Author Index

Home , Hosohedron, Petersen graph, Star polyhedron, Triangle Group

Cambridge University Press 0521814960 - Abstract Regular Polytopes Peter McMullen and Egon Schulte Index More information

Author Index

Aethelard of Bath, 5 Gauss, C. F., 6 Plato, 2 Altshuler, A., 407 Gordan, P. A., 346 Poinsot, L., 6 Archimedes, 4 Gr¨unbaum,B., xiii, 7, 17, Proclus, 2 Aristotle, 3 21, 226, 360 Pythagoras, 2

Bieberbach, L., 150 Hartley, M. I., 58 Reinhardt, C., 17 Boetius, 4 Havas, G., 376 Ronan, M. A., 21 Bokowski, J., 140 Hess, E., 6, 16, 208 Bracho, J., 220 Hurwitz, A., 180 Schläfli,L., 6, 16, 85 Bradwardine, T., 5, 16 Schulte, E., xiii, 21 Brehm, U., 140 Jamnitzer, W., 5 Shephard, G. C., 289 Buekenhout, F., 7, 191 Sloane, N. J. A., 170 Kantor, W. M., 21 Sommerville, D. M. Y., 86 Kato, M., 152 Cauchy, A. L., 6, 16 Stringham, W. I., 6 Kepler, J., 5 Cohen, A. M., 21, 289 Klein, F., 18 Conway, J. H., 170 Theaetetus, 3 Kühnel, W., 162 Coxeter, H. S. M., xiii, 6, 17, Tits, J., 7, 21, 36, 79 21, 226, 289 Todd, J. A., 289 Leo “the mathematician”, 5 Császár, A., 268 Leytem, Ch., 376 Uccello, P., 5 Danzer, L., xiii, 21, 259 McMullen, P., xiii, 21 Davis, M. W., 152 Monson, B., xiii Vinberg, E. B., 78 Dress, A. W. M., 7, 21, 226 Du Val, P., 9 van Oss, S. L., 6, 208 Weiss, A. I., xiii, 38 Dyck, W., 18 Wills, J. M., 140 Pasini, A., 21 Witt, E., 71 Euclid, 3 Petrie, J. F., 7, 17, 226 Wythoff, W. A., 11, 124

543

Subject Index

24-cell, 11, 100, 165, 254, 450 amalgamation hemi-24-cell, 163, 502 FAP, 112–115, 119, 187 realizations of, 138 ﬂat amalgamation property, 112–115 120-cell, 11 free product with, 98 hemi-120-cell, 163 of polytopes, 96–101 600-cell, 11, 212 angle, 85 hemi-600-cell, 163 angle graph, 86 2K, 255–257 angle-sum relations, 86 realizations of, 259–261 apeirogon, 15, 25, 27 2K,G(s), 256–259 regular, see regular apeirogon realizations of, 261–264 apeirohedron, 25 Ln k , 461, 463, 467, 503–509 regular, see regular apeirohedron L3 m;H, 471–484 apeirotope, 25 realizations of, 484–490 n-apeirotope, 25 L4 m;I , 490–501 regular, see regular apeirotope LK,G , 247–255, 264–272 attached (interior mark), 323 T 5 automorphism s , 450–459 T 6 group, 11, 27 1 s , 459–462 T 6 of polytope, 27 2 s,t, 462–465 T 6 convex, 11 3 s,t, 466–470 T 4 1 s , 363–369, 376–377, 382–386 T 4 2 s,t, 369–386 barycentric subdivision, 40 T 4 3 s , 387–389, 392–409, 423–425, basic 429–430, 437–444 2-generator subgroup, 300 T 4 4 s , 387–389, 392–409, 423–425 3-generator subgroup, 300 T 4 5 s , 387–389, 392–409, 423–425 operation, 308 T 4 6 s,t, 388–389, 400–417, 426–429, Bieberbach theorem, 143, 150, 220 443 complex analogue, 310 T 4 7 s,t, 417–423 blend of realizations, 122, 125 branch, 65, 84 action improper, 65 discrete, 149 of geometric diagram, 308 free, 149 Brianchon–Gram theorem, 86 properly discontinuous, 149 buildings, 7, 21 adjoint hermitian form, 303 C-group, 49–60 matrix, 303 string, 50

544

Subject Index 545

canonical map bilinear form, 67 between polytopes, 43 expression for P, 132 topological, 149 matrix, 130 of polytopes, 43 projection, 44 Coxeter complex, 42, 69 canonical projection, 149 euclidean, 73 Cayley–Menger matrix, 128 finite, 71 cell, 15 Coxeter diagram, 65, 84 centrally symmetric euclidean, 73 abstract polytope, 249, 255 spherical, 72 spherical tessellation, 163 string, 66 centre, 206 Coxeter element, 163 centroid, 206 Coxeter group, 10, 64–94 of polygon, 16 canonical bilinear form, 67 chain, 22 compact hyperbolic, 77 length of, 22 crystallographic, 89 type of, 23 distinguished generators of, 65 chamber distinguished subgroup of, 65 closed, 69 euclidean, 71–76 fundamental, 66, 69 finite, 71, 83–94 open, 69 hyperbolic, 76–78 chamber complex, see Coxeter complex irreducible, 66 character, 131 reducible, 66 character norm, 131 representation chiral, 38 canonical, 64–70 map, 18 contragredient, 68–70 polytope, 38 spherical, 71–72 tessellation, 178 string, 10, 66 toroid, 177 Coxeter matrix, 64 circuit criterion, 63 Coxeter system, 65 circuit diagram, 297, 345, 351 cross-polytope, 11 3-circuit, 323 hemi-cross-polytope, 164, 508 4-circuit, 345 Császártorus, 268 n-circuit, 351 cube diagonal-free, 315 3-cube, 3, 11, 200, 365, 369 interior mark of, 297, 323 n-cube, 11 turn of, 313 complex, 263, 292 with tails, 347 realizations of, 136 circuit matching, 314 hemi-cube, 96, 137, 164, 508 class, 96 cut, 168, 172, 201–206, 378–383, 423–431, classification 445–447, 451, 459, 462, 466 combinatorial, 102 universal, 169, 205, 451, 459, 462, topological, 360 466 co-face, 23 cycle, 312 co-i-face, 23 m-cycle, 312 cone, 127 length of, 313 connected, 9, 23 flag-connected, 24 degenerate case, 302 strongly, 9, 24 diagonal, 126, 249 strongly connected, 9, 23 class, 126, 249 contragredient, 68, 302 asymmetric, 134 convex hull, 7 symmetric, 134 convex set, 7 directed, 134 covering, 43 equivalent, 249 k-covering, 43 vector, 126

546 Subject Index

diagram, 65 co-face, 23 K-extendable, 265 co-i-face, 23 K-admissible, 247 face lattice, 8 abstract, 356 face-set, 22 circuit, see circuit diagram face-vector, 30 circuit matching of, 314 improper, 8, 23 Coxeter, see Coxeter diagram of abstract polytope, 22 for [klmp]q , 295 of convex polytope, 8 geometric, 308 of honeycomb or tessellation, 152 basic equivalence, 308 proper, 8, 23 of finite unitary group, 294 face-to-face, 15, 152 tetrahedral, 338, 342 facet triangle with a tail, 333 of abstract polytope, 23 triangular, 294, 323 of convex polytope, 8 of unitary reflexion group, 293, 294 of honeycomb or tessellation, 15, trivial, 246 152 diagram geometries, 7, 21, 191, 509 subfacet, 23 diamond net, 241 facetting operation, 194, 214, 215, 432 dihedron, 153, 161 FAP, 112–115, 119, 187 dimension vector, 225 flag, 9, 15, 22 distinguished generators, 33, 36, 38 adjacent, 9, 24 of Coxeter group, 65 j-adjacent, 9, 24 distinguished subgroup, 33 base flag, 33, 38 of Coxeter group, 65 flag-connected, 24 ditope, 158, 161 strongly, 9, 24 dodecahedron, 3, 11, 200 flag-cycle, 61 great stellated, 137 flag-sequence, 61 hemi-dodecahedron, 138, 163, 164, 193, 502 free product, 98 realizations of, 137 with amalgamation, 98 small stellated, 207 normal form theorem, 98 dual, 28 torsion theorem, 99 self-dual, 28 fundamental region duality, 28 for convex polytope, 11, 66 operation, 192 for Coxeter group, 66, 70 Dyck’s map, 18, 140, 399, 474 for group of isometries, 151

edge Gauss–Bonnet theorem, 182 of abstract polytope, 23 Gaussian integers, 328 of convex polytope, 8 generators of map, 17 basic change of, 308 edge-graph of polytope, 29 basically equivalent sets of, 308 dual, 29 distinguished, 33, 36, 38 Eisenstein integers, 328 of Coxeter group, 65 enantiomorphic, 226 irreducible system of, 298 equivelar reducible system of, 298 n-polytope, 14 ggi, 35 map, 20 string ggi, 35 of type, 30 Gordan’s equation, 346 Euler’s theorem, 88 graph even (element), 357 angle graph, 86 comparability graph, 39 face complete, 268 k-face, 8, 23 edge-graph, 29 base face, 33 dual, 29

Subject Index 547

locally H, 165 regular, see regular honeycomb locally icosahedral, 165 with finite cells, see tessellation locally Petersen, 509 hosohedron, 161 group Hurwitz theorem, 180 affinely reducible, 126 Coxeter, see Coxeter group icosahedron, 3, 11, 200, 212, 218, 481 dihedral, 65 great, 30, 137, 481 extended group, 28, 37 hemi-icosahedron, 96, 101, 137, 163, 164, 268, Hurwitz, 180 404, 474, 502 locally finite (reflexion group), locally icosahedral, 165 306 realizations of, 137 strongly locally finite, 327, 334 identification vector, 274, 276, 283 locally unitary, 306 imprimitive, 291 modular, 106 incidence, 9, 22 of polytope, 27 initial vertex, 124 Picard, 491 intersection property, 34, 65 reflexion, see reflexion group isomorphic, 27 residually finite, 103, 452 combinatorially, 9 rotation (sub)group, 38 isomorphism simple, 7, 191 between polytopes, 27 sparse, 58, 155, 175 class, 27 special, 146, 220, 311 join symmetry, see symmetry group in lattice, 8 triangle, 77 of simplicial complexes, 39 generalized, 320–332 unitary, 290 K-extension, 265 real, 294 universal, 265 Kepler–Poinsot polyhedra, 5, 16, 212, halving operation, 197, 383 217–220 skew, 199 Klein’s map, 18, 474 Hasse diagram, 26 realizations of, 139 Heegaard splitting, 162 helix, 217 lattice hemi- cubic, 167 120-cell, 163 body-centred, 167 24-cell, 163, 164, 502 face-centred, 167 600-cell, 163 identification, 166 {p1,...,pn}, 163 reciprocal (or dual), 167 cross-polytope, 163, 164, 508 line segment, 7, 11, 221 cube, 96, 137, 163, 164, 508 link, 39 dodecahedron, 163, 164, 193, 502 locally toroidal regular polytope realizations of, 138 of rank 4, 360–444 icosahedron, 96, 101, 137, 163, 164, 268, of rank 5, 361, 450–459 404, 474, 502 of rank 6, 361, 459–470 hermitian form, 298–305 of type {3, 3, 3, 4, 3}, 363, 459–462 adjoint, 303 of type {3, 3, 4, 3, 3}, 363, 462–465 degenerate case, 302–305 of type {3, 4, 3, 3, 4}, 267, 269, 275, 281, 284, hole, 18, 196, 236 363, 465–470 k-hole, 195 of type {3, 4, 3, 4}, 269, 270, 272, 275, 281, 285, deep, 213 363, 450–459, 516, 517 homomorphism between polytopes, 27 of type {3, 6, 3}, 363, 387–392, 417–423, 430, honeycomb 437–441 cell of, 15 of type {4, 4, 3}, 162, 363–369, 376–377, euclidean, 15 382–386, 451, 459

548 Subject Index

locally toroidal regular polytope (contd.) polygon, see Petrie polygon of type {4, 4, 4}, 267, 269, 275, 276, 363, property, 17 369–386, 462, 466 Petrie polygon, 7 exceptional case, 376 of map, 17, 196 of type {6, 3, 3}, 102, 363, 387–409, 423–425, of polytope, 163, 184 429–430, 437–444 convex, 94, 236 of type {6, 3, 4}, 268, 363, 387–409, 423–425, Platonic solids, 2, 11 443 polarity, 28, 37 of type {6, 3, 5}, 363, 387–409, 423–425 polygon, 16, 25 of type {6, 3, 6}, 363, 387–392, 400–417, abstract, 25 426–429, 441–443 convex, 8 luggir, 306 generalized polygons, 27 infinite, see apeirogon map, 17 regular, see regular polygon chiral, 19 star-polygon, 16 equivelar, 20 polyhedron, 25 regular, see regular map abstract, 25 meet, 8 convex, 8 mirror, 10, 83, 124 infinite, see apeirohedron mixing regular, see regular polyhedron mix toroidal, 18 of C-groups, 125, 185–192 polytopal, 26 of polytopes, 116, 125, 186–192, 479–480, polytope, 22 501 n-polytope, 25 operation, 183–192 abstract, 22, 25 on polyhedron, 192–201 convex, 7 Mostow rigidity, 182 abstract, 22–31 chiral, 38 node, 65, 84 complex regular, 293 initial, 247 convex, 7 cubical, 268 octahedron, 3, 11, 200, 218, 452 dual, 9, 28 operation finite, 25 basic, 308 locally, 25 duality, 192 flat, 109, 115–120, 275, 372, 508, facetting, 194, 214, 215, 432 514 halving, 197, 383 (k, m)-flat, 109 skew, 199 geometric, 122 mixing, 183–192 group of, 27 on polyhedron, 192–201 incidence polytope, 22 Petrie, 192, 196 infinite, see apeirotope skewing, 199 locally of topological type X, 361 twisting, 244–246 neighbourly, 249, 268, 271, 404 orbit space, 149 weakly, 267 order complex, 39–42, 192 projective, 162–165, 287, 502 augmented, 39 locally, 284, 461, 463, 467, 502–509, 516 pentagram, 2 regular, see regular polytope Petersen graph, 138, 165, 502, self-dual, 28 509 simplicial, 268 locally, 509 spherical, 153 Petrial, 192 locally, 152–162 Petrie symmetric, 123 dual, 192 symmetry group of, 9, 123 operation, 192, 196 topological model of, 161, 367

Subject Index 549

toroidal, 154, 165–175 reduced sequence, 98 higher, 445, 450–470 reflexion group, 64, 290, 298 locally, see locally toroidal regular complex polytope locally finite, 306 poset, 22 locally unitary, 306 pre-polytope, 43 strongly locally finite, 327, 334 primitive, 291 with involutory generators, 298–319 proper central involution, 249, 255 Coxeter, see Coxeter group euclidean quotient, 42–49 discrete infinite, 73, 89 criterion, 56 finite, 71–72, 83–94 lemma, 56 irreducible, 73 polytope, 44, 58–60, 272 reducible, 75 space, 149 preserving figure, see symmetry group unitary radical, 298 n-generator, 293 rank, 22 blend of, 415 co-rank, 23 completely reducible, 290 of Coxeter group, 65 finite, 290–298, 321–327, 334–336, 339–341, rank function, 23 343–344, 347–355, 357–359 rap-map, 43 imprimitive, 291 realization, 121–127 infinite discrete, 309–312, 327–331, 344, 350, blend, 122, 125 354–355 centre of, 127 irreducible, 290 cone, 127–140 primitive, 291 cross-polytope realization, 135 real, 294 degenerate, 122 with involutory generators, 294–298 dimension of, 122 Weyl, see Weyl group discrete, 142 reflexion in hyperplane faithful, 122 complex, 298 linear combination, 122, 126 unitary, 290 of 24-cell, 138 real, 68 of 2K,G(s), 261–264 K euclidean, 10 of 2 , 259–261 reflexion in subspace, 10, 123 L3 of m;H, 484–490 regular, 9, 31 T 4 of 6 s,t, 417 combinatorially, 11, 31, 152 T 4 = , , of p (s,0) (p 3 4 5), 396–398 directly, 38 of n-cube, 136 geometrically, 156 of dodecahedron, 137 semi-regular, 15 of finite regular polytope, 127–140 regular apeirogon, 15, 25, 27 of hemi-dodecahedron, 138 helical, 217, 222 of icosahedron, 137 linear, 217, 221 of Klein’s map, 139 realizations of, 140 of regular apeirogon, 140 zigzag, 217, 222 of regular apeirotope, 140–147 regular apeirohedron of regular polygon, 135 in E3, 220–236 pure, 126 blended, 221–223, 226–229 scalar multiple of, 122, 126 planar, 221 simplex realization, 128 pure, 223–226, 230–236 symmetries of, 123 regular apeirotope symmetry group of, 123 4-apeirotopes in E3, 236–243 translation-free, 143 realizations of, 140–147 trivial, 122 regular honeycomb vertex-faithful, 122 euclidean, 15, 81, 254 vertex-set of, 121 symmetry group of, 67, 73

550 Subject Index

regular honeycomb (contd.) quotient polytopes of, 58–60 hyperbolic, 78, 81, 204, 254, 362, 384, 431–437, realizations of, see realization 445–449 related to a linear group, 490–501 symmetry group of, 67, 78, 431–432, 445–449 self-dual, 37 with finite cells, see regular tessellation spherical, 153 regular map, 17–20, 179 locally, 152–162 Dyck’s map, 18, 140, 399, 474 star-polytope, 16, 31, 206–217 Klein’s map, 18, 139, 474 symmetries of, see symmetry group non-orientable, 180 topological model of, 161, 367 of hyperbolic type, 179 toroidal, see regular toroid polyhedral models of, 140, 268 locally, see locally toroidal regular polytope polytopal, see regular polyhedron universal, 78–83 toroidal, 18, 364, 390–391 in class P1, P2 , 96–101 regular polygon of type {p1,...,pn−1}, 78–83 blended, 135, 217, 222 with given facet and vertex-figure, 97 convex, 3, 6, 11, 27 regular solids, 2, 11 helical, 217 regular tessellation in E3, 217 euclidean, 15, 81, 254 infinite, see regular apeirogon planar, 2, 15, 221–223 pure, 217 symmetry group of, 67, 73 realizations of, 135 hyperbolic, 81 skew, 217, 222 symmetry group of, 67, 78 star-polygon, 5, 16, 135 on space-form, 151 regular polyhedron euclidean, 165–177 convex, 1–4, 11, 217–220 hyperbolic, 178–182 Coxeter’s {4, 2p |4p−2, 2s}, 256, 259, 264 spherical, 162–165 Coxeter’s {4, p |4[p/2]−1}, 140, 184, 196, 256, on surface, see regular map 260 polytopal, see regular polyhedron Coxeter’s skew, 140, 196 on torus, see regular toroid history of, 1–7 spherical, 162–165 in E3, 217–236 with finite or infinite cells, see regular infinite, see regular apeirohedron honeycomb Kepler–Poinsot, 5, 16, 212, 217–220 regular toroid, 154 on surface, see regular map cubic, 165–169, 172–175, 258, 269 Petrie–Coxeter, 18, 196 on 2-torus, 18, 364, 390–391 polyhedral models of, 140, 268 other, 170–175, 275 related to a linear group, 471–490 representation skew, 7, 140, 196 character norm, 131 star-polyhedron, 5, 16, 212, 217–220 character of, 131 toroidal, 18, 364, 390–391 complex conjugate, 130 regular polytope contragredient, 68, 302 abstract, 31–38 degree of, 129, 133 group of, 31–38, 49–58 isomorphic, 131 amalgamation of, 96–101 of Coxeter groups, 64–70 classical, 206 ridge complex regular, 293 of abstract polytope, 23 convex, 7–15, 31, 81, 208 of convex polytope, 8 symmetry group of, 10, 67, 71 root, 75 free extension of, 106–109 lattice, 75 higher toroidal, 445, 450–470 Dn, 167, 170 history of, 1–7 system, 75 infinite, see regular apeirotope locally of topological type X, 361 Schläflideterminant, 208, 306, 314 projective, 162–165, 287, 502 Schläflisymbol locally, 284, 461, 463, 467, 502–509, 516 of abstract polytope, 30

Subject Index 551

of complex regular polytope, 293 hyperbolic, 67, 78, 431–432, 445–449 of convex polytope, 11 on space-form, 156 of map, 17 of regular polyhedron in E3, 217–220, of star-polytope, 207 226–236 section, 9, 23 k-section, 23 tessellation proper, 23 chiral, 178 self-Petrie, 193 cubic, 166, 258 semi-regular, 15 euclidean, 15 sggi, 35 face-to-face, 15, 152 simplex, 8, 11 on manifold, 152 adjacent, 39 regular, see regular tessellation base, 40 with finite or infinite cells, see honeycomb dissection, 432, 505 tetrahedron, 3, 11, 200, 218 type of, 39 thin, 21, 27 simplicial complex tile, 152 of chains, see order complex Tits cone, 70 of chambers, see chamber complex Todd–Coxeter algorithm, 272, 282, 451, 460, 463, k-skeleton, 29 467, 502 dual, 29 toroid, 154 skewing operation, 199 chiral, 177 space regular, see regular toroid complex projective, 291 torus, 150 euclidean, 148 triangle group, 77 hyperbolic, 148 generalized, 320–332 real elliptic, 150 turn, 313 real projective, 149 twisting operation, 244–246 spherical, 148 type unitary, 290 of map, 17 space-form, 148–152 of polytope, 30 affinely equivalent, 151 euclidean, 149, 175–177 universal polytope, see regular polytope, hyperbolic, 149, 178–182 universal isometric, 150 isometry type of, 150 vertex orientable, 150 antipodal, 249, 255 similar, 151 of abstract polytope, 23 spherical, 149, 162–165 of convex polytope, 8 tessellation on, 151 of map, 17 volume of, 151 vertex-figure star, 113 of abstract polytope, 23 starry, 207, 213 of convex polytope, 8 subfacet, 23 replacement, 210 supporting hyperplane, 8 symmetries of geometric polytope, 123 wall, 40 symmetry group Weyl group, 75 of polytope affine, 76 complex regular, 293 Wythoff convex regular, 9, 67, 71 construction, 11, 124, 192, 207 geometric, 123 space, 124 of realization, 123 essential, 131 of regular 4-apeirotope in E3, 236–243 of regular honeycomb or tessellation zigzag, 192 euclidean, 67, 73 k-zigzag, 196