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Regular Complex Polytopes
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CONTENTS (vii to ix)
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xii |
Preface to the first edition (xi to xii)
[xi]
Chapter 3 includes practical instructions for making an icosahedral kaleidoscope: a fascinating toy that shows 12o replicas of any small object, the way Sir David Brewster's ordinary kaleidoscope shows 6. Chapters 4 and 5 provide a new approach to the sixteen regular polytopes in real space of four dimensions. In Chapters 6 and 7, geometric ideas are used for enumerating the finite multiplicative groups of quater-nions, and these groups are seen to have elegant presentations in terms of two generators and two relations. There is an interesting connection with groups of 2 x 2 matrices over the field of residues modulo p. Chapter 8 is an introduction to the analytic geometry of unitary n-space, using vectors, Hermitian forms, and inner products. Two-dimensional unitary transformations are investigated more intensively in Chapter 9, with the aid of quaternions, in a manner suggested by Patrick Du Val and D. W. Crowe. A simple criterion (9.42) is found to distinguish reflections from other unitary transformations. This leads to concise presentations for the groups generated by two or three unitary reflections. The complete list of finite reflection groups in unitary n-space was compiled in 5954 by Shephard and Todd, who found that there are many more of them in the plane than in any higher space. Chapter so checks their results (in the two-dimensional case) by a new method : examining all the finite groups of unitary transformations and picking out those that are generated by reflections. In particular, those that are generated by two reflections are the symmetry groups of the regular complex polygons. These (including star polygons)g' are enumerated in Chapter I I. Somewhat surprisingly, it is possible to make real drawings of these imaginary figures, and in many cases such a drawing of one complex polygon serves as a Cayley diagram for the symmetry group of another. Chapters 12 and 13 deal with regular polytopes and honeycombs, using definitions suggested by Peter McMullen. There are interesting connections with certain projective configurations such as the 27 lines on the cubic surface. A remarkable presentation (13.83) is found for the simple group of order 25920. My own chief contribution was, perhaps, the extended Schlafli symbol
[xii] where the p's arc integers and the q's arc rational numbers (usually 'small' integers). For instance, in this notation fig is 2{3}2 {3}2{4}p. Most of the sections end with Exercises, some easy, some challenging. Answers are sketched at the end of the book. I am grateful to many universities on several continents for giving mc opportunities to lecture on various portions of this material between 1966 and 1972. I offer most cordial thanks to J. H. Conway, D. W. Crowe, P. Du Val, G. C. Shephard and J. A. Todd for help and inspiration, and above all to P. McMullen and J. G. Sunday for reading the whole manuscript and making valuable suggestions. McMullen's M.S. thesis (Birmingham, 1966) was closely related to this work. He and B. B. Phadke kindly drew most of the figures. In particular, the frontispiece is McMullen's drawing of the four-dimensional complex polytope 3{3}3{3}3{3}3, which has 240 vertices and 2160 '3-edges' (appearing as equilateral triangles). Du Val kindly allowed me to reproduce the three parts of Figure 2.4 A from his monograph of 1964. Finally, I would express my gratitude to the staff of the Cambridge University Press for their courtcous and efficient handling of many problems that inevitably arose. University of Toronto H. S. M. Coxeter August 1973 |
| xiii | Preface to the second edition (xiii)[xiii] Although this book is entitled Regular Complex Polytopes, nearly half of it deals with real geometry. The convenience of complex numbers is gently introduced in §4.4, and the first mention of a 'complex polygon' occurs in §4.8. A careful reader may be wondering why Figure 4.7A (on page 42) is said to depict not only the convex 600-cell {3, 3, 5} but also three of the ten regular star polytopes (see page 46). This happens because such a drawing is the projection of a 'skeleton', that is, of the graph formed by the vertices and edges of the original polytope. In fact, as we see in Regular Polytopes (Coxeter 1973,' p. 303, Table VI (iii)), those three star polytopes can be placed so as to have the same skeleton as {3, 3, 5} but different faces and facets. Their projections would have a far more complicated appearance if we included their 'false edges' where faces intersect facets (analogous to the 'false vertices' in the pentagram (l}, and the 'false edges' drawn as thin lines in Figure 2- ij on page 12). Similar considerations apply to Figures 4.7B and c. Most of these figures have recently become amenable to the technique of computer graphics. However, the skilful hand of Peter McMullen, using simply a steel ruler and a pen, can be seen in the delicate intricacy of Figure 4.70 (page 45) and the frontispiece. Chapters 8, 9, 10 provide a complete enumeration of finite reflection groups in the unitary plane, carried out by a faster method than that of Shephard and Todd (1954). The reader may be interested to compare this with the entirely different method of Springer (1974) and Cohen (1976). Many authors follow Bourbaki (1968, pp. 1 i -12) in using the name 'Coxeter group' for a group with the presentation (12, = where qpp = I, so that the generators are involutory. The group so presented is conveniently associated with a graph whose vertices (or 'nodes') stand for the generators while its edges (or 'branches') are marked with those numbers q„. which are greater than 3. When two vertices are not directly joined, the corresponding generators commute For references, see page 203. (so that gm,. = 2). An unmarked edge is understood to indicate qm, = 3 (which happens to be the most prevalent value). This idea (see pages 14, 34) was first proposed in my 'Groups whose fundamental regions are simplexes' (journal of the London Mathematical Society 6 (1931), p. 133). Ten years later it was employed by Witt (1941), who proposed a slightly modified version in which an edge marked q is replaced by an unmarked (q - 2)-fold edge (agreeing neatly with the absence of an edge when q = 2). This version of the graph was quite independently rediscovered by E. B. Dynkin ('Classification of simple Lie groups', Mat. Sbornik 18 (6o) (1940), 347-52) and became known as a 'Dynkin diagram'. Such diagrams play a role in the investigation of many important phenomena, such as the Leech lattice and the Fischer-Griess monster group. See, for instance, Carter (1972), Voskuil (1975), Hazewinkel, Hesselink, Siersma and Veldkamp (1977), Schellekens and Warner (1988), and the compre-hensive treatise by Conway and Sloane (1988, pp. 95-6, 456-65, 481, 492-509, 520-9, 552-3, 570). A. J. Coleman, in the Mathematical Intelligencer 11 (1989), 29-38, referred to an 1890 paper by Wilhelm Killing as 'The Greatest Mathematical Paper of All Time'. Among his many achievements, Killing listed the periods of the 'Coxeter transformation' (R1 R„ in §13.6 on page 15o) '19 years before Coxeter was born!' According to Mostow (1980, pp. 174-8), when Shephard and Todd (1954) decided to modify the 'Coxeter group' by admitting generators of period p > z, they were giving a metrical interpretation to G. Bagnera's projective enumeration of groups generated by homologies. Such a generator of period p appears in the generalized 'Dynkin diagram' as a vertex marked p or, in Mostow's notation, a small circle enclosing the number p. However, when p > 2, it is no longer appropriate to mark an edge with the period of the product of two generators: the relation R1 R3 = R3 R1 can no longer be expressed as (R1 R3)2 = 1. Vertices not directly joined still indicate commutative generators, but now an unmarked edge means 11,11,11, = and an edge marked q means Rµ 9 R5 in the notation of page 148 (in the middle of the right column). Examples can be seen on pages 117, 137-40. The appearance of the mark on page 137 has, of course, no such algebraic interpretation: it merely refers to two real mirrors inclined at the angle 2n/5. [xiv] The geometry of §9•8 (on page 94) can be elucidated, when q is an integer, by applying Wytholl's construction to the group 1), or Pi[q]P2 so as to obtain the 'non-starry' regular polygon • or PI {q}P2• Pi ( In particular, if p, = p, = 2, the geometry is real, 2[q]2 is the dihedral group [q] of order 2q, and 2{q}2 is the regular q-gon, {q}.) The `polygon' has p, vertices on each 'edge', p2 edges through each vertex. The pi vertices on an edge, cyclically permuted by R1, behave like the pith roots of unity and thus can be pictured as the vertices of a real p1-gon. In 2{413 (see Figure i1•5B on page 108), the edges are ordinary because p, = 2, and the 3 edges through a vertex are plainly visible. In 3{4}2 (Figure II -5A) the value p2 = 2 appears as the number of equilateral triangles sharing a vertex. One of the innovations introduced into this Second Edition (pages 97 and 114) is J. G. Sunday's simple interpre-tation for the `middle' number q as the `girth' of the complex polygon. Regular complex polygons paqi 1p, occur as 'faces' in higher-dimensional complex polytopes Pi{41},02{42}P3 • • • • In a drawing, the `edges' continue to appear as regular p1-gons; for instance, equilateral triangles in Figures 12.3A, B, C on pages 120-2. And the number qi, being the girth of a face, is the number of vertices of a minimal cycle such as one of the many irregular quadrangles in Figure 12.4A on page 125. (We must not be confused by the equilateral triangles based on the sides of the peripheral {18}; the third vertex of such a triangle is an accident of the projection, not a vertex of the complex polyhedron.) §12.5 (pp. 132-4) describes the group 3[3]3[3]3[3]3 of order 155,520 with the presentation = 1, 12, H R2HR3HR4, R2 41+ 114 ?Ri 2 R3. It is interesting to observe that the two 'rotations' P=RI R3, Q = R2 R4 generate a subgroup isomorphic to 3[5]3. The commutativity of R, and R3 shows that P3 = 1, and we can verify (in about 14 steps) that P 43-;) (see Coxeter 1984, p. 187). The consequent projection (described on page 140) of the skeleton of the Witting polytope 3{3}3{3}3{3}3 into the combined skeletons of the polygons 3{5}3 and 3{1}3, agrees with the verifiable fact that a transparent copy of Figure 4.8B, and such a copy of Figure 4.8D reduced in the 'golden ratio', will exactly fit over a substantial part of the frontispiece. Readers may have noticed, in these investigations, a preponderance of figures having fivefold symmetry, like the periwinkle or the hoya flower. Interest in such symmetry has increased since the discovery of quasi-crystals. In fact, pentagonal (and even icosahedral) symmetry can be seen not only in plants and marine animals but also in some minerals. According to Peter A. Bancel of IBM (Yorktown Heights, N.Y.), one of the best instances is the quasicrystal A165Cu20Fe1 5 Associated with these ideas is the algebraic fact that the first non-Abelian simple group (see Klein 1913) is the icosahedral group PSL(z, 5). In contrast, our new Chapter 14, beginning on page 156, introduces figures having sevenfold symmetry (such as Figure 14.5D, which illustrates the second simple group PSL(2, 7)). The number 7 has a certain air of mystery, as in the musical fact that the seventh harmonic is dissonant (going up from C to somewhere between B flat and B natural). Accordingly, such symmetry seems appropriate for the most interesting 'almost regular' polyhedra, which share with the Platonic solids the property of having regular faces, arranged the same way at all their vertices. I take this opportunity to express my gratitude to the Syndics of the Cambridge University Press for their decision to keep this book in print by publishing a second edition, and to many friends and colleagues who have drawn attention to misprints and other imperfections in the first edition. I would thank especially John F. Rigby, who corrected Figure 2.4c, and Peter McMullen, who added one more (Figure 14•7A) to his spectacular series of drawings. In addition to their mathematical meaning, those pictures can be appreciated as abstract art. University of Toronto H. S. M. Coxeter August 1990 |
(Created September 30, 2024)