Regular complex polytopes:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Cambridge [u.a.]
Cambridge Univ. Press
1991
|
| Ausgabe: | 2. ed. |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | XIV, 210 S. graph. Darst. |
| ISBN: | 0521394902 |
Internformat
MARC
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| 100 | 1 | |a Coxeter, Harold S. M. |d 1907-2003 |e Verfasser |0 (DE-588)118522507 |4 aut | |
| 245 | 1 | 0 | |a Regular complex polytopes |c H. S. M. Coxeter |
| 250 | |a 2. ed. | ||
| 264 | 1 | |a Cambridge [u.a.] |b Cambridge Univ. Press |c 1991 | |
| 300 | |a XIV, 210 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 650 | 4 | |a Polytopes | |
| 650 | 4 | |a Polytopes | |
| 650 | 0 | 7 | |a Regelmäßiges Polytop |0 (DE-588)4177373-1 |2 gnd |9 rswk-swf |
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Datensatz im Suchindex
| _version_ | 1805083573668020224 |
|---|---|
| adam_text |
Contents
Frontispiece page ii
Preface to the First Edition xi
Preface to the Second Edition xiii
1. Regular polygons
i i Isometries 1
i 2 The cyclic and dihedral groups 1
i "3 The theorem of Leonardo da Vinci 3
1 4 The product of two involutory isometries 3
1*5 Regular polygons in n dimensions 3
i 6 Straight and circular polygons 4
1*7 Zigzags and antiprismatic polygons 6
i 8 Helical polygons 7
1 9 Remarks 8
2. Regular polyhedra
21 Spherical tessellations 9
2 2 Flags and Petrie polygons 12
2 3 Reflection groups and rotation groups 14
2 4 Wythoff's construction 16
2 5 The Schwarz triangles 19
2 6 Remarks 20
3. Polyhedral kaleidoscopes
3 1 The characteristic orthoscheme 21
3 2 The icosahedral kaleidoscope 23
3 3 Cayley diagrams and presentations 24
3 4 Finite groups generated by half turns 26
3 5 Remarks 28
4. Real four space and the unitary plane
4 1 Spherical honeycombs 29
4 2 The crystallographic regular polytopes 30
4 3 Flags and orthoschemes 32
4 4 The spherical torus 35
4 5 Double prisms 37
4 6 The 6oo cell and the 120 cell 38
4 7 The ten star poly topes 46
4 8 A family of regular complex polygons 46
4 9 Remarks 52
5. Frieze patterns
5 1 Some examples 55
5 2 Proof of the periodicity 56
5 3 Ptolemaic patterns 57
5 4 Real polytopes in four dimensions 57
5 5 Different patterns for the same poly tope 60
5 6 Patterns of order 6 and period 3 60
5 7 Real polytopes in n dimensions 62
5 8 Remarks 63
6. The geometry of quaternions
6 1 Pairs of complex numbers 64
6 2 Quaternions of real numbers 65
6 3 Reflections 66
6 4 Rotations 66
6 5 Finite groups of quaternions 67
6 6 Generators for (p, q, 2) 68
6 7 Screws in Euclidean 4 space 69
6 8 Rotatory reflections 72
6 9 Remarks 72
7. The binary polyhedral groups
7 1 The cyclic and dicyclic groups 74
7 2 The binary tetrahedral group 75
7 3 The binary octahedral group 77
7 4 The binary icosahedral group 78
7 5 Finite groups generated by pure quaternions 78
7 6 Representation by matrices 80
7 7 The unimodular group 81
7 8 A representation using residues modulo h + i 81
7 9 Remarks 82
vii
Contents
8. Unitary space
8 i Affine coordinates 83
8 2 Hermitian forms 83
8 3 Inner products 84
8 4 Lengths and angles 84
8 5 Unitary transformations 85
8 6 Dual bases 86
87 Reflections 87
8 8 A complex kaleidoscope 88
8 9 The two dimensional case 88
9. The unitary plane, using quaternions
9 1 Unitary groups 89
9 2 A combination of cyclic groups 90
9 3 An extension of the binary polyhedral groups 90
9 4 Reflections 91
9 5 Groups generated by involutory reflections 91
9 6 Other groups generated by three reflections 93
9 7 Two generator subgroups 93
9 8 The group px [q] p2 and its invariant Hermitian form 94
9 9 Remarks 96
10. The complete enumeration of finite reflection
groups in the unitary plane
io 1 The finite unitary groups in the plane 98
10 2 Reflection groups of type 1 98
10 3 Reflection groups of types 2 and 3 99
10 4 Reflection groups of types 3' and 4 99
10 5 Reflection groups of type 5 100
io 6 Reflection groups of type 6 100
10 7 Reflection groups of type 7 101
io 8 Reflection groups of type 8 101
10 9 Reflection groups of type 9 101
11. Regular complex polygons and Cayley diagrams
hi Regular complex polygons 103
11 2 Real representations 104
11 3 Petrie polygons 105
11 4 Some useful subgroups of p [zq] r 106
11 5 Cayley diagrams for reflection groups 108
11 6 Apeirogons in
viii
117 A general treatment for the binary polyhedral groups 112
11 8 Remarks 113
12. Regular complex polytopes defined and described
12 1 Definitions 115
12 2 Hermitian forms 117
12 3 The Hessian polyhedron 119
12 4 Other complex polyhedra 124
12 5 The Witting polytope 132
12 6 The honeycomb of Witting polytopes 135
12 7 Cartesian products of apeirogons 135
12 8 Cycles of honeycombs 136
12 9 Remarks 140
13. The regular complex polytopes and their
symmetry groups
13 1 The regular polytopes and their van Oss polygons 141
13 2 The regular honeycombs 144
13 3 Cycles and frieze patterns 146
13 4 Presenting the symmetry groups 147
13 5 A historical digression 149
13 6 Petrie polygons and exponents 150
13 7 Numerical properties of the non starry polytopes 153
13 8 Presenting the collineation groups 154
139 Invariants 154
14. Almost regular polytopes
14 1 A complex polyhedron with zp2 triangular faces 156
142 Other complex polyhedra with triangular faces 158
14 3 The exponents 159
14 4 The groups G3a"aq 160
145 The Kleinian polyhedron with 112 triangular faces 161
14 6 McMullen's two polyhedra with 84 square faces 166
14 7 Unitary 4 space 172
14 8 Epilogue 174
Tables
Table i The Schwarz triangles 176
Table ii The finite groups generated by two reflections 176
Table hi The two dimensional reflection groups and their
reflection subgroups 177
Tablk iv The regular polygons 178
Table v The non starry polyhedra and four dimensional
polytopes 180
Table vi The regular honeycombs 181
Contents
Answers l82
References 203
Index 207
ix |
| any_adam_object | 1 |
| author | Coxeter, Harold S. M. 1907-2003 |
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| author_facet | Coxeter, Harold S. M. 1907-2003 |
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| author_sort | Coxeter, Harold S. M. 1907-2003 |
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| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 516 - Geometry |
| dewey-raw | 516.3/6 |
| dewey-search | 516.3/6 |
| dewey-sort | 3516.3 16 |
| dewey-tens | 510 - Mathematics |
| discipline | Physik Mathematik |
| edition | 2. ed. |
| format | Book |
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| id | DE-604.BV004812075 |
| illustrated | Illustrated |
| indexdate | 2024-07-20T07:50:43Z |
| institution | BVB |
| isbn | 0521394902 |
| language | English |
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| physical | XIV, 210 S. graph. Darst. |
| publishDate | 1991 |
| publishDateSearch | 1991 |
| publishDateSort | 1991 |
| publisher | Cambridge Univ. Press |
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| spelling | Coxeter, Harold S. M. 1907-2003 Verfasser (DE-588)118522507 aut Regular complex polytopes H. S. M. Coxeter 2. ed. Cambridge [u.a.] Cambridge Univ. Press 1991 XIV, 210 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Polytopes Regelmäßiges Polytop (DE-588)4177373-1 gnd rswk-swf Regelmäßiges Polytop (DE-588)4177373-1 s DE-604 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=002961327&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Coxeter, Harold S. M. 1907-2003 Regular complex polytopes Polytopes Regelmäßiges Polytop (DE-588)4177373-1 gnd |
| subject_GND | (DE-588)4177373-1 |
| title | Regular complex polytopes |
| title_auth | Regular complex polytopes |
| title_exact_search | Regular complex polytopes |
| title_full | Regular complex polytopes H. S. M. Coxeter |
| title_fullStr | Regular complex polytopes H. S. M. Coxeter |
| title_full_unstemmed | Regular complex polytopes H. S. M. Coxeter |
| title_short | Regular complex polytopes |
| title_sort | regular complex polytopes |
| topic | Polytopes Regelmäßiges Polytop (DE-588)4177373-1 gnd |
| topic_facet | Polytopes Regelmäßiges Polytop |
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