P. SCHOLL, A. SCHURMANN, J. M. WILLS

Polyhedral Models of I--elix Klein's Group "I have the polyhedron on m y desk. I love it?" John H. Conway, Aug. 19, 1993

f

elix Klein's group (which comes accompanied by Klein's curve, Klein's regular map, and Klein's quartic) is one of the most f a m o u s mathematical objects; in A. M. Macbeath's words ([L], p. 104), "It is a truly central piece of mathematics."

Felix Klein discovered this finite group PSL(2,7) of order 168 in 1879 [K], and since then its properties have been investigated, generalized, applied, and discussed in hundreds of papers. The recent book The Eightfold Way [L] contains several survey articles by prominent experts, which collect and discuss the essentials of Klein's group from various aspects. This book was issued on the occasion of the installation at the Berkeley campus of a nice geometric model of Klein's group made of Carrara marble by the artist H. Ferguson. The idea of visualizing Klein's group by geometric models is not new. Felix Klein himself gave a planar and a 3-dimensional model. The planar one is the unsurpassable Poincar6 model (Figure 2), well known from classical complex analysis. Klein's 3-dimensional model consists of three hyperboloids whose axes meet at right angles. In this paper we consider 3-dimensional models which are as close as possible to the Platonic solids, built up of planar polygons with or without self-intersections and with maximal possible symmetry. Polyhedral realizations of groups or regular maps can be considered as contributions to H.S.M. Coxeter's general concept of "groups and geometry." We will show polyhedral realizations of Klein's group, two of them "old" and two new. For this we need to review some basic properties of Klein's group. For more details we refer to [C], [CM], [K], [L], [MS] or [SWl].

Maps, Flags, and Symmetries First we consider the icosahedral group and its polyhedral realizations, the regular icosahedron and dodecahedron

(Figure 1). The 60 elements of the icosahedral group can be represented by the 60 black (or white) triangles of the pattern on the sphere in Figure 1. Such a pattern is called a "regular map," and the 60 black (or white) triangles are indistinguishable under rotations of the sphere. A reflection transposes the black triangles into the white ones and vice versa, giving the extension to the full icosahedral group of order 120. Now the 120 triangles of this regular map on the sphere can be collected in two dual ways to build up a convex regular polyhedron. Either one collects the 3 white and 3 black triangles around each 6-valent vertex of the map, which yields the icosahedron with 20 triangles and 12 5-valent vertices; or the 5 white and 5 black triangles are collected around the 10-valent vertices, which yields the dodecahedron with 12 pentagons and 20 3-valent vertices. Each black or white triangle of the map corresponds to an ordered triplet of a vertex, an edge, and a face of the icosahedron or of the dodecahedron; these triplets are called "flags." So the flags (or the black and white triangles) correspond to the elements of the group; i.e., they represent the elements of the icosahedral group. In the same way, the 168 black (hatched) and 168 white triangles in the Poincar6 model (Figure 2) represent the elements of Klein's group. Again the triangles of the map can be collected in two dual ways as for the icosahedral group. If black triangles may be interchanged with white, we have a group of 336 elements. If one considers the 6-valent vertices in Figure 2, then again 3 black and 3 white triangles fit together to form one

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Figure 2. Klein's g r o u p as a r e g u l a r map.

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large triangle, and one obtains 56 (= 336/6) triangles, but seven of them meet at a common vertex, 24 altogether. From the 14-valent vertices in Figure 2, one obtains 24 heptagons, which meet at 56 3-valent vertices. The only problem is to make a polyhedron with flat faces. There are some differences between the icosahedral group and Klein's group, which cause difficulties.

So {3, 5}10 denotes the regular icosahedron and {5, 3}10 the regular dodecahedron. For Klein's map the Petrie-polygons have length 8, and so the two dual representations are {3, 7}s and {7, 3}s. This explains the title of the book The Eightfold Way. Finally we sketch that a polyhedral realization of Klein's group has genus 3, i.e., it is topologically equivalent to a sphere with three handles. For regular maps of genus g --> 2 with p-gons and q-valent vertices, there is the famous Riemann-Hurwitz identity, which relates all relevant numbers, in particular the genus g and the order A of the automorphism group:

Hidden S y m m e t r i e s and P e t r i e - P o l y g o n s The main difference is that the icosahedral group is one of the rotation groups in Euclidean 3-space, and so the relation between the map on the sphere and the polyhedra is natural and obvious. In fact the geometric objects--the PlaA = 2 ( g - l ) ( l + -l - 1 l) q\ p tonic solids---existed long before the mathematical background (groups, maps) was understood. One had a vague From p = 3, q = 7 (or vice versa) and A = 168 follows g = idea of their deeper importance. 3. As a consequence, such groups have maximal order Klein's group of order 168 (or 336 for the full group) is 84(g - 1), and Klein's group is the first one of these rare much larger than the icosahedral group of order 60 (full: "Hurwitz groups." 120) and is not a symmetry group in Euclidean 3-space. We can create a polyhedron and consider a group of 168 transPolyhedral Models with T e t r a h e d r a l S y m m e t r y formations of it, but they can't all be congruences! It can easily be shown that any 3-dimensional model of Klein's group does contain a geometric subgroup, Klein's group with maximal--i.e., octahedral--symmetry namely the "octahedrai rotation group" of order 24, which has self-intersections, so in order to avoid self-intersecwas of course known to Felix Klein. We seek, then, a model tions, polyhedral embeddings have at most the next lower of Klein's group with octahedral symmetry, the elements symmetry, tetrahedral rotation symmetry of order 12. (or flags) fall into seven orbits of 24 elements each as 168 = 7 . 2 4 (or 14 orbits for the full group). Hence not all automorphisms of the group can be seen, and those which do not occur as geometric symmetries are called "hidden symmetries." These hidden symmetries, though not given by Euclidean motions, are combinatoriai and geometric automorphisms of the polyhedron. For example, the hidden symmeh--ies are shown by the fact that all faces are of the same type (triangles or heptagons), and so are the vertices (7-valent or 3-valent). Another tool to discover hidden symmetries are Petrie-polygons. A Petrie-polygon is a skew polygon (or zigzag line) where every two but no three consecutive edges belong to the same face of the polyhedron. On a regular figure all possible Petrie-polygons have the same length, and for the icosahedral map and hence for the regular icosahedron and dodecahedron, this is 10. The length r of the Petrie-polygons, together with p, the number of sides of a face, and q, the valence of the vertices, characterizes a regular polyhedron {p, q}r. Figure 3. Polyhedral embedding of {3, 7]8 with tetrahedral symmetry.

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In 1985 E. Schulte and J. M. Wills gave such a polyhedral embedding of {3, 7}s in [SWl], built up of 56 triangles, which meet at 7-valent vertices, 24 altogether (Figure 3). Each of the four holes of the model has a strong twist, and it is not clear a p r i o r i that this can be done without self-intersections. The 24 vertices split into two orbits of 12 vertices under the tetrahedral rotation group. The outer orbit of 12 vertices can be realized by the vertices of an Archimedean solid, namely the truncated tetrahedron. Several cardboard and metal models and c o m p u t e r films were made of this realization. (See also [BW], and Conway's comment at the head of this article). In its s y m m e t r y and embedding properties, it corresponds to Ferguson's model, but it is 8 years older. H.S.M. Coxeter's c o m m e n t (Dec. 3, 1984) on this model: " . . . a wonderful result." The constructions and incidences can be found in detail in [SWl] and [SW2]. For more details see [SSW], where one can find also models with integer coordinates. We now c o m e to the dual map {7, 3}s of Klein's group, built up of heptagons. Ferguson's model is the realization of {7, 3}8 on the standard model of an oriented smooth surface of genus 3 with tetrahedral symmetry. It shows the 24 heptagons, and, hence it corresponds to the regular dodecahedron {5, 3}10. It is a help in understanding Klein's group. Ferguson's model is curved, so the heptagons are nonplanar and the model is not a polyhedron.

The construction of a polyhedral model of {7, 3}8 is more difficult, but it can be done with m o d e m c o m p u t e r programs. The result is s h o w n in Figure 4 (for details of construction see [SSW]). The bizarre model is complicated, and is of no help in understanding Klein's group. This underlines the simplicity of its dual polyhedral embedding of {3, 7}s. In the next section we explain why dual polyhedral realizations of the same group can differ so much. Polyhedral M o d e l s w i t h O c t a h e d r a l S y m m e t r y

As already mentioned, any 3-dimensional model of Klein's group with maximal (octahedral) symmetry has self-intersections; in particular this is true of Klein's curved model of three intersecting hyperboloids. So it is a bit surprising that the simplest polyhedral model of Klein's group is a polyhedral immersion with octahedral symmetry. It was found by E. Schulte and J.M. Wills in 1987 [SW 2] and is shown in Figure 5. Its octahedral s y m m e t r y implies that the s y m m e t r y group acts transitively on its 24 vertices: the vertices are all alike. The vertices can be chosen so that their convex hull is the snub cube, h e n c e one of the 13 Archimedean solids. As a consequence, 32 of the 56 triangles are even regular. The three intersecting tunnels of this model correspond to Klein's three intersecting hyperboloids, and the Petrie polygons can easily be seen. Altogether this polyhedral model provides the easiest w a y to understand the structure of Klein's group PSL (2,7). In sharp contrast to this simple model, its dual {7, 3}s is extremely bizarre (see Figure 6). Although its octahedral symmetry group acts transitively on its 24 congruent heptagons, the model is complicated. Again, the model was constructed by computer; for more details, refer to [SSW]. The heptagons have self-intersections, so the model is related to the classical Kepler-Poinsot polyhedra and to Coxeter's regular c o m p l e x polyhedra. It might be surprising that the realizations of a pair of dual maps of the same group can be so different. But the reason is quite simple: In the triangulations the facets are, by definition, triangles, so they are convex and free of selfintersections. All topological complications, twists, and curvature are hidden in the vertices, whose shape is flexible. In the dual, with 3-valent vertices, all complications have to be stored in the heptagons, which makes the models Figure 4. Kepler-Poinsot-type realization of {7, 3}8 with tetrahedral symmetry. star-shaped and bizarre. So this

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Figure 5. Polyhedral immersion of {3, 7}8 with octahedral symmetry.

Figure 6. Kepler-Poinsot-type realization of {7, 3}8 with octahedral symmetry.

model is not a conceptual tool to understand Klein's group, in sharp contrast to its dual. But all these realizations of Klein's group may qualify as contributions to "Art and Mathematics"--and as contributions to Felix Klein's and H.S.M. Coxeter's general idea of bringing algebra and geometry closer together. REFERENCES

[BW] J. Bokowski and J.M. Wills, Regular polyhedra with hidden symmetries. Mathematical Intelligencer 10 (1988), no. 4, 27-32. [C] H.S.M. Coxeter, Regular Complex Polytopes, Cambridge University Press, Cambridge, 2nd edit. 1991. [CM] H.S.M. Coxeter and W.O.J. Moser, Generators and Relations for Discrete Groups, Springer, Berlin 1980 (4th edit.) [G] J. Gray, From the History of a Simple Group, The Mathematical Intelligencer 4 (1982), no. 2, 59-67 (reprint in [L]).

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[GS] B. Gr0nbaum, G. Shephard, Duality of polyhedra, in: Shaping Space, eds. G. Fleck and M. Senechal, Birkh&user, Boston 1988. [K] F. Klein, Ueber die Transformation siebenter Ordnung der elliptischen Funktionen, Math. Ann. 14 (1879), 428-471 (English transl, by S. Lew in [L]). [L] S. Levy (edit.), The Eightfold Way, MSRI Publ., Cambridge Univ. Press, New York 1999. [SSW] P. Scholl, A. SchQrmann and J.M. Wills, Polyhedral models of Klein's quartic, http://www.math.uni-siegen.de/wills/klein/ [SWl] E. Schulte, J.M. Wills, A polyhedral realization of Felix Klein's map {3, 7}8 on a Riemann surface of genus 3, J. London Math. Soc. 32 (1985), 539-547. [SW2] E. Schulte, J.M. Wills, Kepler-Poinsot-type realization of regular maps of Klein, Fricke, Gordon and Sherk, Canad. Math. Bull. 30 (1987), 155-164.