In 7-dimensional geometry, there are 71 uniform polytopes with A7 symmetry. There is one self-dual regular form, the 7-simplex with 8 vertices.

Each can be visualized as symmetric orthographic projections in Coxeter planes of the A7 Coxeter group, and other subgroups.

Graphs

Symmetric orthographic projections of these 71 polytopes can be made in the A7, A6, A5, A4, A3, A2 Coxeter planes. Ak has [k+1] symmetry. For even k and symmetrically ringed-diagrams, symmetry doubles to [2(k+1)].

These 71 polytopes are each shown in these 6 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
A7
[8]
A6
[7]
A5
[6]
A4
[5]
A3
[4]
A2
[3]
1
t0{3,3,3,3,3,3}
7-simplex
2
t1{3,3,3,3,3,3}
Rectified 7-simplex
3
t2{3,3,3,3,3,3}
Birectified 7-simplex
4
t3{3,3,3,3,3,3}
Trirectified 7-simplex
5
t0,1{3,3,3,3,3,3}
Truncated 7-simplex
6
t0,2{3,3,3,3,3,3}
Cantellated 7-simplex
7
t1,2{3,3,3,3,3,3}
Bitruncated 7-simplex
8
t0,3{3,3,3,3,3,3}
Runcinated 7-simplex
9
t1,3{3,3,3,3,3,3}
Bicantellated 7-simplex
10
t2,3{3,3,3,3,3,3}
Tritruncated 7-simplex
11
t0,4{3,3,3,3,3,3}
Stericated 7-simplex
12
t1,4{3,3,3,3,3,3}
Biruncinated 7-simplex
13
t2,4{3,3,3,3,3,3}
Tricantellated 7-simplex
14
t0,5{3,3,3,3,3,3}
Pentellated 7-simplex
15
t1,5{3,3,3,3,3,3}
Bistericated 7-simplex
16
t0,6{3,3,3,3,3,3}
Hexicated 7-simplex
17
t0,1,2{3,3,3,3,3,3}
Cantitruncated 7-simplex
18
t0,1,3{3,3,3,3,3,3}
Runcitruncated 7-simplex
19
t0,2,3{3,3,3,3,3,3}
Runcicantellated 7-simplex
20
t1,2,3{3,3,3,3,3,3}
Bicantitruncated 7-simplex
21
t0,1,4{3,3,3,3,3,3}
Steritruncated 7-simplex
22
t0,2,4{3,3,3,3,3,3}
Stericantellated 7-simplex
23
t1,2,4{3,3,3,3,3,3}
Biruncitruncated 7-simplex
24
t0,3,4{3,3,3,3,3,3}
Steriruncinated 7-simplex
25
t1,3,4{3,3,3,3,3,3}
Biruncicantellated 7-simplex
26
t2,3,4{3,3,3,3,3,3}
Tricantitruncated 7-simplex
27
t0,1,5{3,3,3,3,3,3}
Pentitruncated 7-simplex
28
t0,2,5{3,3,3,3,3,3}
Penticantellated 7-simplex
29
t1,2,5{3,3,3,3,3,3}
Bisteritruncated 7-simplex
30
t0,3,5{3,3,3,3,3,3}
Pentiruncinated 7-simplex
31
t1,3,5{3,3,3,3,3,3}
Bistericantellated 7-simplex
32
t0,4,5{3,3,3,3,3,3}
Pentistericated 7-simplex
33
t0,1,6{3,3,3,3,3,3}
Hexitruncated 7-simplex
34
t0,2,6{3,3,3,3,3,3}
Hexicantellated 7-simplex
35
t0,3,6{3,3,3,3,3,3}
Hexiruncinated 7-simplex
36
t0,1,2,3{3,3,3,3,3,3}
Runcicantitruncated 7-simplex
37
t0,1,2,4{3,3,3,3,3,3}
Stericantitruncated 7-simplex
38
t0,1,3,4{3,3,3,3,3,3}
Steriruncitruncated 7-simplex
39
t0,2,3,4{3,3,3,3,3,3}
Steriruncicantellated 7-simplex
40
t1,2,3,4{3,3,3,3,3,3}
Biruncicantitruncated 7-simplex
41
t0,1,2,5{3,3,3,3,3,3}
Penticantitruncated 7-simplex
42
t0,1,3,5{3,3,3,3,3,3}
Pentiruncitruncated 7-simplex
43
t0,2,3,5{3,3,3,3,3,3}
Pentiruncicantellated 7-simplex
44
t1,2,3,5{3,3,3,3,3,3}
Bistericantitruncated 7-simplex
45
t0,1,4,5{3,3,3,3,3,3}
Pentisteritruncated 7-simplex
46
t0,2,4,5{3,3,3,3,3,3}
Pentistericantellated 7-simplex
47
t1,2,4,5{3,3,3,3,3,3}
Bisteriruncitruncated 7-simplex
48
t0,3,4,5{3,3,3,3,3,3}
Pentisteriruncinated 7-simplex
49
t0,1,2,6{3,3,3,3,3,3}
Hexicantitruncated 7-simplex
50
t0,1,3,6{3,3,3,3,3,3}
Hexiruncitruncated 7-simplex
51
t0,2,3,6{3,3,3,3,3,3}
Hexiruncicantellated 7-simplex
52
t0,1,4,6{3,3,3,3,3,3}
Hexisteritruncated 7-simplex
53
t0,2,4,6{3,3,3,3,3,3}
Hexistericantellated 7-simplex
54
t0,1,5,6{3,3,3,3,3,3}
Hexipentitruncated 7-simplex
55
t0,1,2,3,4{3,3,3,3,3,3}
Steriruncicantitruncated 7-simplex
56
t0,1,2,3,5{3,3,3,3,3,3}
Pentiruncicantitruncated 7-simplex
57
t0,1,2,4,5{3,3,3,3,3,3}
Pentistericantitruncated 7-simplex
58
t0,1,3,4,5{3,3,3,3,3,3}
Pentisteriruncitruncated 7-simplex
59
t0,2,3,4,5{3,3,3,3,3,3}
Pentisteriruncicantellated 7-simplex
60
t1,2,3,4,5{3,3,3,3,3,3}
Bisteriruncicantitruncated 7-simplex
61
t0,1,2,3,6{3,3,3,3,3,3}
Hexiruncicantitruncated 7-simplex
62
t0,1,2,4,6{3,3,3,3,3,3}
Hexistericantitruncated 7-simplex
63
t0,1,3,4,6{3,3,3,3,3,3}
Hexisteriruncitruncated 7-simplex
64
t0,2,3,4,6{3,3,3,3,3,3}
Hexisteriruncicantellated 7-simplex
65
t0,1,2,5,6{3,3,3,3,3,3}
Hexipenticantitruncated 7-simplex
66
t0,1,3,5,6{3,3,3,3,3,3}
Hexipentiruncitruncated 7-simplex
67
t0,1,2,3,4,5{3,3,3,3,3,3}
Pentisteriruncicantitruncated 7-simplex
68
t0,1,2,3,4,6{3,3,3,3,3,3}
Hexisteriruncicantitruncated 7-simplex
69
t0,1,2,3,5,6{3,3,3,3,3,3}
Hexipentiruncicantitruncated 7-simplex
70
t0,1,2,4,5,6{3,3,3,3,3,3}
Hexipentistericantitruncated 7-simplex
71
t0,1,2,3,4,5,6{3,3,3,3,3,3}
Omnitruncated 7-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. "7D uniform polytopes (polyexa)".

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