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Mcmullen P., Schulte E. — Abstract Regular Polytopes
Mcmullen P., Schulte E. — Abstract Regular Polytopes

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Название: Abstract Regular Polytopes

Авторы: 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.

Abstract regular polytopes stand at the end of more than two millennia of geometrical research, which began with regular polygons and polyhedra. The rapid development of the subject in the past twenty years has resulted in a rich new theory featuring an attractive interplay of mathematical areas, including geometry, combinatorics, group theory and topology. This is the first comprehensive, up-to-date account of the subject and its ramifications. It meets a critical need for such a text, because no book has been published in this area since Coxeter's "Regular Polytopes" (1948) and "Regular Complex Polytopes" (1974).


Язык: en

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 2002

Количество страниц: 551

Добавлена в каталог: 23.11.2009

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
$2^{\mathcal{K, G(s)}}$      258
$2^{\mathcal{K, G}}$      256
$2^{\mathcal{K}}$      256
$A_{n}, B_{n}, C_{n}, D_{n}$      72
$d_{G}$      133
$E^{*}$      302
$e^{\perp}$      298
$E_{6}, E_{7}, E_{8}$      72
$F\cdot\Sigma$      44
$F_{4}, G_{3}, G_{4}$      72
$G(m, p, n)$      292
$G(P)$      123
$G(\mathcal{P})$      156
$G/F$      23
$GL(E)$      68
$G^{3}(p, s; q, r)$      323
$G^{4}(p, s; q, r; t)$      333
$k$-skeleton      29
$k$-skeleton, dual      29
$L^{\mathcal{G}}_{2}(\mathbb{Z}_{m})$      471
$P#Q$      125
$PL^{\mathcaL{G}}_{2}(\mathbb{Z}_{m})$      472
$PSL^{#}_{2}(\mathbb{D})$      491
$p_{j(1), ..., j(m)}$      313
$P_{n}, Q_{n}, R_{n}, S_{n}$      73
$r_{0}[p_{1}]r_{1}[p_{2}]r_{2}\cdot\cdot\cdot r_{n-2}[p_{n-1}]r_{n-1}$      293
$r_{0}\{p_{1}\}r_{1}\{p_{2}\}r_{2}\cdot\cdot\cdot r_{n-2}\{p_{n-1}\}r_{n-1}$      293
$S(p_{1}, ..., p_{n-1})$      89
$skel_{k}(\mathcal{P})$      29
$SL_{2}^{#}(\mathbb{D})$      492
$SL_{2}^{\mathcal{I}}(\mathbb{Z}_{m})$      472
$st(\Omega,\mathcal{C})$      113
$S_{4}(p, q; r, s)$      342
$T_{7}, T_{8}, T_{9}$      73
$U_{5}, V_{3}, W_{2}$      73
$U_{n}(\mathbb{C})$      290
$W(M)$      64
$W(\mathcal{D})$      65
$w_{G}$      133
$w_{G}^{*}$      133
$W_{I}$      65
$[1 1 1^{p}]^{s}$      320
$[3, 3, 4, 3]_{s}$      171
$[3^{k, l, m}]$      84
$[4, 3^{n-2}, 4]_{s}$      167
$[4, 4]_{s}$      364
$[6, 3]_{s}$      388
$[k l m^{p}]^{q}$      295
$[p,q]_{r}$      193
$[p_{1}, ..., p_{n-1}]$      10 66
$\alpha(p_{1}, ..., p_{n-1})$      89
$\beta#\Gamma$      122
$\Delta(P)$      126
$\Delta(\mathcal{D})$      308
$\Gamma(C)$      313
$\Gamma(\mathcal{P, F})$      29
$\Gamma(\mathcal{P, \textit{\Omega}})$      29
$\Gamma(\mathcal{P})$      27
$\Gamma\lozenge\Delta$      185
$\Gamma^{+}(\mathcal{P})$      38
$\Gamma^{3}(p, s; q, r)$      320
$\Gamma^{4}(p, s; q, r; t)$      337
$\Gamma^{5}_{s}$      450
$\Gamma_{J}$      33
$\hat{P}L^{\mathcaL{G}}_{2}(\mathbb{Z}_{m})$      472
$\hat{P}SL_{2}(\mathbb{Z}_{m})$      472
$\Lambda_{1}*_{\Lambda}\Lambda_{2}$      98
$\langle\cdot.\cdot\rangle$      298
$\langle\cdot.\cdot\rangle_{adj}$      303
$\langle\cdot.\cdot\rangle_{M}$      67
$\langle\mathcal{P_{1}, P_{2}}\rangle$      96
$\mathbb{CP}^{n}$      291
$\mathbb{C}^{n}$      290
$\mathbb{D}$      328
$\mathbb{E}^{n}$      148
$\mathbb{G}$      328
$\mathbb{H}^{n}$      148
$\mathbb{P}^{n}$      150
$\mathbb{S}^{n}$      148
$\mathcal{C(P)}$      39
$\mathcal{C*D}$      39
$\mathcal{D}_{n}(p, q, r)$      349
$\mathcal{F(P)}$      22
$\mathcal{I}(E)$      149
$\mathcal{J}(h, k; l, m; q)$      352
$\mathcal{K}$-extension      265
$\mathcal{K}$-extension, universal      265
$\mathcal{L^{K, G}}$      247
$\mathcal{L}^{3}_{m, \mathcal{H}}$      471-484
$\mathcal{L}^{3}_{m, \mathcal{H}}$, realizations of      484-490
$\mathcal{L}^{4}_{m, \mathcal{I}}$      490-501
$\mathcal{L}^{n}_{k}$      461 503-509
$\mathcal{L}^{\mathcal{K, G}}$      247-255 264-272
$\mathcal{P(\Gamma)}$      51
$\mathcal{P/\Sigma}$      44
$\mathcal{P}\lozenge\mathcal{Q}$      186
$\mathcal{P}\lozenge_{k}\mathcal{Q}$      116
$\mathcal{P}\searrow\mathcal{Q}$      43
$\mathcal{P}^{*}$      28
$\mathcal{P}_{G}$      129
$\mathcal{Q}/\langle\langle s_{1}, ..., s_{k}\rangle\rangle$      62
$\mathcal{Q}^{\mu}$      192
$\mathcal{T}^{5}_{s}$      450-459
$\mathcal{T}_{3}(p, s; q, r)$      323
$\mathcal{T}_{4}(p, s; q, r; t)$      333
$\mathcal{T}_{4}(q_{1}, ..., q_{4})$      338
$\mathdd{D}_{m}$      490
$\Omega(c)$      313
$\Omega(\mathcal{C})$      313
$\overline{\Gamma}(\mathcal{P})$      28
$\Phi^{j(1)---j(r-1)j(r)}$      25
$\Phi_{J}$      33
$\rtimes$      28
$\tilde{A}_{n}, \tilde{B}_{n}, \tilde{C}_{n}, \tilde{D}_{n}$      73
$\tilde{E}_{6}, \tilde{E}_{7}, \tilde{E}_{8}$      73
$\tilde{F}_{4}$      73
$\Varepsilon_{n}(p, r)$      350
$\vartheta(C)$      313
$\vartheta(\mathcal{C})$      313
$\wr$      257
$\{3, 3, 4, 3\}_{s}$      170
$\{3, 4, 3, 3\}_{s}$      170
$\{3, 6\}_{s}$      19
$\{4, 3^{n-2}, 4\}_{s}$      167
$\{4, 4\}_{s}$      18
$\{6, 3\}_{s}$      19
$\{p, q, r, s\}_{t}$      119
$\{p, q, r\}_{s}$      179
$\{p, q\}$      17
$\{p, q\}_{r, *s}$      200
$\{p, q\}_{r_{1}, ..., r_{k}}$      196
$\{p, q\}_{r}$      18
$\{p, q|h, *k\}$      200
$\{p, q|h\}$      18 196
$\{p, q|h_{2}, ..., h_{k}\}$      196
$\{p, q|h_{2}, ..., h_{k}_{r_{1}, ..., r_{k}}\}$      196
$\{p\}$      11
$\{p_{1}, ..., p_{n-1}\}$      11 30 79
$\{p_{1}, ..., p_{n-1}\}_{h/2}$      163
$\{\frac{p}{d}\}$      16
$\{\infty, 3\}^{(a)}$      224
$\{\infty, 3\}^{(b)}$      224
$\{\infty\}$      15
$\{\mathcal{P_{1}, P_{2}}\}$      97
$\{\}$      11
${1}_\Gamma^{4}_{s}$      364
${1}_\Gamma^{6}_{s}$      460
${2}_\Gamma^{4}_{s,t}$      369 462
${3}_\Gamma^{4}_{s,t}$      467
${6}_\Gamma^{4}_{s,t}$      389
${7}_\Gamma^{4}_{s,t}$      417
${p}_\Gamma^{4}_{s}$      388
${}^2{\mathcal{K, G(s)}}$      256-259
${}^2{\mathcal{K, G(s)}}$, realizations of      261-264
${}^2{\mathcal{K}}$      255-257
${}^2{\mathcal{K}}$, realizations of      259-261
${}^{#}M$      490
${}_{1}\mathcal{T}^{4}_{s}$      363-369 376-377 382-386
${}_{1}\mathcal{T}^{6}_{s}$      459-462
${}_{2}\mathcal{T}^{4}_{s,t}$      369-386
${}_{2}\mathcal{T}^{6}_{s,t}$      462-465
${}_{3}\mathcal{T}^{4}_{s}$      387-389 392-409 423-425 429-430 437-444
${}_{3}\mathcal{T}^{6}_{s,t}$      466-470
${}_{4}\mathcal{T}^{4}_{s}$      387-389 392-409 423-425
${}_{5}\mathcal{T}^{4}_{s}$      387-389 392-409 423-425
${}_{6}\mathcal{T}^{4}_{s,t}$      388-389 400-417 426-429 443
${}_{7}\mathcal{T}^{4}_{s,t}$      417-423
${}_{p}\mathcal{T}^{4}_{s}$      387
120-cell      11
120-cell, hemi-120-cell      163
24-cell      11 100 165 254 450
24-cell, hemi-24-cell      163 502
24-cell, realizations of      138
600-cell      11 212
600-cell, hemi-600-cell      163
Action, discrete      149
Action, free      149
Action, properly discontinuous      149
adj      303
Adjoint, hermitian form      303
Adjoint, matrix      303
Aethelard of Bath      5
Aff S      8
Altshuler, A.      407
Amalgamation, FAP      112-115 119 187
Amalgamation, flat amalgamation property      112-115
Amalgamation, free product with      98
Amalgamation, of polytopes      96-101
angle      85
Angle graph      86
Angle-sum relations      86
Apeirogon      15 25 27
Apeirogon, regular      see Regular apeirogon
Apeirohedron      25
Apeirohedron, regular      see Regular apeirohedron
Apeirotope      25
Apeirotope, $n$-apeirotope      25
Apeirotope, regular      see Regular apeirotope
Archimedes      4
Aristotle      3
Attached (interior mark)      323
Automorphism, group      11 27
Automorphism, of polytope      27
Automorphism, of polytope, convex      11
Barycentric subdivision      40
Basic 2-generator subgroup      300
Basic, operation      308
Bd $K$      142
Bieberbach theorem      143 150 220
Bieberbach theorem, complex analogue      310
Bieberbach, L.      150
Blend of realizations      122 125
Boetius      4
Bokowski, J.      140
Bracho, J.      220
Bradwardine, T.      5 16
Branch      65 84
Branch, improper      65
Branch, of geometric diagram      308
Brehm, U.      140
Brianchon - Gram theorem      86
Buekenhout, F.      7 191
buildings      7 21
C-group      49-60
C-group, string      50
Canonical projection      149
Canonical, bilinear form      67
Canonical, expression for P      132
Canonical, matrix      130
Canonical, projection      44
Cauchy, A. L.      6 16
Cayley - Menger matrix      128
cell      15
Centrally symmetric, abstract polytope      249 255
Centrally symmetric, spherical tessellation      163
Centre      206
Centroid      206
Centroid, of polygon      16
Chain      22
Chain, length of      22
Chain, type of      23
Chamber complex      see Coxeter complex
Chamber, closed      69
Chamber, fundamental      66 69
Chamber, open      69
CHARACTER      131
Character norm      131
Chiral      38
Chiral, map      18
Chiral, polytope      38
Chiral, tessellation      178
Chiral, toroid      177
Circuit criterion      63
Circuit diagram      297 345 351
Circuit diagram, $n$-circuit      351
Circuit diagram, 3-circuit      323
Circuit diagram, 4-circuit      345
Circuit diagram, diagonal-free      315
Circuit diagram, interior mark of      297 323
Circuit diagram, turn of      313
Circuit diagram, with tails      347
Circuit matching      314
class      96
Classification, combinatorial      102
Classification, topological      360
Co-face      23
Co-face, co-$i$-face      23
Cohen, A. M.      21 289
Cone      127
Connected      9 23
Connected, flag-connected      24
Connected, flag-connected, strongly      9 24
Connected, strongly connected      9 23
Contragredient      68 302
Conv $S$      7
Convex hull      7
Convex set      7
Conway, J. H.      170
Covering      43
Covering, $k$-covering      43
Covering, map, between polytopes      43
Covering, map, topological      149
Covering, of polytopes      43
Coxeter complex      42 69
Coxeter complex, euclidean      73
Coxeter complex, finite      71
Coxeter diagram      65 84
Coxeter diagram, euclidean      73
Coxeter diagram, spherical      72
Coxeter diagram, string      66
Coxeter element      163
Coxeter group      10 64-94
Coxeter group, canonical bilinear form      67
Coxeter group, compact hyperbolic      77
Coxeter group, crystallographic      89
Coxeter group, distinguished generators of      65
Coxeter group, distinguished subgroup of      65
Coxeter group, euclidean      71-76
Coxeter group, finite      71 83-94
Coxeter group, hyperbolic      76-78
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