# A7 polytope

 7-simplex

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
Error creating thumbnail: Unable to save thumbnail to destination
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
Error creating thumbnail: Unable to save thumbnail to destination
22
t0,2,4{3,3,3,3,3,3}
Stericantellated 7-simplex
Error creating thumbnail: Unable to save thumbnail to destination
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
Error creating thumbnail: Unable to save thumbnail to destination
50
t0,1,3,6{3,3,3,3,3,3}
Hexiruncitruncated 7-simplex
Error creating thumbnail: Unable to save thumbnail to destination
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
Error creating thumbnail: Unable to save thumbnail to destination

## 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
•

## Notes

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 OctahedronCube Demicube DodecahedronIcosahedron
Uniform 4-polytope 5-cell 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