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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

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}
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

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

Klitzing, Richard. "8D uniform polytopes (polyzetta)".

Notes

Wiley::Kaleidoscopes: Selected Writings of H.S.M. Coxeter

Fundamental convex regular and uniform polytopes in dimensions 2–10
Family An Bn I2(p) / Dn E6 / E7 / E8 / F4 / G2 Hn
Regular polygon Triangle Square p-gon Hexagon Pentagon
Uniform polyhedron Tetrahedron Octahedron • Cube Demicube Dodecahedron • Icosahedron
Uniform 4-polytope 5-cell 16-cell • Tesseract Demitesseract 24-cell 120-cell • 600-cell
Uniform 5-polytope 5-simplex 5-orthoplex • 5-cube 5-demicube
Uniform 6-polytope 6-simplex 6-orthoplex • 6-cube 6-demicube 122 • 221
Uniform 7-polytope 7-simplex 7-orthoplex • 7-cube 7-demicube 132 • 231 • 321
Uniform 8-polytope 8-simplex 8-orthoplex • 8-cube 8-demicube 142 • 241 • 421
Uniform 9-polytope 9-simplex 9-orthoplex • 9-cube 9-demicube
Uniform 10-polytope 10-simplex 10-orthoplex • 10-cube 10-demicube
Uniform n-polytope n-simplex n-orthoplex • n-cube n-demicube 1k2 • 2k1 • k21 n-pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds

Mathematics Encyclopedia