a discussion of the theorem using the 3-4-5 triangle as an example—Figure 5(b). The oldest surviving manuscript is a 13th-century copy in Shanghai library, but much of the content predates Islam by hundreds of years. There is Chinese influence in Islamic art so it is possible that this proof was known to medieval Islamic scholars, too.

¨ zdural suggests [13] that figures such as those in Figure 5 O may have inspired the artistic imagination of the craftsmen. Once the periodic pattern is known, it is a simple matter to extract the Whirling Kites motif. He cites the Whirling Kites pattern on the inner wall of the west iwan of the Friday Mosque as a possible application. Chiral motifs such as the swastika are widespread and found in many cultures; the Whirling Kites motif seems to be unique to Islamic ornament. Perhaps some mathematical input was required for its discovery.

Variation 1 Figure 6(c) shows a geometric design based on the upper panels in the front face of the west iwan of the Friday Mosque. Both panels are counter-clockwise. Here the kites are decorated with a section of a periodic pattern constructed on a triangular grid. Figure 6(a) shows a hexagonal repeat unit for the pattern. The black motif is related to a common square Kufic representation of the name ‘Ali’. Here the text is truncated and reflected; a well-formed hexagonal treatment of the text appears in the centre of panel 91 of the Topkapi Scroll [12]. Figure 6(b) shows how hexagons can be used to fill a kite whose small angle is 60. While Figure 6(c) is not a true reproduction of the panel on the mosque (the mosaic is not laid out so accurately), this method or something similar clearly underlies its construction. Here we pause for a few comments on terminology. The lines in the figures drawn in red show the underlying geometric structure of a design but are not apparent in the finished product. We shall use this convention throughout. We shall also refer to the shapes outlined by red lines as tiles, and to a collection of tiles as a tiling. This is to distinguish them from the individual ceramic shapes, which we shall call tesserae, that are assembled to form a panel or mosaic. Figure 6(c) is a simple example of a 2-level design: two geometric patterns of different scales used in a single design. Many examples of the interplay of patterns on multiple scales can be found in Islamic ornament. In the early works, voids in the background of a large-scale pattern are progressively filled with floral or geometric motifs to leave a design with no vacant spaces. In some of the finest examples of 2-level geometric design, mathematical processes such as subdivision were

(a)

(b)

(c)

Figure 6. A 2-level design from the Friday Mosque, Isfahan.

applied to generate large- and small-scale patterns that are intimately related [3, 4, 6, 10]. In the patterns featured in this article, the Whirling Kites figure establishes the basic framework of a large-scale pattern, and it is embellished with secondary decoration in the following ways: • calligraphy, arabesques or geometric patterns are used to fill the interiors of the kites and central square • bands of arabesques or geometric patterns may be used to outline the compartments, thickening the lines of the frame and rotor. These two techniques (filling and outlining) correspond to Type A and Type B, respectively, in the classification of 2-level designs introduced by Bonner [3]. In the best examples of 2-level designs, the large- and small-scale patterns are complementary in the sense that prominent features of one are highlighted or supported by the other. This is not achieved in Figure 6(c): the hexagonal subdivision of the kite provides a good basis for the construction of a small-scale pattern, and two of the directions within the small-scale pattern are aligned with the long sides of the kite, but the focal points of the pattern are not strong enough to add emphasis where it is needed. Also, the kites are treated independently rather than as parts of a composite figure, so there is no continuity across their boundaries. Other examples we shall analyse reveal that finding a smallscale pattern that is compatible with the features of the Whirling Kites figure is a challenging problem.

Variation 2 Figure 7 shows a Type A 2-level Whirling Kites design from the Madar-i Shah Madrasa (Mother of the Shah or Royal Theological College), also known as the Chahar Bagh Madrasa. Each corner of the large central courtyard is canted with an arch leading to a small octagonal courtyard giving access to the rooms of the college. See [15, p. 293] for a general view. The Whirling Kites design is repeated just below roof level around the small courtyards. The design is used in both mirror-image forms, and the composition of the small-scale pattern varies. The mosaic is made using the ‘cut tile’ technique: large ceramic tiles with a single colour glaze are cut into small tesserae, which are then assembled to make the mosaic panel. Here, the yellow star-shaped tesserae mark out the shapes of the compartments and manifest the 2:1 ratio of the long and short sides of the kites. The kites are filled with a seemingly random arrangement of black and turquoise tesserae. This small-scale pattern is based on a modular design system that underlies many Islamic patterns [6, 7, 10, 11]. The basic system comprises the three equilateral tiles shown in Figure 8: a regular decagon decorated with ten small kites arranged to form a {10/3} star motif, a hexagon shaped like a bow-tie decorated with two kites congruent to those on the decagon, and a convex hexagon with a bobbin-shaped motif. The boundaries of these underlying tiles are not apparent in the finished mosaic but they can be recovered from the design: the black tesserae are the foreground motifs on the tiles, the yellow tesserae are the  2011 Springer Science+Business Media, LLC, Volume 33, Number 3, 2011

87

Figure 9. First-stage analysis of Figure 7.

Figure 7. A Type A 2-level design from the Madar-i Shah Madrasa, Isfahan. (Photograph reproduced courtesy of Brian McMorrow).

decagons have the same orientation (vertex at the top), and this alone reduces the symmetry of the design as a whole to 2-fold rotation. The complete design is asymmetric because each kite has its own irregular filling. The lines in the large-scale pattern fall into two categories according to whether they connect two vertices or two edges of the decagon tiles. The lines connecting two vertices are covered by the diagonals of two bobbin tiles and a tile edge; the other lines (except for the bottomcentre) are covered by various sequences of two bow-ties and two bobbins. In fact, the two lengths produced by these combinations of tiles are not quite equal and the construction illustrated in Figure 9 is a geometric fallacy. This is made clear in Figure 10 which shows the small-scale

Figure 8. Elements in a common modular design system.

centres of the decagons, and the turquoise tesserae are formed by fusing the background regions at the edges of the tiles. The arrangement of the tiles is a typical application of the modular system to this style of 2-level pattern: decagons are placed so their centres coincide with prominent features of the large-scale pattern, other tiles are placed so that their edges or mirror lines are aligned with the outlines of the kites—see Figure 9. The interiors of the compartments are then infilled with more tiles. In this case the decagons are centred on the corners and junctions of the lines in the large-scale pattern and also divide the long sides of the kites. The centres of the decagons on the frame divide each side into three equal parts. If the long and short sides of a kite are in the ratio 2:1 then h, the small angle in a kite, is about 53.13. In a mosaic context, this angle cannot be distinguished from 54—an angle compatible with the 10-fold geometry of the modular system. However, it is not compatible with the 4-fold symmetry of the Whirling Kites pattern. Observe that all the 88

THE MATHEMATICAL INTELLIGENCER

Figure 10. Part of the small-scale pattern from Figure 7 when it is not constrained to fit a kite.

apothem (centre to edge mid-point) of the decagon is ks. (h3) Using Figure 11(c) with relations (h1) and (h2) we can deduce that the length of the short mirror line of the bobbin is 2k(s - 1).

(a)

(b)

(c)

Figure 11. Properties of the tiles in Figure 8.

pattern in the top left kite with its natural geometry. Notice the two half bow-tie tiles at the inner end of the fracture. The gap and the misalignment of the boundary at the topcentre are small and, with minor adjustments to the size and shape of a few tesserae, the small-scale pattern can be made to fit the available space without drawing attention. Similar adjustments are made in the fillings of the other kites—in the bottom right kite, the ‘problem’ is pushed into the small angle of the kite where it affects the frame (as indicated by the discontinuity in Figure 9). In fact, it is impossible to cover a square frame exactly with the tiles using this strategy. We shall now sketch a proof of this—readers who are not interested in the technicalities can skip to the next section. In any tiling composed of the three tiles in Figure 8 all the decagon tiles have the same orientation, and the bow-tie and bobbin tiles can both occur in five orientations aligned at multiples of 36 to each other. We have the requirement that a tile which intersects a line in the square must do so in an edge or a mirror line of the tile. We take the edge length of the tiles to be 1. We shall express some distances across the tiles in terms of the parameters d and k shown in Figure 11(a). Recall that the lengths of a diagonal and an edge of a regular pentagon are in the golden ratio, s. We have pffiffiffi pffiffiffi 5þ1 51 ; d ¼ cosð72 Þ ¼ and s¼ 2 4 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi 5þ 5 : k ¼ sinð72 Þ ¼ 8 First we consider distances that are vertical in Figure 11.

The vertical lines of the square in Figure 9 must be covered by rational combinations of the distances (v1)– (v4). These are parametrised by d and spso ffiffiffi the length of the side of the square must belong to Q½ 5: The horizontal lines of the square must be covered by rational combinations of the distances (h1)–(h3); these are parametrisedpby ffiffiffi k and s. The double radical k is not in the field Q½ 5: Therefore the vertical and horizontal distances covered by the tiles are incommensurable.

Variation 3 Figure 12 shows one of the lower pair of Type A 2-level Whirling Kites designs from the front face of the west iwan of the Friday Mosque. As with the upper pair (Variation 1), both panels are counter-clockwise. The mosaic is predominantly in black and gold with the kites outlined in white. The problems in the Madrasa design (Variation 2) arising from the use of 10-point stars are avoided here by using 12-point stars. These stars are compatible with the 4-fold symmetry of the whole design and its 90 angles at the corners of the frame and the inner square. Figure 13 shows the underlying structure of the design. The 12-point stars are represented by circles. Using the distance between adjacent centres as the unit, we see that the long and short sides of the kites are in the ratio 4:2 so h & 53.13. The panel is subdivided into 20 unit squares, and 8 small kites with sides in the ratio 2:1. To form the mosaic each small square is filled with a standard star pattern that has the centres of 12-point stars at the corners

(v1) The edge length is 1. (v2) The pentagon in Figure 11(a) shows that the radius (centre to vertex) of the decagon is s times its edge length. (v3) Figure 11(b) shows that the distance across the waist of the bow-tie is 1 - 2d. (v4) Figure 11(c) with relation (v2) shows the long diagonal of the bobbin is 2(s - d). Now we consider some horizontal distances in Figure 11. (h1) Figure 11(b) shows the length of the long mirror line of the bow-tie is 2k. (h2) Recalling that the lengths of the two red lines in Figure 11(b) are in ratio s, we can deduce that the

Figure 12. A Type A 2-level design from the Friday Mosque, Isfahan. (Reproduced from [15, p.220] courtesy of Henri and Anne Stierlin, Geneva).  2011 Springer Science+Business Media, LLC, Volume 33, Number 3, 2011

89

Figure 13. Analysis of Figure 12.

Figure 14. A Type B 2-level design from the Friday Mosque, Isfahan. (Photograph reproduced courtesy of Daniel Sanderson).

and an 8-point star in the centre. This pattern covers over half the panel. The decoration in the small kites is based on tiles analogous to the bow-tie and bobbin tiles of Figure 10, but adapted to the angles of a dodecagonal tiling scheme. The 12-point stars are not compatible with the local geometry in the shaded circles (if the spikes are equally spaced, they cannot align with the sides of the kite), but this does not intrude on the eye. We have seen three examples of Type A (filling); we now consider examples of Type B (outlining).

Variation 4 Figure 14 shows the famous 2-level Whirling Kites design from the inner wall of the west iwan of the Friday Mosque. A wider view and some details are shown in photographs IRA 0520, IRA 0604, and IRA 0605 in [17]. The bands outlining the frame and rotor are bordered by fragments of 10-point stars. Connecting the centres of these stars divides the bands into strips of approximately square cells, as shown in Figure 15(a). Using the side of a square as a unit, and measuring along the centre-line of the band, we see that the frame is 15 units along each side, and the central square is 5 units. Therefore, the sides of each kite are (again) in the ratio 2:1. The small-scale design is created by filling each square cell with a pattern based on the template shown in Figure 15(b). This pattern is constructed using another modular system, this time having four decorated tiles: a regular decagon with a {10/4} star motif, a regular pentagon with a {5/ 2} star (or pentagram) motif, an isosceles triangle with sides in the golden ratio decorated with a kite, and a trapezium decorated with an arrowhead. The template can be repeated to form periodic star patterns—see photograph IND 0705 in [17] for an example. Applying the template to the square cells of Figure 15(a) is problematic as the template itself is not 90

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(a)

(b)

Figure 15. Analysis of Figure 14.

square (the height is about 95% of the width). Hence, some juggling of the tesserae is required to make things fit. The pentagrams are most affected by the deformation—they are noticeably irregular in the mosaic. Even though the large-scale pattern has 4-fold symmetry, and is decomposed into squares, the design on the template has only 2-fold symmetry. In Figure 15(a) the orientation of the template is indicated with the double arrow motif from the centre of the template; the copies around the frame are vertically aligned and those in the rotor are aligned top-right to bottom-left. The angle between the bands in the rotor and those in the frame is approximately 54 so it is compatible with the 10-fold geometry underlying the template. This means that it is possible for the stars and other motifs in the small-scale pattern to be aligned consistently throughout the design (as in

Variation 2). However, if the craftsmen who made the mosaic recognised this, either they did not consider it important or they have made a mistake in laying out the design. In the mosaic, the stars in the frame have a vertical spike while the stars in the rotor have a horizontal spike. If the rotor were rotated by 90, all the stars would have the same alignment, and the small-scale designs would be compatible at the junctions where the rotor meets the frame. In the mosaic this is not the case and further juggling is required to disguise it.

Star Placement When the length parameters x, y, s and t are integers, discrete motifs such as flowers or stars can be placed on the Whirling Kites figure so their centres lie on the figure, some coincide with the corners and intersections of the lines, and they are equally spaced along all its lines. Figure 16(a) shows a template for a Whirling Kites design with x = 11 and y = 3. This implies h & 59.49, an angle that is indistinguishable from 60 for practical

(a)

(b)

(c)

(d) Figure 16. A new Type B 2-level design with 12-point star motifs. Design Copyright P.R. Cromwell 2010.

purposes. The figure is formed from strips of squares; where the rotor meets the frame the strips are simply overlaid and the two squares that meet at each junction are concentric. To create a Type B pattern we need to find a star pattern that has a square repeat unit and is compatible with 90 and 60 junctions. We shall not explain the construction of star patterns here—see [5, 8, 9] for information. However, it is clear that a 12-point star is a good candidate for a motif that meets these requirements. Figure 16(c) shows four repeat units of a traditional pattern taken from Plate 94 of Bourgoin [2]. It is constructed using another modular system of decorated tiles: regular polygons of 3, 4 and 12 sides and a shield-shaped tile formed by erecting a right isosceles triangle on each side of an equilateral triangle [16, p. 18]. Figure 16(d) shows the result of placing this repeat unit in each square of Figure 16(a); the junctions between the rotor and the frame are made using the simple mitre joint shown in Figure 16(b). The result is a Type B 2-level Whirling Kites pattern with 12-point stars (coloured yellow in the figure) equally spaced along the centre-line of the bands. Even though it would have been possible for medieval artists to construct patterns like this, we are not aware of a Whirling Kites example in which the principal star motif runs along the centre-line of the bands. The closest we have come is the border pattern shown in photograph EGY 1609 of [17], which shows a band with 12-point stars turning a 90 corner. In most 2-level designs where the small-scale design is a star pattern, the defining features (corners and intersections) of the large-scale design are located in the centres of stars in the small-scale design. In Type B designs, the defining features of the large-scale design are the corners in the boundary of the band. In the example of Figure 14 the centres of the 10-point stars are evenly spaced along the band edges as far as possible. The exterior corners of the frame, the corners of the central square, and the 90 corners of the kites are all located at star centres. The obtuse and acute angles of the kites do not coincide naturally with star centres, although stars have been placed at the acute angles in the top and bottom sides of the frame. Figure 17 shows that it is possible to create a Whirling Kites design in which all the band boundaries have integer length. A 3-4-5 triangle is placed at each junction of the rotor and the frame, and the bands are 4 units wide. Measuring along the centre-line of the bands we have x = 36 and y = 12. This means that the long and short sides of the kites are in the ratio 2:1 and h & 53.13. When trying to select a star whose geometry is compatible with a Whirling Kites design, it is useful to find a fraction of 360 that approximates h. The denominator gives an indication of the number of points in a suitable star, either directly or via simple relationships. In this case 3/20 is a good candidate. However, it is difficult to make patterns from stars with as many as 20 points. The most natural choice for a small-scale geometric pattern is a star pattern with 10-fold motifs. As we have seen, this is far from easy. Motifs with 10fold symmetry are not compatible with the 4-fold symmetry of the whole design: 10-point stars will have the same  2011 Springer Science+Business Media, LLC, Volume 33, Number 3, 2011

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Our final example shows another approach to the problem of star placement.

Variation 5

Figure 17. A band network with integer boundaries.

orientation throughout the pattern, so some band boundaries will pass through opposite spikes, and others will pass between the spikes. There are also the problems of producing a square template to cover the band, and covering the 3-4-5 triangles in both vertical and horizontal alignments. Even if this were done, the small-scale pattern would probably appear too busy and intricate to be effective as ornament—the difference in scale and apparent complexity between the large and small patterns is too great.

Figure 18. A Type B 2-level design from the Friday Mosque, Isfahan. (Photograph reproduced courtesy of Steven Achord). 92

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Figure 18 shows another Type B 2-level design from the Friday Mosque. Other views are shown in photographs IRA 0721 and IRA 0722 in [17]. In this example, placing the stars at prominent points has taken precedence over equal spacing. The underlying structure of the small-scale pattern is shown in Figure 19. The lines dividing the band into cells connect the star centres. Left to right along the bottom of the frame we find four squares, four rectangles, and a final square. The width of each rectangle is determined by the equilateral triangle it contains. This arrangement is repeated around the otherpsides of the frame. The ratio of the ffiffiffi lengths is AB : BC ¼ 3 : 1 so h = 60. To construct the rotor, erect a line from A making an angle of 60 with the bottom of the frame. Repeat on each side of the frame and extend the four lines until they meet. For example, the line starting at A meets the line starting at D in the point E. These four lines bound a square in the middle of the figure, which is subdivided into a 3 9 3 array of congruent squares. These squares are smaller than those in the frame: EF is about 94% of CD. The kite in the lower right is completed with the line CF. Note that CF and DE are not parallel but diverge away from the frame. Label the midpoint of CF as G. This cellular structure provides a framework for laying out the small-scale pattern. The principal star motifs have 12 points and so are compatible with the 90 and 60 angles at the corners of the band. The 16 stars in the central array are aligned so that their spikes lie on the cell boundaries; the tips of spikes of adjacent stars touch. The 12-point stars in the frame are aligned so that the cell boundaries pass between the spikes—this difference may help to disguise

Figure 19. Analysis of Figure 18.

the fact that they are further apart than the others. The star at G marks the transition between the two orientations and has 13 points. The square cells in the frame contain 8-point stars at their centres. Triangle CDG is almost equilateral (the angle at C is about 62.19) and this is close enough for the decoration used in the other triangles to be applied. Each kind of cell has its own filling, and these are consistently applied. The pattern has no awkward juxtapositions or abrupt changes, it is a masterly display of apparently effortless transitions between a progression of patterns.

Conclusion The examples discussed above have highlighted some of the problems encountered in trying to design and fabricate 2-level Whirling Kites designs. The mathematics required to create designs with discrete motifs such as flowers evenly spaced along the centre-lines of the band in a Type B pattern is straightforward and could have been understood by medieval craftsmen. Working with star patterns is more difficult, but it is possible to discover by experiment some configurations of the Whirling Kites figure whose angles are compatible with the geometry of stars. Even so, applying a star pattern to cover the bands presents theoretical as well as practical challenges and the medieval artists produced ingenious and attractive solutions.

Bridges: Mathematical Connections in Art, Music and Science, (Granada, 2003), eds. R. Sarhangi and N. Friedman, 2003, pp. 1–12. [4] P. R. Cromwell, ‘The search for quasi-periodicity in Islamic 5-fold ornament’, Math. Intelligencer 31 no 1 (2009) 36–56. [5] P. R. Cromwell, ‘Islamic geometric designs from the Topkapi Scroll I: Unusual arrangements of stars’, J. Math. and the Arts 4 (2010) 73–85. [6] P. R. Cromwell, ‘Islamic geometric designs from the Topkapi Scroll II: A modular design system’, J. Math. and the Arts 4 (2010) 119–136. [7] E. H. Hankin, The Drawing of Geometric Patterns in Saracenic Art, Memoirs of the Archaeological Society of India, no 15, Government of India, 1925. [8] C. S. Kaplan, ‘Computer generated Islamic star patterns’, Proc. Bridges: Mathematical Connections in Art, Music and Science, (Kansas, 2000), ed. R. Sarhangi, 2000, pp. 105–112. [9] C. S. Kaplan, ‘Islamic star patterns from polygons in contact’, Graphics Interface 2005, ACM International Conference Proceeding Series 112, 2005, pp. 177–186. [10] P. J. Lu and P. J. Steinhardt, ‘Decagonal and quasi-crystalline tilings in medieval Islamic architecture’, Science 315 (23 Feb 2007) 1106–1110. [11] E. Makovicky, ‘800-year old pentagonal tiling from Maragha, Iran, and the new varieties of aperiodic tiling it inspired’, Fivefold

ACKNOWLEDGEMENTS

We are grateful to everyone who has given us permission to reproduce their images, and to Mamoun Sakkal for explaining the Kufic basis of the decoration in Figure 6.

Symmetry, ed. I. Hargittai, World Scientific, 1992, pp. 67–86. [12] G. Necipog˘lu, The Topkapi Scroll: Geometry and Ornament in Islamic Architecture, Getty Center Publication, Santa Monica, 1995. [13] A. O¨zdural, ‘Mathematics and arts: connections between theory and practice in the medieval Islamic world’, Historia Mathematica

REFERENCES

[1] N. Assarzadegan, ‘Dividing and composing the squares’, Lamar University Electronic Journal of Student Research, Fall, 2008. Also in History and Pedagogy of Mathematics Newsletter 68 (July 2008) 13–20. [2] J. Bourgoin, Les Ele´ments de l’Art Arabe: Le Trait des Entrelacs, Firmin-Didot, Paris, 1879. Plates reprinted in Arabic Geometric Pattern and Design, Dover Publications, New York, 1973. [3] J. Bonner, ‘Three traditions of self-similarity in fourteenth and

27 (2000) 171–201. [14] G. Potter, http://www.kufic.info/. [15] H. Stierlin, Islamic Art and Architecture from Isfahan to the Taj Mahal, Thames and Hudson, London, 2002. [16] D. Sutton, Islamic Design: A Genius for Geometry, Wooden Books Ltd, Glastonbury, 2007. [17] D. Wade, Pattern in Islamic Art: The Wade Photo-Archive, http://www.patterninislamicart.com/.

fifteenth century Islamic geometric ornament’, Proc. ISAMA/

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Meissner’s Mysterious Bodies BERND KAWOHL

AND

CHRISTOF WEBER

ertain three-dimensional convex bodies have a counterintuitive property: they are of constant width. In this particular respect they resemble a sphere without being one. Discovered a century ago, Meissner’s bodies have often been conjectured to minimize volume among bodies of given constant width. However, this conjecture is still open. We draw attention to this challenging and beautiful open problem by presenting some of its history and recent development.

C

A Century of Bodies of Constant Width In §32 of their famous book ‘‘Geometry and the Imagination’’, Hilbert and Cohn-Vossen list eleven properties of the sphere and discuss which of these suffices to determine uniquely the shape of the sphere [22]. One of those properties is called constant width: if a sphere is squeezed between two parallel (supporting) planes, it can rotate in any direction and always touch both planes. As the reader may suspect, there are many other convex sets with this property of constant width. To indicate that they have this property in common with spheres, such three-dimensional objects are sometimes called spheroforms ([8, p. 135], [36], [7, p. 33]). Some of the three-dimensional convex sets of constant width have a rotational symmetry. They can be generated by rotating plane sets of constant width with a reflection symmetry about their symmetry line. The drawing in Figure 1 is taken from a catalogue of mathematical models produced by the publisher Martin Schilling in 1911 [34, p. 149]. Under the influence of mathematicians such as Felix Klein, such models were produced for educational purposes, many of them made of plaster. Figure 1 appears to be the earliest drawing showing a nontrivial three-dimensional body of constant width. This body is generated by rotating the Reuleaux triangle around its axis of symmetry. The Schilling catalogue also advertises another rotational as well as a nonrotational body of constant width. The author of its mathematical

94

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DOI 10.1007/s00283-011-9239-y

description is Ernst Meissner, with the help of Friedrich Schilling, not to be confused with Martin Schilling, the editor of the catalogue ([34, p. 106f.], and for a slightly expanded version see [28]). Because Meissner seems to have discovered this body, it is called a Meissner body. Although it is obvious that its construction can lead to two noncongruent bodies of constant width, Meissner explicitly describes only one of them, MV (for details we refer to the paragraph ‘‘Identifying the Suspect’’ below). Because the construction of both bodies follows similar principles, one often speaks of ‘‘the’’ Meissner body. The earliest printed photograph of a plaster Meissner body, the one described in the Schilling catalogue MV, can be found in the 1932 German version of ‘‘Geometry and the Imagination’’, shown in Figure 2 [22, p. 216]. Photographs of all three bodies of constant width mentioned by Meissner can be found in more recent publications ([7, p. 64ff.], [16, p. 96– 98]). The mathematical models must have sold well, for they can still be found in display cases of many mathematics departments. For instance, they can be found not only at many German universities (for the plaster model of the Meissner body MV at the Technical University of Halle in Germany, see http://did.mathematik.uni-halle.de/modell/ modell.php?Nr=Dg-003) but also at Harvard University in the US and even at the University of Tokyo (http://www.math. harvard.edu/~angelavc/models/locations.html). Certainly there are many more bodies of constant width than the four mentioned so far. A very nice collection is displayed in the exhibit ‘‘Pierres qui roulent’’ (‘‘Rolling Stones’’) in the Palais de la De´couverte in Paris (see Figure 8 at the end of the paper). In addition to some rotated Reuleaux polygons (two triangles, four pentagons) it shows two Meissner bodies, both of the same type MF. The exhibit offers the visitor a hands-on, tactile experience of the phenomenon of constant width. Sliding a transparent plate over these bodies of the same constant width causes the bodies to roll, while the plate appears to slide as if lying on balls.

Figure 1. Rotated regular Reuleaux between a gauging instrument.

triangle,

squeezed

Figure 2. Plaster Model of Meissner body MV.

Of course there are also many other, nonrotational bodies of constant width. For their construction see [36], [24], [31], and [2]. In this article we restrict our attention for the most part to the three-dimensional setting. The reader can find more material in excellent surveys on plane and higher-dimensional sets of constant width, for example, in Chakerian & Groemer [12], Heil & Martini [21], or Bo¨hm & Quaisser [7, ch. 2]. As already mentioned, there are two different types of Meissner bodies MV and MF (their construction will be described in the following text). They not only have identical volume and surface area, but are conjectured to minimize

volume among all three-dimensional convex bodies of given constant width. We could not find a written record by Meissner himself that explicitly states the conjecture, but he seems to have guessed that his bodies are of minimal volume [7, p. 72]. Whereas Hilbert and Cohn-Vossen in their book of 1932 do not comment in this direction, Bonnesen and Fenchel mention the conjecture two years later. In the German edition of their ‘‘Theory of Convex Bodies’’, they write, ‘‘es ist anzunehmen’’ which still reads ‘‘it is to be assumed’’ in the English edition of 1987 [8, p. 144]). Since then the conjecture has been stated again and again. For example, Yaglom and Boltyansky make it in all editions of their book ‘‘Convex Figures’’, from the Russian ‘‘predpolagaiut’’ in 1951, via the German ‘‘es ist anzunehmen’’ in 1956, to the English ‘‘we shall assume without proof’’ in 1961 [39, p. 81]. On the other hand, there was the belief that the body that minimizes volume among all three-dimensional bodies of constant width must have the symmetry group of a regular tetrahedron, a property not displayed by the Meissner bodies. This belief was first expressed by Danzer in the 1970s, as Danzer has confirmed to us in personal communication ([19, p. 261], [13, p. 34] and [7, p. 72]). In 2009 an attempt was made to arrive at a body of full tetrahedral symmetry and minimal volume via a deformation flow argument [17]. Incidentally, the Minkowski sum 12 MV  12 MF ; which one obtains halfway through the process of morphing MV into MF, would provide a body with tetrahedal symmetry (see Figure 7). It actually has the same constant width as MV and MF. Its volume, however, is larger than that of the Meissner bodies, due to the Brunn-Minkowski inequality. It can be shown that the increase in volume is slightly more than 2% of the volume of the Meissner bodies [32].

Generating Constant-Width Bodies by Rotation Every two-dimensional convex set can be approximated by convex polygons. Similarly, every two-dimensional convex set of constant width can be approximated by circular arcs

AUTHORS

......................................................................................................................................................... BERND KAWOHL teaches applied mathe-

maticians at the University of Cologne. Most of his research deals with shapes of solutions to partial-differential equations, and related questions from the calculus of variations. For a hobby he sings in a choir whose repertoire emphasizes music of the Comedian Harmonists. Mathematisches Institut Universita¨t zu Ko¨ln D–50923 Ko¨ln Germany e-mail: [email protected]

is concerned with the visualization of mathematical phenomena, especially seemingly paradoxical phenomena such as the Meissner bodies. He does pedagogical research aiming to reconstruct students’ mental processes while solving problems. On the practical side, he develops visualization exercises to help students understand and do mathematics. He teaches both at a teachers’ college and at a secondary school.

CHRISTOF WEBER

Pa¨dagogische Hochschule Nordwestschweiz CH–4410 Liestal Switzerland e-mail: [email protected]

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Figure 3. Nonregular Reuleaux tetragon with four circular arcs.

Figure 4. Figure 3 rotated (side and top view).

and thus by Reuleaux polygons of constant width. If the arcs are all of the same length, one has regular Reuleaux triangles, pentagons, and so on. But to generate a plane convex set of constant width, it is not necessary that all circular arcs be of the same length. Figure 3 shows a plane set of constant width, a Reuleaux tetragon, which is constructed along the lines of [9, p. 192f.]. Note that it is bounded by four circular arcs. Whenever a plane set of constant width is reflection symmetric with respect to some axis, it can be rotated around that axis to generate a three-dimensional set of constant width. A rotated regular Reuleaux triangle leads to the body shown in Figure 1, and if an appropriate nonregular Reuleaux tetragon or a Reuleaux trapezoid is rotated, one will obtain a body similar to the one in Figure 4. Both are not only bodies of revolution but are three-dimensional sets of constant width [9, p. 196f.]. According to the theorem of Blaschke-Lebesgue, the Reuleaux triangle minimizes area among all plane convex domains of given width. Thus one could expect that the rotated Reuleaux triangle in Figure 1 would minimize volume among all rotational bodies of given width. It was not until recently (1996 and 2009) that this long-standing conjecture was confirmed ([10, 25] and [1]).

which is centered at a vertex of a regular tetrahedron with side length d. It consists of four vertices, four pieces of spheres, and six curved edges each of which is an intersection of two spheres. Whenever this Reuleaux tetrahedron is squeezed between two parallel planes with a vertex touching one plane and the corresponding spherical surface touching the other, their distance is d by construction. However, the distance pffiffi planes must be slightly enlarged by a factor of up pffiffiffiof the to 3  22  1:025 when the planes touch two opposite edges of RT. This means that the width of RT is not constant but varies depending on its direction up to 2.5%. Incidentally, as Meissner mentioned in [27, p. 49], the ball is the only body of constant width that is bounded only by spherical pieces. Thus RT, which is bounded only by spherical pieces and is different from a ball, ought not to be of constant width, and indeed it just fails to be. Nevertheless, RT can be used as a starting point for a set of constant width. According to Meissner, some edges must be rounded off by the following procedure ([28], [8, p. 144], [39, p. 81], [6, p. 54f.]): a) Imagine two planes bounding adjacent facets of the underlying tedrahedron. Remove the wedge located between the two planes and containing the curved edge of the Reuleaux tetrahedron RT (see Figure 5 from [39, p. 81]). b) The intersection of the planes with RT contains two circular arcs that meet in the two ends of the wedge. Rotate one of these arcs around the corresponding edge of the tetrahedron. This generates a spindle-shaped surface, a spindle torus. c) Notice that now the sharp edge has become a differentiable surface even across the boundary between spindletorus and spherical piece. After rounding off three edges of RT that meet in a vertex, according to this procedure, one obtains the first type of Meissner body, MV (see Figure 6, left). The second Meissner body MF is obtained by rounding off three edges surrounding one of the faces of RT (see Figure 6, right). Either Meissner body features four vertices, three circular edges, four spherical surfaces, and three toroidal surfaces. Both bodies have identical volume and surface area, and they are invariant under a rotation of 120 around a suitable

Identifying the Suspect: Meissner Bodies The plane Reuleaux triangle of constant width d is constructed as the intersection of three discs of radius d, each centered at a different corner of an equilateral triangle. In an analogous way, a Reuleaux tetrahedron RT can be constructed by intersecting four balls of radius d, each of 96

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Figure 5. Replacing three wedges (left, one shaded gray) by pieces of spindle tori (right).

Figure 6. Meissner body MV with rounded edges meeting in a vertex (left bottom) and Meissner body MF with rounded edges surrounding a face (right top).

axis. A computer animation showing both bodies MV and MF from all sides can be viewed under [38]. Meissner bodies touch two parallel planes between which they are squeezed always in one of two possible ways: either one contact point is located in a vertex and the antipodal contact point is located on a spherical piece of the body, or one contact point is located on a sharp edge and the antipodal contact point is located on a rounded edge of the body. Their constant width becomes obvious if one intersects a sharp, nonrounded edge opposite the rounded edge with a plane orthogonal to the sharp edge. In this plane the sides of the original tetrahedron form an isosceles triangle similar to the one in Figure 3. The line segment passing from the sharp edge of RT through the opposite sharp edge of the regular tetrahedron varies in length and is generally shorter than the width d. If its length is extended to d, one arrives at the boundary of the edge that has been rounded off. Meissner showed the constant width of his bodies using Fourier series [27, p. 47ff.]. Like Hurwitz, he originally studied convex closed curves inscribed in a regular polygon, which remain tangent to all the sides of the polygon during rotations of the curve. Nowadays such curves are called rotors. Following Minkowski, Meissner characterized the curves by their support functions (length of the polar tangents). These are periodic and thus can be expanded in Fourier series. Using this technique, he finally succeeded in describing all rotors of regular polygons analytically [26]. With the analogous technique in three dimensions, he was able to determine the rotors of the cube as bodies of constant width.

He even proved that non-spherical rotors exist not only for the cube, but also for the regular tetrahedron and octahedron. In contrast, there exist no non-spherical rotors for the regular dodecahedron and icosahedron ([30]; for some mechanical adaptions of Meissner’s technique see [9, p. 213ff.]).

Volume and Surface Area of Meissner bodies In this section we give some numerical results on the volume and surface area of the Meissner body of constant width d. The volume VMV and VMF of the two Meissner bodies is identical and is given by pffiffiffi   3 2 1  arccos   p  d 3  0:419860  d 3 ; VMV ¼ VMF ¼ 4 3 3 see [12, p. 68], [7, p. 71], [35, A137615], [31, p. 40–43]. Therefore we will not distinguish between MV and MF. The volume of the Meissner body is approximately 80% of the volume p/6 of a ball of diameter 1 and it is considerably smaller (by about 6%) than the volume of the rotated Reuleaux triangle R3, which is given by   2 p   p  d 3  0:449461  d 3 VR3 ¼ 3 6 in [10] and [35, A137617]. As far as we know, the highest lower bound for the volume of a body K of constant width 1 is the one given by Chakerian, et al. in 1966,  p  pffiffiffi VK >  3 6  7  d 3  0:364916  d 3 ; 3 see [11] and [24].

Figure 7. Morphing MV to MF including the Minkowski mean (second frame).  2011 Springer Science+Business Media, LLC, Volume 33, Number 3, 2011

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Figure 8. Various Bodies of Constant Width (Palais de la De´couverte, Paris).

The surface area SMV and SMF of the two Meissner bodies is identical, as well, and is given by pffiffiffi   3 1  arccos  p  d 2  2:934115  d 2 ; SMV ¼ SMF ¼ 2  2 3 see [12, p. 68], [7, p. 71], [35, A137616]. This follows from the remarkable fact that in three dimensions the volume VK and surface area SK of a convex body K of constant width d are related through Blaschke’s identity ([5, p. 294], [12, p. 66]) 1 p VK ¼  d  SK   d 3 : 2 3 Since VK is monotone increasing in SK, the question of finding the set that minimizes volume is equivalent to finding the set that minimizes surface area (or generalized perimeter) of K among all convex sets of constant width. Incidentally, this is in sharp contrast to the two-dimensional case, in which, according to a theorem of Barbier, all sets of constant width d are isoperimetric, that is they have the same perimeter p  d ([3], [8, p. 139]). The major part of the surface of the Meissner body consists of pieces of a sphere of radius d. The rounded edges (or spindle tori) have an angle of rotation of arccosð1=3Þ; and their smaller principal curvature is constant and has the value 1/d. Their part of the surface area is Z1 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi! arccosð13Þ 3 3 2  2p  d   dx þ x  x2  SSp ¼ 3  2p 2 4 0

 0:334523  d 2 : In other words, the non-spherical pieces of the surface of a Meissner body make up about 11% of the total surface area.

Circumstantial Evidence, but No Proof Why do we believe that Meissner bodies minimize volume among all three-dimensional convex bodies of constant width? There are more than a million different reasons for 98

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it. Clearly the fact that the conjecture has remained unsolved for so long shows that a counterexample is hard to come by. But there is more than this one reason supporting the conjecture. In 2007 Lachand-Robert and Oudet presented a method for constructing a large variety of bodies of constant width in any dimension; see [24]. For plane domains, this construction boils down to a method of Rademacher and Toeplitz from 1930 [33, p. 175f.]. Their algorithm begins with an arbitrary body Kn-1 of constant width in (n - 1) dimensions and arrives at a body Kn of constant width in n dimensions with Kn-1 as one of its cross-sections. It was used in 2009 to generate randomly one million different three-dimensional bodies of constant width [31]. None of them had a volume as small as that of a Meissner body. It should be noted, however, that while the algorithm can generate every two-dimensional set of constant width from a one-dimensional interval, it cannot generate all threedimensional sets of constant width, but only those that have a plane cross-section with the same constant width. In [14], Danzer describes a set K3 of constant width d for which each of its cross-sections has a width less than d. Analysts have recently tried to identify the necessary conditions that a convex body M of minimal volume and given constant width must satisfy. The existence of such a body follows from the direct methods in the calculus of variations and the Blaschke selection theorem. Let us mention in passing that the boundary of M cannot be differentiable of class C 2. If it were, one could consider Me :¼ fx 2 Mjdistðx; oMÞ [ e [ 0g; that is the set M with a sufficiently thin e-layer peeled off and with a volume less than that of M. According to the Steiner formula, its volume VMe can be expressed in terms of the volume VM of M, the mean width dMe of Me ; and the surface area SMe of Me as follows: VMe ¼ VM  e  SMe  dMe  e2 

4p 3 e : 3

By construction, and because of our smoothness assumption, Me is a body of constant width d  2e: Therefore its mean width is dMe ¼ d  2e: If one blows Me up by a linear ~ its volume is given by factor of d=ðd  2eÞ to a set M;  3 d VM~ ¼  VMe ; d  2e ~ is of constant width d again. It will now be shown and M that VM~ \VM can occur. In fact, VM~ \VM occurs when ðd  2eÞ3  VM~ ¼ d 3  VMe \d 3  VM ; or equivalently   4p d 3  VM  e  SMe  ðd  2eÞ  e2   e3 \ d 3  VM : 3 Thus for sufficiently small e the volume VM~ stays below the original volume VM of M, contradicting the minimality of M’s volume. That is, no body of class C 2 can minimize volume. In 2007 a stronger result was shown: Any local volume minimizer cannot be simultaneously smooth in any two antipodal (contact) points [4]. In other words, squeezed between two parallel plates, one of the points of contact

Figure 9. Ernst Meissner, 1883–1939 (ETH-Bibliothek Zu¨rich, Bildarchiv).

with the plane must be a vertex or a sharp-edge point. As already pointed out, Meissner bodies have this property. Because rotated Reuleaux polygons possess this property as well, this result also supports the conjecture without proving it. Finally, in 2009, it was shown by variational arguments that a volume-minimizing body of constant width d has the property that any C2 part of its surface has its smaller principal curvature constant and equal to 1/d [1]. Again, Meissner bodies meet this criterion as well, because they consist of spherical and toroidal pieces with exactly this smaller principal curvature. After this paper was accepted for publication we learned from Qi Guo in personal communication, that Qi Guo and Hailin Jin had just observed another remarkable property of Meissner bodies. It is well known that the inradius r and circumradius R of a body of constant width add up to d. The ratio R/r of these two radii is a measure of asymmetry for a set, and the observation of Guo and Jin is that it is maximized (among all three-dimensional bodies of constant width) pffiffiffi by Meissner bodies. For those bodies R=r ¼ ðð3 þ 2 6Þ=5Þ  1:5798: In fact, R is maximized, given d, by a Meissner body (see, for example, [18]), and so r is minimized and a fortiori R/r is maximized. It is in this sense that the Meissner bodies are more slender and should have less volume than others of constant width d. All these results seem to suggest that another century will not pass before the conjecture is confirmed.

Appendix: CV of Ernst Meissner Who was the man who discovered the body that is presumed to minimize volume? Ernst Meissner was born on 1 September 1883 as the son of a manufacturer in Zofingen, Switzerland. He attended secondary school in Aarau, where he had the same teacher in mathematics as Albert Einstein had had earlier, Heinrich Ganter. Ganter’s style of teaching is described as follows. ‘‘He was a good mathematician but not, in his own reckoning, good enough to pursue a career in higher mathematics. But he could teach, something that many speculative gentlemen cannot do. […] Ganter never treated us demeaningly, but taught us as men.’’ Meissner

Figure 10. Meissner 1931 teaching students suffering from tuberculosis in the ‘‘sanatorium universitaire’’ in Leysin operated by Swiss universities (Conservatoire Nume´rique des Arts et Me´tiers, Bibliothe`que du CNAM).

himself described him ‘‘as a teacher who, far from transmitting mere information to prepare a pupil for a career, educated the heart and character and truly civilized his charges. If all teachers were like Ganter, […] there would be no need for school reform.’’ [15, p. 91] Meissner’s own dedication to teaching is evident from a public lecture that he gave on 18 November 1915 in the town hall of Zurich. The renowned newspaper ‘‘Neue Zu¨rcher Zeitung’’ (NZZ) dedicated half a page to his lecture ‘‘Why does mathematics appear difficult and boring to some while not to others?’’ [29]. For Meissner, grasping a mathematical concept is more than passively understanding its logic. In fact, many people are capable of logical thinking without appreciating mathematics. The deeper understanding of mathematics is rather connected to the creation of one’s own mental images and concepts. Meissner promotes the idea that mathematical education should not confront pupils with abstract and fully matured facts. Instead it should enable them to construct and connect mental images in several ways. To a great extent his criticism still applies to contemporary teaching. After graduating from school, Meissner studied from 1902 to 1906 at the Department of Mathematics and Physics of the Swiss Polytechnic, which was later to become the Swiss Federal Institute of Technology (ETH) in Zurich. He was awarded a doctorate there on the basis of a thesis in number theory. After two semesters at the University of Go¨ttingen, where he studied with Klein, Hilbert, and Minkowski, he returned to the ETH. There he qualified as a professor (Habilitation) in 1909 in mathematics and mechanics. A year  2011 Springer Science+Business Media, LLC, Volume 33, Number 3, 2011

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later he was offered the chair of technical mechanics, which he held until 1938 (see Figures 9 and 10 [37, p.459]). Ernst Meissner died on 17 March 1939 in Zollikon (near Zurich). Meissner’s scientific achievements were extraordinarily diverse (for a list of his publications see [23, p. 294f.]). In his earlier works, he dealt with questions in pure mathematics (geometry, number theory). Not only his dissertation but also his investigations on sets of constant width fall within this period. During the years between 1910 and 1920 he turned increasingly toward applied mathematics (graphic integration of differential equations, graphic determination of Fourier coefficients), and then to mechanics (geophysics, seismology, theory of oscillations). It is in these applied papers that Meissners true scientific achievements lie because, like Franz Reuleaux, the originator of theoretical kinematics, before him, he always sought his models in pure, strict mathematics. In an obituary from 1939, Meissner is depicted as a person who not only expected much from himself but also from those around him. ‘‘Ernst Meissner demanded the most from himself and others. His intense sense of duty and professional ethics made him seem strict and reserved. However, those who knew him better, his nearest friends and his students, were allowed the unforgettable experience of his extraordinarily comprehensive knowledge and his deep perception, a truly classical appreciation of beauty, touching kindness and finely honed wit.’’ [40].

Note added in Proof Recently we learned from Chris Sangwin that bodies similar to the ones depicted in Figure 4 can be purchased via the Internet under http://www.grand-illusions.com/acatalog/ Solids_of_Constant_Width.html.

[9] J. Bryant and C. Sangwin, How Round Is Your Circle? Where engineering and mathematics meet, Princeton University Press, Princeton, NJ, 2008. [10] S. Campi, A. Colesanti and P. Gronchi, Minimum problems for volumes of constant bodies, in: Partial Differential Equations and Applications, Marcellini, P., Talenti, G., and Visintin, E., (eds.), Marcel-Dekker, New York, 1996, 43–55. [11] G. D. Chakerian, Sets of Constant Width, Pacific J. Math., 19 (1966), 13–21. [12] G. D. Chakerian and H. Groemer, Convex bodies of constant width, in: Convexity and its Applications, Gruber, P. M., and Wills, J. M., (eds.), Birkha¨user, Basel, 1983, 49–96. [13] H. T. Croft, K. J. Falconer, and R. K. Guy, Unsolved Problems in Geometry, Springer Verlag, New York, 1991. [14] L. Danzer, U¨ber die maximale Dicke der ebenen Schnitte eines konvexen Ko¨rpers, Archiv der Mathematik, 8 (1957), p. 314–316. [15] K. Datta, The early life of Albert Einstein: Seeking the mature Einstein in his youth, Resonance, 10 (2005), 85–96. [16] G. Fischer, Mathematical Models—Photograph Volume, Vieweg, Braunschweig, 1986. [17] B. Guilfoyle and W. Klingenberg, On C2-smooth surfaces of constant width, Tbil. Math. J., 2 (2009), 1–17. [18] P. Gritzmann and V. Klee, Inner and outer j-radii of convex bodies in finite-dimensional normed spaces. Discrete Comput. Geom., 7 (1992), 255-280. [19] P. M. Gruber and R. Schneider, Problems in Geometric Convexity, in: Contributions to Geometry (Proc. Geom. Sympos. Siegen), To¨lke, J., and Wills, J. M., (eds.), Birkha¨user, Basel, 1979, 255–278. [20] Q. Guo, personal communication, March 2011. [21] E. Heil and H. Martini, Special convex bodies, in: Handbook of Convex Geometry, Gruber, P. M., and Wills, J. M., (eds.), Elsevier, Amsterdam, 1993, 363–368.

REFERENCES

[1] H. Anciaux and N. Georgiou, The Blaschke-Lebesgue problem for constant width bodies of revolution, Preprint, March 2009, 14 pages; http://arxiv.org/abs/0903.4284. [2] H. Anciaux and B. Guilfoyle, On the three-dimensional BlaschkeLebesgue problem, Preprint, June 2009, 10 pages; http://arxiv. org/abs/0906.3217. [3] E. Barbier, Note sur le probe`lme de l’aiguille et le jeu du joint couvert, J. Math. Pures Appl. Ser II, 5 (1860), 273–286. [4] T. Bayen, T. Lachand-Robert and E´. Oudet, Analytic parametrization of three-dimensional bodies of constant width, Arch. Ration. Mech. Anal., 186 (2007), 225–249. [5] W. Blaschke, Einige Bemerkungen u¨ber Kurven und Fla¨chen von konstanter Breite, Ber. Verh. Sa¨chs. Akad. Leipzig, 67 (1915), 290– 297. [6] J. Bo¨hm, Convex Bodies of Constant Width, in: Mathematical Models—Commentary, Fischer, G., (ed.), Vieweg, Braunschweig, 1986, 49–56. [7] J. Bo¨hm and E. Quaisser, Scho¨nheit und Harmonie geometrischer Formen – Spha¨roformen und symmetrische Ko¨rper, Akademie Verlag, Berlin, 1991.

[22] D. Hilbert and St. Cohn-Vossen, Geometry and the Imagination, AMS Chelsea, Providence, R.I., 1952 (transl. from the German: Anschauliche Geometrie, Springer, Berlin, 1932). [23] L. Kollros, Prof. Dr. Ernst Meissner, Verh. Schweiz. nat.forsch. Ges., 1939, 290–296. [24] T. Lachand-Robert and E´. Oudet, Bodies of constant width in arbitrary dimension, Mathematische Nachrichten, 280 (2007), 740–750. [25] F. Malagoli, An Optimal Control Theory Approach to the Blaschke Lebesgue Theorem, J. Convex Anal., 16 (2009), 391–407. [26] E. Meissner, U¨ber die Anwendung der Fourier-Reihen auf einige Aufgaben der Geometrie und Kinematik, Vierteljahrsschr. Nat. forsch. Ges. Zu¨r., 54 (1909), 309–329. (http://www.archive.org/ stream/vierteljahrsschr54natu#page/308/mode/2up). [27] E. Meissner, U¨ber Punktmengen konstanter Breite, Vierteljahrsschr. Nat.forsch. Ges. Zu¨r., 56 (1911), 42–50. (http:// www.archive.org/stream/vierteljahrsschr56natu#page/n53/mode/ 2up). [28] E. Meissner and Fr. Schilling, Drei Gipsmodelle von Fla¨chen konstanter Breite, Zeitschrift fu¨r angewandte Mathematik und Physik, 60 (1912), 92–94.

[8] T. Bonnesen and W. Fenchel, Theory of Convex Bodies, BCS

[29] E. Meissner, Warum erscheint Mathematik einigen schwer und

Associates, Moscow ID, 1987, 135–149 (transl. from the German: Theorie der konvexen Ko¨rper, Springer, Berlin, 1934, §15).

langweilig und anderen nicht? Neue Zu¨rcher Zeitung, 1596

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(26.11.1915), 1–2.

[30] E. Meissner, U¨ber die durch regula¨re Polyeder nicht stu¨tzbaren Ko¨rper, Vierteljahrsschr. Nat.forsch. Ges. Zu¨r., 63 (1918), 544–551. [31] M. Mu¨ller, Konvexe Ko¨rper konstanter Breite unter besonderer Beru¨cksichtigung des Meissner-Tetraeders. Diplomarbeit, Universita¨t zu Ko¨ln, 2009. [32] E´. Oudet, A convex Minkowski-combination of MF and MV and its volume (2010) (http://www.lama.univ-savoie.fr/~oudet/Illustrations/ Meissner_files/meissner1to2.gif and http://www.lama.univ-savoie. fr/~oudet/Illustrations/Meissner_files/meissner1to2.png).

[36] G. Tiercy, Sur le surfaces sphe´riformes, Toˆhoku Math. J., 19 (1921), 149–163. [37] L. Vauthier, Du Sanatorium universitaire suisse au Sanatorium universitaire international, Bulletin de la Socie´te´ d’Encouragement pour l’Industrie Nationale, 1931, 453–471. (http://cnum. cnam.fr/fSYN/BSPI.145.html). [38] Chr. Weber, Gleichdick – Ko¨rper konstanter Breite, ‘‘Film – Erster Meissnerscher Ko¨rper’’ and ‘‘Film – Zweiter Meissnerscher Ko¨rper’’, 2007; http://www.swisseduc.ch/mathematik/material/

[33] H. Rademacher and O. Toeplitz, The Enjoyment of Mathematics, Princeton University Press, Princeton, NJ, 1994. (transl. from the

gleichdick/index.html. [39] I. M. Yaglom and V. G. Boltyansky, Convex Figures, Holt,

second German edition: Von Zahlen und Figuren, 1933). [34] M. Schilling, Catalog mathematischer Modelle fu¨r den ho¨heren

Rinehart and Winston, New York, 1961 (transl. from the German:

mathematischen Unterricht, Leipzig, 1911. (http://uihistories

der Wissenschaften, Berlin, 1956, transl. from the Russian:

project.chass.illinois.edu/cgi-bin//rview?REPOSID=8&ID=7970).

Vypuklye Figury, Gosudarstvennoe izdatel’stvo tekhniko-teore-

[35] N. Sloane, The On-Line Encyclopedia of Integer Sequences, A137615–A137618, 2008; http://oeis.org/.

Jaglom and Boltjanski, Konvexe Figuren, VEB Deutscher Verlag

ticheskoi literatury, Moscow and Leningrad, 1951). [40] H. Ziegler, Ernst Meißner, Z. angew. Math. Mech., 19 (1939), 192.

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Mathematical Entertainments

Michael Kleber and Ravi Vakil, Editors

Creating Clueless Puzzles GERARD BUTTERS, FREDERICK HENLE, JAMES HENLE, AND COLLEEN MCGAUGHEY1

his is an invitation and a report. The report is of our efforts to create puzzles of a special kind. The invitation is to join us in that merry task. The report will cover the history of the genre, something of the mechanics we are using, and examples of our work. As we worked (and played), the variety of clueless puzzles grew. We’ll talk a little at the end about aesthetics and where the field might go.

T

How It Started

This column is a place for those bits of contagious mathematics that travel from person to person in the

Of course the puzzles aren’t actually clueless. The project began when one of us looked at a standard sudoku puzzle and thought there was something inelegant about the numbers. Would it be possible to create a sudoku-like puzzle in which there were no numerical clues? There would have to be something else, of course. Perhaps there could be odd-shaped regions . . .? After some thought and fruitless attempts to create a 4 9 4 or a 5 9 5 puzzle, the first ‘‘clueless’’ was discovered.

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

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]

1

With thanks to the editor and reviewers!

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DOI 10.1007/s00283-011-9204-9

The instructions are to place the digits 1, 2, 3, 4, 5, 6 in the cells to form a Latin square (no digit appearing twice in a row or column) in which the numbers in each region add to the same sum. ‘‘Clueless’’ is now clear. We mean the absence of numerical clues. The puzzle above seems impossible at first. But the sum of the numbers in any row is 1 + 2 + 3 + 4 + 5 + 6 = 21. That means the total sum of the numbers in the entire square is 21 9 6 = 126. Since there are nine regions, each region must sum to 126/9 = 14. Now look at the straight, three-cell region at the bottom left. A little thought tells you that the numbers in this region could only be 6, 5, and 3—no other triple of distinct digits sums to 14. The three-cell region just above it can’t have the same three numbers. A little more thought shows you it must be 6, 4, and 4. Thus, the numbers at the top of the left column must be 1 and 2. That tells us that the remaining numbers in the top left region must be 6 and 5.

12

5 6 4 6 4

653

It’s a pleasant task filling in the rest of the puzzle. A catalog of the four-number combinations that add to 14 is useful.2 All the puzzles in this paper will require taking a set of initial (non-zero) natural numbers and placing copies in a square, with all the numbers distinct in every row and column. Thus, we can actually eliminate all numbers from the description of the puzzle.

4 9 4 puzzle with r regions each summing to s. The sum s must be at least 4, since 4 will be in the square. But s = 4 is not possible. Every region with a 3 would also have to have a 1. So any solution would have four two-cell regions with 3 and 1. Then switching the 3s and 1s would give us a second solution to the puzzle. For a similar reason, s can’t be 5. Clearly s = 40 is absurd. We suspected none of the others (s = 8, 10, 20) were possible (we couldn’t find any). But then to our great surprise we happened upon this,

Finding Small Puzzles We looked first for smaller puzzles of this type. Of course there is a 1 9 1 puzzle. At the very least, an elegant puzzle must have a unique solution. We quickly convinced ourselves there were no 2 9 2 or 3 9 3 puzzles. We did find a 5 9 5 puzzle.

a 4 9 4 clueless with just two regions. Try it. There’s only one solution! We thought we found a different 6 9 6 clueless puzzle, one with 14 regions, each adding to 9. But then a computer program we wrote to solve puzzles found that our puzzle had more than one solution.

Larger Puzzles

Now that you know the trick of finding what the sum of each region is, we write that sum (15, in this case) next to the square. But this is just because we’re friends. Underneath, the puzzles are clueless. We looked hard at the 4 9 4 square. The numbers would add to 40. In theory, if rs = 40 there might be a

The puzzles are enjoyable in themselves, but the real entertainment is finding them. They aren’t easy to find. Ken Ken puzzles, which these resemble, are ubiquitous. And a grid of numbers satisfying the sudoku rules can be turned into a puzzle by adding sufficient clues. Clueless puzzles, on the other hand, are rare. We tried writing programs to find clueless puzzles. We looked at the case of 6 9 6 squares with each region summing to 9. We examined all 812,851,200 Latin squares. For each square, we found all clueless partitions for which that square was a solution. We used this information to find

AUTHORS

......................................................................................................................................................... GERARD BUTTERS taught economics at

FREDERICK HENLE is a lead software devel-

Princeton, worked on consumer protection issues for the Federal Trade Commission, and now plays and teaches piano. This project was a welcome return to his mathematical roots, which include an M.S. from the University of Chicago.

oper at athenahealth. He was a violinist in the Maryland Symphony Orchestra and taught computer science at Mercersburg Academy.

The Federal Trade Commission (retired) Washington, DC USA e-mail: [email protected] 2

Technology athenahealth, Inc. Watertown, MA USA e-mail: [email protected]

The answers to all the puzzles in this paper can be found at http://www.math.smith.edu/*jhenle/cluelessanswers/

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all partitions with only one solution. It took an iMac an hour to find them. There are only 640. Here’s one.

Quite early we imagined a puzzle in which the sums of the regions, rather than being all the same, had to be all different. We haven’t found a puzzle with that simple clue; getting a puzzle with a unique solution appears difficult. But with an additional condition we found something nice.

9

In the case of 9 regions, where the region sums are 14, it took our machine 3 days to find the 989,720 clueless boards. To resolve the 6 9 6 case of 7 regions adding up to 18 would take our program hundreds of years. Puzzles on the 7 9 7 square are even worse; there are over 61 trillion Latin squares of order 7. Using similar methods we proved that there are no 4 9 4 puzzles except variations of the one we found earlier.

For this puzzle, the challenge is to fill in numbers so that the region sums form a sequence of consecutive numbers. We also have a 5 9 5 of this sort.

New Ideas We thought of putting blanks in the square, one in each row and column. The puzzles are still clueless. In the 6 9 6 square, for example, each row and column would have the digits 1, 2, 3, 4, 5. We have found, so far, examples of four species of 6 9 6 squares:

Then one of us imagined a square in which the clue was only that if one region had more cells than another, then the sum of the larger region had to be less than the sum of the smaller. We have, so far, one example.

......................................................................................................................................................... JAMES HENLE is a professor at Smith Col-

COLLEEN MCGAUGHEY is an undergradu-

lege. He has worked in set theory, geometry, nonstandard analysis, combinatorics, economics, and finite games. He is the author, with Tom Tymoczko and Jay Garfield, of Sweet Reason: A Field Guide to Modern Logic and, with David Cohen, Calculus: The Language of Change.

ate at Smith College; she is majoring in marine biology with a minor in music. Although her intended career field is oceanography, she very much enjoyed contributing to the theory of clueless puzzles.

Department of Mathematics and Statistics Smith College Northampton, MA USA e-mail: [email protected] 104

THE MATHEMATICAL INTELLIGENCER

Smith College Northampton, MA USA

That would guarantee that all same-sized regions have the same sum and all same-summed regions have the same size. But we haven’t found a puzzle like this. Symmetric patterns are more elegant than asymmetric patterns. There are some nice symmetric clueless puzzles.

The four puzzles we showed earlier with blocks could be called ‘‘5 in 6’’ puzzles (the digits are 1-5 and they are set in a 6 9 6 square). That name prompted us to imagine what a ‘‘6 in 5’’ puzzle might be. After some fooling around, we found a 7 in 5:

On the other hand, we found a lovely symmetric 5 9 5 which unfortunately has more than one solution. But there are only two solutions and they’re symmetric.

The instructions are to place digits from 1 to 7 in the square, never using the same digit in a row or column, in such a way that the sums of the regions are all the same. There is, of course, only one solution. Finally, there is the reverse problem. These could be called ‘‘clueful’’ puzzles. Here’s one. 4 1 5 2 6 3

3 5 4 6 1 2

1 4 3 5 2 6

5 2 6 4 3 1

2 6 1 3 4 5

6 3 2 1 5 4

We like the puzzle, and we don’t as yet know if a clueless 5 9 5 with 3 regions (and a unique answer) is possible. Finally, one could argue that an elegant puzzle should have an elegant solution, or at least a nice, deductive path to a solution. The largest clueless puzzle we have found,

The challenge is to divide the square into fourteen regions, each with the same sum. There is a unique answer and by happy chance it is itself a clueless puzzle.

Degrees of Elegance The motivation for clueless puzzles was elegance. Elegance remains an issue in subtle ways. The ‘‘anti-monotonic’’ puzzle was slightly inelegant in that two same-sized regions have different sums.

1 2 4 5 3

5 4 3 2 1

2 5 1 3 4

3 1 2 4 5

4 3 5 1 2

It would be nice to have a puzzle with the requirement, Size(A)  Size(B) , Sum(A)  Sum(B):

doesn’t have such a path, at least, not one we have found. The puzzle was discovered by computer. That it has a solution and a unique one was also discovered by computer.

What Next? The reader must have questions in mind already. Of the sorts of puzzles described here, what else is possible? Are there more species of 6 9 6 puzzles, with and without blocks? Are there larger clueless puzzles? The inventing of puzzles has just begun. What about puzzles on cylinders, Mo¨bius strips, doughnuts, and Klein bottles? What about puzzles with more than one block per row and column? If you have ideas, discoveries, opinions, let us know. Join us! Visit our website: http://www.math.smith.edu/ *jhenle/clueless/.

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Statistical Lament (after Joni Mitchell) Robert J. MacG. Dawson

Rows and flows of lines and spots, Histograms and digidots, Box-and-whiskers, quantile plots: I’ve looked at stats that way. But is it just a picture show? If something’s there how will you know? And if they say it isn’t so What is there left to say? I’ve looked at plots from both sides now, Transformed and spun them: still, somehow, It’s plots’ illusions I recall, I really can’t trust plots at all. Dainty ladies tasting teas Rejecting null hypotheses At standard probabilities: I’ve looked at stats that way. But is it just a lottery? At four percent they all agree; At six percent the referee Says ‘‘take this junk away.’’ I’ve looked at tests from both sides now: Retain, reject – and still somehow I look at tests and I recall I really can’t accept at all. Ways of Bayes and likelihood, We’ll teach the things we know we should, The True, the Beautiful, the Good And blow the clouds away. But now my class are acting strange, They tell the Dean that I’m deranged And that they need (till customs change) Their tests and EDA. From both sides now I’ve looked Bayes o’er, Both prior and posterior: It’s mass confusion I recall, I really can’t teach Bayes, at all. Department of Mathematics & Computing Science, St. Mary’s University Halifax, NS B3H 3C3 Canada e-mail: [email protected] This is reprinted with permission from the site http://www.lablit.com/article/646.

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DOI 10.1007/s00283-011-9234-3

Mathematically Bent

Colin Adams, Editor

The Book COLIN ADAMS aul Erdo¨s liked to speak of ‘‘The Book,’’ which was where God had recorded the most elegant mathematical proofs. In 1985, he said, ‘‘You don’t have to believe in God, but you should believe in The Book.’’

P 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]

Next. May I help you? Yes, I believe so. I hope I have been in the right line. I waited for almost a year. What’s your hurry? Well, I’m sorry if I seem impatient, but I want to get access to The Book. You’re a mathematician, I take it? Well, yes. I was told that upon reaching heaven, we would get access to The Book. We could look up all of the most elegant solutions to the difficult math problems that we devoted our careers to solving. I don’t know who starts these rumors. But it is not quite that simple. It’s not? First of all, do you have any idea how big The Book is? I don’t. It contains every elegant proof of every result that has ever been found or ever will be found, or even will not be found. So it’s big. Yes. Well, do you have it on disk? Do you see any computers here? Do you see any outlets to plug the computers into? Do you see any walls on which the outlets could sit? We don’t have computers. We’re old school. It’s not an e-Book. It is a book Book. That’s okay. Can I see the volume on group theory? It is a single book. Not a collection of volumes. It isn’t The Books. It is The Book, a single book that is three miles thick. That sounds a bit unwieldy. But nevertheless, I would like to look up a particular result. For instance, can I see how to prove that the only subgroup of order 4 in the Rachland group is cyclic? Are you sure you want to see it? Yes, I want to see it. But then you will have lost your opportunity to think about it yourself. I already spent 27 years thinking about it. Oh, 27 years. Well, why didn’t you say so? That is a long time. Especially when you consider you will be here for eternity. Look, I am tired of thinking about it. Can I just see the solution?  2011 Springer Science+Business Media, LLC, Volume 33, Number 3, 2011

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Very well. That is the book there lying on its side. It’s page 3,734,322. Come back when you’re done. (Returns two hours later.) That was kind of disappointing. I tried to tell you. The satisfaction factor isn’t high when you see how someone else did it. The most disappointing part was that Rulenko solved it. I hate that guy. Oh, well. But maybe I could work on the related problem. I could show that the subgroups of order 6 are cyclic. There’s no time limit here. Eventually, I would get it. Yes, but you would not be given credit in The Book. What? Do you mean to say that if I prove something in my lifetime on earth, I get credit in The Book, but if I prove it once I arrive in heaven, I don’t get credit, even if no one has already proved it previously? That’s right. But that isn’t fair. It is fair. If someone in heaven solves a problem, the people on earth will never know. So they work for years and years and solve a problem and then get to heaven to find out that their result was previously solved by some dead guy. Now how would that look? And anyway, once you get to heaven, you have unlimited time to work on a problem. Do you really think it is fair that mortals should have to compete with immortals? The mortals get fewer than 100 years to solve a problem. While immortals can work on it for 1000 or more if they want. Okay, okay. Never mind about that. Can I just look up the elegant proof of the four-color theorem? I spent ten years trying to find one, and I would love to see that proof. You know, that is the single most common request. I’m not surprised. But we cannot grant that request. Why not? There isn’t one. What do you mean there isn’t one? Just that. There isn’t one. About as elegant as you are ever going to get is the original Appel-Haken proof. But that’s not elegant at all. It involves a computer proof that checks 10,000 cases. Right. So it’s not in The Book. But why isn’t there an elegant proof? It’s not my decision. Okay, then why didn’t God make an elegant proof? God doesn’t make up mathematics. He can’t just make things true, make them logically follow from a set of axioms, if they do not. Then it would be logically inconsistent. So God isn’t omnipotent? Watch your tongue, buddy. Of course he is omnipotent, but if he makes 1+1 = 3, then the universe, earth, everything else disappears in a logical inconsistency. So what would be the fun in that? Okay, so now I want to look up the proof of the Witherspoon Conjecture. I want to see the elegant proof of it. And you don’t want to think about it yourself? It’s already solved, I just want to see it. 108

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Okay, hold on, Whiteside, width of a knot, Willing cohomology. Ah, here it is. Witherspoon Conjecture. Formulated by Charles Witherspoon, February 12, 1979. Actually, I formulated it February 11, but just told people about it February 12. Oh, so you are Charles Witherspoon. Yes, I am. I made up the conjecture. Ah. I see. Congratulations. Well, here it is, page 6,734, 321. Knock yourself out. Go ahead. Take a look. (Returns an hour later.) It said it was solved by Karen Suinkletter. Yes, that is correct. A 12-year-old girl from Toledo. But that’s wrong. I solved the Witherspoon Conjecture. I published the result in the Journal of Number-Free Theory, December, 1993. And the elegant solution the book had was the way I did it. My name should be there. I’m sorry, but it says here that Karen Suinkletter solved it. She wrote it down on a gum wrapper in July, 1992, and then threw it away. And that counts? Yes, I am afraid it does. She published it, just not in a conventional journal. I protest. That’s nice. No, I mean I want to lodge a formal protest. I want The Book to be fixed. The Book is already fixed. We cannot change anything. But you do change things. Every time someone proves a new elegant result, the book is changed to contain it. No, The Book already contains it. All right then, you change the credit for the result. No, we already know who is going to prove which result. It is already listed back to the beginning of time and forward to the end of time. Well, I’m not happy about that. Sorry. Are we through here? There are a lot of people waiting in line. No, I would like to see the proof that semi-upper left coset multiplicity implies devolutory minor self-multiplication. Page 23,672,445. Help yourself. (An hour later.) That wasn’t the proof. It was a counterexample. Yes, turns out it isn’t true. And it said the counterexample was discovered by Wafflepunklem. What kind of a name is Wafflepunklem? Wafflepunklem is a fipplepicker from the planet Turlemonde. Wait a minute. You mean The Book contains results proved by aliens on other planets? Of course. Why would we only include results proved on earth? Math is math, wherever it is done. But then why, if you include their results in The Book, don’t you include the aliens in heaven? Have you noticed a few blue residents? Perhaps some furniture that moves? Now that you mention it… There you go. Anyway, it’s been nice talking to you. Next. Wait a minute. I want to see the proof of the Riemann Hypothesis. Sorry. You only get three looks a visit.

What? Nobody told me that. You have to read the Welcome Manual. What Welcome Manual? It’s in the desk drawer by the phone in your room, right next to the Gideon Bible.

But I didn’t know that. Please, I really would like to see this one. Sorry. You have to get in line again, fella. And while you’re waiting, take the opportunity to try to solve it yourself. It’ll help pass the time. Next.

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Geometry Where Direction Matters— Or Does It? HORST MARTINI, MARGARITA SPIROVA

uclidean space is uniform and ‘‘isotropic’’, that is, distances are equally determined for all directions. However, there are many phenomena leading to non-isotropic situations. The taxi driver in New York needs such a non-Euclidean unit circle for measuring distances around blocks correctly. Such a circle turns out to be a square with diagonals parallel to the streets and avenues, inducing the so-called Manhattan norm. A good example from physics is crystal growth; crystals’ shapes are mostly polyhedral, not spherical. One can model such situations by the assumption that units for measuring lengths are different in different directions. Then ‘‘unit spheres’’ are no longer Euclidean, but can even have a polygonal or polyhedral shape. Such an (often natural) assumption extends our wellstudied Euclidean space to the notion of finite-dimensional real normed spaces or, as we will call them, Minkowski spaces. These spaces are, in general, non-Euclidean but have the same homogeneous linear structure as their Euclidean subcase; for example, all translations are isometries. It clearly turns out that the shape of the unit circle is important, and that the geometry of such a space depends on this shape. (We remind the reader of the famous quote from the geometer Alfred Clebsch [14] to the effect that ‘‘it is the enjoyment of shape in the higher sense that makes the geometer’’.) Although the foundations of these geometries were introduced by Hermann Minkowski [44] in connection with problems from number theory, Bernhard Riemann, in his famous Habilitationsschrift [47], already suggested one such geometry (the finite-dimensional ‘p norm for p = 4) 30 years earlier. Nevertheless, the term Minkowski Geometry is common today, as attested by the monograph of Tony Thompson [57]. Note that this theory is clearly different from the Minkowskian Space-Time Geometry!

E

AND

KONRAD J. SWANEPOEL

We present some results from Minkowski Geometry that have not yet been mentioned in recent surveys or monographs. There are many hundreds of papers on Minkowski Geometry, widespread in different fields, but not really ordered and summarized. Thompson’s monograph [57] mainly covers the differential and integral geometric aspects. The survey chapters of Giannopoulos and Milman [20], Johnson and Schechtman [25], and Mankiewicz and Tomczak-Jaegermann [33] in the Handbook of the Geometry of Banach spaces deal mostly with asymptotic results, known as the local theory of Banach spaces. Our previous surveys [34, 39, 40], while attempting to give a broader scope for the term Minkowski Geometry, can only be taken as starting points to collect and order material from other parts of this field. Our main aim is to demonstrate the power of geometry as method in this discipline. In particular we consider the following three topics, all in the setting of Minkowski spaces: Elementary Geometry, Discrete Geometry, and Location Science. Terminology. The setting will be an n-dimensional real vector space, which we usually identify with Rn for convenience. Its zero or neutral element will be denoted by o. Given a subset S  Rn and a vector v 2 Rn ; the translate of S by v is the set S þ v ¼ fs þ v: s 2 Sg: Given an additional scalar factor k [ 0, the set kS þ v ¼ fks þ v: s 2 Sg is called a homothet of S. If S1 is a homothet of S2, then S1 and S2 will be called homothetic. A hyperplane is a translate of any (n - 1)-dimensional subspace of Rn : Any hyperplane cuts Rn into two closed half spaces. A convex body is a bounded, closed, and convex subset of Rn with interior points. A hyperplane H is a supporting hyperplane of the convex

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body B if H intersects B and B is contained in one of the closed half spaces determined by H. In the following text we use Mn to denote an n-dimensional normed or Minkowski space. This consists of an n-dimensional vector space, usually Rn ; together with a fixed convex body B that is centered, that is, B = -B, called the unit ball of Mn. The norm jjjj: M n ! R of Mn is induced by the unit ball according to the formula jjxjj ¼ inffk1 : kx 2 Bg: The unit ball can then be described in terms of the norm as expected: B ¼ fx 2 M n ; kxk  1g: The boundary of B, denoted by bd B, is the unit sphere of Mn. Any homothetical copy of B or bdB is called Minkowskian ball or sphere, respectively. We say that two balls B1 and B2 touch if they have non-empty intersection, but the intersection is contained in their boundaries: [ 6¼ B1 \ B1  bdB1 \ bdB2 : If there passes exactly one supporting hyperplane through each boundary point of B, the Minkowski space is called smooth. If bd B does not contain a non-degenerate line segment, then Mn is strictly convex. The most familiar examples of Minkowski spaces are given by the p-norms, p 2 ½1; 1: These spaces, denoted by ‘np , are defined to be Rn with norm !1=p n X p jjxjjp ¼ jjðx1 ; x2 ; . . .; xn Þjjp :¼ jxi j i¼1

if p 2 ½1; 1Þ; and jjxjj1 ¼ jjðx1 ; x2 ; . . .; xn Þjj1 :¼ maxfjx1 j; jx2 j; . . .; jxn jg: The space ‘n2 is the familiar n-dimensional Euclidean space. The unit ball of ‘n1 is the n-dimensional hypercube, and the

unit ball of ‘n1 the n-dimensional cross polytope (octahedron when n = 3). The Minkowski plane ‘21 gives us the Manhattan distance.

Some Elementary Geometry in Minkowski Spaces Some results from the Elementary Geometry of Minkowski planes are well known. For example, the standard construction of an equilateral triangle via two intersecting and equally sized circles, with the center of each lying in the other, still works, but one has to be careful when the circles contain segments (see also Section below for more general equilateral sets). On the other hand, the ‘‘side-side-side’’ congruence theorem for triangles no longer holds, and the Theorem of Pythagoras holds if and only if the plane is Euclidean [27, p. 182]. However, many Euclidean theorems have never been examined for the purpose of finding their Minkowskian analogues. Below we list some inspiring results in this direction.

Geometry of Circles in Minkowski Planes. Which properties of Euclidean circles remain valid in a plane with an arbitrary norm, or a strictly convex norm? One such property is the so-called ‘‘re-entrant property’’. Here is how it works in the Euclidean plane. Let there be given a circle C and a circle C0 of the same radius centered at p0 on C. If p1 is one of the intersection points of C and C0 (Figure 1, left), and C1 is the circle with the same radius as C centered at p1, then C and C1 intersect in p0. Again, let p2 be the second intersection point of C and C1. If C2 is the circle with center p2 and the same radius as C, then C and C2 intersect in p1. The fact that in this process the new circle always passes through a previous intersection point is called the reentrant property of a circle. This can be extended to strictly

AUTHORS

......................................................................................................................................................... HORST MARTINI studied in Dresden, Germany, near his hometown, and then taught mathematics, geography, and astronomy in a secondary school. After receiving his Ph.D., also in Dresden, he obtained his Habilitation from the University of Jena, and in 1993 he became Full Professor at the Chemnitz University of Technology. His research fields are classical and discrete geometry, convexity, and finite-dimensional real Banach spaces. His passions include travelling to places of geographical and historical significance and studying different genres of good music.

Fakulta¨t fu¨r Mathematik Technische Universita¨t Chemnitz D-09107 Chemnitz Germany e-mail: [email protected] 116

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MARGARITA SPIROVA was born in Bulgaria;

she earned her bachelor’s degree at the University of Shumen and graduate degrees at the University of Sofia. For the last three years she has been working in Germany, where, in 2010, she received her Habilitation. Her research interests are mainly concentrated in the areas of discrete geometry, convexity, and computational geometry. Most recently she has coauthored ‘‘Regular Tessellations in Normed Planes’’ with Horst Martini in Symmetry, Culture and Science (2011). Fakulta¨t fu¨r Mathematik Technische Universita¨t Chemnitz D-09107 Chemnitz Germany e-mail: [email protected]

Figure 1. The re-entrant property of the Euclidean circle (left), the Euclidean Flower of Life (middle), and a Minkowskian Flower of Life (right).

convex Minkowski planes, as shown by the following theorem.

T H E O R E M 1 (Kelly [26]). Every circle in a strictly convex Minkowski plane is re-entrant and goes around itself six times. One of the beautiful arrangements of Euclidean circles, the so-called Flower of Life (Figure 1, middle), is based on this re-entrant property. Due to the theorem of Kelly one can construct a Minkowskian Flower of Life (Figure 1, right). We continue with T¸ it¸ eica’s theorem, also called the three-circles theorem [6].

T H E O R E M 2 Let p1, p2, p3 be three distinct points on the unit circle C of a strictly convex, smooth Minkowski plane, and let xi + C, i = 1, 2, 3, be three circles different from C, each of which contains two of the three points pi. Then T3 j¼1 ðxj þ CÞ consists of precisely one point p. See Figure 2 (left) for the Euclidean version. This theorem still holds without any smoothness assumption [35] (Figure 2, right). The point p is called the C-orthocenter of the triangle p1p2p3, since in the Euclidean case it is the (classical) orthocenter of this triangle (i.e., the intersection point of its altitudes). This is not the only reason for retaining this term for Minkowski planes. In the more general framework of

......................................................................... originally from South Africa, earned his Ph.D. at the University of Pretoria. After two years in Germany during which he obtained his Habilitation, he moved to the London School of Economics. He is interested in any geometrical problem with a combinatorial flavour and likes neat proofs. Apart from geometry, he is fascinated by logic and set theory. In his spare time he is catching up on English literature such as Robinson Crusoe, the Harry Potter series and China Mie´ville’s wonderful novels. KONRAD J. SWANEPOEL,

Department of Mathematics London School of Economics London WC2A 2AE United Kingdom e-mail: [email protected]

Figure 2. The three-circles theorem in a strictly convex Minkowski plane.

Theorem 2, the vector from any vertex of p1p2p3 to the Corthocenter is orthogonal to the opposite side of the triangle. There exist various orthogonality concepts for Minkowski planes [3, 4, 5] which coincide with the usual orthogonality in the Euclidean plane. The notion we need here is called James orthogonality. Two vectors x; y 2 M n are called James orthogonal if jjx þ yjj ¼ jjx  yjj: In Theorem 2 and Figure 2 the vector p - pi is James orthogonal to the triangle side pj pk, {1, 2, 3} = {i, j, k} [35, Theorem 3.2]. Moreover, the points p1, p2, p3 and p (as well as the points x1, x2, x3, and o) form a C-orthocentric system, that is, each point of such a system is the C-orthocenter of the triangle formed by the other three [6, Remark 2]. Many interesting properties of orthocentric point systems in the Euclidean plane can also be extended to Minkowski planes [35, 42]; here is one of them [35].

T H E O R E M 3 Let p1, p2, p3, p4 be four distinct points in a strictly convex Minkowski plane lying on the circle x + kC. Let hi be the C-orthocenter of the triangle pj pk pl, where {i, j, k, l} = {1, 2, 3, 4}. Then (i) the points hi, hj, hk, pl form a C-orthocentric system; (ii) the points hi, hj, pk, pl lie on a circle with radius k. (See Figure 3.) In the Euclidean plane it is well known that for any triangle p1 p2 p3, its orthocenter p, centroid g ¼ 13 ðp1 þ p2 þ p3 Þ; and circumcenter x (the center of the unique circle through the vertices of the triangle) lie on the so-called Euler line of the triangle. In any Minkowski plane we may still define the centroid as before, as this is defined by the vector space (affine) structure. If the plane is smooth, any

Figure 3. Theorem 3 (left) and Miquel’s theorem (right). Ó 2011 Springer Science+Business Media, LLC, Volume 33, Number 3, 2011

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p1234, p1235, p1245, p1345, p2345 lie on a circle C12345; and so on ad infinitum.

Figure 4. In- and excircles of triangles.

triangle has a circumcenter x. (In Figure 4 (right) it can be seen that not every triangle in a Minkowski plane with a non-smooth unit ball has a circumcircle.) In general the circumcenter is not necessarily unique. If we assume that the plane is strictly convex, then for any triangle that has a circumcenter, the circumcenter is unique, and the following linear dependence holds as in the Euclidean case: 1 1 p þ x ¼ g: 3 3 Thus every triangle still has an Euler line [6, Remark 1]. The following phenomenon can be observed repeatedly: Statements true in the Euclidean plane for circles of different sizes have Minkowskian analogues only for congruent circles. A remarkable theorem from Euclidean geometry, true for circles of arbitrary sizes, is the so-called Miquel theorem or six-circles theorem. It plays an essential role in the Foundations of Geometry [8, p. 131]. The following is its Minkowskian analogue [6].

T H E O R E M 4 Let C be the unit circle in a strictly convex, smooth Minkowski plane, and let four points xi of C be given. If Ci, i = 1, 2, 3, 4, are the four translates of C (different from C) determined by pairs of neighboring points, then either there exists a proper translate of C passing through the four points yi, where yi 2 Ci \ Ciþ1 ; C5 ¼ C1 ; and yi 62 C; or yi = xi for i = 1, 2, 3, 4. A configuration of circles as in Theorem 4 is called an (83, 64)-configuration because it is formed by 8 points and 6 circles, with each point lying on 3 circles and every circle passing through 4 points. Theorem 4 remains valid without the assumption of smoothness [37, Theorem 4.2] (Figure 3, right). Moreover, by defining the points yi suitably, Theorem 4 can be extended to arbitrary Minkowski planes [53]. Theorem 2 has a natural extension to more than three initial circles, called Clifford’s chain of theorems [15, p. 262]. This extension also holds for all strictly convex Minkowski planes, but again only for circles of equal radii [36].

T H E O R E M 5 Let Ci = xi + kC, i = 1, 2, 3, 4, be four circles passing through a point p in a strictly convex Minkowski plane with unit circle C. Let pij be the second intersection point of the circles Ci and Cj. Then each triangle pij pjk pki, where i; j; k 2 f1; 2; 3; 4g with i = j, j = k, k = i, has a circumcircle Cijk. Furthermore, the four circles C123, C234, C341, C241 all pass through the point p1234. Let C5 = x5 + kC be a fifth circle through p. Then the five points 118

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Theorems related to circles having equal radii are also useful for solving certain covering problems in Minkowski planes. As an example, we point out a theorem whose configuration can be considered as a degenerate case (when the points yi coincide) of the configuration in Theorem 4. Note that such a regular 4-covering of a convex body K in the plane by four homothetical copies, each of the same size, is a covering of K by homothetical copies of the smallest possible ratio [30].

T H E O R E M 6 In a strictly convex Minkowski plane with unit circle C and unit disc B, let there be given four circles Ci ¼ xi þ kC; i ¼ 1; . . .; 4; passing through a point p such that Ci and Ci+1 do not touch each other, whereas Ci and Ci+2 touch each other (x5: x1, x6: x2). If pi+1 (p5: p1) is the second intersection point of Ci and Ci+1 and p1, p2, p3, p4 lie S on the same circle of radius l [ k, then 4i¼1 Bi is a regular 4-covering of p + lB, where Bi is the disc bounded by the circle Ci [37]. We mention one more result related to Minkowskian circles. The Apollonius problem is the problem of finding all circles touching three given circles. This problem has been solved for strictly convex, smooth Minkowski planes by the second author [52, Section 6].

Results Related to Triangle Geometry in Minkowski Planes. There are various ways to extend the concept of angle-bisector to Minkowski planes [39, x 4.8]. We mention here Glogovskii’s definition, where the angle-bisector of the convex angle ^pxq is the unique ray formed by the points each having the same Minkowskian distance to the rays ½x; pi and ½x; qi forming the angle ^pxq: According to this definition the three angle-bisectors of a triangle p1 p2 p3 intersect in a common point interior to the triangle, called the incenter of p1 p2 p3 [40, x 7.1]). Therefore, every triangle in a Minkowski plane possesses a Minkowskian incircle (touching all three sides of it). Figure 4 shows the construction of the three Glogovskii bisectors and the incircle of triangles in the smooth and in the strictly convex case. The radius of the incircle is said to be the inradius of p1 p2 p3. Moreover, every triangle p1 p2 p3 in a Minkowski plane has at least two Minkowskian excircles, that is, for at least two vertices pi the bisector of ^ pj pi pk ; fi; j; kg ¼ f1; 2; 3g; the bisector of the angle formed by the ray ½pk ; pj i and the ray opposite to ½pk ; pi i; and the bisector of the angle with ray ½pj ; pk i and the ray opposite to ½pj ; pi i intersect in a point that is called an excenter of the triangle p1 p2 p3 [7, Theorem 5]; see again Figure 4. The radii of the corresponding excircles are said to be the exradii of p1 p2 p3. The ith height of the triangle p1 p2 p3 is the minimal distance (in the norm) from pi to the line passing through pj and pk, where {i, j, k} = {1, 2, 3}. The following theorem directly extends results well known for the Euclidean plane [7].

T H E O R E M 7 In a Minkowski plane let p1 p2 p3 be a triangle with inradius k, exradii k1, k2, k3 (at most one of them can be 1), and corresponding heights l1, l2, l3. Then 1 1 1 1 ¼ þ þ ; k l1 l2 l3 1 1 1 1 ¼ þ þ ; k k1 k2 k3 2 1 1 ¼  ði ¼ 1; 2; 3Þ; li k ki   1 1 1 1 1 1 1 4 2þ 2þ 2 ¼ 2þ 2þ 2þ 2: l1 l2 l3 k k1 k2 k3 All these facts have generalizations for n-dimensional Minkowski spaces [7]. The concept of the nine-point (or Feuerbach) circle also extends nicely to Minkowski planes. In the Euclidean case it is well known that this is the circle which, for any triangle, passes through the feet of the three altitudes, the midpoints of the three sides, and the midpoints of the segments from the three vertices to the orthocenter. The next theorem clarifies the analogous situation in strictly convex Minkowski planes [6].

T H E O R E M 8 Let p1 p2 p3 be a triangle in a strictly convex, smooth Minkowski plane with circumcircle x + kC and orthocenter p. The circle 12 ðx þ pÞ þ 12 C (called the Feuerbach circle of the triangle p1 p2 p3) passes through six ‘‘remarkable’’ points, namely the midpoints of the sides of p1 p2 p3, and the midpoints of the segments ppi, i = 1, 2, 3; see Figure 5 (left). The smoothness assumption is unnecessary here [35]. Moreover, the Feuerbach circle of any triangle passes through three additional points (the intersections of a side and the line determined by the opposite vertex and the orthocenter) if and only if the plane is Euclidean [6, Theorem 6]. This is why the term ‘‘six-point circle’’ instead of ‘‘ninepoint circle’’ is used for strictly convex Minkowski planes. On the other hand, the Feuerbach circle is the locus of the midpoints of the segments py, where y moves through x + kC [35, Theorem 4.6]; see Figure 5 (left). The three circles with radius 12  k and centers at the midpoints of the triangle sides intersect in the center of the Feuerbach circle [35]. In the next theorem the notion of Feuerbach circle is extended to quadrilaterals [35].

T H E O R E M 9 Let p1, p2, p3, p4 be four pairwise distinct points in a strictly convex Minkowski plane lying on the circle x + kC. Then the Feuerbach circles of all four triangles

obtainable from {p1, p2, p3, p4} pass through a common point p, and their centers lie on the circle p þ 12  kC; see Figure 5 (right). We see that it is natural to call the circle p þ 12  kC the Feuerbach circle of the quadrilateral p1 p2 p3 p4.

Equilateral Sets We now discuss the problem of finding equilateral sets of points in a Minkowski space. This problem nicely illustrates the variety of mathematics needed, while at the same time it indicates the difficulties that are often faced in Minkowski Geometry. The arguments needed to prove the theorems that follow are mostly very geometrical and come from affine and convex geometry, the local theory of Banach spaces, as well as topology. Equilateral sets are not only of intrinsic interest; they have been used in Differential Geometry to find minimal cones in Minkowski spaces [31, 45]. For a longer exposition including proofs, see the third author’s survey [55]. A set S of points in an n-dimensional Minkowski space Mn is equilateral if, for some k [ 0; jjx  yjj ¼ k for all x; y 2 S; x 6¼ y: Let e(Mn) denote the largest cardinality of an equilateral set in Mn (by a simple compactness argument, there is a finite maximum). Theorem 1 already shows that e(M2) C 3 (see Figure 1, left). One may expect this argument to extend to higher dimensions, leading it is hoped to e(Mn) C n + 1 for all n-dimensional Minkowski spaces. Surprisingly, as we will see later in this text, this is only known for n B 4. Before discussing lower bounds, we first gain some perspective by looking at upper bounds for e(Mn).

Upper Bounds. It is well known (and an easy exercise in linear algebra) that e(‘n2 ) = n + 1: the maximum cardinality of an equilateral set in n-dimensional Euclidean space is n + 1. On the other hand, note that {±1}n, the set of the ± 1-vectors, is an equilateral set in ‘n1 containing 2n vectors, thus eð‘n1 Þ  2n : It is not so difficult to see that equality holds here. Petty [46] showed more generally that e(Mn) B 2n for all Minkowski spaces by observing that equilateral sets are also antipodal. Here a set S of points is called antipodal if for any two points p and q in S there exist two parallel hyperplanes, one passing though p, the other passing through q, such that S is contained in the closed slab bounded by the two hyperplanes. (For more on antipodality, see the survey of the first author and V. Soltan [38].) Using a result of Danzer and Gru¨nbaum on the maximum number of points in an antipodal set in an ndimensional vector space, Petty deduced the upper bound of 2n. It was rediscovered by P. S. Soltan [51]. The characterization of the equality case was given by Groemer [21].

T H E O R E M 10 For any Minkowski space Mn, e(Mn) B 2n. Equality holds if and only if Mn is isometric to ‘n1 [46, 51, 21].

Figure 5. Feuerbach circles.

Conversely, Petty showed that for any antipodal set S there exists a norm for which the set becomes equilateral. In fact, Ó 2011 Springer Science+Business Media, LLC, Volume 33, Number 3, 2011

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the norm can be taken to be the one for which the unit ball is the convex hull of the difference set S  S ¼ fx  y : x; y 2 Sg: If the norm is strictly convex, then an equilateral set is strictly antipodal. Here, the condition of strict antipodality on S means, as before, that for any two points p and q in S, there exist two parallel hyperplanes, one passing though p, the other passing through q, but now the remaining points S n fp; qg are in the open slab bounded by the hyperplanes. Gru¨nbaum [23] showed that a strictly antipodal set in 3-space has cardinality at most 5. From this, Petty deduced the first part of the following theorem. Lawlor and Morgan [31] proved the second part by giving an explicit construction [46, 31] .

THEOREM 11 For any strictly convex 3-dimensional Minkowski space M3, e(M3) B 5. There exists a strictly convex, C 1 Minkowski space M3 of dimension 3 for which e(M3) = 5. The same type of converse holds as in the case of antipodal sets: For any strictly antipodal set there exists a strictly convex norm in which the set becomes equilateral, since each nonzero point in S - S is now a vertex of its convex hull. Such a norm can even be chosen to be C 1 away from o. It is therefore of interest to find good upper bounds for the number of points in a strictly antipodal set in Rn : However, no upper bound better than the one of 2n - 1 that follows from Theorem 10 is known! Gru¨nbaum [23] conjectured an upper bound of 2n - 1. Surprisingly, Erd} os and Fu¨redi [18] constructed strictly antipodal sets of cardinality growing exponentially in the dimension. The best current lower bound of 3n/3 is due to Talata [11, Section 9.11]. Thus the following conjecture [19] becomes very interesting.

C O N J E C T U R E 12 There exists some constant e [ 0 such that if Mn is a strictly convex n-dimensional Minkowski space, then eðM n Þ  ð2  eÞn : Lawlor and Morgan [31] were interested in differentiable norms. They conjectured that, similar to the strictly convex case, e(M3) B 5 if M3 is smooth. This turned out to be false [49].

T H E O R E M 13 For any smooth 3-dimensional Minkowski space M3, e(M3) B 6. There exists a C 1 Minkowski space M3 of dimension 3 such that e(M3) = 6. The proof of the upper bound of 6 in the above theorem consists essentially of a classification of all 3-dimensional antipodal sets. No short or intuitive proof is known. On the other hand, the second part of the above theorem can be explained in an intuitively appealing way: Geometrically, the existence of an equilateral set of size 6 for some C 1 norm means that for some 3-dimensional C 1 convex body, there are 6 of its translates that pairwise touch each other. By Theorem 11 the unit ball of such a space necessarily contains line segments on its boundary. The following describes the construction of a C 1 unit ball that admits 6 pairwise touching translates. For the construction we need a Reuleaux triangle. This is the intersection of three (Euclidean) circular discs with

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Figure 6. The mutual supporting plane of 6 pairwise touching translates of the C 1 unit ball of Theorem 13.

centers on the vertices of an equilateral triangle, and radii equal to the side length of the equilateral triangle. Figure 6 demonstrates that there exist three pairwise disjoint translates D þ vi ; i ¼ 1; 2; 3; of a Reuleaux triangle D in the plane as well as three pairwise disjoint translates D  vi ; i ¼ 1; 2; 3; of D such that each translate of D intersects each translate of D: The vectors vi are three Euclidean vectors at 120° angles. To construct the unit ball B, start with a Reuleaux triangle D in a (horizontal) plane P in R3 that misses the origin, and put its negative D in the plane P: Let v be a vector normal to P such that P þ v ¼ P: In the parallel plane P0 ¼ P þ 12 v through the origin, we need the vectors 12 vi ; i ¼ 1; 2; 3; to be on the boundary of the unit ball B. This will ensure that the translates B + vi end up pairwise touching. It is then possible to close up the two pieces D and D into a C 1 body B such that its boundary surface passes through 12 vi : The six required translates of B will then be B + vi, B + v + vi, i = 1, 2, 3. These 6 translates all have P as a supporting plane. Any Bi touches any B þ v þ vj ði; j 2 f1; 2; 3gÞ in P; as in Figure 6.

Lower Bounds. It is not difficult to find n + 1 equilateral points in ‘np : The standard unit vector basis e1 ; e2 ; . . .; en together with an appropriate multiple of the all-one vector ð1; 1; . . .; 1Þ will clearly be equilateral. In ‘n1 one can even find 2n equilateral points: ei ; i ¼ 1; . . .; n: However, for an arbitrary space it seems surprisingly difficult to find a large equilateral set. In general one would hope for the following conjecture [22, 46, 57, 45]. C O N J E C T U R E 14 If Mn is an n-dimensional Minkowski space, then e(Mn) C n + 1. We have already observed the truth of this conjecture for n = 2. Petty [46] proved the conjecture for n = 3, and Make’ev [32] for n = 4. In both cases non-trivial topological results were needed. Indeed, in all lower bounds for e(Mn) that have been found, some topological result occurs in the proof, starting with the intermediate-value theorem for the case n = 2 (implicitly used in Kelly’s Theorem 1). The case n = 3 uses the fact that the plane with a point removed is not simply connected [46]. The topological tools for n = 4 are even more sophisticated [32].

pffiffiffiffiffiffiffiffi c log n follows that for any M n ; eðM n Þ  log log n for some constant c [ 0. In [56] the lower bound was improved further by using, instead of Dvoretzky’s theorem, a theorem of Alon and Milman [1] asserting that any Minkowski space contains a large subspace that is either close to Euclidean space or close to some ‘k1 : More precisely, for any e [ 0; there exists cðeÞ [ 0 such that any n-dimensional Minkowski space X n pffiffiffiffiffiffiffiffi contains a Y k where k  e cðeÞ log n ; and either dðY k ; ‘k2 Þ\1 þ e or dðY k ; ‘k1 Þ\1 þ e: This result is applied for some fixed e [ 0; and then combined with further arguments (including an analogue of Theorem 15 for ‘n1 ), to obtain the following result. Figure 7. Petty’s maximal pairwise touching packing of double cones over the Euclidean ball.

Petty discovered the following obstruction to any attempt at a simple-minded induction proof [46]. He constructed, for each dimension n C 4, a norm such that there exists an equilateral set of 4 points that is maximal with respect to inclusion. The norm is the following: sffiffiffiffiffiffiffiffiffiffiffiffiffi n X xi2 : jjxjj ¼ jjðx1 ; . . .; xn Þjj :¼ x1 þ i¼2

The unit ball is a double cone over a Euclidean ball of dimension n - 1. In Figure 7 a maximal packing of four pairwise touching unit balls is shown, where it can clearly be seen that the packing is maximal. Another result obtained by topological means was found independently by Brass [12] and Dekster [16]. It asserts that spaces sufficiently close to the Euclidean space behave exactly as the Euclidean space as far as equilateral sets are concerned. To quantify ‘‘sufficiently close’’, we introduce a well-known distance notion between Minkowski spaces of the same dimension. The BanachMazur distance d(X n, Y n) between two n-dimensional Minkowski spaces X n and Y n is the infimum of all c C 1 such that there exists a linear isomorphism T: X n? Y n satisfying jjT jjjjT 1 jj  c: Since the spaces X n and Y n are finitedimensional, this infimum is of course attained. Equivalently, d(X n, Y n) equals the smallest c C 1 such that for some linear bijection T: X n? Y n, the unit balls B(X n) and B(Y n) satisfy T ðBðX n ÞÞ  BðY n Þ  cT ðBðX n ÞÞ:

T H E O R E M 15 Let Mn be an n-dimensional Minkowski 1 : space with Banach-Mazur distance dðM n ; ‘n2 Þ  1 þ nþ1 n Then any equilateral set in M of at most n points can be extended to an equilateral set of n + 1 points. In particular, e(Mn) C n + 1 [12, 16].

T H E O R E M 16 [56] There is an absolute constant c [ 0 such that forpffiffiffiffiffiffiffi any n-dimensional Minkowski space ffi M n ; eðM n Þ  e c log n : Kusner’s Problems. As observed earlier, e(‘n1 ) C 2n and e(‘np ) C n + 1 for 1\p\1: Kusner [24] asked whether equality holds in both cases, that is, whether e(‘n1 ) = 2n and e(‘np ) = n + 1 holds for all 1\p\1: It was shown by Bandelt, Chepoi, and Laurent [13] that e(‘31) = 6, and by Koolen, Laurent, and Schrijver [28] that e(‘41) = 8. For p an even integer it is easy to show an upper bound of about pn using linear algebra (observation of Galvin; see Smyth [50]). The currently best known upper bound for even integers is given in the following theorem [54].

T H E O R E M 17 For p an even integer and n C 1 we have eð‘np Þ 



ðp2  1Þn þ 1 if p 0 ðmod 4Þ; p if p 2 ðmod 4Þ: 2n þ 1

In particular, e(‘n4 ) = n + 1. As observed by Smyth [50], Theorem 15 implies the following result.

C O R O L L A R Y 18 (Smyth [50]) There exists c [ 0 such that c e(‘np ) = n + 1 for all p with 2  p\ n log n:

Smyth [50] combined a linear algebra method with the Jackson theorems from approximation theory to show that e(‘np ) \ cp n(p+1)/(p-1) for all 1\p\1: Alon and Pudla´k [2] combined Smyth’s approach with a result on the ranks of approximations of the identity matrix to deduce the following improved bound [2].

T H E O R E M 19 For each p 2 ½1; 1Þ there exists cp [ 0 By the celebrated theorem of Dvoretzky, any Minkowski space has a relatively large almost Euclidean subspace. The following precise formulation states the currently best known bounds due to Schechtman [48]. There exists a positive constant c [ 0 such that for any e [ 0; any n-dimensional Minkowski space X n contains a k-dimensional subspace Y k where k  ceðlog 1=eÞ2 log n; such that dðY k ; ‘k2 Þ  1 þ e: Combining this bound with Theorem 15, it

such that e(‘np ) \ cp n(2p+2)/(2p-1). They also combined the rank method with a probabilistic argument (randomized rounding) to prove the following result [2].

T H E O R E M 20 For any odd integer p there exists cp [ 0 such that e(‘np ) \ cp n log n.

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Perhaps surprisingly, for 1 \ p \ 2 there are examples showing e(‘np ) [ n + 1 if n is sufficiently large (necessarily depending on p, as Theorem 15 shows) [54]. The simplest such construction is as follows. Note that the set of four ± 1-vectors in R3 with an even number of -1’s, that is,

Given an n-dimensional Minkowski space Mn with norm jjjj and unit ball B, the dual norm, on the dual space ðM n Þ ; is defined as jj/jj ¼ maxjjxjj¼1 /ðxÞ for any functional / 2 ðM n Þ : If we identify Mn and ðM n Þ with Rn ; then the dual unit ball B is the polar body of B:

fð1; 1; 1Þ; ð1; 1; 1Þ; ð1; 1; 1Þ; ð1; 1; 1Þg;

B ¼ fy : hx; yi  1 for all x 2 Bg:

is equilateral in ‘3p for any p. Now consider R6 to be the direct sum R3  R3 ; and place the above four vectors in each copy of R3 : This gives the following 8 vectors in R6 :

Given a functional / 2 ðM n Þ of norm 1 and a point x 2 M n ; define the cone Cðx; /Þ ¼ x  fa : /ðaÞ ¼ jjajjg; that is, C(x, /) is the translate by x of the union of all rays from the origin through the exposed face /-1(- 1) \ B of the unit ball B of Mn. A norming functional of x 2 M n is a / 2 ðM n Þ such that jj/jj ¼ 1 and /ðxÞ ¼ jjxjj: The hyperplane /1 ð1Þ ¼ fy 2 M n : /ðyÞ ¼ 1g is then a supporting hyperplane of the unit ball at x. By the separation theorem, each x 2 M n has a norming functional. A Minkowski space is smooth iff each x = o has a unique norming functional. It is known that ðM n Þ is isometric to Mn, and that Mn is smooth iff ðM n Þ is strictly convex. A point x0 is a Fermat-Torricelli point (or FT point) of distinct points xP 1 ; . . .; xm in a Minkowski space if x = x0 minimizes x 7! m i¼1 jjx  xi jj: The Fermat-Torricelli locus (or FT locus) FTðx1 ; . . .; xm Þ of x1 ; . . .; xm is the set of all FT easily implies points of x1 ; . . .; xm : The triangle inequality P that FT(x, y) = [xy]d. Since the mapping x 7! m i¼1 jjx  xi jj is a convex function that increases without bound as jjxjj ! 1; the FT locus of any finite set is always non-empty, compact, and convex. A star configuration (of degree m) in a Minkowski space is a set of segments fxxi : i ¼ 1; . . .; mg emanating from the same point x, with xi = x for all i. Such a configuration {xxi} is pointed if there is a hyperplane H through x such that the interior of each segment xxi is in the same open half space bounded by H. A floating Fermat-Torricelli configuration (or floating FT configuration) is a star configuration fx0 xi : i ¼ 1; . . .; mg such that x0 is an FT point of fx1 ; . . .; xm g; and an absorbing Fermat-Torricelli configuration (or absorbing FT configuration) is a star configuration fx0 xi : i ¼ 1; . . .; mg such that x0 is an FT point of fx0 ; x1 ; . . .; xm g: The following characterization of FT points extends the classical characterization in the Euclidean situation [17].

fð1; 1; 1; 0; 0; 0Þ; ð1; 1; 1; 0; 0; 0Þ; ð1; 1; 1; 0; 0; 0Þ;

ð1; 1; 1; 0; 0; 0Þ;

ð0; 0; 0; 1; 1; 1Þ; ð0; 0; 0; 1; 1; 1Þ;

ð0; 0; 0; 1; 1; 1Þg:

ð0; 0; 0; 1; 1; 1Þ;

It is easily checked that this set is equilateral when p ¼ log 3= log 2 ¼ 1:5849. . .: A generalization of this, using Hadamard matrices, provides the following result [54].

T H E O R E M 21 For any k 2 N; if log2 ð4  24k Þ\p 

4 Þ; then log2 ð4  2kþ1

eð‘np Þ 

2kþ1 n  1: 1

2kþ1

3 n For example, if 1\p  log log 2 ; then e(‘p ) C 4n/3 - 1. For p 2 ½1; 2Þ close to 2 this theorem implies that e(‘np ) [ n + 1 if p \ 2 - c/n for some constant c [ 0 (cf. Corollary 18).

The Fermat-Torricelli Problem in Minkowski Spaces The famous Fermat-Torricelli problem (in Location Science also, historically incorrectly, called the Steiner-Weber problem) asks for the unique point x minimizing the sum of distances to arbitrarily given points x1 ; . . .; xm in Euclidean n-dimensional space [29], [10, Chapter II]. Since in arbitrary Minkowski spaces the solution set (called the Fermat-Torricelli locus) to this problem is not necessarily a singleton, one is motivated to study geometrically also the shape of this set in terms of the shape of the unit ball. To this end, we introduce the following terminology. We denote the dsegment [9, x 9] from x to y by ½xyd :¼ fz 2 M n : jjx  zjj þ jjz  yjj ¼ jjx  yjjg: A d-segment is a convex set centered at 12 ðx þ yÞ; and is not necessarily a segment (see Figure 8 below). The notion of d-segment was introduced by Menger [43, p. 80], who gave with it a not-well-known, historically early contribution to Location Science.

a

x

b

B o

T H E O R E M 22 Let x0 ; x1 ; . . .; xm be points in a Minkowski space. (1) If x0 6¼ x1 ; . . .; xm ; then fx0 xi : i ¼ 1; . . .; mg is a floating FT configuration iffP each xi - x0 has a norming functional /i such that m i=1/i = o. (2) If x0 = xj for some j ¼ 1; . . . m; then fx0 xi : i ¼ 1; . . .; m; i 6¼ jg is an absorbing FT configuration iff each xi - x0 (i = j) has a norming functional /i such that    X   m  /i    1:    i6¼j i¼1

−x Figure 8. A d-segment in a Minkowski plane. 122

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From this it follows that if fox1 ; . . .; oxm g is a floating FT configuration, then fox1 ; . . .; oxm1 g is an absorbing FT configuration.

T H E O R E M 23 In any Minkowski space Mn, suppose we are given a point p 2 FTðx1 ; . . .; xm Þ n fx1 ; . . .; xm g: Let /i be a norming functional of xi - p for each i ¼ 1; . . .; m; such P that m i=1/i = o. Then FTðx1 ; . . .; xm Þ is the intersection of the cones Cðx1 ; /1 Þ; . . .; Cðxm ; /m Þ [17]. Comparing this with the notion of d-segment, we obtain the following result [41].

P R O P O S I T I O N 24 For any x; jjxjj ¼ 1; in a Minkowski plane we have that FT(x,-x) is the segment from x to -x whenever x is not in the relative interior of a segment on the boundary of the unit ball, whereas if x is in the relative interior of a maximal segment ab on the boundary of the unit ball, then FT(x,-x) is the (unique) parallelogram with sides parallel to oa and ob having x and -x as opposite vertices; see Figure 8. It easily follows that the FT locus of a finite set of points in a Minkowski plane is always a convex polygon (possibly degenerating to a segment or a point). The following examples from [41] suggestively illustrate how the shape of the FT locus in Minkowski planes can be described by using this proposition, Theorem 23 and the notion of d-segments.

E X A M P L E 1 Let the unit ball of the Minkowski plane M2 be the parallelogram with vertices {±x, ±y}, where x and y are any two linearly independent vectors. If x and y form the standard basis of R2 ; then we obtain the Manhattan norm ‘21, for which the unit ball is a square (see Figure 9). Let A ¼ f 12 ðx þ yÞ; 12 ðx  yÞg: We now use Theorem 22 to find FT(A). We let /1 be the norming functional of 12 ðx þ yÞ; that is, the (unique) functional in (M2)* of norm 1 for which /-1 1 (1) is the line through x and y. Similarly, we let /2 be the norming functional of 12 ðx  yÞ; that is, the functional of norm 1 for which /-1 2 (1) is the line through x and -y. Then the norming functionals of  12 ðx þ yÞ and  12 ðx  yÞ are -/1 and -/2, respectively. By Theorem 22 we then have that o is an FT point of A (since the sum of the norming functionals is o). By Theorem 23 we have

   1 þ yÞ; /1 \ C ðx  yÞ; /2 2     1 1 \ C  ðx þ yÞ; /1 \ C  ðx  yÞ; /2 :

FTðAÞ ¼ C



1 ðx 2

2

2

The union of the rays from the origin through the side /1 1 (- 1) \ B of the unit ball is the whole third quadrant ^ðxÞoðyÞ: Thus Cð12 ðx þ yÞ; /1 Þ is the translate of this quadrant by 12 ðx þ yÞ; that is, the angle ^ 12 ðx þ yÞ 12 ðx þ yÞ 12 ðx  yÞ: The other cones are similarly found, and their intersection is exactly the square conv A (the shaded part in Figure 9). The d-segments [aibi]d are said to be d-concurrent if their intersection is non-empty.

C O R O L L A R Y 25 If fx1 ; . . .; x2k g can be matched up to form k d-segments ½xi xkþi d ; i ¼ 1; . . .; k; that are d-conT current, then FTðx1 ; . . .; x2k Þ ¼ ki¼1 ½xi xkþi d : E X A M P L E 2 In Figure 10 we first apply Proposition 24 to get [x1x3]d and [x2x4]d. We obtain that [x1x3]d is the usual segment x1x3, since the exposed faces of the unit ball B containing x1 and x3, respectively, are both singletons (x1 and x3 are not in the interiors of segments on the boundary of B). Also, [x2x4]d is the parallelogram with opposite vertices x2 and x4 and sides parallel to the vectors from the origin to the endpoints of the segment on the boundary of B containing x4. By Corollary 25, FT(x1, x2, x3, x4) = [x1x3]d \ [x2x4]d (since the intersection is non-empty), which is the diagonal of the parallelogram [x2x4]d indicated as a bold segment in Figure 10. E X A M P L E 3 In Figure 11 the Minkowski plane has an affine regular hexagon as unit ball B. The set A consists of the midpoints of the sides of B. If we now apply Proposition 24 to pairs of points on opposite edges, we obtain that the d-segments of these pairs of points have non-empty intersection, which is the shaded hexagon in Figure 11. By Corollary 25, this hexagon is FT(A). A metric line in a Minkowski space is a subset isometric to the real line. A set in a Minkowski space is d-collinear if it is contained in some metric line.

y

x4 B

B −x

ft( A)

x

x3

x1

−y Figure 9. FT(A) = conv(A) is possible in the rectilinear norm.

x2 Figure 10. d-concurrent d-segments.

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[2] Alon, N., and Pudla´k, P.: Equilateral sets in lpn , Geom. Funct.

B

Anal. 13 (2003), 467–482. [3] Alonso, J., and Benitez, C.: Orthogonality in normed linear spaces: a survey, I. Main properties, Extracta Math. 3 (1988), 1–15.

ft( A)

[4] Alonso, J., and Benitez, C.: Orthogonality in normed linear spaces: a survey, II. Relations between main orthogonalities, Extracta Math. 4 (1989), 121–131. [5] Amir, D.: Characterizations of Inner Product Spaces, Birkha¨user, Basel, 1986. [6] Asplund, E., and Gru¨nbaum, B.: On the geometry of Minkowski planes, Enseign. Math. 6 (1960), 299–306.

Figure 11. d-concurrent d-segments.

[7] Averkov, G.: On the geometry of simplices in Minkowski spaces, Stud. Univ. Zˇilina Math. Ser. 16 (2003), 1–14.

C O R O L L A R Y 26 If fx1 ; . . .; x2k g is a d-collinear set of even cardinality in its natural order, then

[8] Benz, W.: Vorlesungen u¨ber die Geometrie der Algebren,

FTðx1 ; . . .; x2k Þ ¼

k \

Springer-Verlag, Berlin, Heidelberg, New York, 1973. [9] Boltyanski, V., Martini, H., and Soltan, P. S.: Excursions into

½xi x2ki d ¼ ½xk xkþ1 d :

i¼1

If fx1 ; . . .; x2kþ1 g is a d-collinear set of odd cardinality in its natural order, then FTðx1 ; . . .; x2kþ1 Þ ¼ fxkþ1 g: How many points of A can be contained in FT(A)? If A consists of two points, then A  FTðAÞ; and the previous examples show that it is possible for three and four points of A to belong to FT(A).

Combinatorial Geometry, Springer, Berlin, 1997. [10] Boltyanski, V., Martini, H., and Soltan, V.: Geometric Methods and Optimization Problems. Kluwer Academic Publishers, Dordrecht, 1999. [11] Bo¨ro¨czky, Jr, K.: Finite Packing and Covering, Cambridge Tracts in Mathematics, 154, Cambridge University Press, Cambridge, 2004. [12] Brass, P.: On equilateral simplices in normed spaces, Beitra¨ge Algebra Geom. 40 (1999), no. 2, 303–307. [13] Bandelt, H.-J. , Chepoi, V., and Laurent, M.: Embedding into rectilinear spaces, Discrete Comput. Geom. 19 (1998), 595–

T H E O R E M 27 Let M2 be a Minkowski plane, and let A  M 2 : Then |A \ FT(A)| B 4. If |A \ FT(A)| = 4, then M2 has a parallelogram as unit ball and A contains a homothet of {±x ±y}, where x and y are two consecutive vertices of the unit ball. If |A \ FT(A)| = 3, then M2 has an affine regular hexagon as unit ball, and A contains a homothet of {o, x, y}, where x and y are two consecutive vertices of the unit ball. In both cases, [41] FTðAÞ ¼ convðA \ FTðAÞÞ: For higher dimensions we have the following result [41].

T H E O R E M 28 Let Mn be a Minkowski space, and let A  M n : Then |A \ FT(A)| B 2n. Furthermore, if |A \ FT(A)| = 2n, then Mn is isometric to ‘n1 , with A \ FT(A) corresponding to a homothet of the Hamming cube {0,1}n. Higher-dimensional analogues of Proposition 24 can be easily obtained by using the respective generalizations of d-segments [9, Chapter II]. In particular, a Minkowski space Mn is strictly convex iff FT(A) is a singleton for all noncollinear subsets A. ACKNOWLEDGMENTS

We thank the referees for their valuable suggestions for making the paper more accessible.

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Spaces, Vol. I, Johnson, W. B., and Lindenstrauss, J. (eds.), North-Holland, Amsterdam, 2001, pp. 837–870. [26] Kelly, P. J.: A property of Minkowskian circles, Amer. Math. Monthly 57 (1950), 677-678. [27] Klotzek, B., and Quaisser, E.: Nichteuklidische Geometrie, Deutscher Verlag der Wissenschaften, Berlin, 1978. [28] Koolen, J., Laurent, M., and Schrijver, A.: Equilateral dimension of the rectilinear space, Designs Codes Cryptogr. 21 (2000), 149–164. [29] Kupitz, Y. S., and Martini, H.: Geometric aspects of the generalized Fermat-Torricelli problem, In: Intuitive Geometry, Bolyai Society Mathematical Studies, Vol. 6 (1997), 55–127. [30] Lassak, M.: Covering a plane convex body by four homothetical copies with the smallest positive ratio, Geom. Dedicata 21 (1986), 157–167.

Theory Appl. 115 (2002), 283–314. [42] Martini, H., and Wu, S.: On orthocentric systems in strictly convex Minkowski planes, Extracta Math. 24 (2009), 31–45. [43] Menger, K.: Untersuchungen u¨ber allgemeine Metrik, Math. Ann. 100 (1928), 75–163. [44] Minkowski, H.: Sur les proprie´te´s des nombres entiers qui sont de´rive´es de l’intuition de l’espace, Nouvelles Annales de Mathe´matiques, 3e Se´rie 15 (1896); also in: Gesammelte Abhandlungen 1, Band XII, pp. 271–277. [45] Morgan, F.: Minimal surfaces, crystals, shortest networks, and undergraduate research, Math. Intelligencer 14 (1992), 37–44. [46] Petty, C. M.: Equilateral sets in Minkowski spaces, Proc. Amer. Math. Soc. 29 (1971), 369–374. [47] Riemann, B.: U¨ber die Hypothesen, welche der Geometrie zu Grunde liegen, Abh. Ko¨nigl. Ges. Wiss. Go¨ttingen 13 (1868).

[31] Lawlor, G., and Morgan, F.: Paired calibrations applied to soap

[48] Schechtman, G.: Two observations regarding embedding sub-

films, immiscible fluids, and surfaces or networks minimizing

sets of Euclidean spaces in normed spaces, Adv. Math. 200

other norms, Pacific J. Math. 166 (1994), 55–83. [32] Make’ev, V. V.: Equilateral simplices in a four-dimensional

(2006), 125–135. [49] Schu¨rmann, A., and Swanepoel, K. J.: Three-dimensional antipodal and norm-equilateral sets, Pacific J. Math. 228

normed space, J. Math. Sci. (N. Y.) 140 (2007), 548–550. [33] Mankiewicz, P., and Tomczak-Jaegermann, N.: Quotients of

(2006), 349–370.

finite-dimensional Banach spaces; random phenomena, Hand-

[50] Smyth, C.: Equilateral sets in ‘np, manuscript, 2002.

book of the Geometry of Banach Spaces, Vol. I, Johnson, W. B.,

[51] Soltan, P. S.: Analogues of regular simplexes in normed spaces,

and Lindenstrauss, J. (eds.), North-Holland, Amsterdam, 2001.

Dokl. Akad. Nauk SSSR 222 (1975), no. 6, 1303–1305, English

pp. 1201–1246.

translation: Soviet Math. Dokl. 16 (1975), no. 3, 787–789.

[34] Martini, H., and Spirova, M.: Recent results in Minkowski geometry, In: Proc. Internat. Conf. Math. Appl. (Mahidol Univer-

[52] Spirova, M.: Circle configurations in strictly convex normed planes, Adv. Geom. 10 (2010), 631–646.

sity of Bangkok, Thailand, 2007), Mahidol University Press,

[53] Spirova M. (2010) On Miquel’s theorem and inversions in normed

Bangkok, 2007, pp. 45–83. (also appeared in a special volume of the East-West J. Math.: Contributions in Mathematics and Applications II, (2007), 59–101.)

planes. Monatsh. Math. 161, 335–345. [54] Swanepoel, K. J.: A problem of Kusner on equilateral sets, Arch. Math. (Basel) 83 (2004), 164–170.

[35] Martini, H., and Spirova, M.: The Feuerbach circle and ortho-

[55] Swanepoel, K. J.: Equilateral sets in finite-dimensional normed

centricity in normed planes, Enseign. Math. 53 (2007), 237–258.

spaces, In: Seminar of Mathematical Analysis, Daniel Girela A´lvarez, Genaro Lo´pez Acedo, Rafael Villa Caro (eds.), Secretariado de Publicationes, Universidad de Sevilla, Seville, 2004,

[36] Martini, H., and Spirova, M.: Clifford’s chain of theorems in strictly convex Minkowski planes, Publ. Math. Debrecen 72 (2008), 371– 383.

pp. 195–237.

[37] Martini, H., and Spirova, M.: On regular 4-coverings and their

[56] Swanepoel, K. J., and Villa, R.: A lower bound for the equilateral

applications for lattice coverings in normed planes, Discrete

number of normed spaces, Proc. Amer. Math. Soc. 136 (2008),

Math. 309 (2009), 5158–5168. [38] Martini, H., and Soltan, V.: Antipodality properties of finite sets in Euclidean space, Discrete Math. 290 (2005), 221–228.

127–131. [57] Thompson, A. C.: Minkowski Geometry, Cambridge University Press, Cambridge, 1996.

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125

Oscillatory Sums VOLKER BETZ, VASSILI GELFREICH

e all know about oscillatory integrals, where the positive and negative part of the integrand gives rise to cancellations that result in a value of the integral much smaller than the values of the integrand, or in the integral being finite even though the integrand is not Lebesgue-integrable. The basic mechanism of the cancellations can be analysed and understood quite easily, and rigorously. But what about the integrals’ discrete cousin, the series? As a first example, lookk at the sequence of finite sums ðsn Þn2N P ðnÞ with sn ¼ 3n k¼0 k! . The sum is alternating all right, but otherwise behaves rather badly: the terms quickly grow in absolute value, reach an exponentially large maximum at k = n, and then decay to zero, being of order one around k = en (all of this can be seen easily using Stirling’s formula). As for the cancellation that we might hope for, the ratio between two consecutive terms with k & n is e, so their difference is of the order en. In conclusion, when we just look at the terms of sn, we have a hard time believing that this sum could give any meaningful value. Of course, we do know that sn converges to zero as n ! 1, as it is just the truncated Taylor series of e-n, the latter being the value that sn has when the upper summation index is taken to be infinity. By the Leibnitz criterion, e-n - sn is bounded by the first omitted term, which for k = 3n + 1 is already exponentially small. Thus, after a second look there is nothing mysterious or even especially interesting about sn, aside from the sums being small for reasons that are not very obvious from the terms. Let us make things slightly more exciting and add a perturbation. We define

W

Sn ¼

n X ðk  nÞk k¼0

k!

:

ð1Þ

The terms show the same qualitative behaviour as those of sn, with two important differences: this time, we do not have a Taylor series that relates Sn to a well-known function; and taking the upper summation index to be infinity renders the expression for Sn meaningless. So, should we still expect Sn to have any sort of reasonable behaviour, or even converge as n ! 1? If so, why?

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AND

FLORIAN THEIL

Let us investigate the first question by actually calculating Sn for the first few hundred n. Surprisingly, Sn converges to zero, and very quickly: a logarithmic plot (Figure 1a) shows that this convergence is exponential! But that is not all. By Figure 1a, it appears that the exponential rate of convergence of Sn is just under 1/3; so we will take a brave guess and say it is 1/p. Let us cancel this exponential decay and see what we get. Plotting S~n ¼ ð1Þn Sn en=p (Figure 1b) makes things appear even more mysterious: now we not only have a sequence that has no reason to converge, but nevertheless does, but when the exponential decay rate is taken out, it displays a complicated and rather beautiful structure: There are seven sine curves hidden in the values of S~n , each of them with a period of just under 700, and a very slight overall increase in amplitude. (The factor (-1)n is there for cosmetic reasons: without it, we would see not seven but fourteen sine curves, in pairs such that for every curve the negative curve is also present. The image would not look nearly as nice.) These images make the second question about the reason for the behaviour of Sn all the more interesting. In addition, further questions arise: why is the renormalized sequence displaying a beautiful pattern of interlaced sine curves? What have the numbers 1/p and 7 to do with it? In the case of (1), we will be able to answer all these questions analytically, and by doing it find a very good (but hidden) reason for Sn to converge. Before doing this, let us see whether (1) is an isolated freak phenomenon, or whether there are more of those sequences that converge without having any obvious reason to do so. It turns out that there are many, and we have listed some of those that we found in Table 1. Let us call them oscillatory sums, for their formal similarity with oscillatory integrals. Not all oscillatory sums converge, but all seem to be much smaller than their maximal element, or even the difference between the maximal element and the next smaller one. Also, some oscillatory sums show the interlaced sine curves when renormalized, whereas some do not, see Figure 2. There are many oscillatory sums where no analytic trick like the one that we will use to treat (1) seems to work, but which nonetheless behave in a very regular way. At present, we have no idea how general this behaviour is. Have we maybe

(b)

(a) 100

200

300

400

500

600

1.5 1.0

50 0.5 100

200

400

600

800

1000

0.5 150

1.0 1.5

Figure 1. a) ln jSn j for n ¼ 1; . . .; 600. b) The first 1000 values of (- 1)n Snen/p, coloured modulo seven.

unconsciously picked sequences that are similar enough to (1) in order to behave similarly? Also, we do not know whether there is a common reason for all of them to behave as they do, or whether one has to study them case by case. It is essentially only in the case of (1) that we have a good idea of what is going on, which we will now present. Let us view Sn not as a sequence, but as the values f(n) that some function f attains at the integers. We define f ðtÞ ¼

bt c X ðk  tÞk k¼0

k!

;

ðt [ 0Þ

of all, f(n) = Sn for all n 2 N. Next, f is piecewise polynomial, being a polynomial of degree n in the interval [n,n + 1]. Then f is differentiable on ð1; 1Þ, indeed f 2 C n ðn; 1Þ for any n 2 N. To see this, note that when t increases and crosses an integer value n 2 N, the sum in the definition of f gains the additional term (n - t)n/n! This term vanishes at t = n together with its first (n - 1) derivatives. But most importantly, f is the unique solution of the equation1 f 0 ðtÞ ¼ f ðt  1Þ ðt 62 f0; 1gÞ

ð2Þ

where bt c denotes the integer part of t 2 R. For t \ 0 we set f(t) = 0. The function f has some interesting properties: First

ð3Þ

subject to the initial condition f ðtÞ ¼ 1

for t 2 ½0; 1:

Table 1. Four oscillatory sums: max elem denotes the element of maximal absolute value, D max elem the difference between the absolute values of the latter and the next smaller one, taken at n = 600. The exponential rate of growth or decay is estimated from the numerics; the value in parentheses indicates that in this case the growth is not actually exponential sn n P ðk 2 n2 Þk ð2kÞ!

k¼0 n P

2

2 k

ðk n Þ ð3kÞ! k¼0 n pffiffi pffiffiffi k P ð k  nÞ k! k¼0 2n P n1 ðknÞ2

e

max elem (n = 600)

D max elem (n = 600)

|s600|

exponential rate c

graph

1.8 9 10235

9.6 9 10232

1.0 9 1033

0.1266

Fig. 2 a)

3.5 9 1029

5.0 9 1026

6.5 9 1014

(0.0569)

Fig. 2 b)

0

Fig. 2 c)

-1

Fig. 2 d)

5.1 9 10 ð1Þk

1

7

9.1 9 10

5

1.6 9 10-3

1.9 9 10

-3

6.3 9 10-262

k¼0

AUTHORS

......................................................................................................................................................... VOLKER BETZ obtained his Ph.D. from

VASSILI GELFREICH studied at the Lenin-

the Technical University of Munich in 2002. He works on the mathematics of quantum theory and statistical mechanics—on molecules, on interaction of particles with scalar fields, and on Bose-Einstein condensation. He likes best problems that offer some sort of surprising insight, without overly technical details.

grad (later St. Petersburg) State University, in the Department of Mathematical Physics, obtaining his Candidate degree in 1992. He also received the Doctor of Sciences degree in 2000 from the St. Petersburg branch of the Steklov Institute. He works on Hamiltonian dynamical systems and bifurcation theory. He has been at Warwick since 2002, and as of autumn 2011 he will become a professor.

Department of Mathematics University of Warwick Coventry, CV4 7AL UK e-mail: [email protected]

1

Mathematics Institute University of Warwick Coventry, CV4 7AL UK e-mail: [email protected]

The authors thank D. Turaev for pointing out that f(t) satisfies equation (3).

Ó 2011 Springer Science+Business Media, LLC, Volume 33, Number 3, 2011

111

(a)

nγ

sn

(b)

nγ

sn

200 1.0 100

0.5 n 100

200

300

400

500

n

600

0.5

100

200

300

400

500

600

100

200

300

400

500

600

100

1.0 200

(c)

(d)

sn

n

sn

0.240 0.005 0.239 100

200

300

400

500

600

n 0.238

0.005 0.237 0.010 0.236 0.015 0

n

Figure 2. Plots of the renormalized oscillatory sums from Table 1. We plotted sne-cn for n ¼ 1; . . .; 600, where c is the exponential rate given in the next to last column of Table 1.

Indeed, if t 62 N we can differentiate the definition of f to obtain bt c bX t1c X ðk  tÞk1 ðk þ 1  tÞk ¼ ¼ f ðt  1Þ: f 0 ðtÞ ¼  ðk  1Þ! k! k¼1 k¼0 For integer t C 2, the equation follows from the continuity of f and f 0 . Existence of a unique solution to (3) follows from considering the integral form of the equation, Z t1 f ðsÞds; f ðtÞ ¼ 1 

be the Laplace transform of f. Since we have just seen that f (t) B et, the integral is finite at least for Re(p) [ 1. In order to derive the equation for u we multiply equation (3) by e-pt and integrate from 1 to 1. Integrating the left-hand side by parts we obtain Z 1 Z 1 ept f 0 ðtÞdt ¼ e p f ð1Þ þ p ept f ðtÞdt 1 1 Z 1 ept dt þ puðpÞ ¼ 1 þ puðpÞ: ¼ e p  p 0

0

for t C 1: existence is obtained by induction on the intervals [n,n + 1], and for uniqueness note that the difference g(t) of R t1 two solutions satisfies jgðtÞj6 0 jgðsÞjds; gð0Þ ¼ 0, and apply the Gronwall lemma. By a similar argument, or by a direct estimate on (2), we can see that f (t) O et for all t > 0. So f satisfies a linear delay-differential equation with constant coefficients, which can be solved using the Laplace transform. Let Z 1 e pt f ðtÞdt uðpÞ ¼

Integrating the right-hand side we get Z 1 Z 1 e pt f ðt  1Þdt ¼ e pðtþ1Þ f ðtÞdt ¼ e p uðpÞ: 1

0

Therefore 1 þ puðpÞ ¼ ep uðpÞ and consequently uðpÞ ¼

1 : p þ ep

0

......................................................................... FLORIAN THEIL received his Ph.D. at Hannover in 1997. He has been at Warwick since 2001 and is now a senior lecturer. His work is in partial-differential equations and the theory of solids.

Mathematics Institute University of Warwick Coventry, CV4 7AL UK e-mail: [email protected] 112

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To get back to the function f, we need to invert the Laplace transform. For this, let us first study the analytic continuation u(z) of u. It has simple poles where the denominator is zero, namely, where z þ ez ¼ 0;

ð4Þ

and it is analytic otherwise. Since (4) is equivalent to z ez = -1, the poles closest to the real axis are W (-1) and W ð1Þ, where W is the Lambert function, that is, the inverse function of zez. This function is multivalued, and in order to get error estimates, we want the location of the next pair of poles, too.

(a)

(b)

1.0

1.0

0.5

0.5

0.5

0.5

1.0

1.0

Figure 3. Guess what sequence you are seeing in each of these pictures!

We transform (4) into the pair of real equations x ¼ e

x

cos y

y ¼ ex sin y;

ð5Þ

ð6Þ

with z = x + iy. Since f is real on the real line, singularities come in complex conjugate pairs, and we restrict to y [ 0. Equation (6) implies that ex ¼ sinðyÞ=y 6 1, so x \ 0, which means that all the singularities are on the left of the real axis. Squaring and adding (5) and (6) yields y2 = e-2x - x2, so for any solution |y| grows monotonically with |x|, and viceversa. Finally, rearranging (6) and taking the logarithm yields x ¼ lnðsinðyÞ=yÞ whenever sinðyÞ [ 0, with no real solution otherwise. Inserted into (5), this leads to   sin y sin y ¼ y cos y: ð7Þ ln y The left-hand side of (7) is defined for (kp, (k + 1) p) with k even, and the equation has exactly one solution in each of these intervals. The first two of them are y1 = 1.33724 and y2 = 7.58863, leading to z1 = W(-1)&-0.31832 + 1.33724i and z2 & -2.06228 + 7.58863i. All the other solutions have even larger negative real part. The inverse Laplace transform can now be done using the Bromwich integral: Z i1 1 e pt dp; f ðtÞ ¼ 2pi i1 p þ e p where as a path of integration we choose the imaginary axis, which is right of all singularities of u, as is required. Now, shifting the path of integration to the left and using residues, we get ez1 t þ OðeReðz2 Þt Þ: f ðtÞ ¼ 2Re 1 þ z1 With zj = xj + iyj this gives f ðtÞ ¼

2 ex1 t cosðy1 t þ argð1 þ z1 ÞÞ þ Oðe x2 t Þ: j1 þ z1 j

ð8Þ

So it turns out that f is a decaying exponential times a cosine function, and we are in a position to answer all of the questions that we asked before. First, what does 1/p have to do with Sn? The answer is, nothing at all, except that it happens to agree with the -Re(W(- 1)) in the first four valid digits. The discrepancy is small enough so it does not lead to

an exponential growth in Figure 1b, although a slight increase of amplitude is indeed visible. Second, what has seven to do with it, and why do we see the interlaced sine functions? Given (8), the answer becomes obvious, although it may have been obvious form the start for people who are working in signal processing. Have a look at Figures 3a and b, and guess what you are seeing. The surprising answer is that both the plots actually depict the same sequence, namely ðcosðnÞÞn2N . The only difference is that the first plot contains the first 1000 elements, whereas the second one contains the first 3000. If you want to verify this, hold Figure 3a in front of your eyes at an acute angle, and look at it from the side! Apart from telling us that we might want to be careful when drawing conclusions from looking at plots, Figure 3b could have suggested from the start what we have seen in (8), namely that Sn is an undersampling of a trigonometric function at a frequency that is incommensurable to the period of that function. And now the significance of the number 7 (or rather 14, as discussed previously) for Sn is not hard to understand any longer: If we consider the defect of diophantine approximations   Imðz1 Þ p  ; dðnÞ ¼ min  16p;q6n 2p q then the numbers n where d jumps correspond to sampling frequencies that will give the illusion of periodic behavior. If N points are plotted then we need that N d(n) = O(1). The first solution for 1000 d(n)O3 is obtained for n = 14. Let us finally note that the trick of writing an oscillatory sum as the values of a function at integer points, and deriving a differential equation for that function, works for a few other sums: for example, for the truncated negative expoPbcnc nential sn ðcÞ ¼ k¼0 ðnÞk =k!, with c [ 0, we introduce Pbct c sðtÞ ¼ k¼0 ðtÞk =k! and obtain the equation s0 ðtÞ þ sðtÞ ¼ ðtÞbct c =bct c!, with the solution ! Z t bcr c t r ðrÞ dr : sðtÞ ¼ e 1þ e bcr c! 0

ð9Þ

A simple estimate of the integrand using Stirling’s formula shows that s(t) diverges when c \ e, decays exponentially but not as quickly as e-t when e \ c \ 1/W(1/e) & 3.591 (with W again the Lambert function); in the latter case the integral in (9) still diverges. For c [ 1/W(1/e), s(t) converges like e-t. We could have had the same insight by Ó 2011 Springer Science+Business Media, LLC, Volume 33, Number 3, 2011

113

estimating the first remainder term ncn/(cn)!. On the other k Pbn=ac k , when transhand, a sum of the form sn ¼ k¼0 b ðaknÞ k! formed to a function s(t) in the obvious way, fulfills s0 ðtÞ ¼ bsðt  aÞ for every a, b [ 0, and can be treated in the same way as Sn. But this seems to be pretty much the furthest we can stretch the idea of solving a simple

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differential equation to treat an oscillatory sum: in particular, we re-emphasize that for the many types of oscillatory sums that we tried, including those in Table 1, we have found no way of understanding the highly regular behaviour they seem to exhibit.

Paul Cohen and Forcing in 1963 REUBEN HERSH

recently became aware of some writings of the prominent French metaphysical philosopher, Alain Badiou. His book Number and Numbers [3] actually proposes John Conway’s surreal numbers as the fundamental components or descriptors of Being itself! An earlier work of his, Being and Event [2], used Paul Cohen’s famous invention of ‘‘forcing’’ to explain how true novelty and free choice are possible in the Universe. (I published a skeptical review of Number and Numbers [9], and I was recently consulted by a colleague from the University of New Mexico philosophy department regarding Badiou’s pretentious misuse of Cohen’s forcing.) These experiences bring back vividly my interactions with Paul in the early 1960s. He had just astounded the mathematical world by proving the independence of the Continuum Hypothesis and the Axiom of Choice from the system of Zermelo-Fraenkel. I was a lowly instructor at Stanford University on a two-year appointment, having completed my Ph.D. at New York University, and I had not yet attained my tenure-track post here in New Mexico. In my present mature years, I look back at those interactions in a riper perspective. But at the time, I found interaction with Paul Cohen difficult and troubling. My job at Stanford was a result of the beneficence of my NYU mentor Peter Lax and his Stanford collaborator Ralph Phillips. Although I had written a creditable doctoral thesis, I was acutely conscious of how little I knew or understood in the broad field of contemporary advanced mathematics. When encountering other mathematicians, my goal was self-protection—to cover up my ignorance and to try to pass for a real mathematician. Paul’s aggressive, brash way of coming on to people was the utter opposite of my own. In social origin, Paul was not that different from me—a tallish, nerdish New York Jew. In fact, Paul’s family origin was in poverty and deprivation. But he had always been triumphant in competitive mathematical environments— Stuyvesant High in Manhattan, Brooklyn College, the

I

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University of Chicago graduate school. He approached a new relationship by demonstrating that he was the smartest person in the room. When we met, my timidity and his aggressiveness led to my immediate withdrawal, verbally if not physically. Yet we did develop a sort of relationship. I think Paul was lonesome. He wanted to impress and control people, not necessarily drive them away. In fact, he would work out his ideas by talking to some willing listener, often a qualified grad student or postdoc (or, on occasion, one of the mature mathematicians Paul considered worth listening to). I didn’t serve Paul’s needs, either as a sounding board or as a playful competitor. But we happened to be neighbors in Menlo Park, an unpretentious suburb of Palo Alto, the site of Stanford University. Paul was beginning to struggle to assimilate his new status as a superstar. (He had already been well established as a promising young star.) He asked my wife Phyllis for advice on buying better clothing in order to look like more of ‘‘a gentleman.’’ (She was glad to help him, of course.) He even enjoyed bits of conversation with our son Daniel, then aged 6 or 7. On an extended visit to Sweden, a great change occurred in Paul’s life. He met his future wife, Christina, and she accompanied him home to Menlo Park. As I knew her back then, Christina was an attractive, cheerful, friendly, bright young woman, not particularly inclined to the academic or intellectual style. Perhaps that was why Paul was reluctant, hesitant, to let people know about her. Even her name ‘‘Christina’’ was an issue for a Jewish boy from Brooklyn. How and when would Paul tell his mother about Christina? His mother would certainly have to come from Brooklyn for the wedding. But first, he somehow had to find a rabbi willing to perform the ceremony. Not an easy task, even in the Bay area of Northern California! But it all worked out, they did get married, and they had three beautiful children. The reason I ultimately established a solid connection with Paul Cohen was that, years before I even entered graduate school in New York, I had worked at Scientific

American magazine as an editorial assistant. Paul had been invited by Scientific American to write about his work on the Continuum Hypothesis. Scientific American was then still following a policy of publishing articles written by scientists themselves. Such manuscripts often required heavy rewriting in order to be accessible to the Scientific American readership. But Paul’s manuscript had actually been rejected. Not only was it not publishable, it was impossible for the editors of Scientific American to edit it so as to make it publishable. (Of course they did pay Paul the usual fee for his work.) That was why Paul asked me if I would like to try and rewrite his manuscript to make it publishable in Scientific American. Of course I was flattered and delighted at this invitation to connect with the peak of mathematical achievement. In the manuscript and in talks he had given, Paul had tried to explain his work by analogy with the history of geometry. Just as the fifth postulate of Euclid, the parallel postulate, had long been troublesome, and finally, by the establishment of non-Euclidean geometry, had been proved to be independent of the other axioms, so the Axiom of Choice, in set theory, was long troublesome. Paul had created a model for the Zermelo-Fraenkel (ZF) axioms of set theory, in which the axiom of choice failed; therefore that axiom is independent of the other axioms, just as Euclid’s parallel postulate is independent of Euclid’s other axioms. That analogy became the outline of my article, taking ideas from Paul’s rejected manuscript, his other writings on the subject, and his oral presentations that I had heard. When I showed him my rewrite, he questioned my slight attempts at a touch of humor, and suggested I publish it under my own name. Of course I preferred to be a coauthor with the great and famous Paul Cohen, not just the solo author of a Scientific American article. And so the collaboration came about [6]. It was already evident that the technique of ‘‘forcing,’’ which Paul used on the Continuum Hypothesis and the Axiom of Choice, is a powerful method to show that other open problems are undecidable. Within a few years many

AUTHOR

......................................................................... REUBEN HERSH had a fairly long life as a

research mathematician, much of it at the University of New Mexico. He never lost his interest in writing about mathematics and mathematicians, and in recent years that is his primary activity. Readers of The Mathematical Intelligencer have seen his name in our pages and are likely to know his book The Mathematical Experience with Philip J. Davis, and his recent book Loving and Hating Mathematics with Vera JohnSteiner. 1000 Camino Rancheros Santa Fe, NM 87505 USA e-mail: [email protected]

open problems were shown by others to be independent of ZF (indeed, of ZFC: Zermelo-Fraenkel plus the Axiom of Choice) by the method of forcing. And logicians have developed and elaborated forcing in forms and guises far beyond Cohen’s original version. Given some statement formulated in an axiom system, can you create one model of that system in which the statement is provable, and another model in which it is disprovable? If so, by the Completeness Theorem, which equates consistency with having a model, it will follow that both the given statement and its negation are consistent with that axiom system. (For example, the pseudosphere, a model for non-Euclidean geometry, shows that non-Euclidean geometry is consistent, and therefore that Euclid’s parallel postulate is not provable.) Cohen (following on Go¨del) had achieved that result for the Zermelo-Frankel axiom system of set theory, with regard to both the Continuum Hypothesis (even assuming the Axiom of Choice) and the Axiom of Choice. ‘‘Forcing’’ quickly became a necessary topic in any second-year (advanced) course in logic or metalogic. Logicians modified it, generalized it, reworked it, and applied it. On the other hand, mathematicians who are not logicians sometimes find it obscure and difficult to grasp. I can offer a few helpful words. Of course they will just be rewordings of what Cohen himself and subsequent logicians have already said in more technical or less readable ways. The first essential thing to grasp is the famous Lo¨wenheimSkolem theorem, which says: ‘‘If your axiom system is consistent (if it has any model at all) then it has a countable model—a countable set in which the relations of your axiom system are defined and satisfied.’’ This is paradoxical and confusing, for it means, in particular, that the system of real numbers, which is uncountable, can be modeled by a countable set. In this countable model of the real number system, how can the theorem ‘‘the real numbers are uncountable’’ still hold true? The paradox is explained simply. The statement ‘‘the real numbers are uncountable’’ merely says that there is no one-to-one mapping between the real numbers and the natural numbers. So, in the model, there cannot be any such mapping between whatever is interpreted as the set of natural numbers and whatever is interpreted as the set of real numbers. The expression ‘‘set of real numbers’’ is being interpreted by a set that is actually countable. But that set can’t be ‘‘counted’’ (mapped onto the natural numbers) in the model. The Lo¨wenheim-Skolem theorem is essential in Cohen’s treatment, because it gives him a countable model in which to carry out the construction called ‘‘forcing,’’ and thereby analyze the Continuum Hypothesis and the Axiom of Choice. In fact, since we can label everything in any countable set by using sequences of zeroes and ones, everything we do can be done just by working with sequences of zeroes and ones. Given, then, the countable model of the ZermeloFraenkel axioms, which Lo¨wenheim-Skolem provides, what is our task—or rather, what was Cohen’s task? Simply to enlarge this countable model—add more elements to it, each of which is just a sequence of zeroes and ones—in such a way that the enlarged countable set is still a model of ZFC, but in which the Continuum Hypothesis is false. This sounds confusing, I admit. The Continuum Hypothesis is a Ó 2011 Springer Science+Business Media, LLC, Volume 33, Number 3, 2011

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statement about the continuum, about an uncountable set. How can you claim to prove anything about this uncountable set without having an uncountable set to talk about? The answer has two parts. First, the method of forcing is a way to add a new sequence of zeroes and ones, one step at a time, in such a way as to preserve the ZFC axioms. This is not so surprising, after all. There are only a countable number of such axioms, and so it should be possible sequentially to check that they are all obeyed as we introduce a new element into the model. However, to write down exactly how to achieve this was a great technical feat, and it’s still quite a chore to read through. As we add each new bit to the sequence of zeroes and ones we are introducing into the model, the difficulty is in ensuring continually that no previously chosen bit has to be rejected. The rules to accomplish this are precisely what we mean by ‘‘forcing.’’ The new sequence is restrained, ‘‘forced’’ to satisfy certain conditions, in order not to spoil the model, in order to keep the validity of the ZF axioms. Cohen showed how to do this, while keeping his freedom to impose some new conditions on the new elements, for the sake of making some desired statement provably true in the new model. A key point is to adjoin new reals that are ‘‘generic,’’ that is, do not code any special information about the model we started with. In fact, there are plenty more new sequences of zeroes and ones available to insert into the model. There are actually uncountably many such sequences available (since, as Cantor proved, the set of infinite sequences of zeroes and ones is an uncountable set.) Cohen was able to use his forcing method to introduce into the countable Lo¨wenheimSkolem model a number of generic reals that are in one-toone correspondence with an ordinal in the original model, which is larger than countable infinity in that model, but still without introducing the complete set of sequences of zeroes and ones (which would have represented the set of all subsets, the ‘‘power set,’’ of the natural numbers). In doing so, he was creating a model of ZFC in which there is an uncountable set strictly smaller than the continuum—that is, in which the Continuum Hypothesis is provably false. Cohen achieved his results after a determined, year-long struggle. Solomon Feferman, one of the up-and-coming logicians at Stanford, served as a principal sounding board. Naturally, Paul went through many wrong ideas before he came up with the right ones. And then getting it all down in correct form was a major job. Logicians were impressed, and they needed a little time to absorb Paul’s radical new method before they could confidently endorse his result. But he was impatient with people who weren’t convinced immediately. So he went straight to the top—to Go¨del himself. As he prepared the written version to show to Go¨del, Feferman and others brought problematic points to his attention and suggested how they might be dealt with.

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I’ve been told there were two episodes. The first time Cohen went to Go¨del’s house in Princeton, Prof. Go¨del came to the door, took the manuscript without a word, and shut the door. After a few days, Paul was invited back, and this time was welcomed inside. Still, Go¨del needed a little time to edit Paul’s article before he sent it to the Proceedings of the National Academy of Sciences. Paul found it hard to be patient, even with Go¨del. He wrote to Go¨del pleading with him to hurry up with a public endorsement of Paul’s achievement. Go¨del answered reassuringly. After I left Stanford for New Mexico, I had one more major encounter with Paul Cohen. He came to UNM to lecture on his work.Stan Ulam was living in Santa Fe and was working in Los Alamos. He sent word that he wished to meet Paul Cohen. I had the honor of introducing Cohen and Ulam to each other. After that introduction, Ulam always referred to me as ‘‘Paul Cohen’s friend,’’ and would ask me about Paul. I usually had no news to tell him. It’s well known that Paul’s only concern became Riemann’s conjecture on the zeroes of the zeta function. For someone who had solved problem number one on Hilbert’s list, nothing short of the Riemann conjecture was worthy of his time. At one point a rumor flew around the world—‘‘Cohen has proved Riemann’s conjecture!’’ It was the irksome duty of the chairman of Stanford’s math department to answer a deluge of queries, over and over, ‘‘No, it’s not true.’’

REFERENCES

[1] Peter Sarnak. Remembering Paul Cohen (1934-2007), Notices Amer. Math. Soc. 57 (2010), 824-838. [2] Alain Badiou. Being and Event, Continuum, London, 2007. [3] Alain Badiou. Number and Numbers, Polity Press, Cambridge, 2008. [4] Paul Cohen. Set Theory and the Continuum Hypothesis, W. A. Benjamin, New York, 1966. [5] Paul Cohen. The discovery of forcing, Rocky Mountain J. Math. 32(4) (2002), 1071-1100. [6] Paul Cohen, Reuben Hersh. Non-Cantorian set theory, Scientific American (December 1967), 104-116. [7] Solomon Feferman. Some applications of the notions of forcing and generic sets, Fundamenta Mathematicae 56 (1965), 325345. [8] Kurt Go¨del. Collected Works, Vol. IV, Correspondence A-G, in S. Feferman et al. (eds.), ‘‘Paul J. Cohen,’’ pp. 375-387. [9] Reuben Hersh. Review of Number and Numbers by Alain Badiou, Math. Intelligencer 31(3) (2009), 67-69. [10] Gregory H. Moore. The origins of forcing, in F. R. Drake, J. K. Truss, (eds.), Logic Colloquium ‘86, 143-173.

Years Ago

David E. Rowe, Editor

Gustav Theodor Fechner: Pioneer of the Fourth Dimension HANS G. FELLNER,

AND

WILLIAM F. LINDGREN

he essays translated below, ‘‘Der Schatten ist lebendig’’ and ‘‘Der Raum hat vier Dimensionen,’’ are in all likelihood the earliest popular writings on the fourth dimension. They are valuable not only as historical documents but also as a window into the mind of one of the most important German philosophers of the nineteenth century, Gustav Theodor Fechner (1801–1887). In these essays, Fechner describes a shadow-man capable of moving about on surfaces and interacting with other shadows. He draws an analogy between the shadow-man’s inability to perceive anything of a dimension perpendicular to his surface and our difficulty in imagining a fourth dimension of space. This ‘‘dimensional analogy’’ has played a fundamental role in numerous subsequent introductions to higher-dimensional space, most notably Edwin A. Abbott’s Flatland. Nevertheless, Fechner did not intend to write a primer on geometry, but to use geometric space as the setting for his essays, and in doing so he anticipated both Abbott and Lewis Carroll.

T

Years Ago features essays by historians and

Gustav Theodor Fechner mathematicians that take us back in time. Whether addressing special topics or general trends, individual mathematicians or ‘‘schools’’ (as in schools of fish), the idea is always the same: to shed new light on the mathematics of the past. Submissions are welcome.

â

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

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DOI 10.1007/s00283-011-9214-7

Fechner was born near Halle, Germany, in 1801. At the age of sixteen, he matriculated at the University of Leipzig, where he remained for the rest of his life. By the time he received a bachelor’s degree in medicine and passed the practical physician’s examination, he had developed an aversion to medicine; this is evident in two satirical essays published under the pseudonym, Dr. Mises: ‘‘Proof that the moon is made of iodine’’ and ‘‘A panegyric for today’s medicine and natural history.’’ He abandoned medicine and took up the study of the philosophy of nature, in which subject he earned a master’s degree in 1823 and completed a habilitation thesis the following year. Although he was granted permission to lecture at the university, he was so dissatisfied with what he had accomplished that he quit the field. Subsequently, he earned a living by translating French science books, such as Louis J. The´nard’s six-volume chemistry textbook and Jean B. Biot’s four-volume physics textbook, as well as by editing an eight-volume home encyclopedia and a pharmaceutical journal. He also began conducting investigations in electricity, and in 1831 he published an important paper on quantitative measurements of the galvanic battery, which provided experimental confirmation of Ohm’s Law (Jungnickel and McCormmach 1986). This paper secured his reputation as a physicist, and he was appointed professor of physics in 1834. In late 1839, Fechner developed an acute neurotic illness, which forced him to discontinue lecturing. The illness began with a partial blindness brought on by looking at the sun through colored glasses in experiments on after-images and contrast phenomena. It deepened into an inability to eat and various psychotic symptoms. In October 1843, his sight abruptly returned, and he began a slow and progressive

Gustav Theodor Fechner (1801–1887).

recovery. He retained the title of professor of physics and a modest salary; however, except for occasional lectures, he was no longer active in university life. In 1846, he published U¨ber das ho¨chste Gut (On the greatest good) in which he sketched a naturalistic ethics. In the same year, under the pseudonym Dr. Mises, he published Vier Paradoxa (Fechner 1846), which contains the two essays translated below. His mystic experience in October 1843 is reflected in Nanna oder u¨ber das Seelenleben der Pflanzen (Nanna, or the Soul Life of Plants, 1848), also by Dr. Mises. In this book as well as Zend-Avesta oder u¨ber die Dinge des Himmels und des Jenseits (Zend-Avesta, or Concerning Matters of Heaven and the World to Come, 1851), he develops his theory that everything in the world has some degree of

Fechner’s essays were a possible influence on Edwin A. Abbott, the author of Flatland.

consciousness. According to Fechner, there is one basic stuff, which when viewed externally is called ‘‘physical’’ and when viewed internally is called ‘‘mental.’’ Just as Johannes Kepler was gradually led from mystical speculations to the discovery of the laws of planetary motion, ‘‘so Fechner’s bold analogies led him to the conviction that there is a definite quantitative relation between the mental and the material’’ (Ho¨ffding 1908). As he described it in his classic work, Elemente der Psychophysik: ‘‘Originally, the task did not at all present itself as one of finding a unit of mental measurement; but rather one of searching for a functional relationship between the physical and the psychical that would correctly express their general interdependence’’ (translated from Fechner 1860, p. 559). In developing psychophysics, Fechner introduced

AUTHORS

......................................................................................................................................................... HANS G. FELLNER received his Ph.D. from

WILLIAM F. LINDGREN received his Ph.D.

Kent State University in 1973. Until his retirement in 2008, he was a professor of physics at Slippery Rock University, where he conducted experimental investigations of phase transitions in liquid crystals. In his retirement he has enjoyed traveling and entertaining his new grandson.

from Southern Illinois University in 1971. His published research is in general topology. He is the author (with Peter Fletcher) of Quasiuniform spaces (1982) and (with Thomas F. Banchoff) Flatland: An edition with notes and commentary (2010). He is an avid walker who has walked long-distance trails in the United States and Europe.

608 West Prairie St. Ext. Harrisville, PA 16038 USA e-mail: [email protected]

Department of Mathematics Slippery Rock University Slippery Rock, PA 16057 USA e-mail: [email protected] Ó 2011 Springer Science+Business Media, LLC, Volume 33, Number 3, 2011

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Fechner was well acquainted with A. F. Mo¨bius, one of the first mathematicians to write about the fourth dimension.

mental measurement into psychology, and for that reason he is regarded as one of the founders of experimental psychology. As Henri Ellenberger put it: ‘‘A large part of the theoretical framework of psychoanalysis would hardly have come into being without the speculations of the man whom Freud called the great Fechner’’ (Ellenberger 1970, p. 218). Our brief biographical sketch has done scant justice to Fechner’s significant impact on contemporary science and philosophy. This, however, has been admirably described and demonstrated in Michael Heidelberger’s exhaustive critical examination of Fechner’s life and work, Nature from within: Gustav Theodor Fechner and his psychophysical worldview (Heidelberger 2004).

Fechner and the Fourth Dimension Of the lecture courses Fechner attended at Leipzig, he valued most highly those of Ernst Weber on physiology and those of Karl Mollweide on mathematics. He was convinced that without mathematics a meaningful study of physics was impossible, and not long before writing Vier Paradoxa he engaged in a serious study of mathematics including the ‘‘most difficult matters of Cauchy’’ (Kuntze 1892, p. 106). (Cauchy’s first published paper on higher-dimensional geometry, ‘‘Sur quelques the´ore`mes de la ge´ome´trie de position,’’ appeared in 1846.) At this time Fechner might well have encountered the notion of time as a fourth dimension in the Encyclope´die (1754) edited by Denis Diderot and Jean d’Alembert. In an article on ‘‘dimension,’’ d’Alembert writes: ‘‘I said above that it is impossible to conceive of more than three dimensions. A clever acquaintance of mine believes that one might nevertheless regard duration as a fourth dimension, and that the product of time with solidity 128

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would in a way be a product of four dimensions; this idea may be contested, but it seems to me to have some merit other than that of mere novelty.’’ (translated from Archibald 1914) Fechner was well acquainted with August Ferdinand Mo¨bius, a mathematician and professor of astronomy at Leipzig University who reflected on the fourth dimension. In his Barycentric Calculus (1827), Mo¨bius points out that a three-dimensional solid could be transformed into its mirror image by a rotation in four-dimensional space. However, he adds that no such rotation is possible because ‘‘ein solcher Raum nicht gedacht werden kann’’ (such a space cannot be conceived). Another possible influence on Fechner was the work of Hermann Grassmann. Grassmann’s Ausdehnungslehre (Extension Theory 1844) presents an abstract foundation for geometry, which is not limited to three dimensions. He once visited Leipzig to consult with Mo¨bius, the German mathematician most likely to appreciate his ideas. But even Mo¨bius was daunted by the high level of philosophical abstraction in Grassmann’s work, and like everyone else he failed to recognize its profound significance. By coincidence, the Jablonowskian Society of Science in Leipzig had recently posed a prize question, based on a fragment of Leibniz touching on the geometry of position. Mo¨bius saw a connection between Extension Theory and Leibniz’s remarks, and he urged Grassmann to submit an essay. Grassmann did so and won the prize. We do not know whether Fechner read Extension Theory, but he was one of the members of the Society who judged the competition (Petsche 2009, pp. 37–43). Philosophical speculations about space and spatial perception were a prominent theme during Fechner’s life, and in Leipzig an important influence stemmed from the ideas of the Go¨ttingen philosopher Johann Friedrich Herbart (1776– 1841). Today Herbart is mainly remembered as a philosopher of education and an early psychologist, but for much of the nineteenth century his metaphysics influenced science and mathematics. After his death, Leipzig became the center of the Herbart School; among the notable Herbartians there was Moritz Drobisch, a professor of mathematics and friend of Fechner. Heidelberger notes that Herbart was an early role model for Fechner, but he nonetheless ‘‘exploited every opportunity to demonstrate how his own position deviated from Herbart’s’’ (Heidelberger 2004, p. 31). Herbart’s writings on the philosophy of space are a likely target of the satire in ‘‘Space has four dimensions.’’ The essence of Herbart’s views on space is found in Psychologie als Wissenschaft (1825) and Allgemeine Metaphysik (1829). In the first of these works he investigates ‘‘sensible space,’’ the space of the world as given by our senses. In the second, he proposes a higher form of geometry, ‘‘intelligible space,’’ which derives its concepts from certain primitive notions such as ‘‘position,’’ ‘‘between,’’ ‘‘inside,’’ and ‘‘outside,’’ together with rules for various logical relations. His successive construction of the line, the plane, etc., is reminiscent of the derivation sequence: A point moves to determine a line segment, a segment moves to determine a surface, and a surface moves to determine a solid. References to point, line, plane surface, and solid as the constituent elements in ideal and physical magnitudes first occur in the fourth century

An illustration from Karl Friedrich Zo¨llner’s Transcendental Physics. Zo¨llner was convinced that the medium Henry Slade had tied four overhand knots in a closed cord by gaining access to a fourth dimension.

B.C. (Philip 1966). Aristotle denies the possibility of extending this sequence beyond the third dimension (On the heavens 268a,b). Herbart notes that in principle it could be continued to any number of dimensions, yet he concludes that intelligible space, like sensible space, cannot have more than three dimensions (Lenoir 2006, pp. 148–154). The essays below were translated from Kleine Schriften (1875), where they were reprinted with minor changes from their original form in Vier Paradoxa (1846). In Kleine Schriften, Fechner attached an addendum to ‘‘Space has four dimensions’’ in which he notes that Immanuel Kant, G. F. Bernhard Riemann, Hermann von Helmholtz, and Felix Klein also considered spaces of more than three dimensions. Kant was the first philosopher to consider seriously the idea of higher-dimensional spaces. In ‘‘Thoughts on the true estimation of living forces’’ (1747), he says that the threedimensionality of space seems to be a consequence of the inverse-square law, and he finds it probable that spaces of other dimensions exist. Riemann’s inaugural lecture on the foundations of geometry, ‘‘On the hypotheses which lie at the foundation of geometry,’’ is the most important nineteenthcentury paper on the fourth dimension. Although delivered in 1854 at the request of Gauss, who was duly impressed, it only

became known posthumously after Dedekind published it in 1868. According to Riemann, investigations of the geometry of space should be based on a prior theory of manifolds, or what came to be called Riemannian spaces, which can have any number of dimensions. His ideas provided the mathematical foundation for the four-dimensional geometry of space-time in Einstein’s theory of general relativity. The most intriguing part of Fechner’s addendum is his remark that Prof. Dr. Zo¨llner1 had described to him an ingenious way of explaining phenomena that appear in three-dimensional space yet are caused by forces from a fourth dimension. It appears that the source of this ‘‘ingenious’’ method was Felix Klein, who had given Zo¨llner an account of some of his recent work on space-curves. In particular, he mentioned that a knotted, closed curve cannot be untied in three-dimensional space, but in four-dimensional space it could be unknotted by deformations. Klein relates that Zo¨llner ‘‘took up this remark with an enthusiasm that was unintelligible to me. He thought he had a means of experimentally proving the ‘existence of the fourth dimension’’’ (Klein 1926, p. 157). By this time, Zo¨llner had become a devout spiritualist, and he believed that spiritual manifestations could be explained as interventions from a fourth dimension of space. In 1877–1878, he held experimental se´ances in Leipzig with the American medium Henry Slade designed to demonstrate the existence of a fourth dimension. Zo¨llner was fully convinced that he had succeeded, and he published the results of his ‘‘experiments’’ with Slade in Zo¨llner (1976). He reports that in December 1877, Slade tied four overhand knots in a closed cord by gaining access to a fourth dimension. Fechner attended two of these sessions, and his ‘‘almost involuntary’’ involvement in this infamous affair severely damaged his scientific reputation at the time (Wundt 1913, pp. 338–343; Staubermann 2001).

Dr. Mises, Fechner’s Alter Ego Beginning in 1821, Fechner wrote thirteen essays and books under the pseudonym Dr. Mises. Fechner often puts a cherished idea that might be unpopular into the mouth of Dr. Mises in order to avoid the scorn that might ensue if he advanced it directly. The message behind the jest of ‘‘The shadow is alive’’ is Fechner’s conviction that all nature is besouled (beseelt), but he has cast his panpsychism in a form that enables him to avoid being held responsible for it. Fechner believed that we cannot know whether space has only three dimensions: ‘‘Even for geometry, there are matters of faith in the number of dimensions and the theory of parallels. Indeed, everything is strictly a matter of faith, except that which is experienced directly or logically determined’’ (Fechner 1879, p. 17). Nonetheless, his purpose in ‘‘Space has four dimensions’’ is not to mock philosophers such as Herbart who insist that space is three dimensional, but to satirize the methods of reasoning that these philosophers use to ‘‘prove’’ whatever they please. The pseudo-arguments advanced for the four-dimensionality of space in ‘‘Space has four dimensions’’ are a parody of such methods.

1

Johann Carl Friedrich Zo¨llner (1834–1882) was professor of physical astronomy at the University of Leipzig and a friend of Fechner. He made his academic reputation in 1858 with the invention of a new type of astronomical photometer.

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Helmut Adler, the translator of Elemente der Psychophysik, observes that Fechner’s difficult style and his peculiar use of the German language prevented his ideas from being widely known (Fechner 1966, p. xix). Indeed, his sentences are long and involved and his style is diffuse; nonetheless, his writing expresses important ideas with a good deal of charm. In our translations, we have attempted to achieve clarity without destroying the appeal of the original. We thank Andreas Grotewald and Bernard Freydberg for reading these translations and making many helpful suggestions. After we completed our translation, we became aware of an earlier translation of ‘‘The shadow is alive’’ (Fechner 1991).

‘‘The Shadow is Alive’’ by G. T. Fechner There is actually nothing new in regarding a shadow as living. The ancients did so when they declared that after death the soul is a shadow and thereby ascribed a kind of life to it. According to the Greeks, just as a person casts a shadow that walks beside him, he also casts a shadow that will walk ‘‘after’’ him.2 Sunlight produces the former, and one’s own lifelight the latter. But why should it be necessary that a person die in order to bring his shadow to life? Shouldn’t a person rejoice if his most faithful companion under the sun walks with him as a living being rather than as a corpse? The legends foretell horror for those who have sold their shadows, for they have sold their twin brother.3 Once the devil has the shadow-soul, he will soon retrieve the light-soul as well. Although the shadow has many apparent similarities with us living beings, he also has many characteristics that we lack, and generally these are advantages. We are smallest at birth, grow, and then shrink with age. The shadow begins his life-day long, shrinks by noon, and lengthens again in the evening. He wants to show plainly that he doesn’t do everything as we do. He can determine the time of day simply by noting his own size. We live in three dimensions; he is satisfied with two, but that only makes him less cumbersome. The third dimension, which renders us thick and rigid, is the biggest obstacle in the way of all our attempts to make something else of ourselves. However we may turn, our pigtail always hangs behind us and our nose remains in front. If the shadow doesn’t like his pigtail any more, he pushes it inside himself and it’s gone. If he becomes displeased with his nose, he moves it within himself and it’s gone. Sometimes he grows his arms long, then he tucks them into his body, as in a pocket, and they are gone; in the next moment he makes them reappear. One moment he walks upright on a wall, the next he flits smoothly across the floor, and the next he bends himself like a T-square. He runs through thick and thin, while we choose our way carefully. He never gets muddy boots, never hurts himself on a stone, nor drowns in water, yet he fears fire even more than we do. He even passes through others of his own kind. When two shadows meet, they make 2

themselves somewhat blacker; when two people meet, they tend to turn somewhat pale. During all these changes, every shadow retains his peculiar characteristics. Neither a genius nor a fool can behave differently than his shadow. A person’s silhouette can even be used to represent his character. One sees in all this that the shadow not only differs from us living beings, but he is also an independent form of life. Meanwhile, man imagines that God has endowed only a few things with life, and he is so proud to be among these few that he strives to retain this special distinction. He does not accept the assertions made above that the shadow is alive, but objects: ‘‘All that is not enough. In order to live, one must first exist. A shadow has no substance at all; it is an illusion. It is not merely nothing, it is less than nothing.’’ What can the shadow reply? He begins by returning the same or equivalent arguments. Just as a man may say that he doesn’t believe in the life of his shadow, with equally good reasons the shadow can reply that he doesn’t believe in the life of his man. Because I am not a shadow, I don’t know exactly what the shadow thinks of his relation to man; in any case, two ways of looking at things are open to him. According to the first, he regards himself as a spirit and the person as his body. He is sure that this body exists merely to provide his otherwise immaterial existence with a connection to earthliness, just as a human considers his mortal body to be only a medium for his soul. The difference is that the shadow exists as a spirit beside his body, whereas our spirit resides within its body. Either of these spatial relations of spirit to body is quite possible. It is just as easy for a spirit to hang its jacket next to itself as to put it on. Indeed, don’t we believe that at death the soul will take off its ‘‘body jacket’’? And if we insist on a particular way of uniting spirit and body, how can we object when the shadow maintains that another way is the only way? If one observes the sharp contrast that the most learned philosophers make between body and spirit, then one might conclude that only the shadow’s point of view is true. However, the philosophers cannot be considered impartial on this matter, because they are themselves inspired by the shadow realm (the underworld). Why else would their theses be so little able to tolerate a well-focused illumination?4 I readily grant the shadow his rights, but don’t want to abridge any of ours. Nature endeavors to realize all possibilities, and so it has permitted both relations simultaneously. Thus one spirit exists in the body and another walks next to it, and so that they won’t quarrel, it has been arranged that each thinks that he alone has the body. It’s well known that nature uses one means to many ends. As it says in the Bible, ‘‘No one can serve two masters.’’5 By this is meant: No spirit can serve two masters. On the other hand we see everywhere that the same material can serve very different spirits. The Moon must give us light and at the same time provide a place of residence and nourishment for its inhabitants.6 Why shouldn’t our body

Plato records Socrates’ arguments for the immortality of the soul in Phaedo. Adelbert von Chamisso’s classic, Peter Schlemihl’s wundersame Geschichte (1814), is the story of a man who sold his shadow to the devil in return for unlimited wealth. 4 Fechner makes a pun on the word Beleuchtung, which can also mean examination. 5 Matthew vi.24. 6 Marilyn Marshall suggests that Fechner’s ideas on the moon’s habitability may have come from Franz von Paula Gruithuisen’s U¨ber die Natur der Kometen mit Reflexionen auf ihre Bewohnbarkeit und Schicksale (1811) (Corbet and Marshall 1969, p. 151). 3

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serve both a spirit which is in it and another which is beside it? Like the Moon, our body must provide residence and nourishment for one spirit (the soul) and give light to another (the shadow)—to be sure, not positively but negatively, that is, to supply the necessary absence of light. Our body is purposefully constructed, and it is inconceivable that nature should allow this purpose to be lost. It would be lost, however, if the shadow had no use for it. After all, I imagine that the shadow will express the following opinions about the body: Without this body I could not exist in this world; therefore it exists for my sake. Of course, it not only sustains me in this vale of tears, but also binds me to it. I hope that it will not always be necessary to carry around this heavy burden, which hangs on my heels, nor to walk in a world where there is more evil, that is, light, than goodness. If only I strive here to be as black as possible, then some day I will be taken up into a higher shadowkingdom, a kingdom of pure night, where I and other equally good shadows will walk blissfully without body or light. Obviously it is only my body that prevents me from gazing upon the great Primeval Shadow in the heavens, who created me and all other shadows. My body stands like a wall separating him from me, but one day it will fall.’’ The shadow may be wrong in thinking that the best and ultimate in the world could only be something similar to himself, and perhaps we are justified in making fun of him for this vanity. In turn, he would be justified in ridiculing us who believe that the best and ultimate in the world must be similar to ourselves. For both the truth remains that there is something after this life, which cannot be seen; although the shadow is better situated in this regard than we, who are completely enclosed by the same wall that confronts the shadow on only one side. There is yet another way that one may conceive of the shadow. On one hand, we regard this black neighbor as a constant companion dependent upon us; on the other, we see the reverse of our positive being. We stand in the opposite relation to our shadow. Thus, my shadow can regard me as his shadow just as I regard him as mine.7 However much I consider him to be lacking, my shadow regards me as having that amount too much. Whether it is a deficiency or an excess depends on the point of view. When he reflects on his mysterious being, the shadow will say, ‘‘What does this coarse clod that follows me have in common with the true sphere of being? He is only excrement, which has fallen from the region of reality; he has no existence of his own. He is an imaginary being entirely dependent on me, he must do everything that I do, yet he certainly does not share the freedom and agility of my movement but only affects a crude imitation. While I turn now right, now left, according to the time of day and year, he remains a rigid, upright stick and must always assume exactly the position determined by my position and that of the sun. Where is there any evidence of freedom or independence? If I disappear, he disappears as well, for a shadow has never perceived the existence of his human being any longer than he has perceived his own existence. A positive being is inconceivable without the

7

contrast provided by a negative being; it owes its illusory existence to this antithesis.’’ Now a person says, ‘‘I know perfectly well that I really exist, here I am.’’ The shadow replies, ‘‘I am here as well. One sees me, one senses my coolness. If I did not exist, how could one talk about me?’’ Meanwhile nothing will be clear to mankind. Naturally a shadow cannot provide enlightenment, and so I will bring my lamp. If the shadow were merely nothing, then I would not be inclined to defend his existence; however, he is less than nothing, and this proves useful to him. What does one perceive more strongly, being full or being hungry? Children and nations are at peace when they have the necessities, but they scream about anything that is lacking. Less has an even greater effect than more. Why shouldn’t nature feel a hunger for light wherever it is missing just as keenly as we do for food, freedom of the press, etc., whenever they are lacking? One may respond, ‘‘Not nature but the shadow should feel. Even if nature should perceive something at the location of the shadow, the shadow would still be no more an independent, living being than the chill, which I feel on my leg.’’ But what is all of mankind other than tissue and a consequence of the feeling of nature, yet detached from the rest of nature? Isn’t the shadow just as distinct from the rest of nature as are human beings? What is more distinct than the contrast between a shadow and the surrounding light? If the shadow is merely feelings, he feels at least as independent as a human, because he is detached from the rest of nature to the same extent. Until he can grasp a shadow in his hands, man will continue to demand tangible evidence of these perceptions and to refer to the shadow’s sense of perception as a shadow of perception. Ironically, he will insist that a spirit is an immaterial being until its immateriality has been confirmed in materiality. Analogously, the shadow confines himself to tangible surfaces of bodies. He dislikes more than two dimensions. Those who attribute no sense of perception to the void that the shadow makes in the light can at least grant this capability to the surface over which the void passes. The surface can perceive whatever is missing at any moment. It changes constantly, of course, yet the shadow retains his individuality. The material that makes up our bodies changes constantly as well. The matter in an old person is entirely different from that of the same individual as a child. With a person, as time goes by various materials pass through his form; with a shadow, as time goes by he passes over various material—the result is fundamentally the same. In any case, we observe that an individual’s identity depends on neither substance nor form. The form of an oldster is different from that of a child; it is enough that the later form be connected continuously through time with the earlier form. This principle applies to a person’s shadow just as it does to the person himself. Finally it is said, why prove that the shadow can feel, since he doesn’t feel anyway?—And how does anyone know that?—Just because one doesn’t have knowledge of the existence of something, he shouldn’t conclude that it doesn’t

See Hans Christian Andersen’s ‘‘The Shadow’’ (1847).

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exist.—But it is just the same with the shadow as with us. Would we think or feel any less because the shadow knows nothing of our thoughts and feelings? Then how can we be so unfair as to use such an argument against the shadow? I can neither affirm nor deny that the shadow experiences as great a diversity of thoughts and feelings as we do. Since many extraordinary events occur in our lives, which go unnoticed by others, even more extraordinary things may happen to the shadow, which escape our notice. In any case, if (like us on earth) fluctuations in the shadow’s feelings and thoughts are connected with coarser and finer variations in the body, then we must conclude that the shadow is hardly the lowest of creatures. He constantly changes not only his form, but also his shading. Other shadows and sidelights are always playing inside him as he moves this way and that. What aspect of a changeable life of feelings and thoughts does he lack? Can one really say that this play is meaningless? What is the meaning of the play of colors that catches our eye? The play of the shadows is analogous to the play of colors in our eyes; why should it have any less meaning for the shadow than when we see mountains, trees, and lakes? In short, I regard the shadow as a flat, black person, and see only reasons for his (living) existence, but none against it. If nothing else, this observation can be useful in reducing the number of superfluous dogs, for one usually keeps a dog only to have a living being with which to go for a walk or to conduct a one-sided conversation. Because one finds such a being in his shadow, it won’t be necessary to resort to a stranger. A shadow costs nothing to maintain, he is as obedient as a dog, and he will never be unfaithful. ‘‘Space has Four Dimensions’’ by G. T. Fechner A fourth dimension of space, one may say, is the fifth wheel of a wagon. No, I reply, it is only the fourth wheel of a wagon, without which the wagon would be of little use. A wagon needs a fourth wheel in order to move; we shall see that the fourth dimension of space serves the same purpose. From the outset, I certainly cannot hope to prevail in altering the view of four dimensions of space held by two classes of people: those who believe only what they perceive, and those who perceive only what they believe. In the first class I include the natural scientists who rely only on their senses and in the latter class the philosophers. In order to examine this matter thoroughly, the former class will walk the length and breadth of their rooms and then declare, ‘‘We have two dimensions. Exactly where is the third?’’ Because they are not accustomed to looking toward the sky (except for the astronomers who, of course, are always looking at the sky but see it inverted because of the nature of their telescopes), the idea of a third dimension does not occur to them so readily as the other two. Perhaps, the existence of a third dimension appears to them barely acceptable in the first place. Eventually they will admit that falling bodies require that a third dimension be admitted at least as a hypothesis; finally they confirm this hypothesis by climbing a stepladder. Then they say, ‘‘Alright, there is a third dimension, but where is the fourth?’’8 8

Having looked in vain for the fourth dimension in their rooms, they will go outside to repeat the experiment on a larger scale. They will go straight ahead, turn right, look up, and be finished. Looking for a fourth dimension, they examine crystals, plants, and animals, first with the naked eye, then with a microscope, then dissect them in order to search in the interior. They induce the government to send an expedition to the North Pole to find it; finally they have buildings constructed that avoid the familiar dimensions as much as possible in order to make unbiased observations of the unknown dimensions. Once all these approaches prove futile, they will be satisfied and declare that there is no fourth dimension, just as others have proclaimed for similar reasons that there is no God. The philosophers behave differently. Whereas the scientists actually walk around and look at the thing they want to examine, the philosophers retreat from it as much as possible and look away in the belief that the best way to discover the essence of a thing is to turn their backs on it, and the surest way to avoid a clash with reality is to abstain from troubling themselves with the matter. Accordingly they carefully avoid all measuring, weighing, dissecting, and observing. They sit in their easy chairs and expect that the pure idea of space will come to them. It comes. But as this idea is about to enter the mind, a door-keeper stops it and declares: ‘‘My master never counts past three; whatever exceeds that must remain outside.’’ Space, because its nature is to find its place everywhere, would dearly like to find a spot in the philosopher’s head, and so, satisfied that it has four dimensions, it leaves the fourth outside and is admitted. The philosopher counts: perception is threefold, God is threefold, man is threefold, and likewise space is threefold. There is no number in the world except three; one itself comprises only three thirds; likewise each unit can be broken down into three thirds. Since each third is equal to three thirds, one equals nine, and by the same reasoning it equals 81. Consequently, one equals every power of three. Surprised by how quickly progress is made in this way, he derives from this the principle of absolutely true, pure, higher mathematics, and in the process finds enough to refute Newton and Gauss, who rigidly insist that one equals one. The philosopher forgets so thoroughly about time and space that they in turn forget about him. In short, a fourth dimension of space will meet with very little acceptance from these two groups. More is to be expected from those who base everything on practical applications once we show them all that can be accomplished with an additional dimension of space. Currently the population is crowded and the land is insufficient to feed it adequately. If we had another dimension, then the fields would extend not only in width but also in height, and the square of the yield would increase to at least a cube of the yield. Of course the heavens, which now lie above us in the direction of the third dimension, would have to be shifted to the fourth dimension—where, in any case, it already finds itself with most of mankind. One would no longer have to emigrate to find whatever he might need or

For a discussion of arguments for the three-dimensionality of space, see Max Jammer, Concepts of space; the history of theories of space in physics. 2d ed. (1969), pp. 172–185 and Peter Janich, Euclid’s heritage: Is space three-dimensional? (1992).

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desire. Bread, money, happiness, freedom, and equality—all that one seeks would be in the new fourth dimension, which naturally would start just above every one’s head exactly as the third dimension does now. All current ‘‘castles in the air’’9 would then be transformed into actual castles, and building new castles in the air in the higher regions of the fourth dimension would always be permitted, because some people cannot live except in castles in the air. For the life of me, I can’t understand why the common people, who have heard many stories of this fourth dimension by other names, should suddenly cease to believe in it once it is called by its proper name. The benefits of the fourth dimension are so great and so evident that there is absolutely no reason why it should not be accepted just as readily as other ideas that promised less but were accepted for what they promised. From whence we get the fourth dimension is a separate matter, which does not diminish in the least the benefits that can be expected. In any case, it will be most convenient to derive the fourth dimension from the original three. Indeed, since the blessings only begin with the fourth dimension, it may be best to eliminate the old three entirely in constructing the fourth and thereby make the fourth dimension the only one. A single dimension has great practical advantages. One need not look around but rather always head toward a goal. Once an impetus is given, everything must proceed in the direction of common progress, and no longer can anyone stand idly on the sidelines. Universal freedom will be reached most easily; for everyone can do as he wants, since the only thing one can want is to follow the one and only direction that exists. Universal equality is achieved at the same time, for there is no longer any difference between thick and thin, because everything is as thin as possible. Certainly all friends of progress, freedom, and equality will recognize their ideal in this world of one dimension. Meanwhile others will prefer the old three dimensions with their comfortable breadth and thickness in spite of all their inherent shortcomings (indeed even because of these shortcomings) to the new dimension of sheer progress toward improvement. They will value the room in threedimensional space that allows them to get out of the way of those who are dedicated to unconditional progress. Above all, they will be vexed by the question: Where in the world of one dimension will they find room for their belly and how thin must the sausages be, if the entire pig is only as thick as a (mathematical) line? It would be of little consolation to tell them that in the world of progress there is neither time nor leisure for eating, and so it is not possible to acquire a thick belly. If I add a fourth dimension for those satisfied with three, they will once again make frowning faces. ‘‘It is always something new, they will say, and no end is in sight. Let us remain in our good old three dimensions, into which we were born and educated; every day there is too much that is

new.’’ I hope to win them over by promising to employ the new dimension to restore the pigtails that were cut off along with the heads in the French Revolution, although they will be less concerned about the heads. Also I will remind them of the advantage of a four-dimensional stomach. Then I believe that this entire worthy class will come over to my side. But the men of progress we will set free. In truth, those whom I pity, should a fourth dimension be added to the three, are the students who are already frightened when they have to advance from plane geometry to the mountain of solid geometry. They would face a geometry of four dimensions—a Pelion upon Ossa.10 What kind of perspective drawings will be necessary to prove that a fourdimensional prism can be dissected into four pyramids of equal volume? However, for the able geometers, who have already crawled into every angle and corner of the old dimensions, this will be a new treat. Once they are confronted with entirely new corners, the old candy will seem stale. Nevertheless they may hold their spherical trigonometry ready for spheres of four dimensions, because I am about to introduce the fourth dimension. The manner in which I shall introduce a fourth dimension to space is admittedly peculiar; namely, I begin by removing one dimension from the three. Imagine a small, multicolored little man, who moves around on a piece of paper in a camera obscura11; such a being exists in two dimensions. Nothing prevents us from regarding this being as living. In the previous essay, we saw that even a shadow-man can be thought of as a living being. Here we will not argue again that he is alive; it is enough to have done it once. Since all of his vision, audition, thoughts, and endeavors are confined to two dimensions, he would of course know as little of a third dimension as we, who live in three dimensions, know of a fourth. The experimenting shadow- or color-man would scurry around on his plane and search in vain for a third dimension and employ microscopes and telescopes just as vainly as does our scientist in searching for the fourth. He cannot lift his gaze above the plane, but can look only in the direction of the plane. Since his comprehension is formed in the context of his mode of viewing, the philosophizing shadow man will be no more able to escape two dimensions than our philosophers are able to escape three. Both the scientists and philosophers (in two dimensions) will declare that is impossible that a third dimension exist, since no more than two (straight) lines can be drawn at right angles through a point. They would have no idea where to put the third dimension, yet it exists. It obviously exists for us who live in three dimensions. We are merely color- and shadow-men in three dimensions instead of two. As we see that dimensions do not end at two except for beings who limit themselves to two, should we not conclude that dimensions do not end at three except for beings who limit themselves to three? Perhaps the world

9

A castle in the air (Luftschloss) is a visionary project or scheme, a daydream, or an idle fancy. Mount Pelion and Mount Ossa together form the Magnissia peninsula in the Aegean Sea. The phrase ‘‘to pile Pelion upon Ossa’’ means to add to what is already great, or to add difficulty to difficulty (from Virgil Georgics I. 281, imponere Pelio Ossam). 11 A camera obscura was the precursor of the modern photographic camera. The earliest versions, dating to antiquity, consisted of small darkened enclosures with light from external objects admitted through a single tiny hole to form an image of the objects on the opposite surface. 10

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cannot count past three? There is not the slightest reason why the world should be limited to three dimensions; therefore I conclude according to the Principle of Sufficient Reason12 that it is not. Let us consider whether the third dimension looks any different from the first or second. If no greater skill is required to create the third than the second or the first, then likewise no greater skill will be required to create the fourth and fifth than the third and second. Nature only ceases what it has begun when it loses strength. But the third dimension is no shorter than the other two. Once we have the fourth dimension, we also have the fifth, sixth, seventh, and so on through an infinite dimension. We can truly revel in dimensions, manufacture them like stickpins, and erect a framework as elaborate as we wish. Formerly, a dimension may have seemed to be a strange thing; but now dimensions will prove to be commonplace. Indeed, if one turned each hop-pole13 in Bavaria, each tollgate in Austria, and each knout14 in Russia into a new dimension, there would be no lack of material for many more. Of course, the philosophers will say: ‘‘We philosophers are the brain of the world; if nothing higher than three goes into our heads, which are the best of all, then it’s amply demonstrated that absolutely nothing in the world goes beyond three.’’ I consider the world to be a huge chicken, from which philosophy together with the philosophers come as a mere wind egg.15 As is well known, the egg wants to be more clever than the chicken; but since the egg can only count to three and the chicken is certainly smarter than the egg, we have a proof that the chicken can count still further. Meanwhile even a nonphilosopher may say: ‘‘Three is a nice round number. Perhaps in its youth, space learned the proverb, ‘‘All good things come in threes.’’ Then when it reached the third dimension, it restrained the natural desire to go further thereby setting a good example for mankind, who should likewise restrain themselves.’’ But that is circular reasoning, because in a space of four dimensions it will be said that all good things must come in fours, and in a space of five dimensions it will be said that all good things must come in fives. Moreover, we want to be modest and for the present we will consider only the fourth dimension, which we have almost in our grasp. Let the tenth or hundredth fly to the roof. If a person cannot refute me, then he will say: No refutation is needed. The proofs with shadow- and pseudo-men are only shadow- and pseudo-proofs. Show us a mere onehundredth of a line from the fourth dimension and we will concede one hundred miles or however much you want. Now it is reasonable that a person doesn’t want to buy a cat in a sack unless he can see at least a small part of the tail. To the philosophers I could reply: A cat in the sack is better than

a sack without a cat, which is what you want to sell to people. To the scientists I would say: It is better to take the cat in the sack, because if you let it out, it will probably escape. In order to do my best, I shall reconsider the colored man in two dimensions. If I can lay hold of a third dimension in a space of two dimensions, then it must be easy to come to grips with a fourth dimension in a space of three dimensions. This is merely a specific application of a method, which has been fruitful from time immemorial: Whatever cannot be found in three-dimensional reality can be found in two dimensions, that is, on paper. To come to the point, I take the plane surface where my pseudo-man is located and move it throughout the third dimension. Then the pseudo-man will experience everything that exists in the third dimension; he will even be altered by coming into other spaces where the light rays are ordered and colored differently. At the end of his journey he may appear pale and wrinkled; whereas at the beginning he appeared ruddy and smooth. Of course the little man never has a piece of the third dimension all at once and therefore believes that he has always remained in his two dimensions. All he understands of his traversal of the third dimension is the passage of time and the concomitant changes he undergoes. Accordingly, he says, ‘‘There is time and with time everything changes, myself included.’’ Like the shadow-man, we say, ‘‘There is time and with time everything changes, ourselves included.’’ What is the basis for this observation? As our space of three dimensions moves through the fourth dimension, we perceive only the passage of time and the changes that occur. Nothing is fundamentally simpler or more natural. Our three-dimensional universe is an immense ‘‘primary sphere’’ (Urkugel), which comprises a set of individual spheres.16 Each of these spheres moves in an orbit thus the primary sphere also moves, but where can it move if there is no fourth dimension? As the primary sphere moves through the fourth dimension, all the embedded spheres and all that is alive on them is carried through the fourth dimension. The foregoing opens the way for a beautiful observation. Everything that we will experience is already here, and all that we have experienced is still here. Our three-dimensional surface (nothing prevents us from speaking of a three-dimensional ‘‘surface’’ in four-dimensional space) has already passed through the one but not yet through the other. To describe the life of a person, we imagine a long beam that extends in the direction of the fourth dimension; in the beginning the beam exhibits the form of a child, in the middle that of an adult, and toward the end an elderly person. The configuration of the person at any moment of life can be imagined as a three-dimensional cross-section of this beam. In order to understand this properly, consider how certain

12 Leibniz held that the Principle of Sufficient Reason (das Gesetz des zureichenden Grundes) is fundamental to all reasoning. ‘‘There can be found no fact that is true or existent, or any true proposition, without there being a sufficient reason for its being so and not otherwise, although we cannot know these reasons in most cases.’’ That is, nothing exists without a reason for its being and for being as it is. (The Monadology 1714). 13 A hop-pole (Hopfenstange) is a tall pole on which hop-plants are trained. 14 A knout (Knutenstrick) is a whip for flogging criminals. 15 A wind egg (Windei) is an imperfect or unproductive egg, especially one with a soft shell. 16 Compare: ‘‘Das Universum ist eine Kugel, und alles, was im Universum ein Totales ist, ist eine Kugel.’’ This figurative model of the universe is given in Lehrbuch der Naturphilosophie (1810) by Lorenz Oken, a naturalist whose philosophy influenced Fechner as a student.

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cute little inlaid pieces are manufactured to be used for adorning broaches, rings, etc. To begin, one cements long colored sticks to one another in a suitable sequence and cuts the assembled sticks crosswise to obtain easily a collection of identical mosaics. Similarly, at each moment the threedimensional cutting surface will cut a person’s ‘‘beam of life’’ to produce a new person. There are two differences: The inlaid work is cut in two dimensions, whereas the cutting surface for a person is three-dimensional, and for each subsequent cut the person is slightly different, while the shape of the mosaic is replicated exactly. One could even obtain mosaics in the form of a child on the first cut and the shape of an old person with the last cut if one were to use suitable rods, which progress lengthwise instead of remaining uniform. The preceding method promises very useful consequences if one could only discover a means of cutting a person’s beam of life by crosscuts into discs or short cylinders and then setting these sections beside one another instead of behind one another in sequence. Then one could cut an entire army of soldiers from a single person, and each soldier would not only have the same uniform, but also the same face, the last face a bit older than the first. If one were clever enough to train the ‘‘soldier-pole’’ before cutting, then after cutting one would have an entire army uniformly trained. The officers would be cut from a special pole, as is current practice. Of course, each soldier would live only a short time because a mere fraction of the entire lifetime of the person is available to him. But that doesn’t matter for soldiers, who only exist to be killed in order to provide a place for new people. They would fulfill their purpose all the more quickly. Another important application of this invention would allow us to dispense with the printing trade. Every book, which an author writes, extends like a beam into the fourth dimension; it does not disappear from the earth at the moment the author finishes writing it. Using the previous method, we can cut as many copies as desired from this beam; these copies have the additional merit of being written in the author’s own hand. Of course, each of them would last only a short time, but that wouldn’t matter, for books only exist so that new ones can be written (on the same subject). Our ‘‘books’’ would serve their purpose and make room for new ones even a bit more quickly. Accordingly, I would recommend that there be an open competition in relation to this matter. Of course, no one would find the solution but that doesn’t matter with contest problems, which are not there to be solved but only to be posed in order to make room for new ones. There is another matter worth noting: We can spare ourselves all of our own movements by employing the motion of the surface of three dimensions through the fourth. In order to establish this beautiful proposition and secure the eternal rest toward which all pious people strive steadfastly, one will first have to endure a mental exercise.

Consider two beams of light, one red and one yellow, that both strike a horizontally oriented sheet of white paper at its center (labeled O) creating an orange spot. Suppose that the red beam is perpendicular to the paper and the yellow beam is directed obliquely toward O. If the paper is raised vertically keeping the angle of incidence of both beams unchanged, a red spot (labeled R) will appear and remain at O as the paper continuously intercepts different cross-sections of the red beam. There will also be a yellow spot (labeled Y) that stays to one side of the red spot and moves away from it, even though the yellow beam remains rigidly directed toward the paper. Now suppose that the angle between the beams is increased; that is, the yellow beam is directed more obliquely. If the paper is raised just as before, the distance between the yellow and red will be greater and the yellow spot will move away from the red spot more rapidly. These differences must be caused by the angle of the yellow beam, for the motion of the paper has not changed.17

A′

R

red

A

Y

B′

yellow

O

B

In the same way, when something seems to move in our three dimensions, it is only because the beam that extends into four-dimensional space is directed obliquely against the three-dimensional surface, and as the surface moves, it cuts the beam at different places. The more oblique the angle, the more rapid the motion appears. If the motion is along a curve, then it is simply because the beam is curved. This leads to several new fruitful observations. First of all, we see that the mathematician no longer has any cause to complain about the additional work that the fourth dimension entails, since he can omit the entire theory of motion. Everything remains as it is and it is not necessary that he calculate the ‘‘primary path’’ (Urgang) of the world, it will continue on its course. To determine the shape of fourdimensional space, one need only consider the (temporal) variable t as a fourth spatial coordinate. Second, the scientist will gain new ways of looking at nature. For example, if we see a planet moving around its orbit, it is only because the planet extends on a spiral- or corkscrew-shaped beam into the fourth dimension. At any given moment, the threedimensional surface on which the planet is located cuts through this spiral beam just as the two-dimensional paper

17

Let x denote the length of the segment RY, let y denote the length of the segment OR, and let h denote the angle between OR and OY. Then x = y tan h. If y0 denotes the rate at which the paper is being raised and x0 denotes the rate at which the yellow spot is moving away from the red spot (i.e., x0 = dx/dt and y0 = dy/dt, then x0 = y0 tan h. If h = 45°, then x0 = y0 ; that is, the yellow spot will move away from the red one at the same rate at which the paper is being raised. If h is increased to 76°, the yellow spot will move away from the red one approximately four times as fast as the paper is being raised.

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cut through the beams of light, thus it appears that the planet is moving around an orbit.18 It is illuminating to regard the entire universe as a large growth with spiral fibers and all of astronomy as merely a microscopic part of botany. Most important are the practical consequences. Only now will man clearly recognize that with all his movement he gains nothing at all; to all intents and purposes, he does not move. As the Bible says, ‘‘The race is not to the swift.’’19 He achieves nothing but to be inclined more obliquely, and a person who follows a crooked path will only turn himself into a screw. Mankind is now spared all worry. All bread that he will eat has already been baked, he need not even open his mouth to eat it, his mouth will be opened for him once the world reaches a certain location, and a bit further his mouth will be closed. Any bruise which one will suffer has already been bandaged and has healed. The money that one will collect already lies counted and will only be collected in the crossing of three dimensions, and if a Jew races from house to house for the sake of a penny, he can be certain that if the penny is not destined to be in his purse, all his racing will be futile.20 In short, henceforth mankind can live the most comfortable life in the world; it will always arrive where it is destined to arrive. It remains to answer the question, ‘‘Where will all this motion in the direction of the fourth dimension lead?’’ One can pose two hypotheses: According to the first, we will be led in the most natural way to the fulfillment of every hope that mankind has for the future, namely the resurrection of the dead, the rejuvenation of the body, the return to God in paradise, where the Jews will end up in the bosom of Abraham,21 to a life with an entirely new manner of being, which in every respect can be regarded as a fulfillment of the present life and in which the fairest system of judgment imaginable obtains. What more could anyone want? Nonetheless, I worry that man in his usual arrogant way, once he is entirely sure of everything that he earlier only wished for or hoped for, will begin to wish for still more or perhaps something entirely different. For the satisfaction of such arrogance, it is always good to have anther hypothesis ready. Note that almost all motion in nature is back and forth. The pendulum swings back and forth, the strings of a musical instrument go back and forth, when light passes, the ether22 goes back and forth, man also runs back and forth, indeed each leg swings back and forth individually. It therefore appears more than likely that at a certain time in the future, the order of events in the world will be reversed. Everything that has already happened will happen in reverse order, since otherwise one could accuse nature of pursuing a single direction while two were available to her. Every wheel that rolls forward can also roll backward, and it is strange that

while one speaks of the wheel of time, one never considers its backward motion. Now suppose that such a backward motion begins at a certain moment. Then it seems logical that all graves will open and all men who have died will rise again and if someone’s bones lie widely dispersed, they will be reassembled as a living being; each person will become younger day by day; aging will cease to exist and life will consist of rejuvenation; finally everyone will return to his mother’s womb and the same goes for the mother and so on further back until each set of parents collects its children, the Jews will again be collected in the bosom of Abraham, until finally all the seeds of mankind have been gathered again in Adam and Eve as in two sacks and will be returned into paradise where Eve will crawl back inside Adam and be changed into his rib, Adam will be taken by God and molded into a ball of clay; then God will take up all the earth, the ocean, the sun, and the stars into oneness with Himself. In the course of this retrogressing, each person will now receive from others exactly what he gave them. The shoemaker will receive from me exactly the same shoes that he delivered to me, and I will receive from him the same money that I paid him; the ox will receive the leather from the shoemaker that he gave him, and the man will get fodder back from the ox; the field will recover the grain that the farmer harvested, and the farmer will get back the seed and the fertilizer he brought to the field. In short, no one will be able to say that he received any more or less, or any better or worse, than he gave, since he will get back exactly what he gave. Certainly such a retrogressing would meet the highest standards of justice. In this way all that was promised and hoped for will be completely fulfilled. All will be judged fairly—each will be measured with exactly the same standards that he used to measure others; nonetheless, as noted above, mankind will not be satisfied. Instead, each person will seek to be judged by more generous and more favorable criteria in order to get a little more than he has earned. Since I am one of these unreasonable souls, I leave the second hypothesis to those who wish to let the wheel of time run backward. I adhere to the first hypothesis that there is an infinite progression of the world in a single direction from worse to better. I see a whip behind each being driving him in this direction, whether or not he wishes to go this way. Therefore the entire world with its four dimensions may very well be nothing but a large four-legged creature, which is being driven by such a whip. Ahead stands a feeding trough of eternal bliss where the world, if it is weary, may rest and gorge itself. But sooner or later the big teamster will lift his whip once again and drive the world forward to another manger, which is even more full of eternal bliss.

18 To illustrate this point, we consider the analogous construction in three-dimensional space. The position of a point moving around the unit circle, x2 + y2 = 1, may be represented by the parametric equations, x = cos t, y = sin t. Then the equations, x = cos t, y = sin t, z = t, determine a helix (‘‘a corkscrew-shaped beam’’) extending into the third dimension. As a plane surface is raised, the point of intersection of the helix and the surface will move around the unit circle. 19 Ecclesiastes ix.11. 20 Apart from this portrayal of a Jew as a miser, we are not aware of any evidence that Fechner was an anti-Semite. We do have the following slight evidence to the contrary. The Jewish writer and poet, Sigfried Lipiner, was a student and admirer of Fechner. Lipiner introduced his friend Gustav Mahler to Fechner’s works, which Mahler read ‘‘with engrossing interest.’’ For Fechner’s influence on Mahler’s music, see Hefling (2000). 21 Abraham’s bosom was a figurative expression in the Jewish oral tradition for the blissful state of the righteous (souls) after death (cf. Luke xvi.22, 23). 22 During the 19th century, physicists theorized the existence of ether, a universal substance that acted as the medium for transmission of electromagnetic waves. The ether was assumed to be weightless, transparent, frictionless, undetectable chemically or physically, and permeating all matter and space.

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Now it is obvious what would be lost if one would amputate one of the legs of this animal; at the same time it is easy to understand why the fourth leg has been overlooked. Indeed, because the animal always keeps this leg elevated in order to make progress, we creatures standing on earth think that it has only three legs and if we should look forward we would perceive the elevated leg only as a finger pointed upward. All the while this elevation is essential in order that progress may continue. Addendum 1875: At the time of the writing of this essay (1846), I was unaware that Kant had already considered the possibility of spaces of more than three dimensions. Notable mathematicians such as Riemann, Helmholtz, and Klein have speculated on the subject. Further, I remember reading an announcement of a paper (whose title I do not recall) published several years ago by Kirchmann.23 In this paper, Kirchmann, doubtless without knowledge of my essay, described change in the world in a way similar to that given above, but with more philosophical seriousness. Finally, in several conversations, Prof. Dr. Zo¨llner has described to me a very ingenious way of explaining wonders that appear in three-dimensional space yet are caused by forces from a fourth dimension. If a demonstration of the facts of these wonders could be found, it would provide an empirical proof of the existence of a fourth dimension. He might include this demonstration in a presentation of more general considerations in which this notion occurs.

Fechner, Gustav T. 1879. Die Tagesansicht gegenu¨ber der Nachtansicht. Leipzig: Breitkopf and Ha¨rtel. Fechner, Gustav T. 1966. Elements of Psychophysics. Translated from vol. 1 of Elemente der Psychophysik by Helmut E. Adler. New York: Holt, Rinehart and Winston. Fechner, Gustav T. 1991. The shadow is alive. Translated from Der Schatten ist lebendig by Stephen C. Simmer. Spring 51, pp. 80–85. Heidelberger, Michael. 2004. Nature from within: Gustav Theodor Fechner and his psychophysical worldview. Translated by Cynthia Klohr. Pittsburgh: University of Pittsburgh Press. Hefling, Stephen E. 2000. Mahler, Das lied von der Erde. Cambridge: Cambridge University Press. Ho¨ffding, Harald. 1908. A history of modern philosophy. Trans. by B. E. Meyer. London: Macmillan and Co. Jungnickel, Christa, and Russell McCormmach. 1986. Intellectual mastery of nature, vol I. Chicago: The University of Chicago Press, pp. 58–61. Klein, Felix. 1979. Development of mathematics in the 19th century. Translated by M. Ackerman from Vorlesungen u¨ber die Entwicklung der Mathematik im 19. Jahrhundert (1926). Brookline, MA: Math Sci Press. Kuntze, Johnannes E. 1892. Gustav Theodor Fechner (Dr. Mises). Ein deutsches Gelehrtenleben. Leipzig: Breitkopf and Ha¨rtel. Lenoir, Timothy. 2006. Operationalizing Kant, manifolds, models, and mathematics in Helmholtz’s theories of perception in The Kantian legacy in the nineteenth century, Michael Friedman and Alfred Nordmann (eds.). Cambridge, MA: The MIT Press.

REFERENCES

Petsche, Hans-Joachim. 2009. Hermann Grassmann: biography. Basel: Birkha¨user.

Archibald, R. C. 1914. Time as a fourth dimension. Bull. Amer. Math.

Philip, J. A. 1966. The ‘‘Pythagorean theory’’ of the derivation of

Soc. 20, pp. 409–412. Corbet, Hildegard, and Marilyn E. Marshall. 1969. The comparative

Staubermann, Klaus B. 2001. Tying the knot: skill, judgement and

anatomy of angels: a sketch by Dr. Mises: 1825. J. Hist. Behav. Sci.

authority in the 1870s Leipzig spiritistic experiments. Br. J. Hist. Sci.

5, pp. 135–151. Ellenberger, Henri F. 1970. The discovery of the unconscious. New York: Basic Books.

magnitudes. Phoenix 20, pp. 32–50.

34, pp. 67–79. Wundt, Wilhelm. 1913. Reden und Aufsa¨tze. Leipzig: Kro¨ner. Zo¨llner, Johann C. F. 1976. Transcendental physics. A reprint of the

Fechner, Gustav T. 1846. Vier paradoxa. Leipzig: Leopold Voss.

1888 ed. published by Colby & Rich, Boston. (Translated from vol. 3

Fechner, Gustav T. 1860. Elemente der Psychophysik, vol. 2. Leipzig: Breitkopf and Ha¨rtel, p. 559.

of Wissenschaftliche Abhandlungen with a preface and appendices by Charles C. Massey.) New York: Arno Press.

Fechner, Gustav T. 1875. Kleine Schriften. Leipzig: Breitkopf and Ha¨rtel.

23

J. H. von Kirchmann was a prolific writer and the founding editor of the Philosophische Bibliothek (1868), Kirchmann’s essay in Ueber die Unsterblichkeit; ein philosophischer Versuch (1865), pp. 63–73, is probably the one to which Fechner refers.

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The Mathematical Tourist

Dirk Huylebrouck, Editor

James Stirling: Mathematician and Mine Manager Does your hometown have any mathematical tourist

lthough James Stirling enjoyed a substantial reputation as a mathematician among his contemporaries in Britain and in some other European countries, he published remarkably little after his book Methodus Differentialis: sive Tractatus de Summatione et Interpolatione Serierum Infinitarum in 1730. An article with valuable information on the achievements of James Stirling during his time as mine manager in Scotland appeared in the Glasgow Herald of August 3, 1886, see [8]. The Herald article was reprinted in Mitchell’s The Old Glasgow Essays of 1905 [7]. Detailed accounts of Stirling’s mathematical achievements and correspondences with his contemporaries can be found in [12, 13], and [14].

attractions such as statues, plaques, graves, the cafe´

The Stirlings and Their Estates

P. MARITZ

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.

A

James Stirling was the third son of Archibald Stirling (March 21, 1651 - August 19, 1715) of the estate Garden (about 20 km west of Stirling) in the parish of Kippen, Stirlingshire, Scotland. In 1180, during the reign of King William I of Scotland, a Stirling acquired the estate of Cawder in Lanarkshire, and it has been in the possession of the family ever since. Sir Archibald Stirling (1618 - 1668) was a conspicuous Royalist in the Civil War, and was heavily fined by Cromwell; but his loyalty was rewarded at the Second Restoration (1661), and he ascended the Scottish bench with the title of Lord Garden, a Lord of Session of some distinction in the reign of Charles II (1660 - 1685). On his death in 1668, Lord Garden left Garden to his second son Archibald, the mathematician’s father, who became Laird of Garden in 1668 [14]. The family was a strong supporter of the Jacobite cause; this was to have a significant influence on the life of the mathematician James Stirling. (Jacobitism was the political movement dedicated to the restoration of the Stuart kings to the thrones of England, Scotland, and Ireland.) Figure 1 shows the memorial tablet to the Stirlings of Garden in Dunblane Cathedral, Perthshire. Sir Archibald Stirling, Lord Garden (mentioned above), and some of his descendants, were buried in this cathedral. In January 1686 Archibald Stirling, the mathematician’s father, married Anna, the eldest daughter of Sir Alexander Hamilton of Haggs, near Linlithgow. They had a family of four sons and five daughters. James, the mathematician, was the third son. He was born in the old Tower of Garden, previous to May 11, 1692, and was baptised on August 1 of that year [3].

â

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

James Stirling at Balliol Little is known of James Stirling’s early years. He reached Oxford toward the close of the year 1710. He was Ó 2011 Springer Science+Business Media, LLC, Volume 33, Number 3, 2011

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DOI 10.1007/s00283-010-9193-0

Figure 1. Memorial tablet in Dunblane Cathedral. Photograph: Hendry [3].

nominated Snell Exhibitioner on December 7, 1710, and he matriculated on January 18, 1711, at Balliol College. The Snell Exhibitions to Balliol College were established by the will of an Ayrshire man, John Snell (*1629 - 1679) [1, 12]. Their main provision was for the maintenance of Scottish students from Glasgow at Oxford. From the Register of Admissions and Degrees, Balliol College (not legible at places) [10]:

AUTHOR

......................................................................... PIETER MARITZ was born in South Africa,

and holds the degree of Dr. Wisk. Nat. from the University of Leiden in the Netherlands. His research interests include general measure theory, the theory and applications of vector measures and multifunctions, the history of mathematics, and mathematics education. He enjoys hiking in the mountains of the Western Cape region, gardening, watching and attending major sport events, and he has an interest in the older philatelic items. Department of Mathematics University of Stellenbosch Private Bag X1, Matieland 7602 South Africa email: [email protected]

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1711. Jan 4 Jacobus Sterling Filius Tertius Archibaldi Sterling in Parochiaˆ sancti Niniani in comitadu Sterling ~ in Scotiaˆ admissus est communarius et deinde sc ill : Jan; 10 exhibitionarius snellianus in locum Jacobi carnegy. Through the interest of the Earl of Mar, James Stirling was also nominated Warner Exhibitioner. No requirement to take Holy Orders (this being frustrated by the 1690 reestablishment of Presbyterianism in Scotland) or to return to Scotland was imposed on the Snell Exhibitioners. Anne, Queen of England, Scotland, and Ireland, died in August 1714, and the German George I of the House of Hanover acceded to the British throne. In 1715 there was another Jacobite Rebellion, which melted away after the Battle of Sheriffmuir on November 13, 1715. The Records of Balliol bear witness to Stirling’s tenure of the Snell and Warner Exhibitions to September, 1716. There is, however, no indication of his expulsion from Oxford, though the last mention of him informs us that he had lost his scholarship for refusing to take ‘The Oaths’ of allegiance to the House of Hanover [3, 14]. Certainly Stirling could not graduate from Oxford, and he left in 1716 [13]. In the minutes of a meeting of the Royal Society of London on April 4, 1717, when Brook Taylor lectured on extracting roots of equations and on logarithms, it was recorded that ‘‘Mr Sterling of Balliol College in Oxford had leave to be present’’ [11, 12]. James Stirling’s reputation among his contemporaries was off to an early start under the guidance of John Keill. In 1694, Keill (from Edinburgh) was admitted at Balliol as a Senior Commoner, and in 1712 he was elected Savilian Professor of Astronomy. He was a frequent correspondent

Figure 2. Balliol College. Print by David Loggan, 1675.

of Newton’s, and became thereby the pivotal figure in a protracted and fierce controversy over the discovery of the differential calculus. As a diversion from the main dispute, in 1715, Leibniz sent a complex problem in the properties of a certain kind of hyperbola to mathematicians in England, by way of a challenge. On February 24, 1715, Keill was able to write to Newton to say that he had a solution, adding that ‘‘Mr Sterling, an undergraduate here, has likewise solved this problem.’’ The precocious undergraduate was James Stirling [4].

First publication At about the same time, Stirling became acquainted with John Arbuthnot, the well-known mathematician, physician, and satirist [13]. Stirling must already by this time have been working on his first publication Lineæ Tertii Ordinis Neutonianae sive Illustratio Tractatus D. Newtoni De Enumeratione Linearum Tertii Ordinis. Cui subjungitur, Solutio Trium Problematum. This work was mainly a commentary on Newton’s enumeration of curves of the third degree. Isaac Newton had in his Tractatus de Quadratura Curvarum and Enumeratio linearum tertii Ordinis (1704) made great advances into the theory of higher plane curves, and brought order into the classification of cubics, but he furnished no proofs of his statements. James Stirling proved all Newton’s theorems up to, and including, the enumeration of cubics. Stirling’s connections with Arbuthnot enabled him to publish this 128-page book in Oxford. Afterwards, Colin Maclaurin and P. Murdoch made major contributions to the organic descriptions of curves in their respective works Geometria Organica (1720) and Genesis Curvarum per Umbras (1740). Another edition of Stirling’s Lineae Tertii Ordinis was published in Paris in 1787 as Isaaci Newtoni Enumeratio Linearum Tertii Ordinis. Sequitur illustratio eiusdem tractatus Iacobo Stirling.

In Venice The Venetian ambassador Nicholas Tron left London to return to Venice in June 1717, and it is almost certain that Stirling accompanied him. Stirling had been offered the Chair of Mathematics at the University of Padua (then in the Republic of Venice), but, for some reason that is not known, the appointment fell through [14]. Not much is known about his stay in Venice, but he certainly continued his mathematical research. In 1718 Stirling submitted, through Newton, his first Royal Society paper Methodus Differentialis Newtoniana Illustrata, Phil. Trans. 30 (No.362) Sept - Oct. 1719, 1050 1070. (Royal Society, June 18, 1719.) The first part of this paper is an explanation of Newton’s treatise Methodus Differentialis (1711) and is devoted to interpolation and quadrature by finite differences, Newton’s forward difference formula, Stirling’s interpolation formula, Bessel’s interpolation formula, and the closed Newton-Cotes quadrature formulae for 3, 5, 7, 9, and 11 ordinates. In the second part of his paper, Stirling pursued the transformation of a slowly converging series into one that converges rapidly to the same sum, and he introduced a process for finding limits, now known as the ‘‘Stirling-Schellbach algorithm’’ [6, pp. 276 - 280]. In August 1719, Stirling wrote from Venice thanking Newton for his kindness and offering to act as intermediary with Nicolaus I Bernoulli [14]. In 1721, Stirling was in Padua where he attended the university and where Nicolaus I Bernoulli occupied the chair from 1716 until 1722. According to Hendry [3], Stirling had to leave Venice in a hurry for fear of assassination because of his discovery of the secrets of the Venetian glass manufacturers, and Tweedle wrote that Stirling had, at the request of certain London merchants, acquired information regarding the manufacture of plate glass [14].

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In London It is not clear what Stirling did between 1722 and late 1724, but it is known that, at least from 1722, he had the intention of becoming a teacher in London: In August 1722, Colin Maclaurin (1698 - 1746) visited Newton in London and Newton showed him a letter from Stirling in which the latter wrote that he intended to set himself up as a mathematics teacher in London. At that stage, Maclaurin was professor of mathematics (since August 1717 already, at the age of 19) at Marischal College, University of Aberdeen, but he moved to Edinburgh in November 1725 as professor of mathematics. The minutes of the meeting of the Royal Society on October 27, 1726, recorded that ‘‘Mr Sterling was proposed for a Fellow by Dr Arbuthnot and recommended by Sir Alexander Cumming.’’ He was elected on November 3, 1726, and admitted on December 8, 1726 (Journal Book of the Royal Society, see [13]). Stirling enjoyed the friendship of Sir Isaac Newton and was corresponding on equal terms with most of the leading European scientists. About this time, Stirling was appointed to succeed Benjamin Worster as one of the partners of William Watt’s Little Tower Street Academy, Covent Garden, London, which was, according to references [2] and [11], ‘‘. . . one of the most successful schools in London; . . . although he had to borrow money to pay for the mathematical instruments he needed.’’ The Academy’s prospectus of 1727 listed ‘A Course on Mechanical and Experimental Philosophy’ given by Stirling and others. The syllabus included mechanics, hydrostatics, optics, and astronomy. While in London, Stirling published his most important work, namely, Methodus Differentialis, sive Tractatus de Summatione et Interpolatione Serierum Infinitarum, G. Strahan, London, 1730.à Regarded as one of the early classics of numerical analysis, this book contains the results and ideas for which Stirling is mainly remembered today: Stirling numbers, Stirling’s interpolation formula, and the speeding up of the convergence of series. Stirling’s formula for ln n! and the asymptotic formula for n!, for which he is best known, appear as Example 2 to proposition 28. After Abraham De Moivre (1667 - 1754) published his Miscellanea Analytica in 1730, Stirling wrote to him, pointing out some errors that he had made in a table of logarithms, and told De Moivre about Example 2 to Proposition 28. De Moivre was able to extend his earlier results using Stirling’s ideas and published a Supplement to Miscellanea Analytica a few months later. The influence of Metodus Differentials has now extended for more than 280 years. A vast number of articles and books have been devoted to ‘Stirling’s formula’ (proofs, extensions, and so on), and the modern development of combinatorial theory has ensured a central place for the Stirling numbers. Stirling showed that       1 1 1 1 ln m þ þ lnð2pÞ  m þ ln m! ¼ m þ 2 2 2 2 1  þ    ð1Þ 24ðm þ 12Þ à

Neglect all the terms involving reciprocal powers of m þ 12 and exponentiate:   1 pffiffiffiffiffiffi m þ 12 mþ2 ð2Þ m!  2p e Formulae (1) and (2) are Stirling’s own versions of the expressions that are generally labelled ‘Stirling’s formula’. De Moivre obtained the expression   1 1 1 ln m þ lnð2pÞ  m þ lnðm  1Þ! ¼ m  2 2 12m 1 1  þ  ð3Þ 360m3 1260m5 Neglect the reciprocal powers of m and exponentiate. Then rffiffiffiffiffiffi  2p m m ðm  1Þ!  m e or m! 

pffiffiffiffiffiffiffiffiffiffimm 2pm e

ð4Þ

Unfortunately, (3) and (4) are also labelled ‘Stirling’s formula’, probably because they are a little simpler than Stirling’s own versions. Maclaurin placed great reliance upon Stirling’s judgement, and frequently consulted Stirling while writing his Treatise of Fluxions [14]. In London, Stirling mixed on easy and familiar footing with Henry St. John, Lord Bolingbroke, and his circle of Tory supporters, which included Jonathan Swift, Alexander Pope, and other influential people. With such people around him, it was expected that he would soon be rewarded with a post of high academic standing. But it was not along such lines that Stirling’s future was to be shaped.

Leadhills At the age of forty-three, James Stirling was to enter upon an entirely different career that in due course would link his name with the precursors of the Scottish Industrial Revolution. In fact, Stirling is often described as one of the forgotten pioneering figures in the opening stages of the Scottish Industrial Revolution [3]. During the summers of 1734 - 1736 he was employed by the Scotch Mining Company at Leadhills, Lanarkshire, in Scotland. Leadhills, today only fifteen minutes’ drive from Abington, was once a busy lead-mining village; now it is a peaceful place of some 300 inhabitants where clusters of white and grey stone cottages nestle on the hillside. This village lies at the head of the barren and heathy valley through which the Sonar Burn flows down to the river Clyde. Lead has been mined in these hills since the thirteenth century; the last shaft in this area was closed in 1959. These hills also yield gold. A ring of Leadhills gold was presented to Queen Mary (1867 - 1953, the Queen Consort of George V), and the mace for the new Scottish Parliament has a thick band of Leadhills-Wanlockhead gold around it [5].

It was re-issued in 1753 by Richard Manby, Ludgate Hill, London, and in 1764 by J. Whiston and B. White, Fleet Street, London. A translation by Francis Holliday was published in 1749 by E. Cape, St. John’s Gate, London. An annoted translation by Ian Tweddle was published by Springer in 2003 [13].

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At the beginning of May 1737, Stirling returned to Leadhills with a permanent resident appointment as Chief Agent at a salary of £210 per year [12]. He proved extremely successful as a practical administrator [11]. There had been little systematic method in the running of the mines and they had been approaching bankruptcy. All this was altered by new methods and rules that he introduced, and with the prosperity of the mines the welfare and amenities for the miners themselves also improved. His measures to improve output and productivity as well as working conditions and social environment made the Leadhills mines not only a profitable undertaking but a model for future social reformers. For example, he divided the workers into four classes, namely, miners, labourers, washers, and smelters, he organised them into decent shifts, encouraged them to cultivate plots of land and crofts, and they were also made to contribute toward the maintenance of the sick and aged of the village [3, 7, 8]. He did not give up mathematics; in fact, there is a discussion of unpublished mathematical work in notebooks of Stirling that were probably written between 1730 and 1745 [12]. Although he continued his mathematical correspondence with Euler, Maclaurin, and others, it is quite clear that most of his energy was spent on mining affairs. On December 6, 1733, Stirling read a paper entitled Twelve propositions concerning the figure of the Earth to the Royal Society of London. This paper could be regarded as the first major contribution by a British scientist to the theoretical study of the figure of the earth and its gravitational forces since the seminal work of Newton and Huygens [12, pp. 102, 169 - 171].§ Stirling was considered the leading British expert on the subject for the next few years by all, including Maclaurin and Robert Simson, who went on to make major contributions themselves. As Stirling’s unpublished manuscripts show, he did go much further than the 1735 paper, but probably the pressure of work at the mining company gave him too little time to polish the work [12]. He explained in a letter to Maclaurin in 1738 why he had not published despite pressure to do so: I got a letter this last summer from Mr Machin wholly relating to the figure of the Earth and the new mensuration, he seems to think this a proper time for me to publish my proposition on that subject when everybody is making a noise about it; but I choose rather to stay till the French arrive from the south, which I hear will be very soon. And hitherto I have not been able to reconcile the measurements made in the north to the theory ... [11]. The French expedition to Ecuador, referred to by Stirling as ‘the south’, left in 1735 but did not return until 1744. To this period there belonged only one paper by Stirling, namely a very short article entitled A Description of a machine to blow Fire by the Fall of Water. This machine is known to engineers as ‘Stirling’s Engine’, and furnishes an ingenious mechanical contrivance to create a current of air,

Figure 3. Stirling’s house in Leadhills. Photograph: P. Maritz.

due to falling water, sufficiently strong to blow a forge or to supply fresh air in a mine.} The Scotch Mining Company built James Stirling a house supplied with a well stocked library and wine cellar, and provided him with a carriage and a pair of horses [3]. The house (also known as Woodlands Hall) still stands in cherished preservation on slightly rising ground above the village. Harriet Martineau, who visited Leadhills many years after Stirling’s time, wrote a delightful description of the house, which subsequently appeared in Dickens’s Household Words; the house was included in a list of the most distinguished gardens of the eighteenth century. Stirling successfully managed the mines for thirty-five years, during which time he transformed them into one of the most profitable industrial enterprises in Scotland. Allan Ramsay, born in Leadhills in 1686, had worked in the mines as a washer’s boy. By 1718 he had become known as a poet, and then started business as a bookseller, adding a circulating library in 1725 in the Luckenbooths of Edinburgh, apparently the first in Britain. Ramsay contributed to the revival of vernacular Scottish poetry. It was through his influence and the vision of the enlightened mine manager of the day, James Stirling, that a ‘subscribed’ library was formed in 1741, allowing for the first time leadminers and their families access to literature. It is the oldest subscription library in the British Isles. Of the 23 founder members at Leadhills, all were miners except the minister and the schoolmaster. The early books were mainly religious in character, and included many volumes of sermons. Today the library contains various relics of the past life of the village and the mines, as well as a book collection [5, 6]. In the year 1752, at the request of the magistrates of Glasgow, James Stirling carried out a survey and submitted proposals for the deepening of the upper reaches of the River Clyde. This survey paved the way for the later schemes of well-known engineers such as Goldburn, James Watt, and others [8]. In that year Stirling was presented with a silver kettle and stand.

§ An extended version of Stirling’s results appeared as the article: Of the figure of the Earth, and the variation of gravity on the surface, Phil. Trans 39 (No. 438), July Sept. 1735 (1735 - 1736), 98 - 105. (Royal Society, 1735.) } This machine is not to be confused with the much better-known hot-air ‘Stirling Engine’ invented in 1816 by the Scottish clergyman Robert Stirling.

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Figure 4. Leadhills Miners Library. Photograph: P. Maritz.

Figure 5. Name-plate of Leadhills Miners Library. Photograph: P. Maritz.

The kettle bears the inscription: ‘‘A compliment made by the Town Council of Glasgow to James Stirling, mathematician, for his services, pains, and trouble in surveying the river (Clyde) towards deepening it by locks. 1st July 1752’’. Stirling’s wife, Barbara (ne´e Watson), died in February 1753. The date of his marriage does not seem to be known, but it was no earlier than 1745. By his marriage with Barbara, Stirling had a daughter, Christian, who married her cousin, Archibald Stirling of Garden, and their descendants retain possession of the estate of Garden [14]. Archibald succeeded his father-in-law as manager of the mines. In 1746, James Stirling was elected to membership of the Royal Academy of Berlin. In 1753, he resigned from the Royal Society of London as he was in debt to the Society and could no longer afford the annual subscriptions. It cost him £20 to resign.

Greyfriars Church In 1638, the National Covenant was signed in the Greyfriars Church in Edinburgh. The covenant rejected the attempts by Charles I to reintroduce episcopacy and a new English prayer book, and affirmed the independence of the Scottish Church. A major Jacobite rebellion took place in 1745, and Maclaurin played an active role in the defence of Edinburgh against the Jacobites. There was no trace of James Stirling being implicated, though his uncle of Cawder was imprisoned by the government [14]. In 1746, Maclaurin died, partly as a consequence of the battles of the previous year, and he was buried in Greyfriars Kirkyard. 146

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Figure 6. Silver tea-kettle presented to James Stirling.

Figure 7. Plaque of James Stirling in Greyfriars Kirkyard. Photograph: P. Maritz.

Stirling was considered for MacLaurin’s chair at Edinburgh. However, his strong support for the Jacobite cause meant that such an appointment was impossible, especially in the year after the Rising [11, 12]. Charles and his Jacobites were finally defeated on April 16, 1746, by Cumberland’s forces at the Battle of Culloden, and effectively ended Jacobitism as a serious force in Britain. James Stirling died in Edinburgh on December 5, 1770, at the age of 78 years, when on a visit there to obtain medical treatment.

The plaque is wall-mounted, deeply incised and in good condition. The inscription reads: PROPE HVNC LOCVM SEPVLTVS JACET JACOBVS STIRLING COGNOMINE VENETIANVS MATHEMATICVS ILLVSTRVS QVI A. D. D. MDCXCII NATVS ARCHIBALDI STIRLING DE GARDEN FILIVS QVARTVS ANNO MDCCLXX MORTVVS EST John Ramsay assesses Stirling thus [9]: ‘‘This gentleman may be regarded as an excellent specimen of the Scotsmen of the last age, who began their course without patrons and without money, yet being well taught, and obliged to avail themselves of time and chance, their spirit of industry and address enabled them to surmount every difficulty, raising them to eminence, and commanding the esteem of all who knew them.’’ ACKNOWLEDGMENTS

Permission to use the photograph in Figure 1 was granted by the Curator of the Scotland’s Magazine Archives. The author is grateful for the opportunity granted by the Librarian of Balliol College, Oxford, to consult the documents on James Stirling. The copyright of the print by David Loggan in Figure 2 belongs to Balliol College, Oxford, and permission to use it was granted by Anna Sander, Lonsdale Curator of Archives and Manuscripts, Balliol College, Oxford. The illustration in Figure 6 is used by the courtesy of Colonel James Stirling of Garden. REFERENCES

[1] W. Innes Addison: The Snell Exhibitions. Founder, Foundation, Foundationers. James MacLehose & Sons, Publishers to the

[2] Dictionary of Scientific Biography. Volume XIII. Editor in Chief C. C. Gillespie. Charles Scribner’s Sons, New York, 1970 – 1990, 67 – 70. [3] W. B. Hendry: James Stirling ‘The Venetian’, Scotland’s Magazine, October 1965, 33 – 35. [4] J. Jones: Balliol College. A History. Second Edition. Oxford University Press, 1997. [5] Leadhills Reading Society. Available at URL http://www.lowther hills.fsnet.co.uk (accessed June 25, 2003). [6] Leadhills Library website. Available at URL http://www.minerslibrary.fsnet.co.uk (accessed March 28, 2008). [7] John Oswald Mitchell: The Old Glasgow Essays. MacLehose, 1905. [8] Muir Collection, MSB 691, 4(10), J. Stirling. The National Library of South Africa, Cape Town. [9] John Ramsay: Scotland and Scotsmen in the eighteenth century. Editor Alexander Allardyce, 2 volumes. W. Blackwood and Sons, Edinburgh and London, 1888. [10] Register of Admissions and degrees, 1682 – 1833. Balliol College, Oxford. [11] James Stirling. Available at

URL

http://www-history.mcs.

st-andrews.ac.uk/history/mathematicians/Stirling.html (accessed March 28, 2008). [12] I. Tweddle: James Stirling: ‘This about series and such things.’ Scottish Academic Press, Edinburgh, 1988. [13] I. Tweddle: James Stirling’s Methodus Differentiales. An Annotated Translation of Stirling’s text. Springer-Verlag. London. 2003. [14] C. Tweedie: James Stirling: a sketch of his life and works along with his scientific correspondence. Clarendon Press, Oxford, 1922.

University of Glasgow, 1901.

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Reviews

Osmo Pekonen, Editor

Le dossier Pythagore. Du chamanisme a` la me´canique quantique by Pierre Bre´maud PARIS: E´DITIONS ELLIPSES, 2010, 336 PP., €24, ISBN: 978-2-7298-6088-2 REVIEWED BY TAKIS KONSTANTOPOULOS

Feel like writing a review for The Mathematical Intelligencer? You are welcome to submit an unsolicited review of a book of your choice; or, if 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] 148

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n the introduction to his essay, Bre´maud observes that quite a few mathematicians show no patience for Pythagoras, frequently ridiculing or ignoring him, or equating his school to humbug and fraud. Many indeed would endorse Eric Temple Bell’s opinion ‘‘The sixth century before Christ was the time, and Greece the place, for human beings to reject once for all the pernicious number mysticism of the East. Instead, Pythagoras and his followers eagerly accepted it all as the celestial revelation of a higher mathematical harmony. Adding vast masses of sheer numerological nonsense of their own to an already enormous bulk, they transmitted this ancient superstition to the golden age of Greek thought, which passed it on in the first century A.D. to the decadent numerologist Nicomachus. He, enriching his already opulent legacy with a wealth of original rubbish, left it to be sifted by the Roman Boethius, the dim mathematical light of the Middle Ages, thereby darkening the mind of Christian Europe with the venerated nonsense, and encouraging the gematria of the Talmudists to flourish like a weed.’’ –Eric Temple Bell. Such contempt for Pythagoras is shared by historians and philosophers as well. Bre´maud quotes Arnold Toynbee’s claim that all the great sages of antiquity, Confucius, Buddha, Lao Tze, Deutero-Isaiah, have influenced posterity with the exception of Pythagoras. The Swiss author Jeanne Hersch (once in charge of the philosophy department of UNESCO) omits to mention in her well-known and influential textbook on the history of philosophy from Thales to Heidegger the name of Pythagoras, as well as the words ‘‘Pythagoreans’’ or ‘‘Pythagoreanism’’, an exploit that the author compares to the avoidance of an indispensable vowel like ‘‘e’’ in writing a novel. Bre´maud presents his work as a reaction against the implicit condemnation of Pythagoras to damnatio memoriae by mathematicians, historians, and philosophers. For most historians of science, Pythagoras played but a minor role in the unfolding story of mathematics (many would be hard-pressed to attribute anything else to

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Pythagoras other than ‘‘his’’ famous theorem, which–as we see in the book–was known before him). Why is it that Pythagoras is so often neglected or associated with theosophy, freemasonry, cheap mysticism, and the worshiping of the Golden Ratio? What are the relations of Pythagoreanism to classical Greek, and, therefore, post-renaissance and modern science? Why does Christianity take issue with Pythagoras? What are, really, the contributions of his school? Should a working mathematician care about his heritage, and do we not have here yet another book on the wise man of archaic Greece? Fortunately, the reader soon realizes that Bre´maud is considering the previous issues from an educated point of view, after examination of a large number of sources. The subtitle of the book, ‘‘from shamanism to quantum mechanics’’, hints at the essence of the essay. It deals more with the spirit, contributions, and influence of Pythagoreanism than the life of Pythagoras himself – if he had one. Even if he hadn’t, the consequences of Pythagorean thought are historically determined. The Pythagoreans formed a sect with initiations and rituals, but also a school with a political and philosophical program that lasted about one millenium, from 6th c. BCE to 6th c. CE when Plato’s Academy was finally dissolved by Justinian. The key notions of the subtitle correspond to the two strands of Pythagoreans, the akousmatikoi (listeners) and the mathematikoi (learners), and also to the interplay between them. Shamanism refers to Orphism (see the following text) as practiced by the Pythagoreans, to supposed journeys between Hades and this world (katabaseis), the purification rituals, the study of music, the reading of Homer, etc. On the other hand, the evocation of quantum mechanics refers to the rationalist influence of the school: For the first time in the history of homo sapiens, it became clear that the use of pure reason could explain, justify, and predict natural phenomena. Even though the famous theorem was certainly known before Pythagoras (the essay contains an account of the Egyptian, Mesopotamian, and Indian sources), it is his school that established pure reason as a framework for knowledge. Bre´maud provides a summary of historical facts, myths, and legends scattered throughout the literature. Pythagoras is supposed to have been born in Samos and to have been influenced by the sages of the East before moving to Magna Grecia in Southern Italy, where he established his famous school. The different viewpoints found in the literature are placed in the context of the transition from archaic Greece to classic Greece. The book also discusses possible connections of Pythagoreanism with Egypt (the Book of the Dead) and India (the ‘‘parallel lives’’ of Buddha and Pythagoras), the rallying of Copernicus, Kepler, and Newton under the banner of Pythagoras, and relations with the Kabbalah. The author devotes an entire chapter to the relationship of Pythagoreanism and Orphism, that ancient religion that later influenced Christianity and its rituals. Pythagoras’s mother is traditionally named Parthenais (the name stems from the word ‘‘virgin’’) whereas his father was supposed to be the god Apollo. He was a reformer of Orphism, which professed an ascetic way of life, the transmigration of souls (metempsychosis), and threatened evildoers with punishment in

afterlife. Orphism was a people’s religion, a far cry from the established religion of the twelve gods of Olympus. Another chapter is devoted to the uneasy relation between Pythagoreanism and Christianity. The two seem to share origins and rituals, but the former places Nature above God, whereas the latter does the opposite. This may explain a centuries-long love-hatred relationship between the two systems: whereas, at times, Christian apologists (such as Justin Martyr) tried to accommodate Pythagorean philosophy, the Church finally made its choice: Plato was eventually considered harmless enough, and even thought to have been prophesying about Jesus, whereas Pythagoras had to be excised from theology. Strong anti-Pythagorean sentiment sometimes persists to this date. The example of the philosopher Jeanne Hersch, who manages to ignore Pythagoras altogether, is striking. Could this be a conspiracy? Bre´maud reviews the fascinating story of the subterranean Pythagorean basilica of the Porta Maggiore in Rome (Bagnani 1919), its ominous resemblance to a Christian church (some, indeed, attribute the origin of Christian basilicas to Pythagorean models), and its continuing inaccessibility to the public. The enduring battle between Pythagoreanism and Christianity can be seen as yet another instance of antagonism between reason and faith. Compare, for instance, the following two quotes (Freeman 2005): ‘‘Blessed is he who learns how to engage in inquiry, with no impulse to harm his countrymen or to pursue wrongful actions, but perceives order of immortal and ageless nature, how it is structured.’’ –Euripides, fragment of unnamed play, 5th c. BCE and, a thousand years later, ‘‘There is another form of temptation, even more fraught with danger. This is the disease of curiosity. It is this which drives us to try and discover the secrets of nature, those secrets which are beyond our understanding, which can avail us nothing and which man should not wish to learn.’’ –Augustine, 5th c. CE The latter part of the book contains a more technical discussion of Pythagorean ideas and mathematics. For instance, there is a simple and accessible exposition of the history and principles of the Pythagorean musical scale. Pythagorean triples are derived, and perfect numbers are discussed. Euclid’s theorem, stating that if 2n - 1 is a prime number then 2n -1(2n - 1) is perfect, is established. In addition, Euler’s proof of the (partial) converse is detailed in the appendix. There are discussions about the Pythagorean significance of specific integers, as well as the meaning of irrational numbers for Pythagoreans. The famous Pythagorean theorem is examined in detail with its three canonical proofs (of which, Euclid’s seems to be the most complex), and Bre´maud argues convincingly that a proof of the already-known theorem had been derived by the Pythagorean school in the first half of the fifth century BCE. There is also a chapter on the Pythagorean astronomy, or rather, astral religion, and on the modern representatives of the Pythagorean tradition such as Kepler and Newton. Last, but not least, there is a chapter on the golden ratio and its adoration by amateur mathematicians and artists alike. Bre´maud is a mathematician. Because he is an amateur historian not subjected to the academic constraints, he Ó 2011 Springer Science+Business Media, LLC, Volume 33, Number 3, 2011

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has been able to approach with a free mind a difficult subject, which, for a number of reasons, is far from being consensual. He often expresses his own opinions, pointing out the omissions and contradictions that have blurred the picture of Pythagoras, but at the same time gives enough material to allow one to reach one’s own conclusion. Obviously Bre´maud has consulted a vast literature, as the bibliography shows, on the subject of Pythagoras and Pythagoreanism. The result is a well-documented historico-philosophical essay, broad in scope and accessible to a wide audience, a welcome and fresh addition to the literature.

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REFERENCES

Bagnani, Gilbert (1919). The subterranean basilica at Porta Maggiore, J. Roman Studies 9, 78-85. Freeman, Charles (2005). The Closing of the Western Mind: The Rise of Faith and the Fall of Reason, Vintage Books, New York. Department of Mathematics Uppsala University P.O. Box 480 751 06 Uppsala Sweden e-mail: [email protected]

The Music of Pythagoras: How an Ancient Brotherhood Cracked the Code of the Universe and Lit the Path from Antiquity to Outer Space by Kitty Ferguson NEW YORK, WALKER & COMPANY, 2008, HARDCOVER, 320 PP., US $25.95, ISBN 978-0-8027-1631-6 REVIEWED BY MEREDITH CLEMENT, LUV GRUMMER, MELINDA LITTLEFIELD, DEVIN SMITH, CHARLOTTE SIMMONS, AND JOHN BARTHELL

ccording to Bertrand Russell, Pythagoras of Samos was the most influential of all western philosophers. Whereas some insist he was more of a mystic than a mathematician, mathematics historians generally agree that Pythagoras’s and subsequent ‘‘Pythagoreans’’ interest in numbers ultimately furthered the development of the branch of mathematics known today as number theory. Pythagoras is most recognized for the so-called Pythagorean theorem, taught in geometry classrooms all over the world. A new book by Kitty Ferguson, The Music of Pythagoras: How an Ancient Brotherhood Cracked the Code of the Universe and Lit the Path from Antiquity to Outer Space (2008), attempts to unravel the legacy of Pythagoras and his followers (Pythagoreans), digging far deeper than this theorem and connecting Pythagoras to a history of ideas that provides one of the strongest attempts to resurrect his influence on philosophy and the sciences since Bertrand Russell attempted to do so in the 1940s. Ferguson contends that the ‘‘Pythagorean vision,’’ the realization that there is an underlying order to nature and that numbers provide a pathway to unraveling the deepest mysteries of the universe, continues to underpin the development of science. Readily admitting that what is known with certainty about Pythagoras fills a mere paragraph, it is the attempts by successive generations to fill in the gaps that interest her most. She invites the reader along as she attempts to understand how and why the Pythagoreans became so influential over such a long period of time, in spite of their valiant efforts to keep their own activities secret. After relating well-known stories about Pythagoras and his secret society of Pythagoreans, she devotes her book to this ambitious goal. Ferguson first leads the reader through the sixth century BC accounts of Pythagoras’s life and his discoveries told by Iamblichus, Porphyry, and Diogenes Laertius. She engages such hypothetical questions as: if Pythagoras did travel to Egypt and Babylon, what could he have learned there? She recounts familiar stories of his miracles and eccentricities, such as his having heard the voice of a friend in the barking of a dog. She also relates the paradoxical revulsion the vegetarian Pythagoreans had for beans; a condition that may have

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even led to the death of Pythagoras himself as he fled mortal enemies near Croton. Followers had to be silent for five years to prove that they were worthy of meeting Pythagoras and being ‘‘admitted to his house.’’ Pythagoras himself did not write anything down, and most of what we know about him today is related through stories that others (such as Iamblichus) have written. Ironically, history (with the early influence of Plato and Aristotle) has taught us that the loss of the written word is the potential loss of ideas themselves, including the origin of those ideas. Modern interpretations of the scientific process, including those of Fleck and Kuhn, emphasize the importance of social factors in scientific change. We will likely never know what ideas actually should be credited to Pythagoras, nor which of his ideas have been lost (or even ‘‘stolen’’) over time. Although the information in Part I appears in many other sources, most readers will find material in Part II (‘‘Fifth Century BC – Seventh Century AD’’) that is new to them. ‘‘Pseudo-Pythagorean’’ books appeared in second-century Rome and Alexandria: ‘‘According to one count, at the height of the era of Pythagorean forgeries, there were eighty works ‘by Pythagoras’ in circulation and two hundred purporting to be by his early followers.’’ As Ferguson notes, without Aristotle we might not have the little information about Pythagoras that is available today. Yet, she cautions that Aristotle did not distinguish between Pythagoreans of Pythagoras’s time and those of Plato’s time. In her discussion of Plato and Aristotle, Ferguson gives us new insight into these very different historical figures. Ferguson tries hard to connect Pythagoras to our current world but in doing so drifts from what is truly Pythagorean. The mad dash from the eighth century to the twenty-first is a whirlwind of dates and names that is difficult to follow (in Chapter 17, for instance, she mentions eighteen people in twenty-one pages). An entire chapter discusses a single historical figure, Johannes Kepler, perhaps because of his fascination with musical harmony in the universe: ‘‘Kepler had one of the truest ears for the harmony of mathematics and geometry.’’ Otherwise, it is difficult to understand why the longest chapter in a book about Pythagoras should be devoted to Kepler. Ferguson’s familiarity with the work of authors such as Thomas Kuhn, Paul Feyerabend, Jacob Bronowski, Carl Sagan, Bertrand Russell, and Arthur Koestler is impressive. Overall, we found her book interesting, informative, insightful, and accessible. But as a scholarly work, compared with Christoph Riedweg’s Pythagoras: His Life, Teaching, and Influence, she includes too vast a cast of characters and topics whose connection to Pythagoras is tenuous at best. It is believed that Pythagoras himself discovered the relationship between musical pitch and the length of a vibrating harp string, the first natural law ever formulated mathematically, and that his musical discoveries led him to believe that ‘‘there was order to the universe, and this order was made of numbers.’’ He identified the pattern behind the ‘‘harmonious’’ tunes that could be heard in music using mathematical ratios and determined that ‘‘surely this pattern must not be an isolated instance.’’ Ferguson aims to show the effects that the integration of math into reason still has on  2011 Springer Science+Business Media, LLC, Volume 33, Number 3, 2011

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society. But, in the end, this historical overview of a man (and his followers) who is known to be steeped in secrecy leaves us hungry for what we cannot really have: a substantive account of a man who kept his thoughts to himself and hidden from others. What, then, was the point of this exercise? Another historical writer (unmentioned by Ferguson) whom Pythagoras influenced was Henry David Thoreau. In his essay ‘‘The Bean-Field,’’ Thoreau, noting ‘‘for I am by nature Pythagorean,’’ describes how he toils to raise beans as a cash crop on barely productive land that adjoined his famous cabin at Walden Pond, recording a profit of only 8.71‘ dollars during his stay there. Other than to be sardonic, why would Thoreau choose this unprofitable and antiPythagorean crop? The image of Thoreau writing in solitude at his dwelling, surrounded by the forbidden bean, is far too conspicious to be coincidence, made all the more so by his strident, Pythagorean-like rejection of a meat diet in the subsequent essay ‘‘Higher Laws.’’ Similarly, Ferguson’s relentless pursuit of the true legacy for Pythagoras, a man whose contributions have never been recorded first-hand in the historical record, could only culminate for us in the form of a paradox. In the end, as Riedweg put it, ‘‘each author to some extent writes his own history…adapting the tradition to his own

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image of Pythagoras.’’ Riedweg and Ferguson handle this task in different ways. Whereas Ferguson employs this ambiguity as fertile ground for speculating on Pythagoras’s contributions to our current state (including studies of extraterrestrial life), Riedweg stays closer to the evidence. Ferguson may have, at times, strayed from biography into reasoned fictionalization by the final pages of her work. Riedweg, on the other hand, recognized his own risks in taking on this elusive subject, providing a much more conservative, though at times digressive, account.

REFERENCE

[1] Riedweg, C. Pythagoras: His Life, Teaching, and Influence, translated by Steven Randell, Ithaca: Cornell University Press, 2008. Departments of Biology and Mathematics & Statistics University of Central Oklahoma Edmond, OK 73034 USA e-mail: [email protected]

Pythagoras: His Life, Teaching, and Influence by Christoph Riedweg, translated by Steven Rendall ITHACA, CORNELL UNIVERSITY PRESS, 2008, PAPERBACK, 198 PP., US $19.95, ISBN 978-0-8014-7452-1 REVIEWED BY MEREDITH CLEMENT, LUV GRUMMER, MELINDA LITTLEFIELD, DEVIN SMITH, CHARLOTTE SIMMONS, AND JOHN BARTHELL

ythagoras: His Life, Teaching, and Influence surveys the essential facts of our current understanding of Pythagoras and the dynamic and intriguing character of Greek history’s intellectual ‘‘hinge’’ period (the sixth century BCE). No other book in the last decade has taken this task on as vigorously and succinctly as Christoph Riedweg’s. The book is divided into four chapters: 1, Fiction and Truth: Ancient Stories about Pythagoras; 2, In Search of the Historical Pythagoras; 3, The Pythagorean Secret Society; and 4, Thinkers Influenced by Pythagoras and his Pupils. The early sections of the first chapter discuss Pythagoras’ birth, appearance, early travels, and his reputation as a ‘‘miracle worker.’’ Our interdisciplinary seminar group of students and faculty from mathematics and the sciences found this material delightful and informative. It weaves together some of the most intriguing aspects of his personality that historical conjecture has to offer. Pythagoras is said to have been ‘‘very tall and of noble stature, and his voice, character, and every other aspect were marked by an exceptional degree of charm and embellishment…He dressed in a white robe, wore trousers [apparently atypical for Greeks of this time period], and crowned his head with a golden wreath, probably as a sign of his elevated status.’’ According to various legends, Pythagoras: 1, whispered in the ear of an ox found feeding on beans and it ‘‘would never touch any thenceforward’’; 2, was spoken to by the river Casuentus (near Mesopotum), which exclaimed loudly, ‘‘Greetings, Pythagoras!’’; and 3, was able to be present in two places, separated by a journey of many days, at once. Additionally, he is said to have ‘‘put an end to an epidemic of plague, stopped wind and hailstorms, and when necessary calmed the waves, or overcame a poisonous snake—allegedly by biting it back.’’ Such outlandish claims are not unlike the claims for Empedocles, Pythagoras’ contemporary, as Riedweg notes. Next we get a glimpse of ‘‘another side of the image of Pythagoras that is less familiar today: Pythagoras as a moral authority whose influence stretched not only to his own pupils but as far as the whole society of the city of Croton.’’ Pythagoras encouraged his followers (the ‘‘Pythagoreans’’) to be gentle, honorable, temperate, and of good repute, to pursue education (which ‘‘distinguishes men from animals, Hellenes from barbarians, freeman from slaves, and also philosophers from ordinary people’’), and to ‘‘even with fire and sword eradicate from the body, sickness; from the soul,

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ignorance; from the belly, luxury; from the city, sedition; from a family, discord; and from all things, excess.’’ These philosophical precepts help us understand the historical aura that extols Pythagorean virtues but simultaneously confuses his actual role in history. When Riedweg embarks on a discussion of Pythagoras’ philosophy, his prose becomes murky: ‘‘Just as the neo-Pythagorean wants to see Pythagoras’ doctrine of numbers as merely a pedagogical-communicative aid, so Xenocrates and other pupils of Plato argue against the literal interpretation (proposed by Aristotle, among others) of the famous account of the creation in this dialogue; according to them, Plato was speaking instead like teachers of geometry, who drew figures about the origin—not that the world ever would have had an origin, but on didactic grounds [!], so that it can be better recognized, as when one observes the origin of a geometrical figure.’’ This may be an artifact of translation, but our enthusiasm for reading onward dwindled at this point, and it is unlikely that any of us would have finished the book had we not been reading it as a group. The temptation to skip digressions recurred frequently; twelve pages, for example, are devoted to tracing the word ‘‘philosophy’’ and a discussion of whether or not the Pythagoreans can legitimately be called a ‘‘sect’’ according to the literal meaning of the word. The reader who perseveres through Riedweg’s asides experiences a day in the life of the typical Pythagorean as told by the ‘‘original’’ sources: ‘‘On awakening, it was the Pythagoreans’ custom to arouse their souls with the sound of the lyre, so that they might be more alert for action, and before going to sleep they soothed their minds by means of the same music in order to calm them down, in case too turbulent thoughts might still inhabit them. …And after the morning walk, they…used this time for instruction and lessons, and for the improvement of their characters. After such study, they turned to the care of their bodies. Most used oil-rubs and took part in footraces; a lesser number wrestled in gardens and groves: some engaged in long-jumping or in shadow boxing, taking care to choose exercises well-adapted to their bodily strength.’’ In addition to the infamous Pythagorean taboo on the consumption of beans, we learn of other puzzling prohibitions: cutting ones’ hair or nails during a festival, poking a fire with a long knife, tearing the wreath to bits, and taking swallows into the house. And then there is the vegetarian diet prescribed by Pythagoras—whose own ‘‘body always maintained the same form, as if on a straight line; he was not sometimes well, sometimes ill, and also not sometimes fattened and sometimes losing weight and getting thinner.’’ Can we ever really know anything at all about Pythagoras? The ‘‘sifting’’ begins in Chapter 2. After describing the rich cultural and intellectual period in Greek history to which Pythagoras belonged, Riedweg presents ‘‘the oldest testimonies’’ individually; these include reports from such prePlatonic ‘‘witnesses’’ of the sixth and fifth centuries BCE as Xenophanes, Heraclitus, Empedocles, Herodotus, and Democritus. Riedweg’s treatment is careful and thorough, and he presents compelling arguments for each of his conclusions. Thus he cites reports by Xenophanes, Heraclitus,  2011 Springer Science+Business Media, LLC, Volume 33, Number 3, 2011

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and Empedocles before concluding that: ‘‘even though the reports regarding Pythagoras’ exceptionally successful political and educational effect on southern Italy may have been transfigured by legend,’’ Pythagoras ‘‘exercised on his contemporaries a strongly polarizing influence’’ and ‘‘cannot have been a ‘normal’ man.’’ At other times, Riedweg raises interesting questions that occur to him as he reflects on the sources (e.g., whether Xenophanes knew Pythagoras personally) but to which he does not claim to have the answer. The third chapter briefly discusses the conflicts of the Pythagoreans following the death of Pythagoras, both internal (between the acousmatics and the mathematicians) and external (culminating in the anti-Pythagorean rebellions that ultimately dispersed the school’s members). Riedweg offers a brief description of the best known Pythagoreans of the fifth and fourth centuries BCE, including a colorful portrait of acousmatic Pythagorean Diodorus of Aspendus, who is said to have returned to Greece where he spread ‘‘the Pythagorean sayings.’’ He was a strict vegetarian whose appearance likely drew attention: he had a ‘‘shabby cloak (tribon), rucksack, and walking stick, long beard, long hair, and bare feet.’’ Also of note is Archytas, a friend of Plato, who introduced the important idea in mathematics of harmonic mean. He is also regarded as the first to have applied mathematical principles to mechanics and is remembered for his invention of a wooden bird (a dove according to Riedweg, a pigeon according to some sources) that could actually fly.W. K. C. Guthrie has written that: ‘‘In general the separation of early Pythagoreanism from the teaching of Plato is one of the historian’s most difficult tasks. …If later Pythagoreanism was coloured by Platonic influences, it is equally undeniable that Plato himself was deeply affected by earlier Pythagorean beliefs…’’ Riedweg offers support for this assessment, contending that the early work of authors such as Porphyry and Iamblichus’ on Pythagoras’ philosophy is ‘‘shot through with Platonic thought’’ and ‘‘can be regarded as far more a summary of Plato’s philosophy than of Pythagoras’s.’’ He revisits this topic at length in the final chapter, emphasizing that the ideas and philosophies of Pythagoras and Plato have become blurred over time.

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In a brief, one-page section entitled ‘‘The Beginnings of Western Mathematics,’’ Riedweg explores the role that Pythagoras and his followers played in the development of Greek mathematics, concluding that we really cannot say with certainty much more than Aristotle did; namely, that the ‘‘Pythagoreans were the first to concern themselves with mathe´mata—that is, with arithmetic, geometry, astronomy, and music,’’ all of which they ‘‘advanced.’’ All things considered, Riedweg has written a comprehensive and well-researched account of the man who has meant many things to many people: ‘‘the philosopher, the astronomer, the mathematician, the abhorrer of beans, the saint, the prophet, the performer of miracles, the magician, the charlatan.’’ Riedweg’s careful and honest commentary about the limitations of our knowledge inspires trust: ‘‘In view of the uncertainties that hinder our reconstruction of his teachings, we can hardly get beyond mere plausibilities in trying to determine his possible influence on other thinkers and writers. A few suggestions may therefore suffice.’’ In our view, this book is a rich resource and belongs on the shelf of anyone desiring to learn as much about Pythagoras and his followers as possible ‘‘given the exceptionally fragmentary records.’’

REFERENCES

[1] Boyer, C. A History of Mathematics, 2nd ed., revised by U. C. Merzbach, New York: John Wiley & Sons, 1991. [2] Guthrie, W. K. C. A History of Greek Philosophy, vol. 1, The Earlier Presocratics and the Pythagoreans, Cambridge: Cambridge University Press, 1965. [3] Riedweg, C. Pythagoras: Leben, Lehre, Nachwirkung, Mu¨nchen: C. H. Beck, 2002.

Departments of Biology and Mathematics & Statistics University of Central Oklahoma Edmond, OK 73034 USA e-mail: [email protected]

Mitt Liv som Pythagoras (‘‘My Life as Pythagoras,’’ a novel published in Swedish) by Fredrik La˚ng HELSINKI: SCHILDTS, 2005, 316 PP., 16 €, ISBN 951-50-1510-3 REVIEWED BY JOHAN STEN

ythagoras still inspires ever-new generations to explore and assess his mystic vision of the universe. For us mathematicians, his intriguing idea of an allembracing harmony of the world governed by numbers [1, 2] seems prophetic. For musicians, Pythagoras is everpresent through his tuning system and a curious comma that bears his name [3]. But Pythagoras the man remains elusive, almost unreal. To write a trustworthy biography appears impossible, as few facts are available. The direct sources are scarce: most of what we have is but hearsay and gossip, as Pythagoras did not trust the written word. The vivid illustration of Pythagoras in Raphael’s fresco ‘‘The School of Athens’’ in the Stanza della Segnatura of the Vatican – a bearded bald man stooping over his writings, surrounded by eager disciples – captures the established image. For a novelist, the lack of precise knowledge is good news. There is ample space to present a fresh interpretation of the saintly figure. Fredrik La˚ng’s novel shows what it takes to bring Pythagoras into life: a balance act intertwining fact and myth. In my opinion, La˚ng succeeds overwhelmingly. And it could not be otherwise: not only is Fredrik La˚ng, born in 1947, a foremost Finnish novelist writing in Swedish, but also a scholar with a Ph.D. in philosophy. Fortunately, La˚ng does not need to show off his learning. There are occasional philosophical digressions in the novel, with moments of dreamlike introspection, but the narrative remains fluent; it is served spiced up with a subtle sense of humour and a dose of intentional anachronisms (such as the nickname ‘‘Megawatt releaser’’ for the thunder god). The story is told mainly in the voice of Zalmoxis, a simple, faithful servant. However, the identity of the storyteller frequently blurs into a previous (or perhaps future) parallel identity or alter ego that confuses both Zalmoxis and Pythagoras with the author himself. A constant allusion is thus made to a distinctive Pythagorean belief, the transmigration of the soul. Pythagoras grew up on the island of Samos in the 6th century B.C. in an Ionian culture of craftsmen and merchants. His markedly intellectual disposition was manifested at an early age. In the novel, a decisive moment in his youth is the day when his earthly father Mnesarchus takes him to the Mint to witness the miracle of value creation – incidentally, one of La˚ng’s favourite philosophical themes. The striking of coins leads the young Pythagoras to realise the importance of Number. At the same time, focusing his attention on the pitch of the blows of different hammers, he discovers that some of them, for some reason, are more pleasing in concert than

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others. This is the germ of Pythagoras’s doctrine of numerical proportion and harmony. Driven by his desire for learning, Pythagoras sails to nearby Miletus to meet with Thales, one of the Seven Sages of Greece, who advises him to proceed to Egypt to learn from its ancient civilization. After a perilous journey, Pythagoras swears obedience to the priests in Memphis and becomes instructed in their esoteric knowledge. After many years, the Persians invade the city and Pythagoras is carried as a prisoner to Mesopotamia. During his captivity, he takes the opportunity to study another ancient culture, that of the Chaldeans, with their peculiar beliefs and remarkable achievements in science. La˚ng even suggests that Pythagoras, while in Babylon, met with Jews in exile, including the prophet Ezekiel. This is one of the most bizarre encounters in the novel. When Pythagoras eventually returns to his native Greece after a decades-long absence, he is as learned in esoteric wisdom as anyone could be and is eager to establish a scientific-religious brotherhood of his own. But the circumstances in Samos prove hostile to his ideas, and he finally settles in Croton in the Greek colonies of Southern Italy (the Greek ‘‘Wild West’’) to retire in peace, quietly transferring to his followers his mathematical knowledge and moral tenets, which include pioneering attempts to reform society by teaching a new, more respectful attitude toward women and slaves (then considered almost nonhuman). He also possesses enough medical knowledge to cure certain illnesses but, on the whole, La˚ng does not present Pythagoras as a performer of miracles. His sober account contrasts with the traditional mystic view. The secretive and exclusive Pythagorean brotherhood was a profitable object for myth-makers since its beginnings, and the Pythagoreans themselves did not hesitate to exploit the credulity of their contemporaries for their own benefit. Developing a semidivine nature for their Master must have been an appealing option. The reader can hardly miss a parallel with another Middle-East legend of even greater magnitude and consequence. La˚ng’s account culminates in the unveiling of the secret of the incommensurability of the edges of a regular pentagram – a well-known Pythagorean symbol – and the circumscribed regular pentagon, and what is more, that of the hypotenuse and the sides of an isosceles right-angled triangle. This dreadful, inexplicable discovery brings into serious doubt the sacrosanct Pythagorean principle ‘‘All is Number’’. Hippasus, the favourite disciple of his Master, receives the blame for revealing the secret and must die for his sin. In his disappointment in the turn of events, Pythagoras descends into the sea to join the island of Delos, the abode of his divine father Apollo. The novel is compelling, and has been favourably reviewed by Holger Thesleff [4], a world authority on the Pythagorean source materials. One should not expect too many scholarly ingredients to be incorporated into a good story. But the author could have mentioned, in the end notes, that the Chaldeans apparently were aware of the Pythagorean theorem in a quite general form [1, 2]. By the same token, perfect and friendly numbers, probably originating in the Orient [2], could have been brought into the story as well. I must applaud La˚ng’s capacity to portray a Pythagoras who bleeds, sweats, and experiences hunger, thirst, or sexual Ó 2011 Springer Science+Business Media, LLC, Volume 33, Number 3, 2011

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arousals. The elusive mystic becomes a human being, physically perhaps more enduring than most of us, but at the same time no less a virtuous and holy man, a subtle thinker and a visionary. This exquisite novel should be translated from the original Swedish to languages accessible to a wider readership. Hitherto, only a translation into Finnish exists [5].

REFERENCES

[1] O. Neugebauer: The Exact Sciences in Antiquity. Copenhagen: Ejnar Munksgaard, 1951. [2] B. L. van der Waerden: Science Awakening. Groningen: P. Noordhoff Ltd, 1954.

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[3] J. Fauvel, R. Flood, and R. Wilson (eds.): Music and Mathematics, from Pythagoras to Fractals. Oxford: Oxford University Press, 2003. [4] H. Thesleff: ‘‘Att leva som Pythagoras’’ (in Swedish), Nya Argus, 9– 10, 2005, pp. 157–159. [5] F. La˚ng: Ela¨ma¨ni Pythagoraana, translated into Finnish by Marja Kyro¨. Helsinki: Tammi, 2005.

Technical Research Center of Finland P.O.B. 1000 (Datava¨gen 3) FI-02044 VTT Finland e-mail: [email protected]

Complex Dynamics: Families and Friends by Dierk Schleicher (ed.) WELLESLEY, MASSACHUSETTS, A K PETERS, LTD., 2009, 650 PP., US $69.00, ISBN 978-1-56881-450-6 REVIEWED BY MARY REES

n 2005 John Hamal Hubbard celebrated his sixtieth birthday. His families and friends organised a conference in his honour for him, for themselves, and for the mathematical community, especially those with an interest in complex dynamics in any shape or form. The principal organisers were Adrien Douady, the man Hubbard occasionally referred to as his father, and Dierk Schleicher, one of the older of Hubbard’s many mathematical children. Dierk has overreaching responsibility for the volume I have before me, which serves as a memento for the conference—and much more. The event was as much a birthday celebration for modern complex dynamics as for Hubbard himself because he, with Douady, Dennis Sullivan, and others, had helped to engineer the rebirth of the subject a quarter of a century earlier. Of course, Hubbard was a mature mathematician by that time with that of many outstanding results to his credit. But since the early 1980s his name has been inseparable from the subject of complex dynamics. Back then his name was usually entwined with that of Douady, who had been his thesis advisor. Clearly they were kindred spirits who understood each other well and worked hand-in-glove. Together they discovered and described the beautiful structure of the Mandelbrot set in the family of quadratic polynomials, which has enraptured and engrossed mathematicians ever since. They proved for the first time that the Mandelbrot set is connected, and also that the multiplier of the attractive periodic point parametrises the hyperbolic component of a quadratic polynomial. In the process they introduced rays and arguments into complex dynamics. Rays and arguments come from the uniformising map, in two important settings: the complement of the filled Julia set of a polynomial, and the complement of the Mandelbrot set: dynamical and parameter rays, respectively. There is a canonical scaling of the uniformising map in any of these cases (for quadratic polynomials, at least: and this is essentially true for higher-degree polynomials also). Rational rays always land, and the landing points can be characterised. This result (or set of results) has huge ramifications. The pattern of landing rays for a filled Julia set determines the filled Julia set up to homeomorphism, using an object called the Hubbard tree, and the dynamics is determined up to topological conjugacy. Parameter rays landing on the Mandelbrot set also give very detailed information about the topological structure of the Mandelbrot set. In both the dynamical and parameter settings coincidences between landings of rational parameter rays can be

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computed algorithmically. This theory has led to the mantra that almost certainly originates from Douady and Hubbard, ‘‘Sow in the dynamical plane and reap in the parameter space.’’ Of course, this is an important basic principle in dynamics in general, but it is especially pertinent to complex dynamics. This theory and the language associated with it has pervaded complex dynamics ever since, and the evidence in this volume of many contributions is plain to see. Few subjects have had such a spectacular and joyous re-emergence as complex dynamics. John Hubbard was one of the first contributors. Many of the others are represented in this volume. We start with Jean-Christophe Yoccoz’s perceptive Foreword. Yoccoz only refers briefly to his own crucial and far-reaching contribution to the story: but refers to it, inevitably, because of John Hubbard’s role in disseminating the properties of the Yoccoz puzzle in an earlier sixtieth birthday volume (Milnor’s). Dierk Schleicher’s introduction follows. Dierk arrived on the scene after the rebirth, but has been an important member of the family for around twenty years. He makes clear the part that personal relations have played in the development of the subject, and the extent to which it has been a family business. With the sudden death of Adrien Douady just over a year after the conference, the responsibility for shaping and editing this volume became Dierk’s alone. The keystone in complex dynamics is the Mandelbrot set, which rightly takes centre-stage here. Thurston’s famous monograph on polynomials and laminations, the remarkably revealing interpretation of the Douady-Hubbard description of quadratic (and higher-degree) polynomials, using invariant laminations instead of landing rays and Hubbard trees, takes pride of place, published here for the first time. This brings sharply into focus what must be one of the most satisfying and enjoyable classifications in mathematics: an easily computable finite-to-one correspondence between rational numbers and critically finite quadratic polynomials, determined by the arguments of the alpha fixed point of the polynomial. This gives a complete description of the topological dynamics of the polynomials and of their hyperbolic components. For nearly thirty years, the preprint produced by Bill Thurston has been repeatedly photocopied (and, more recently, scanned). Here, it has been faithfully reproduced, with discreet, but occasionally significant, well-judged editing by Dierk Schleicher and Nikita Selinger. Schleicher has also provided a crucial missing section: what he reckons would have been Thurston’s II.7. This spells out the way in which the quadratic laminations determine the topological dynamics on the corresponding Julia set in the cases when the Julia set is locally connected. Similarly the connection between the quadratic minor lamination and the Mandelbrot set is ennunciated, as is the way in which this leads very easily to the result that (conjectural) local connectivity of the Mandelbrot set implies density of hyperbolicity. This nontrivial implication, one of Douady and Hubbard’s corollaries, and one which has made MLC the main goal in complex dynamics research ever since, becomes visually almost immediate in Thurston’s laminations setting, as Schleicher explains. Thus, the first article of the book is the first full print version of a monograph that has been in existence since the early days Ó 2011 Springer Science+Business Media, LLC, Volume 33, Number 3, 2011

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of the return to prominence of complex dynamics. Thurston’s monograph starts the first of the book’s four parts, devoted to polynomial dynamics and to mathematics derived from the study of polynomial dynamics. Both the Douady-Hubbard theory of rays and arguments, and Thurston’s lamination theory, extend to higher degrees, but extra issues arise, and the story is still far from complete, as is immediately illustrated in Chapter 2. Here, Alexander Blokh and Lex Oversteegen present their interesting examples of cubic lamination maps with wandering gaps. This contrasts with one of the star results of Thurston’s article, that wandering gaps do not arise for quadratic lamination maps. That result is key to obtaining the main structure results for the combinatorial model for the Mandelbrot set: density of the combinatorial versions of hyperbolic components in the combinatorial Mandelbrot set. It is also not hard to show that wandering gaps do not occur in any degree for lamination maps that represent hyperbolic polynomials. The Blokh-Oversteegen examples are termed weakly hyperbolic because they satisfy the so-called ColletEckmann condition. I think that the implications of these examples for the combinatorial model for the cubic connectedness locus are probably still unclear, although much interesting work has been done in this area. The combinatorial model for this is already a very awkward object to visualise, is of four real dimensions, and is not locally connected. Cubic polynomials are naturally the first case beyond the quadratic. With Bodil Branner, Hubbard carried out a major investigation of the parameter space of cubic polynomials in the 1980s. This study to some extent parallels those of Douady and Hubbard and their group of quadratic polynomials and of the Mandelbrot set. Of course cubic parameter space, being of a bigger dimension, exhibits features that are not present in the quadratic case. For example, the Mandelbrot set for quadratic polynomials is bounded. It can be characterised both as the set of quadratic polynomials for which the Julia set is connected and as the complement of the set of polynomials for which the Julia set is a Cantor set. For cubic polynomials, there is a trichotomy: the Julia set can be disconnected but not a Cantor set. The set for which this happens is unbounded and forms part of the escape locus. Branner and Hubbard completely characterised this part of the escape locus in terms of tableaux. It consists of countably many copies of the Mandelbrot set – both topologically and in a clear dynamical sense—and uncountably many points, a result that was named ‘‘Points are points.’’ The structure that was found has been a source of inspiration for much of the work in this field ever since – including, of course, Yoccoz’s use of tableaux to study non-renormalisable points of the Mandelbrot set. The Yoccoz puzzle was a powerful tool from the outset – but, curiously, only worked in degree two. In the couple of years before Hubbard’s sixtieth, this drawback was overcome, mainly with Jeremy Kahn’s and Mikhail Lyubich’s Quasi-additivity law. This led to a rush of results, some of which were being engendered during Hubbard’s birthday conference. ‘‘Points are points’’ was proved for polynomials of all degrees greater than three in about 2006, independently, by Qiu-Yin and Kozlovski-van Strien. The fourth article of Part I, by Tan Lei and Yin Yongcheng, illustrates the 158

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new technique. The last chapter of Part I is one of the series of papers by Jeremy Kahn and Mikhail Lyubich and collaborators that makes use of the quasi-additive law: in this case, extending local connectivity results to infinitely renormalisable polynomials satisfying the molecule condition. The structure of this book is to start with quadratic polynomials and move out from these key examples. So after Part I, devoted to polynomial dynamics in many shapes and forms, Part II is on rational and transcendental dynamics. Shishikura’s contribution is the revision of a much used preprint from 1990. The result is simply that a rational map with just one fixed point of one of the following two types – repelling, or parabolic with multiplier one – has connected Julia set. An important example is Newton’s method of any polynomial. Shishikura notes in a remark that connectedness of the Julia set for Newton’s method of an entire function is unknown. The next contribution, by Blanchard, Devaney, Garijo, Marotta, and Russell, is on a family of rational maps, and it investigates the topology of the Julia sets for this family. Part II ends with an article by Nuria Fagella and Christian Henriksen on the Teichmu¨ller space of an entire transcendental function, that is, the Teichmu¨ller spaces associated to Fatou domains, including wandering domains. Part III is called ‘‘Two Complex dimensions.’’ This turns out to mean parameter spaces of complex dimension two, as well as dynamical spaces. Thus, this part of the volume gives representation to two quite different directions of study within complex dynamics. In fact, John Hubbard has been much involved in both. His work with Bodil Branner on cubic parameter space illustrates the first, since this is essentially a parameter space of complex dimension two. The first two chapters in Part III, that of John Milnor, and the following chapter by Carsten Petersen and Tan Lei, make use of the framework introduced by Douady and Hubbard and Branner and Hubbard: rays, arguments of rays, Hubbard trees. Complex dynamical systems for which the phase space is of complex dimension two have a quite different character from those with phase space of dimension one. A key example is the complex Henon map, which has been the focus of much of Hubbard’s research over the past twenty years. Although the complex Henon map itself does not feature in this book, this type of example, and these concepts, make an appearance in the article by Eric Bedford and Jeffrey Diller on a certain family of birational maps, studied with one real parameter. This family was studied earlier by Abarenkova, d’Auriac, Boukraa, Hassani, and Maillard. In such examples one starts to see the concepts that appear in studies in real dynamics: splitting of the tangent bundle, stable and unstable manifolds. In the complex setting, these are related to concepts that appear in research in straight complex analysis, such as currents and plurisubharmonic functions. In recent years, these concepts have appeared in higher-dimensional complex parameter spaces also. Dujardin has been an important exponent of this approach, and his chapter in this volume on the parameter space of cubic polynomials is an example of this. As is customary in such work, the two plurisubharmonic functions G ± and two currents T ± that are studied are obtained from the Green’s function on the attractive basin of infinity of the polynomial. Properties of these functions and currents on different subsets of parameter spaces are

related to dynamical properties of the polynomials in these subsets. In particular, properties of the current T + are related to the wringing map coordinates produced by Branner and Hubbard, and it is shown that the point components, which they characterised in the escape locus, have full mass for T +. The article by Xavier Buff and Adam Epstein on Bifurcation Measure and Postcritically Finite Rational Maps provides another example of these techniques. They make use of a plurisubharmonic function and current—called the bifurcation current—associated with the average Lyapunov exponent, as a function of a rational map of degree d. The bifurcation measure is then an exterior power of the bifurcation current. They obtain a characterisation of the support of the bifurcation measure, an extension of an earlier result of Bassanelli and Berteloot. They use varied techniques; Adam Epstein’s fabled transversality result (concerning multipliers at periodic cycles) is stated and proved during the course of this article. The fourth part, ‘‘Making New Friends,’’ is an interesting miscellany. The first article, by Arnaud Che´ritat, is a report on a project very dear to Adrien Douady, the hunt for Julia sets of positive measure. As Che´ritat notes in his abstract, at the time of Hubbard’s sixtieth birthday conference, the groundbreaking result that he and Xavier Buff obtained just a few months later was still work in progress – although the famous analogue for Kleinian groups, the Ahlfors conjecture, had been solved less than two years earlier. The Julia set result is, of course, in the opposite direction. This is a very interesting and illuminating account of how a group of mathematical ideas were considered and turned over, regrouped, and eventually fitted together and used to obtain one of the biggest results in complex dynamics in recent years. Several articles are related to the Thurston pullback map, introduced by Thurston in his important theorem (again from the early 1980s) characterizing those critically finite branched coverings that are equivalent to rational maps, in a natural sense. Douady and Hubbard proved this theorem in a muchquoted paper in Acta Math. in 1993. Two of this book’s chapters related to the pullback are concerned with the

pullback as a dynamical object in its own right. The pullback map is a map on the Teichmu¨ller space of a marked, or punctured, sphere – which is a complex manifold. The article by Nekrashevych is placed in Part I because it is concerned with the pullback map for critically finite polynomials. It represents a connection with combinatorial group theory, which has become prominent in recent years. Nekrashevych and colleagues such as Laurent Bartholdi, in particular, have promoted this approach. The connection with polynomial dynamics is via iterated monodromy groups of postcritically finite backward iterations of topological polynomials. These are characterised in Nekrashevych’s article. The second chapter, in Part IV, by Buff, Epstein, Koch, and Pilgrim, gives various interesting examples of pullback maps with different properties. The other article concerned with this theorem of Thurston, also in Part IV, is by Nikita Selinger. It is a preliminary research announcement of a modified proof of Thurston’s theorem, and of a related result of Kevin Pilgrim, using an extension of the Thurston pullback to an augmented Teichmu¨ller space. The illustrations are not only beautiful but also an important reminder of the role that pictures play in research in this field, and in the work of Hubbard and his students and co-workers in particular. The final article, by Camarena, Maloney, and Roeder, is on a topic outside complex dynamics proper, although it does concern Kleinian groups: computing arithmetic invariants for hyperbolic reflection groups. As Roland Roeder points out in the introduction, although John Hubbard may not have worked on this particular topic, hyperbolic geometry is an important interest of his, and he passes on to all his students his strong enthusiasm for mathematics in general, and experimental mathematics in particular. Department of Mathematical Sciences University of Liverpool Liverpool L69 7ZL England e-mail: [email protected]

Ó 2011 Springer Science+Business Media, LLC, Volume 33, Number 3, 2011

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Fatou, Julia, Montel, The Great Prize of Mathematical Sciences of 1918, and Beyond by Miche`le Audin SPRINGER, 2011, VI + 332 PP., 69,95 EUR, ISBN: 978-3-642-17853-5

Fatou, Julia, Montel, le grand prix des sciences mathe´matiques de 1918, et apre`s… by Miche`le Audin SPRINGER, 2009, VI + 276 PP., 36,97 EUR, ISBN: 978-3-642-00445-2 REVIEWED BY JEAN-PIERRE KAHANE

The war in Europe is raging furiously. Millions of soldiers died in its first months. Among those who fought valiantly, and were gravely injured, was a former student of the Ecole Normale Supe´rieure. He had entered the School in 1911 and was one of the brightest students ever seen, ranking first in the entrance competition both at the Ecole Polytechnique and the Ecole Normale Supe´rieure, showing promise of becoming an important mathematician. In 1915 and during the following years, while undergoing intensive medical treatment—including multiple surgeries— at the military hospital of Val de Graˆce, he wrote hundreds of pages of difficult mathematics on various subjects. This remarkable character was Gaston Julia. Meanwhile, the French Academy of Sciences continued to meet every Monday, to receive and publish scientific contributions as ‘‘Notes aux Comptes Rendus’’, and to accept plis cachete´s (sealed papers). The academy decided to launch a competition for a Grand Prix on a theme in mathematics that seemed ripe and promising: the investigation of global properties of iterations of rational mappings in the plane. Gabriel Koenigs, not yet a member of the Academy, had already investigated the local properties and, more at the heart of the subject, Pierre Fatou had a series of striking examples of exotic behaviour for the closure of the orbits. Fatou had published a Note in 1906 on the subject while writing his dissertation on trigonometric and Taylor series, and he certainly was considered as a natural competitor. The Grand Prix was to be awarded in 1918. Fatou had entered the Ecole Normale Supe`rieure in 1898, four years after Paul Montel, Henri Lebesgue, and Paul

1915

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Langevin. Montel’s most famous achievement is the introduction of normal families (familles normales) of analytic functions, one of the first and best examples of the importance of the notion of compactness in analysis. Both Fatou and Julia used normal families as soon as they appeared in 1917. Given a rational function R and its iterates R n ðn ¼ 1; 2; 3; . . .Þ the points around which the Rn are a normal family constitute an open set; the complementary set is closed and it is called now the Julia set. Fatou had designated it by F (probably for frontier), and Julia by E 0 , the derivative of the set E consisting of the repelling periodic points of all iterates. 1916, 1917, 1918—the war was still raging, and academic life went on. Julia defended his thesis and proceded immediately to work on the iteration problem. Fatou, pushed by Montel, wrote his me´moires on the subject. On March 15, 1915, the Academy expelled Felix Klein and two other German scientists for a text they signed about the German army; Charles de la Valle´e Poussin from Louvain (destroyed by the German army) was elected instead. In 1918 Fatou did not compete for the Great Prize; there were three competitors: Samuel Latte`s, Salvatore Pincherle, and Gaston Julia. On November 11, 1918, Armistice Day, the General-in-Chief, Marshal Foch, was elected as member of the Acade´mie des Sciences. On December 2, Gaston Julia received the Great Prize, his me´moire was ‘‘crowned’’. Most of the book is a description of this period from different angles: the characters, the mathematics involved, and the atmosphere in France and in particular at the Academy at the time. It is a fascinating piece of work. Inside the frame that I have just described, Audin draws the portraits, the environment, the foreground, and the background, and does it with her own views, her feelings, and her passion for mathematics and for the truth. Although the period 1915– 1918 is both the beginning and the crux of the book, Audin continues to explore academic life afterward—how people evolved and how mathematics went on. Audin uses modern terms when necessary to explain the papers of Fatou or Julia, and figures not contained in the original articles. For instance, we see the common boundary of the basins of attraction of the function 12 ðz þ z 2 Þ; one of the first examples considered by Fatou, a Jordan curve with no tangent line at any point. We see also the ‘‘basilica’’, named after the basilica St. Marco in Venice and its reflection in water viewed by Benoit Mandelbrot, as the Julia set for the function z2 - 1. Although Fatou certainly had such figures in mind, there is no trace of them. Julia also had similar figures in mind, and fortunately some of them were kept: handwritten figures were included in plis cachete´s he had sent to the Academy. One of them is the Julia set of the function 12 ð3z  z 3 Þ; reproduced on the cover of the book. We can now make better figures with computers, but clearly all the analytic developments in the papers of Fatou and Julia have geometric roots and, as Audin claims, figures were drawn. There are remarks or comments of this sort throughout the book: about the terms in use (who introduced the term ‘‘re´flexif’’, the term ‘‘Julia set’’?), the relations with other theories (the Newton method for computing the roots of an equation, the Kleinian groups and their study by Poincare´), the normal families and the points J of Julia, the modern developments

(Douady, Sullivan, the Mandelbrot set), and the construction of figures (Che´ritat versus Mandelbrot). There are beautiful mathematical figures in the book, mainly by Arnaud Che´ritat. However, the main interest is the quality of the information and of the exposition. Even specialists in the field will find something new in the book, and nonspecialists will have an excellent introduction to modern complex dynamics. About half of the book consists of mathematics. The collection of mathematical personages includes not only Fatou, Julia, and Montel, but Hadamard, Borel, Lebesgue, Picard, Painleve´, Darboux, Paul Le´vy among the French, and Ritt, Hausdorff, Ostrowski among the foreigners. There are long and important quotations from their writings, and accurate analysis of their works and their attitudes. Audin acts as a historian, consulting archives, reports and letters, and books and journals. All people appear in their historical context, beginning with the First World War. For example, we see Picard and Painleve´ greeting the new member Marshal Foch in a grandiose ceremony at the Institut de France. Painleve´ was an important politician as well as an excellent mathematician, and he was president of the Academy when Julia received the Prize. His address at the time of Foch’s induction is quoted at different places in the book—it expresses the patriotism of the time in a more decent form than that of some other academicians. Hadamard appears in many places, and an important report he made on Fatou is reproduced entirely. He was the person who knew Poincare´ the best, and he certainly was active in the choice of the subject of the Grand Prix. Two of Hadamard’s sons were killed in the First World War, the third in the Second. About Fatou he wrote that he gave up competing for the Prize for health reasons. Fatou had poor health indeed, but also he would have been conscious that the atmosphere of the time gave Julia the edge—or he simply didn’t want to compete. The me´moires of Julia and Fatou were published at the same time, they deal with the same subject, they worked independently, and the Academy had to decide delicate questions of priorities on the basis of published Notes aux Comptes Rendus and unpublished plis cachete´s. Julia’s originality was fully recognized. His career was rapid and bright. He was elected as a member of the Academy in 1934, though he had been ranked second to Paul Montel by the committee of mathematicians—Montel had to wait for a few years. Later the same situation was created in 1952 when Garnier, supported by Julia, was elected instead of Maurice Fre´chet, who had been ranked first by the mathematicians. Fre´chet had to wait, and Paul Le´vy refused to be a candidate before Fre´chet was in. The age of entrance to the Academy increased dangerously. Julia was active in many ways, in particular in the International Union of Mathematicians. Audin is critical of his position in different circumstances, and ends the book with a request for a biography of Gaston Julia. A few readers were shocked by this disrespect, and I shall discuss one of the expressions of this disrespect later. The book also covers the relations between Julia and Montel. Apparently they were quite smooth in 1932 but turned tense later. A report by Paul Le´vy in 1965 provoked

an exchange of letters about the points J of a family of analytic functions compared to the irregular points of Montel—Le´vy showed that these two notions defined the same points and Montel wanted his contribution to be recognized. Paul Montel appears frequently in the book and not only through the ‘‘normal families’’. He was more than just a decent dean of the Faculty of Sciences of Paris in a difficult period, that of the German occupation. In 1965 Montel was 89 and, Audin observes, judging from his writing, far from senile. I can confirm this. The large amphitheater of Orsay, named now amphithe´atre Henri Cartan, had just been constructed, and I heard Paul Montel speaking without microphone with a clear and powerful voice, jokingly describing his life: he had never been ill, but he had recently contracted a serious disease: une incurable longe´vite´. He died in 1975, aged 98. Fatou had died in 1929, and Julia would die in 1978. Between Julia and Fatou there had been no competition. Fatou was an astronomer, more exactly adjoint-astronomer—he was promoted to a full position as astronomer only two years before his death. He was highly appreciated by leading mathematicians, Hadamard, Lebesgue, Montel, to name a few. In his writings he was as unselfish as possible. For example, what Fatou called the Parseval relation is due to him alone. Lebesgue had to push him to write his thesis, on trigonometric and Taylor series; in his books Lebesgue mentions discoveries of Fatou (for instance, the first and best example of a trigonometric series that converges everywhere without being a Fourier-Lebesgue series). Reading Fatou is a pleasure. Comparing the me´moires of Fatou and Julia reveals very different personalities from the very beginning. Posterity has reestablished Fatou’s position as a mathematician; Milnor has said: ‘‘The most fundamental and incisive contributions were those of Fatou himself. However, Julia was a determined competitor and tended to get more credit as a wounded war hero.’’ About half of the book is devoted to Fatou, with unpublished letters and testimonials. Miche`le Audin investigates traces of his life by consulting the archives of the French Mathematical Society (of which he was an active member) as well as collections of letters held by his family. His works are listed in the bibliography, and I know no other place where this list can be found. His letters reveal an exceptional personality, clear and generous. There is plenty of good mathematics in these letters, including problems, and I suggested once that the book should be read starting with these letters in the Appendix. Fatou had no students. Julia however had a late and bright student, Jacques Dixmier, and Alain Connes was a student of Dixmier. Montel was the director of many theses, at a time when the direction was purely formal. The thesis of Fatou, the normal families of Montel, and the contributions of Julia to the theory of functions quickly became classical, but the matter of iteration and the me´moires of Fatou and Julia remained untouched in France for almost 60 years, becoming popular only now with the explosion of complex dynamics. (In Sweden they were better known: Carleson became interested in the 1960s, and his student Hans Brolin wrote his dissertation on iteration of rational mappings in 1965). Why did it take so long in France? The  2011 Springer Science+Business Media, LLC, Volume 33, Number 3, 2011

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book does not answer the question but there are some facts to take into consideration. In France, the war killed half of the possible future mathematicians, and the younger generation reacted against the older one. Moreover Picard was opposed to the theory of sets and all new matters in mathematics, and he had great influence and power at the time. Iterations came back in a different way, through the works of Denjoy in the 1930s. Adrien Douady became interested in the me´moires of Fatou in the late 1970s. Benoit Mandelbrot had been advised by his uncle Szolem Mandelbrojt to read Fatou and Julia in the 1950s, which he did, but he turned to linguistics and other matters before rediscovering the Julia sets and describing the Mandelbrot set in the 1980s. Miche`le Audin makes the cutting comment that Julia’s main contribution to mathematics, apart from his work on iteration and more generally complex analysis, may have been the way he collected money for the publication of the Collected Works of Henri Poincare´. This raises a serious question. The French are slow to publish the collected works of their best mathematicians. Poincare´ is one exception, Julia another. Actually the last volume of the Collected Works of Gaston Julia contains discourses by Julia and on Julia, in particular at the occasion of his Jubilee, and it tends to give a glorious impression. It is worth reading that volume as a complement to Audin’s, and to have a look at the others of a more scientific character. The Collected Works introduce the reader to the personality of Julia as well as to his scientific achievements. The reader is left to judge. For Fatou and Montel this possibility doesn’t exist and it is a pity. Audin has a strong sympathy for Fatou and a definite esteem for Montel. In my opinion this is fully justified, even if she strikes a blow at the glorious figure of Julia. It would be desirable to have the collected works of

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Montel and of Fatou for many reasons, in particular as a complement to this book. The book is impressive in its content, the amount of information it gives, and in the quality of the investigation and of the exposition. For many readers, as for myself, the fact that the author is present on every page with a remark, a comment, and sometimes a mockery, will make it more pleasant and easy to read. In a way you discover the author as well as the characters she paints. I like her judgment and style, but I understand that they may irritate other readers. Mathematics is not out of the world, and things in the world are not a model of smoothness. To express in the same book the beauty of mathematics, the ugliness of the war, and how that interacts with the life of real people, seems to me a remarkable achievement. Note. This report was written after reading the original version of the book, that is, the French version. The English version is more than a translation, it is enlarged and the presentation is different. It should be observed that it is the first book of a new subseries of the Springer Lecture Notes in Mathematics devoted to the history of mathematics. The creation of this new subseries is good news. I only regret that the new version contains no reference to the previous one, except an appreciation on the back of the cover. A book devoted to history should include its own history. I thank Hannah Katznelson for her help in preparing the English version of this text.

Laboratoire de mathe´matique Universite´ Paris-Sud F-91 405 Orsay cedex France e-mail: [email protected]

Esthe´tique et Mathe´matiques: Une Exploration Goodmanienne by Caroline Jullien RENNES, FRANCE: PRESSES UNIVERSITAIRES DE RENNES, 2008, 274 PP., 18.00€, ISBN 978-2-7535-0619-0 REVIEWED BY NATHALIE SINCLAIR AND DAVID PIMM

he difference between art and science is not that between feeling and fact, intuition and inference, delight and deliberation, synthesis and analysis, sensation and cerebration, concreteness and abstraction, passion and action, mediacy and immediacy, or truth and beauty, but rather a difference in domination of certain specific characteristics of symbols. (Goodman 1968)

T

Sometimes a book draws heavily on another book for its ideas or intellectual tools, yet to explain the project of the first book, aspects of the second are needed. Such is the case with Esth e tique et Mathe´matiques. The general project lying behind this book is a study of the relations between art and science, comparing and contrasting what the author terms les belles sciences with les beaux arts, seeking to build bridges between them. Her main goal is to point up the significant role of aesthetics in mathematics, zooming in for extensive discussion of mathematical examples toward the end. This is a worthwhile intellectual enterprise, but, as her subtitle indicates, her primary means for bridge-building is Nelson Goodman’s book Languages of Art: an Approach to a Theory of Symbols. Goodman’s book grew out of lectures he gave at Oxford in 1962. It makes striking reading for the mathematically interested, though the word ‘‘aesthetics’’ does not show up until page 100 and the word ‘‘mathematics’’ is nowhere to be found. Drawing on les beaux arts (painting and sculpture, music, dance, literary texts), Goodman offers a general account of symbols within symbol systems and points out general properties of notation. In response to the question ‘‘What is art?’’, provoked not only by photography, but also by artistic movements such as primitivism and found art, Goodman shifts the question to ‘‘When does this work function as art?’’, stressing the communicational aspects of art and, in particular, its semiotics. Whereas earlier art theorists focused on ‘‘significant form’’ or ‘‘fidelity’’, Goodman draws on exemplification, representation, metaphor and expression, denotation and depiction. In later chapters he focuses on the centrality of notation and scores, sketches and scripts, examining a range of art forms to see whether they meet his criteria for a notational system. He proposes aesthetics as a form of understanding and suggests that ‘‘in aesthetic experience the emotions function cognitively. The work of art is apprehended through the feelings as well as through the senses’’. Goodman concludes his account by offering three symbol processes that ‘‘may

be symptoms of the aesthetic’’, identifying a symptom as ‘‘neither a general nor a sufficient condition for, but merely tend[ing] in conjunction with other such symptoms to be present in, aesthetic experience’’. Goodman’s symptomatic trio he names as: syntactic density, semantic density, and syntactic repleteness (though he later adds a fourth concerned with exemplification and, in a subsequent book, a fifth, multiple and complex reference). He uses these criteria to distinguish among different possible contenders for being a notational system or not, for example, distinguishing pictures from diagrams (a difference Goodman claims involves syntactic repleteness, the extent to which features of the symbol signify within the system). The preceding account brings us exactly up against the tension we identified earlier: in order to explain these ideas in any detail, we would end up undertaking a review of Goodman’s book. Indeed, justifying the choice of and explaining Goodman’s ideas for her own purposes comprises the entire second part of Jullien’s book. Yet, nevertheless, these are the core ideas that Jullien draws on when discussing the mathematics examples.

Focusing in on Jullien’s Book Jullien’s initial approach to her topic is to document historical links between mathematics and art, including a discussion of classical views on aesthetics and mathematics prior to claiming a growing break in the eighteenth century and the two-cultures divide separating science and art. Despite this general observation, she reports that Jean-Pierre Crouzas, in a general treatise on ‘‘the beautiful’’, had a specific chapter dedicated to mathematics and science. All rely on the principle of unity in variation – which she asserts is ‘‘[un] principe qui se trouve applique´ de fac¸on e´clatante en mathe´matiques’’. Much of the second section documents the discussions directly pertaining to the mathematical aesthetic by mathematicians for the most part. Jullien structures this section by opposing the Platonic conception of mathematical beauty, which is deeply rooted in truth (objective, independent, immutable truth), to Poincare´’s, which privileges the human sensibility to beauty, thus offering a more subjective aesthetic. Jullien’s final chapter in this part looks at empirical solutions to the question of the mathematical aesthetic (whether it exists, whether it is more Platonic or more Poincare´-like), drawing on the writings of mathematicians, philosophers, and psychologists. She concludes by proposing a pragmatic approach to the mathematics aesthetic, one that is less concerned with comparing mathematics with the arts than with acknowledging the fact that the development of mathematics requires aesthetic choices. This is an argument made more persuasively by Tymoczko (1993). Jullien also revisits the numerous testimonies of the guiding character of the aesthetic: that is, the way in which aesthetic responses can alert a mathematician to certain promising or alluring relationships (for a more detailed study, see Sinclair 2004). The second part of the book is devoted to justifying the choice of Goodman’s work and then presenting the theoretical apparatus and methodological approach that she will take to analyzing works of mathematics. If Goodman’s theory Ó 2011 Springer Science+Business Media, LLC, Volume 33, Number 3, 2011

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enables one to decide whether or not a work functions as a work of art, because it fulfills certain syntactic and semantic requirements of the notational system that encodes it, Jullien wants to explore the analogous question in mathematics. As such, she is not so much interested in responding to the question ‘‘What is mathematics?’’ as the question ‘‘When is it mathematics?’’ Although Jullien will ultimately focus on objects that are commonly regarded as being works of pure mathematics, it seems to us that the shift in questioning could be especially useful in boundary contexts, such as mathematical physics or experimental mathematics. Given its heavy use of theory, this part of the book is dense and feels often very removed from mathematics. The reader is left to try to exemplify the various constructs herself, and there is little foreshadowing of just what the tools of Goodman will offer in terms of satisfying the double project of ascertaining that mathematics can indeed be analyzed using these tools and that mathematics can be seen as functioning aesthetically.

Part III: Goodmanian Symptoms and Mathematics The final part is really the heart of the book and is where Jullien puts Goodman’s theory to work on mathematical examples. She investigates first the symbolic functioning of algebraic demonstrations (using the standard proof of the irrationality of H2 as a case study), and then, more generally, the symbolic functioning of mathematical figures (which she further subdivides into two sections, one on ‘‘images’’ that are used in visual proofs and the other on diagrams). Her goal is to show that Goodman’s symptoms can be successfully applied to mathematical demonstrations and figures and, of course, that mathematics can thus be shown to be functioning aesthetically. In addition, she wishes to use her Goodmanian analysis in order to shed insight on some characteristic features of mathematics, such as generalization and abstraction. The first mathematical example analyzed is the overexposed and over-familiar proof of the irrationality of H2. Jullien shows how the proof possesses the Goodmanian symptom of exemplification in that the statement a2 = 2b2, which is an algebraic manipulation of the originating equation a/b = H2, exemplifies the notion of being even – and more so than any of the many other equivalent statements one could generate. She argues that this kind of exemplification move is characteristic in mathematics, where, in trying to show that one thing A has the property B, one must find the way of expressing A in terms that serve to define B. In this case, one wants to pull from all the concordances of a/b = H2 the one that shows, reveals, or requires that a is even. This is the symptom of semantic density at work. In a later chapter, Jullien also argues that the proof’s semantic density is what accounts for its generativity, in that it leads to proofs of the irrationality of other numbers. Jullien argues that the proof also possesses syntactic repleteness (which she calls syntactic saturation) by defining repleteness in terms of the lack of superfluous semiotic marks. This too, she argues, is characteristically mathematical, in the sense of the mathematical writing style (le codage mathe´matique) that avoids explaining things that are trivial 164

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and seeks shortcuts that help draw attention to the main insight of the theorem. While Jullien is nonevaluative about this style – and, indeed, does not even question whether it must, in some fundamental way, be the writing style of mathematics – others have been more critical. For example, Henderson and Taimina (2006) point to its tendency to obscure underlying motivations and understandings; Csiszar (2003) sees this as undermining the validity of mathematics; Thurston (1994) has written about the nefarious effects such a style can have on the development of mathematics itself. For Jullien, the two Goodmanian symptoms (syntactic repleteness and semantic density), working in tandem, account for the aesthetic criteria of economy and simplicity offered by G. H. Hardy (1940). While it seems important to her to relate her own analysis to the criteria of mathematicians, Jullien insists that her project – like that of Goodman – is one of assessing the aesthetic value of mathematics and not the aesthetic merit of particular artifacts; in other words, she is not interested in evaluation, only in functioning. Although the distinction seems clear, it is worth asking whether a successful Goodmanian analysis need depend so strongly on the judgments of mathematicians (which, as Wells’s 1990 survey of mathematicians’ views on the most beautiful theorems shows, are not necessarily homogenous). After this relatively short discussion, Jullien undertakes a much more ambitious and complex analysis of the relationship between metaphors and the aesthetic, as well as of the way in which metaphors enable mathematical generalization and abstraction. By thus analyzing the anatomy of mathematical proof in terms of Goodman, Jullien argues that real understanding of a proof requires knowing how to make it function aesthetically, in the sense that one must be able to attend not so much to its denotative function but more to what it exemplifies. (So, a2 = 2b2 can be seen as denoting a particular relationship, but also exemplifying evenness.) Jullien closes the first section of Part III by demonstrating the role of exploiting the aesthetic functioning of mathematics in the particular case of tracing the successive abstractions (which involve, also, successive metaphorical moves) that lead from the idea of an irrational number to that of an algebraic number, and then to that of an integral domain (anneau inte`gre). This analysis allows her to further hypothesize repleteness as the mechanism that enables mathematical generalization. Every stage of abstraction – going, for example, from the notion of ring to that of principal ideal domain (anneau principal) – involves eliminating contingent hypotheses, so that in the final move to abstraction there are only necessary ones. This move to the necessary – that is, to the necessary and sufficient condition – echoes Hardy’s notion of economy as an aesthetic quality in mathematics. Her analysis provides an explanatory mechanism to complement Hardy’s more qualitative observation. The second section of this final part opens with a discussion of the aesthetic value of geometry and a general discussion of the role of figures and diagrams in mathematics. Jullien then turns her attention to Hippocrates’s visual proof of the squaring of the lune. Given the pictorial nature of this artifact, she is now much closer to the domain of analysis undertaken by Goodman (at least in terms of his choices of exemplification!). Her analysis allows her to reinterpret

claims of the beauty of proofs without words – proffered by many, including Delahaye (2005) – in terms of Goodman’s symptomology. The global reading of the proof (seeing it all at once, globally, with each part interdependent) derives from its syntactic density, whereas the capacity to read the image in parts (following the dissection of the areas of the half circles into lunes and a triangle) derives from its relative saturation. Whereas Jullien’s reading of Hippocrates’s proof without words suggests that all such proofs can function aesthetically, her treatment of other mathematical images, such as the mathematical diagram, offers a more nuanced consideration for the role of the image in mathematics. She begins by formulating the mathematical diagram in terms of its intuitive value, and points out that it is this value that underlies the claims made by those who argue that the mathematical aesthetic is closely linked to diagrams and images. Of course, Jullien wants to explore whether these diagrams function aesthetically in the sense of Goodman. She thus opts to analyze the diagram that usually accompanies Rolle’s theorem. Jullien infers that the diagram itself does not function aesthetically, but that the combination of the semantically dense diagram with the syntactically dense algebra are two intensions of a single extension and that the fecundity of reasoning comes from the dialectic interplay between them. One might take issue with Jullien’s claim about the difficulty of exemplifying infinity. Indeed, the book Images of Infinity (Hemmings and Tahta, 1984) is just one among many that do just that. But more to the point, mathematical diagrams can be seen as having a discourse of their own, with an associated grammar, for which many conventions exist to represent mathematical concepts, even continuous and dynamic ones (the use of arrows, dotted lines, stop-frame-like sequences of images, etc.). More radically, perhaps, the contemporary move toward digital dynamic representations and visualizations might put into question Jullien’s claims that the diagram cannot function aesthetically on its own.

In Conclusion As we stated previously, we are highly empathetic to Jullien’s main project and found much to reflect on in terms of the way in which the Goodmanian analysis she proposes illuminates certain aspects of mathematics, highlighting especially discursive features of the formal means of communication of the discipline. That said, however, we had to work hard at coming to grips with Jullien’s book, needing to read Goodman’s book en passant as it were, and at the end we are left unsure to what degree it was worth it. For Mathematical Intelligencer readers mainly focused on the mathematics

itself, her examples offer detailed, analytic accounts of an eclectic range of instances – not mere passing allusions tossed off in a sentence of two or described metaphorically. For those more engaged by the processes of doing, learning, and teaching mathematics (such as ourselves), her lack of attention to these elements, studying only finished textual elements as her mathematical ‘‘artworks’’, contributed to our final lack of satisfaction. What might be the aesthetic dimension of doing mathematics?

REFERENCES

Csiszar, A. (2003) ‘‘Stylizing rigor; or, why mathematicians write so well’’, Configurations 11(2), 239-268. Delahaye, J.-P. (2005) «De´monstrations et certitude en mathe´matiques» , Pour la Science 49, 38-43. Goodman, N. (1968) Languages of Art: an Approach to a Theory of Symbols, Indianapolis, IN, Bobbs-Merrill. Hardy, G. H. (1940) A Mathematician’s Apology, Cambridge, Cambridge University Press. Hemmings, R. and Tahta, D. (1984) Images of infinity, Derby, UK, Association of Teachers of Mathematics. Henderson, D. and Taimina, D. (2006) ‘‘Experiencing meanings in geometry’’, in Sinclair, N., Pimm, D., and Higginson, W. (eds.), Mathematics and the aesthetic: new approaches to an ancient affinity, New York, NY, Springer, pp. 58–83. Sinclair, N. (2004) ‘‘The roles of the aesthetic in mathematical inquiry’’, Mathematical Thinking and Learning 6(3), 261-284. Thurston, W. (1994) ‘‘On proof and progress in mathematics’’, Bulletin of the American Mathematical Society 30(2), 161-177. Tymoczko, T. (1993) ‘‘Value judgements in mathematics: can we treat mathematics as an art?’’, in White, A. (ed.), Essays in Humanistic Mathematics, Washington, DC, MAA, pp. 67-77. Wells, D. (1990) ‘‘Are these the most beautiful?’’, The Mathematical Intelligencer 12(3), 37-41. Faculty of Education Simon Fraser University 8888 University Drive Burnaby, BC V5A 1S6 Canada e-mail: [email protected] 3669 West 18th Avenue Vancouver, BC V6S 1B3 Canada e-mail: [email protected]

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Stamp Corner

Robin Wilson

Recent Mathematical Stamps: 2007–2008 Al-Khwa ¯ rizmı¯ (780–850) The Persian mathematician Muhammad ibn-Mu¯sa¯ (al-) Khwa¯rizmı¯ lived in Baghdad and wrote influential works on arithmetic and algebra. The latter, Kitab al-jabr wal-muqabala, gives us our word algebra, whereas in medieval Europe the word algorism (after his name) came to mean arithmetic. We now use algorithm to mean a finite step-by-step procedure.

L. E. J. Brouwer (1882–1966) Luitzen Egbertus Jan Brouwer was a geometer and a topologist and the founder of intuitionism. He spent most of his life at the University of Amsterdam and is remembered for his invariance and fixed-point theorems in topology and for his refusal to accept proofs by contradiction. The stamp, issued to commemorate the centenary of his Ph.D. thesis, features his formula symbolizing the fallibility of the principle of the excluded middle.

Croatian Arithmetic On 25 January 2008, Croatia issued a stamp to commemorate the 250th anniversary of the publication of the first Croatian mathematical instruction book. Mihaly Sˇilobod Bolsˇic´’s book Arithmetika Horvatszka (Croatian arithmetic), published in Zagreb in 1758, presents the arithmetic of integers and fractions, the rule of three, practical work in accounting, and mathematical tables.

Leonhard Euler (1707–1783) In 2007, Switzerland issued a stamp to celebrate the 300th anniversary of Euler’s birth in Basel. It shows his portrait, with a polyhedron and his polyhedron formula. Euler had introduced the concept of an edge, and writing to Christian Goldberg in 1750, he stated his formula in the form H + S = A + 2, where H = hedrae (faces), S = anguli solidi (vertices) and A = acies (edges).

Metrication Many countries have issued stamps featuring the metric system and metrication. This stamp was issued in 2007 to commemorate the centenary of the introduction of the metric system to Denmark.

Yerevan Institute This stamp was issued by the Armenian Post Office in 2007 to commemorate the 50th anniversary of the founding of the Yerevan Scientific Research Institute of Mathematical Machines, which is involved with the development of highreliability computing systems.

Leonhard Euler L. E. J. Brouwer

Croatian Arithmetic

Al-Khwa¯rizmı¯

â 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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Metrication

Yerevan Institute