In 6-dimensional geometry, there are 35 uniform polytopes with A6 symmetry. There is one self-dual regular form, the 6-simplex with 7 vertices.

Each can be visualized as symmetric orthographic projections in Coxeter planes of the A6 Coxeter group, and other subgroups.

Graphs

Symmetric orthographic projections of these 35 polytopes can be made in the A6, A5, A4, A3, A2 Coxeter planes. Ak graphs have [k+1] symmetry. For even k and symmetric ringed diagrams, symmetry doubles to [2(k+1)].

These 35 polytopes are each shown in these 5 symmetry planes, with vertices and edges drawn, and vertices colored by the number of overlapping vertices in each projective position.

# A6
[7]
A5
[6]
A4
[5]
A3
[4]
A2
[3]
Coxeter-Dynkin diagram
Schläfli symbol
Name
1
t0{3,3,3,3,3}
6-simplex
Heptapeton (hop)
2
t1{3,3,3,3,3}
Rectified 6-simplex
Rectified heptapeton (ril)
3
t0,1{3,3,3,3,3}
Truncated 6-simplex
Truncated heptapeton (til)
4
t2{3,3,3,3,3}
Birectified 6-simplex
Birectified heptapeton (bril)
5
t0,2{3,3,3,3,3}
Cantellated 6-simplex
Small rhombated heptapeton (sril)
6
t1,2{3,3,3,3,3}
Bitruncated 6-simplex
Bitruncated heptapeton (batal)
7
t0,1,2{3,3,3,3,3}
Cantitruncated 6-simplex
Great rhombated heptapeton (gril)
8
t0,3{3,3,3,3,3}
Runcinated 6-simplex
Small prismated heptapeton (spil)
9
t1,3{3,3,3,3,3}
Bicantellated 6-simplex
Small birhombated heptapeton (sabril)
10
t0,1,3{3,3,3,3,3}
Runcitruncated 6-simplex
Prismatotruncated heptapeton (patal)
11
t2,3{3,3,3,3,3}
Tritruncated 6-simplex
12
t0,2,3{3,3,3,3,3}
Runcicantellated 6-simplex
Prismatorhombated heptapeton (pril)
13
t1,2,3{3,3,3,3,3}
Bicantitruncated 6-simplex
Great birhombated heptapeton (gabril)
14
t0,1,2,3{3,3,3,3,3}
Runcicantitruncated 6-simplex
Great prismated heptapeton (gapil)
15
t0,4{3,3,3,3,3}
Stericated 6-simplex
Small cellated heptapeton (scal)
16
t1,4{3,3,3,3,3}
Biruncinated 6-simplex
17
t0,1,4{3,3,3,3,3}
Steritruncated 6-simplex
cellitruncated heptapeton (catal)
18
t0,2,4{3,3,3,3,3}
Stericantellated 6-simplex
Cellirhombated heptapeton (cral)
19
t1,2,4{3,3,3,3,3}
Biruncitruncated 6-simplex
Biprismatorhombated heptapeton (bapril)
20
t0,1,2,4{3,3,3,3,3}
Stericantitruncated 6-simplex
Celligreatorhombated heptapeton (cagral)
21
t0,3,4{3,3,3,3,3}
Steriruncinated 6-simplex
Celliprismated heptapeton (copal)
22
t0,1,3,4{3,3,3,3,3}
Steriruncitruncated 6-simplex
celliprismatotruncated heptapeton (captal)
23
t0,2,3,4{3,3,3,3,3}
Steriruncicantellated 6-simplex
celliprismatorhombated heptapeton (copril)
24
t1,2,3,4{3,3,3,3,3}
Biruncicantitruncated 6-simplex
25
t0,1,2,3,4{3,3,3,3,3}
Steriruncicantitruncated 6-simplex
Great cellated heptapeton (gacal)
26
t0,5{3,3,3,3,3}
Pentellated 6-simplex
27
t0,1,5{3,3,3,3,3}
Pentitruncated 6-simplex
Tericellated heptapeton (tocal)
28
t0,2,5{3,3,3,3,3}
Penticantellated 6-simplex
Teriprismated heptapeton (tapal)
29
t0,1,2,5{3,3,3,3,3}
Penticantitruncated 6-simplex
Terigreatorhombated heptapeton (togral)
30
t0,1,3,5{3,3,3,3,3}
Pentiruncitruncated 6-simplex
Tericellirhombated heptapeton (tocral)
31
t0,2,3,5{3,3,3,3,3}
Pentiruncicantellated 6-simplex
32
t0,1,2,3,5{3,3,3,3,3}
Pentiruncicantitruncated 6-simplex
Terigreatoprismated heptapeton (tagopal)
33
t0,1,4,5{3,3,3,3,3}
Pentisteritruncated 6-simplex
34
t0,1,2,4,5{3,3,3,3,3}
Pentistericantitruncated 6-simplex
tericelligreatorhombated heptapeton (tacogral)
35
t0,1,2,3,4,5{3,3,3,3,3}
Omnitruncated 6-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. "6D uniform polytopes (polypeta)".

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

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