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A8 polytope

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Orthographic projections
A8 Coxeter plane

8-simplex

In 8-dimensional geometry, there are 135 uniform polytopes with A8 symmetry. There is one self-dual regular form, the 8-simplex with 9 vertices.

Each can be visualized as symmetric orthographic projections in Coxeter planes of the A8 Coxeter group, and other subgroups.

Graphs

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Symmetric orthographic projections of these 135 polytopes can be made in the A8, A7, A6, A5, A4, A3, A2 Coxeter planes. Ak has [k+1] symmetry.

These 135 polytopes are each shown in these 7 symmetry planes, with vertices and edges drawn, and vertices colored by the number of overlapping vertices in each projective position.

# Coxeter-Dynkin diagram
Schläfli symbol
Johnson name
Ak orthogonal projection graphs
A8
[9]
A7
[8]
A6
[7]
A5
[6]
A4
[5]
A3
[4]
A2
[3]
1
t0{3,3,3,3,3,3,3}
8-simplex
2
t1{3,3,3,3,3,3,3}
Rectified 8-simplex
3
t2{3,3,3,3,3,3,3}
Birectified 8-simplex
4
t3{3,3,3,3,3,3,3}
Trirectified 8-simplex
5
t0,1{3,3,3,3,3,3,3}
Truncated 8-simplex
6
t0,2{3,3,3,3,3,3,3}
Cantellated 8-simplex
7
t1,2{3,3,3,3,3,3,3}
Bitruncated 8-simplex
8
t0,3{3,3,3,3,3,3,3}
Runcinated 8-simplex
9
t1,3{3,3,3,3,3,3,3}
Bicantellated 8-simplex
10
t2,3{3,3,3,3,3,3,3}
Tritruncated 8-simplex
11
t0,4{3,3,3,3,3,3,3}
Stericated 8-simplex
12
t1,4{3,3,3,3,3,3,3}
Biruncinated 8-simplex
13
t2,4{3,3,3,3,3,3,3}
Tricantellated 8-simplex
14
t3,4{3,3,3,3,3,3,3}
Quadritruncated 8-simplex
15
t0,5{3,3,3,3,3,3,3}
Pentellated 8-simplex
16
t1,5{3,3,3,3,3,3,3}
Bistericated 8-simplex
17
t2,5{3,3,3,3,3,3,3}
Triruncinated 8-simplex
18
t0,6{3,3,3,3,3,3,3}
Hexicated 8-simplex
19
t1,6{3,3,3,3,3,3,3}
Bipentellated 8-simplex
20
t0,7{3,3,3,3,3,3,3}
Heptellated 8-simplex
21
t0,1,2{3,3,3,3,3,3,3}
Cantitruncated 8-simplex
22
t0,1,3{3,3,3,3,3,3,3}
Runcitruncated 8-simplex
23
t0,2,3{3,3,3,3,3,3,3}
Runcicantellated 8-simplex
24
t1,2,3{3,3,3,3,3,3,3}
Bicantitruncated 8-simplex
25
t0,1,4{3,3,3,3,3,3,3}
Steritruncated 8-simplex
26
t0,2,4{3,3,3,3,3,3,3}
Stericantellated 8-simplex
27
t1,2,4{3,3,3,3,3,3,3}
Biruncitruncated 8-simplex
28
t0,3,4{3,3,3,3,3,3,3}
Steriruncinated 8-simplex
29
t1,3,4{3,3,3,3,3,3,3}
Biruncicantellated 8-simplex
30
t2,3,4{3,3,3,3,3,3,3}
Tricantitruncated 8-simplex
31
t0,1,5{3,3,3,3,3,3,3}
Pentitruncated 8-simplex
32
t0,2,5{3,3,3,3,3,3,3}
Penticantellated 8-simplex
33
t1,2,5{3,3,3,3,3,3,3}
Bisteritruncated 8-simplex
34
t0,3,5{3,3,3,3,3,3,3}
Pentiruncinated 8-simplex
35
t1,3,5{3,3,3,3,3,3,3}
Bistericantellated 8-simplex
36
t2,3,5{3,3,3,3,3,3,3}
Triruncitruncated 8-simplex
37
t0,4,5{3,3,3,3,3,3,3}
Pentistericated 8-simplex
38
t1,4,5{3,3,3,3,3,3,3}
Bisteriruncinated 8-simplex
39
t0,1,6{3,3,3,3,3,3,3}
Hexitruncated 8-simplex
40
t0,2,6{3,3,3,3,3,3,3}
Hexicantellated 8-simplex
41
t1,2,6{3,3,3,3,3,3,3}
Bipentitruncated 8-simplex
42
t0,3,6{3,3,3,3,3,3,3}
Hexiruncinated 8-simplex
43
t1,3,6{3,3,3,3,3,3,3}
Bipenticantellated 8-simplex
44
t0,4,6{3,3,3,3,3,3,3}
Hexistericated 8-simplex
45
t0,5,6{3,3,3,3,3,3,3}
Hexipentellated 8-simplex
46
t0,1,7{3,3,3,3,3,3,3}
Heptitruncated 8-simplex
47
t0,2,7{3,3,3,3,3,3,3}
Hepticantellated 8-simplex
48
t0,3,7{3,3,3,3,3,3,3}
Heptiruncinated 8-simplex
49
t0,1,2,3{3,3,3,3,3,3,3}
Runcicantitruncated 8-simplex
50
t0,1,2,4{3,3,3,3,3,3,3}
Stericantitruncated 8-simplex
51
t0,1,3,4{3,3,3,3,3,3,3}
Steriruncitruncated 8-simplex
52
t0,2,3,4{3,3,3,3,3,3,3}
Steriruncicantellated 8-simplex
53
t1,2,3,4{3,3,3,3,3,3,3}
Biruncicantitruncated 8-simplex
54
t0,1,2,5{3,3,3,3,3,3,3}
Penticantitruncated 8-simplex
55
t0,1,3,5{3,3,3,3,3,3,3}
Pentiruncitruncated 8-simplex
56
t0,2,3,5{3,3,3,3,3,3,3}
Pentiruncicantellated 8-simplex
57
t1,2,3,5{3,3,3,3,3,3,3}
Bistericantitruncated 8-simplex
58
t0,1,4,5{3,3,3,3,3,3,3}
Pentisteritruncated 8-simplex
59
t0,2,4,5{3,3,3,3,3,3,3}
Pentistericantellated 8-simplex
60
t1,2,4,5{3,3,3,3,3,3,3}
Bisteriruncitruncated 8-simplex
61
t0,3,4,5{3,3,3,3,3,3,3}
Pentisteriruncinated 8-simplex
62
t1,3,4,5{3,3,3,3,3,3,3}
Bisteriruncicantellated 8-simplex
63
t2,3,4,5{3,3,3,3,3,3,3}
Triruncicantitruncated 8-simplex
64
t0,1,2,6{3,3,3,3,3,3,3}
Hexicantitruncated 8-simplex
65
t0,1,3,6{3,3,3,3,3,3,3}
Hexiruncitruncated 8-simplex
66
t0,2,3,6{3,3,3,3,3,3,3}
Hexiruncicantellated 8-simplex
67
t1,2,3,6{3,3,3,3,3,3,3}
Bipenticantitruncated 8-simplex
68
t0,1,4,6{3,3,3,3,3,3,3}
Hexisteritruncated 8-simplex
69
t0,2,4,6{3,3,3,3,3,3,3}
Hexistericantellated 8-simplex
70
t1,2,4,6{3,3,3,3,3,3,3}
Bipentiruncitruncated 8-simplex
71
t0,3,4,6{3,3,3,3,3,3,3}
Hexisteriruncinated 8-simplex
72
t1,3,4,6{3,3,3,3,3,3,3}
Bipentiruncicantellated 8-simplex
73
t0,1,5,6{3,3,3,3,3,3,3}
Hexipentitruncated 8-simplex
74
t0,2,5,6{3,3,3,3,3,3,3}
Hexipenticantellated 8-simplex
75
t1,2,5,6{3,3,3,3,3,3,3}
Bipentisteritruncated 8-simplex
76
t0,3,5,6{3,3,3,3,3,3,3}
Hexipentiruncinated 8-simplex
77
t0,4,5,6{3,3,3,3,3,3,3}
Hexipentistericated 8-simplex
78
t0,1,2,7{3,3,3,3,3,3,3}
Hepticantitruncated 8-simplex
79
t0,1,3,7{3,3,3,3,3,3,3}
Heptiruncitruncated 8-simplex
80
t0,2,3,7{3,3,3,3,3,3,3}
Heptiruncicantellated 8-simplex
81
t0,1,4,7{3,3,3,3,3,3,3}
Heptisteritruncated 8-simplex
82
t0,2,4,7{3,3,3,3,3,3,3}
Heptistericantellated 8-simplex
83
t0,3,4,7{3,3,3,3,3,3,3}
Heptisteriruncinated 8-simplex
84
t0,1,5,7{3,3,3,3,3,3,3}
Heptipentitruncated 8-simplex
85
t0,2,5,7{3,3,3,3,3,3,3}
Heptipenticantellated 8-simplex
86
t0,1,6,7{3,3,3,3,3,3,3}
Heptihexitruncated 8-simplex
87
t0,1,2,3,4{3,3,3,3,3,3,3}
Steriruncicantitruncated 8-simplex
88
t0,1,2,3,5{3,3,3,3,3,3,3}
Pentiruncicantitruncated 8-simplex
89
t0,1,2,4,5{3,3,3,3,3,3,3}
Pentistericantitruncated 8-simplex
90
t0,1,3,4,5{3,3,3,3,3,3,3}
Pentisteriruncitruncated 8-simplex
91
t0,2,3,4,5{3,3,3,3,3,3,3}
Pentisteriruncicantellated 8-simplex
92
t1,2,3,4,5{3,3,3,3,3,3,3}
Bisteriruncicantitruncated 8-simplex
93
t0,1,2,3,6{3,3,3,3,3,3,3}
Hexiruncicantitruncated 8-simplex
94
t0,1,2,4,6{3,3,3,3,3,3,3}
Hexistericantitruncated 8-simplex
95
t0,1,3,4,6{3,3,3,3,3,3,3}
Hexisteriruncitruncated 8-simplex
96
t0,2,3,4,6{3,3,3,3,3,3,3}
Hexisteriruncicantellated 8-simplex
97
t1,2,3,4,6{3,3,3,3,3,3,3}
Bipentiruncicantitruncated 8-simplex
98
t0,1,2,5,6{3,3,3,3,3,3,3}
Hexipenticantitruncated 8-simplex
99
t0,1,3,5,6{3,3,3,3,3,3,3}
Hexipentiruncitruncated 8-simplex
100
t0,2,3,5,6{3,3,3,3,3,3,3}
Hexipentiruncicantellated 8-simplex
101
t1,2,3,5,6{3,3,3,3,3,3,3}
Bipentistericantitruncated 8-simplex
102
t0,1,4,5,6{3,3,3,3,3,3,3}
Hexipentisteritruncated 8-simplex
103
t0,2,4,5,6{3,3,3,3,3,3,3}
Hexipentistericantellated 8-simplex
104
t0,3,4,5,6{3,3,3,3,3,3,3}
Hexipentisteriruncinated 8-simplex
105
t0,1,2,3,7{3,3,3,3,3,3,3}
Heptiruncicantitruncated 8-simplex
106
t0,1,2,4,7{3,3,3,3,3,3,3}
Heptistericantitruncated 8-simplex
107
t0,1,3,4,7{3,3,3,3,3,3,3}
Heptisteriruncitruncated 8-simplex
108
t0,2,3,4,7{3,3,3,3,3,3,3}
Heptisteriruncicantellated 8-simplex
109
t0,1,2,5,7{3,3,3,3,3,3,3}
Heptipenticantitruncated 8-simplex
110
t0,1,3,5,7{3,3,3,3,3,3,3}
Heptipentiruncitruncated 8-simplex
111
t0,2,3,5,7{3,3,3,3,3,3,3}
Heptipentiruncicantellated 8-simplex
112
t0,1,4,5,7{3,3,3,3,3,3,3}
Heptipentisteritruncated 8-simplex
113
t0,1,2,6,7{3,3,3,3,3,3,3}
Heptihexicantitruncated 8-simplex
114
t0,1,3,6,7{3,3,3,3,3,3,3}
Heptihexiruncitruncated 8-simplex
115
t0,1,2,3,4,5{3,3,3,3,3,3,3}
Pentisteriruncicantitruncated 8-simplex
116
t0,1,2,3,4,6{3,3,3,3,3,3,3}
Hexisteriruncicantitruncated 8-simplex
117
t0,1,2,3,5,6{3,3,3,3,3,3,3}
Hexipentiruncicantitruncated 8-simplex
118
t0,1,2,4,5,6{3,3,3,3,3,3,3}
Hexipentistericantitruncated 8-simplex
119
t0,1,3,4,5,6{3,3,3,3,3,3,3}
Hexipentisteriruncitruncated 8-simplex
120
t0,2,3,4,5,6{3,3,3,3,3,3,3}
Hexipentisteriruncicantellated 8-simplex
121
t1,2,3,4,5,6{3,3,3,3,3,3,3}
Bipentisteriruncicantitruncated 8-simplex
122
t0,1,2,3,4,7{3,3,3,3,3,3,3}
Heptisteriruncicantitruncated 8-simplex
123
t0,1,2,3,5,7{3,3,3,3,3,3,3}
Heptipentiruncicantitruncated 8-simplex
124
t0,1,2,4,5,7{3,3,3,3,3,3,3}
Heptipentistericantitruncated 8-simplex
125
t0,1,3,4,5,7{3,3,3,3,3,3,3}
Heptipentisteriruncitruncated 8-simplex
126
t0,2,3,4,5,7{3,3,3,3,3,3,3}
Heptipentisteriruncicantellated 8-simplex
127
t0,1,2,3,6,7{3,3,3,3,3,3,3}
Heptihexiruncicantitruncated 8-simplex
128
t0,1,2,4,6,7{3,3,3,3,3,3,3}
Heptihexistericantitruncated 8-simplex
129
t0,1,3,4,6,7{3,3,3,3,3,3,3}
Heptihexisteriruncitruncated 8-simplex
130
t0,1,2,5,6,7{3,3,3,3,3,3,3}
Heptihexipenticantitruncated 8-simplex
131
t0,1,2,3,4,5,6{3,3,3,3,3,3,3}
Hexipentisteriruncicantitruncated 8-simplex
132
t0,1,2,3,4,5,7{3,3,3,3,3,3,3}
Heptipentisteriruncicantitruncated 8-simplex
133
t0,1,2,3,4,6,7{3,3,3,3,3,3,3}
Heptihexisteriruncicantitruncated 8-simplex
134
t0,1,2,3,5,6,7{3,3,3,3,3,3,3}
Heptihexipentiruncicantitruncated 8-simplex
135
t0,1,2,3,4,5,6,7{3,3,3,3,3,3,3}
Omnitruncated 8-simplex

References

[edit]
  • H.S.M. Coxeter:
    • H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973
  • Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6 [1]
    • (Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380–407, MR 2,10]
    • (Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559-591]
    • (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
  • N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. Dissertation, University of Toronto, 1966
[edit]

Notes

[edit]
Family An Bn I2(p) / Dn E6 / E7 / E8 / F4 / G2 Hn
Regular polygon Triangle Square p-gon Hexagon Pentagon
Uniform polyhedron Tetrahedron OctahedronCube Demicube DodecahedronIcosahedron
Uniform polychoron Pentachoron 16-cellTesseract Demitesseract 24-cell 120-cell600-cell
Uniform 5-polytope 5-simplex 5-orthoplex5-cube 5-demicube
Uniform 6-polytope 6-simplex 6-orthoplex6-cube 6-demicube 122221
Uniform 7-polytope 7-simplex 7-orthoplex7-cube 7-demicube 132231321
Uniform 8-polytope 8-simplex 8-orthoplex8-cube 8-demicube 142241421
Uniform 9-polytope 9-simplex 9-orthoplex9-cube 9-demicube
Uniform 10-polytope 10-simplex 10-orthoplex10-cube 10-demicube
Uniform n-polytope n-simplex n-orthoplexn-cube n-demicube 1k22k1k21 n-pentagonal polytope
Topics: Polytope familiesRegular polytopeList of regular polytopes and compounds