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Mcmullen P., Schulte E. — Abstract Regular Polytopes
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Название: Abstract Regular PolytopesАвторы: Mcmullen P., Schulte E. Аннотация: geometrical research, which began with regular polygons and polyhedra. They are highly symmetric combinatorial structures with distinctive geometric, algebraic or topological properties; in many ways more fascinating than traditional regular polytopes and tessellations. The rapid development of the subject in the past 20 years has resulted in a rich new theory, featuring an attractive interplay of mathematical areas, including geometry, combinatorics, group theory and topology. Abstract regular polytopes and their groups provide an appealing new approach to understanding geometric and combinatorial symmetry. This is the first comprehensive up-to-date account of the subject and its ramifications, and meets a critical need for such a text, because no book has been published in this area of classical and modern discrete geometry since Coxeter's Regular Polytopes (1948) and Regular Complex Polytopes (1974). The book should be of interest to researchers and graduate students in discrete geometry, combinatorics and group theory.
Язык: Рубрика: Математика /Статус предметного указателя: Готов указатель с номерами страниц ed2k: ed2k stats Год издания: 2002Количество страниц: 551Добавлена в каталог: 23.11.2009Операции: Положить на полку |
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Предметный указатель
Coxeter group, irreducible 66 Coxeter group, reducible 66 Coxeter group, representation, canonical 64-70 Coxeter group, representation, contragredient 68-70 Coxeter group, spherical 71-72 Coxeter group, string 10 66 Coxeter matrix 64 Coxeter system 65 Coxeter, H. S. M. xiii 6 17 21 226 289 Cross-polytope 11 Cross-polytope, hemi-cross-polytope 164 508 Cs 268 Csaszar, A. 268 Cube, 11 Cube, 263 292 Cube, 136 Cube, 3-cube 3 11 200 365 369 Cube, hemi-cube 96 137 164 508 Cut 168 172 201-206 378-383 423-431 445-447 451 462 466 Cut, universal 169 205 451 459 462 466 CYCLE 312 Cycle, length of 313 Cycle, m-cycle 312 Danzer, L. xiii 21 259 Davis, M. W. 152 Degenerate case 302 Diagonal 126 249 Diagonal, class 126 249 Diagonal, class, asymmetric 134 Diagonal, class, symmetric 134 Diagonal, directed 134 Diagonal, equivalent 249 Diagonal, vector 126 Diagram 65 Diagram geometries 7 21 191 509 Diagram, 247 Diagram, 265 Diagram, abstract 356 Diagram, circuit see Circuit diagram Diagram, circuit matching of 314 Diagram, Coxeter see Coxeter diagram Diagram, for 295 Diagram, geometric 308 Diagram, geometric, basic equivalence 308 Diagram, geometric, of finite unitary group 294 Diagram, geometric, tetrahedral 338 342 Diagram, geometric, triangle with a tail 333 Diagram, geometric, triangular 294 323 Diagram, of unitary reflexion group 293 294 Diagram, trivial 246 Diamond net 241 Dihedron 153 161 Dim 122 Dimension vector 225 Distinguished generators 33 36 38 Distinguished generators, of Coxeter group 65 Distinguished subgroup 33 Distinguished subgroup, of Coxeter group 65 Ditope 158 161 dodecahedron 3 11 200 Dodecahedron, great stellated 137 Dodecahedron, hemi-dodecahedron 138 163 164 193 502 Dodecahedron, realizations of 137 Dodecahedron, small stellated 207 Dress, A. W. M. 7 21 226 Du Val, P. 9 Dual 28 Dual, self-dual 28 Duality 28 Duality, operation 192 Dyck's map 18 140 399 474 Dyck, W. 18 Edge of abstract polytope 23 Edge of convex polytope 8 Edge of map 17 Edge-graph of polytope 29 Edge-graph of polytope, dual 29 Eisenstein integers 328 Enantiomorphic 226 Equivelar map 20 Equivelar n -polytope 14 Equivelar of type 30 Euclid 3 Euler's theorem 88 Even (element) 357 Face, 8 23 Face, base face 33 Face, co-face 23 Face, co-face, co- 23 Face, face lattice 8 Face, face-set 22 Face, face-vector 30 Face, improper 8 23 Face, of abstract polytope 22 Face, of convex polytope 8 Face, of honeycomb or tessellation 152 Face, proper 8 23 Face-to-face 15 152 Facet, of abstract polytope 23 Facet, of convex polytope 8 Facet, of honeycomb or tessellation 15 152 Facet, subfacet 23 Facetting operation 194 214 215 432 FAP 112-115 119 187 Flag 9 15 Flag, adjacent 9 24 Flag, adjacent, j-adjacent 9 24 Flag, base flag 33 38 Flag, flag-connected 24 Flag, flag-connected, strongly 9 24 Flag, flag-cycle 61 Flag, flag-sequence 61 Free product 98 Free product, with amalgamation 98 Free product, with amalgamation, normal form theorem 98 Free product, with amalgamation, torsion theorem 99 Fundamental region, for convex polytope 11 66 Fundamental region, for Coxeter group 66 70 Fundamental region, for group of isometries 151 Gauss - Bonnet theorem 182 Gauss, C. F. 6 Gaussian integers 328 Generator subgroup 300 Generators, basic change of 308 Generators, basically equivalent sets of 308 Generators, distinguished 33 36 38 Generators, distinguished, of Coxeter group 65 Generators, irreducible system of 298 Generators, reducible system of 298 Ggi 35 Ggi, string ggi 35 Gordan's equation 346 Gordan, P. A. 346 Gr xiii 7 17 21 226 360 Graph, angle graph 86 Graph, comparability graph 39 Graph, complete 268 Graph, edge-graph 29 Graph, edge-graph, dual 29 Graph, locally 165 Graph, locally icosahedral 165 Graph, locally Petersen 509 Group, affinely reducible 126 Group, Coxeter see Coxeter group Group, dihedral 65 Group, extended group 28 37 Group, Hurwitz 180 Group, locally finite (reflexion group) 306 Group, locally finite (reflexion group), strongly locally finite 327 334 Group, locally unitary 306 Group, modular 106 Group, of polytope 27 Group, Picard 491 Group, reflexion see Reflexion group Group, residually finite 103 452 Group, rotation (sub)group 38 Group, simple 7 191 Group, sparse 58 155 175 Group, special 146 220 311 Group, symmetry see Symmetry group Group, triangle 77 Group, triangle, generalized 320-332 Group, unitary 290 Group, unitary, real 294 Halving operation 197 383 Halving operation, skew 199 Hartley, M. I. 58 Hasse diagram 26 Havas, G. 376 Heegaard splitting 162 Helix 217 Hemi- 163 Hemi-120-cell 163 Hemi-24-cell 163 164 502 Hemi-600-cell 163 Hemi-cross-polytope 163 164 508 Hemi-cube 96 137 163 164 508 Hemi-dodecahedron 163 164 193 502 Hemi-dodecahedron, realizations of 138 Hemi-icosahedron 96 101 137 163 164 268 404 474 502 Hermitian form 298-305 Hermitian form, adjoint 303 Hermitian form, degenerate case 302-305 Hess, E. 6 16 208 hole 18 196 Hole, 195 Hole, deep 213 Homomorphism between polytopes 27 Honeycomb, cell of 15 Honeycomb, euclidean 15 Honeycomb, regular see Regular honeycomb Honeycomb, with finite cells see Tessellation Hosohedron 161 Hurwitz theorem 180 Hurwitz, A. 180 icosahedron 3 11 200 212 218 481 Icosahedron, great 30 137 481 Icosahedron, hemi-icosahedron 96 101 137 163 164 268 404 474 502 Icosahedron, locally icosahedral 165 Icosahedron, realizations of 137 Identification vector 274 276 283 Imprimitive 291 incidence 9 22 Initial vertex 124 Int k 142 Intersection property 34 65 Isomorphic 27 Isomorphic, combinatorially 9 Isomorphism, between polytopes 27 Isomorphism, class 27 Jamnitzer, W. 5 Join in lattice 8 Join of simplicial complexes 39 K 162 Kantor, W. M. 21 Kato, M. 152 Kepler - Poinsot polyhedra 5 16 212 217-220 Kepler, J. 5 Klein's map 18 474 Klein's map, realizations of 139 Klein, F. 18 Lattice, cubic 167 Lattice, cubic, body-centred 167 Lattice, cubic, face-centred 167 Lattice, identification 166 Lattice, reciprocal (or dual) 167 Leo "the mathematician" 5 Leytem, Ch. 376 Lin 299 Line segment 7 11 221 Link 39 Lk 39 Locally toroidal regular polytope of rank 4 360-444 Locally toroidal regular polytope of rank 5 361 450-459 Locally toroidal regular polytope of rank 6 361 Locally toroidal regular polytope of type {3, 3, 3, 4, 3} 363 Locally toroidal regular polytope of type {3, 3, 4, 3, 3} 363 Locally toroidal regular polytope of type {3, 4, 3, 3, 4} 267 269 275 281 284 363 465-470 Locally toroidal regular polytope of type {3, 4, 3, 4} 269 270 272 275 281 285 363 Locally toroidal regular polytope of type {3, 6, 3} 363 387-392 417-423 437-441 Locally toroidal regular polytope of type {4, 4, 3} 162 363-369 376-377 382-386 451 Locally toroidal regular polytope of type {4, 4, 4} 267 269 275 276 363 369-386 462 466 Locally toroidal regular polytope of type {4, 4, 4}, exceptional case 376 Locally toroidal regular polytope of type {6, 3, 3} 102 363 387-409 423-425 429-430 437-444 Locally toroidal regular polytope of type {6, 3, 4} 268 363 387-409 423-425 443 Locally toroidal regular polytope of type {6, 3, 5} 363 387-409 423-425 Locally toroidal regular polytope of type {6, 3, 6} 363 387-392 400-417 426-429 Luggir 306 MAP 17 Map, chiral 19 Map, equivelar 20 Map, regular see Regular map McMullen, P. xiii 21 Meet 8 Mirror 10 83 124 Mixing, mix, of C-groups 125 185-192 Mixing, mix, of polytopes 116 125 186-192 479-480 501 Mixing, operation 183-192 Mixing, operation, on polyhedron 192-201 Monson, B. xiii Mostow rigidity 182 Node 65 84 Node, initial 247 octahedron 3 11 200 218 452 Operation, basic 308 Operation, duality 192 Operation, facetting 194 214 215 432 Operation, halving 197 383 Operation, halving, skew 199 Operation, mixing 183-192 Operation, mixing, on polyhedron 192-201 Operation, Petrie 192 196 Operation, skewing 199 Operation, twisting 244-246 Orbit space 149 Order complex 39-42 192 Order complex, augmented 39 Pasini, A. 21 Pentagram 2 Petersen graph 138 165 502 509 Petersen graph, locally 509 Petrial 192 Petrie polygon 7 Petrie polygon, of map 17 196 Petrie polygon, of polytope 163 184 Petrie polygon, of polytope, convex 94 Petrie, dual 192 Petrie, operation 192 196 Petrie, polygon see Petrie polygon Petrie, property 17 Petrie, X F. 7 17 226 Plato 2 Platonic solids 2 11 Poinsot, L. 6 Polarity 28 37 Polygon 16 25 Polygon, abstract 25 Polygon, convex 8 Polygon, generalized polygons 27 Polygon, infinite see Apeirogon
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