A6 polytope

Orthographic projections
A6 Coxeter plane
6-simplex

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.

## Contents

* 1 Graphs
* 2 References
* 3 External links
* 4 Notes

## 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

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1 |  | 80px | 80px | 80px | 80px |
t0{3,3,3,3,3}
6-simplex
Heptapeton (hop)
2 |  | 80px | 80px | 80px |  |
t1{3,3,3,3,3}
Rectified 6-simplex
Rectified heptapeton (ril)
3 |  | 80px | 80px | 80px |  |
t0,1{3,3,3,3,3}
Truncated 6-simplex
Truncated heptapeton (til)
4 |  | 80px | 80px | 80px |  |
t2{3,3,3,3,3}
Birectified 6-simplex
Birectified heptapeton (bril)
5 |  | 80px | 80px | 80px |  |
t0,2{3,3,3,3,3}
Cantellated 6-simplex
Small rhombated heptapeton (sril)
6 |  | 80px | 80px | 80px |  |
t1,2{3,3,3,3,3}
Bitruncated 6-simplex
Bitruncated heptapeton (batal)
7 |  | 80px | 80px | 80px |  |
t0,1,2{3,3,3,3,3}
Cantitruncated 6-simplex
Great rhombated heptapeton (gril)
8 |  | 80px | 80px | 80px |  |
t0,3{3,3,3,3,3}
Runcinated 6-simplex
Small prismated heptapeton (spil)
9 |  | 80px | 80px | 80px |  |
t1,3{3,3,3,3,3}
Bicantellated 6-simplex
Small birhombated heptapeton (sabril)
10 |  | 80px | 80px | 80px |  |
t0,1,3{3,3,3,3,3}
Runcitruncated 6-simplex
Prismatotruncated heptapeton (patal)
11 |  | 80px | 80px | 80px |  |
t2,3{3,3,3,3,3}
Tritruncated 6-simplex
Tetradecapeton (fe)
12 |  | 80px | 80px | 80px |  |
t0,2,3{3,3,3,3,3}
Runcicantellated 6-simplex
Prismatorhombated heptapeton (pril)
13 |  | 80px | 80px | 80px |  |
t1,2,3{3,3,3,3,3}
Bicantitruncated 6-simplex
Great birhombated heptapeton (gabril)
14 |  | 80px | 80px | 80px |  |
t0,1,2,3{3,3,3,3,3}
Runcicantitruncated 6-simplex
Great prismated heptapeton (gapil)
15 |  | 80px | 80px | 80px |  |
t0,4{3,3,3,3,3}
Stericated 6-simplex
Small cellated heptapeton (scal)
16 |  | 80px | 80px | 80px |  |
t1,4{3,3,3,3,3}
Biruncinated 6-simplex
Small biprismato-tetradecapeton (sibpof)
17 |  | 80px | 80px | 80px |  |
t0,1,4{3,3,3,3,3}
Steritruncated 6-simplex
cellitruncated heptapeton (catal)
18 |  | 80px | 80px | 80px |  |
t0,2,4{3,3,3,3,3}
Stericantellated 6-simplex
Cellirhombated heptapeton (cral)
19 |  | 80px | 80px | 80px |  |
t1,2,4{3,3,3,3,3}
Biruncitruncated 6-simplex
Biprismatorhombated heptapeton (bapril)
20 |  | 80px | 80px | 80px |  |
t0,1,2,4{3,3,3,3,3}
Stericantitruncated 6-simplex
Celligreatorhombated heptapeton (cagral)
21 |  | 80px | 80px | 80px |  |
t0,3,4{3,3,3,3,3}
Steriruncinated 6-simplex
Celliprismated heptapeton (copal)
22 |  | 80px | 80px | 80px |  |
t0,1,3,4{3,3,3,3,3}
Steriruncitruncated 6-simplex
celliprismatotruncated heptapeton (captal)
23 |  | 80px | 80px | 80px |  |
t0,2,3,4{3,3,3,3,3}
Steriruncicantellated 6-simplex
celliprismatorhombated heptapeton (copril)
24 |  | 80px | 80px | 80px |  |
t1,2,3,4{3,3,3,3,3}
Biruncicantitruncated 6-simplex
Great biprismato-tetradecapeton (gibpof)
25 |  | 80px | 80px | 80px |  |
t0,1,2,3,4{3,3,3,3,3}
Steriruncicantitruncated 6-simplex
Great cellated heptapeton (gacal)
26 |  | 80px | 80px | 80px |  |
t0,5{3,3,3,3,3}
Pentellated 6-simplex
Small teri-tetradecapeton (staf)
27 |  | 80px | 80px | 80px |  |
t0,1,5{3,3,3,3,3}
Pentitruncated 6-simplex
Tericellated heptapeton (tocal)
28 |  | 80px | 80px | 80px |  |
t0,2,5{3,3,3,3,3}
Penticantellated 6-simplex
Teriprismated heptapeton (tapal)
29 |  | 80px | 80px | 80px |  |
t0,1,2,5{3,3,3,3,3}
Penticantitruncated 6-simplex
Terigreatorhombated heptapeton (togral)
30 |  | 80px | 80px | 80px |  |
t0,1,3,5{3,3,3,3,3}
Pentiruncitruncated 6-simplex
Tericellirhombated heptapeton (tocral)
31 |  | 80px | 80px | 80px |  |
t0,2,3,5{3,3,3,3,3}
Pentiruncicantellated 6-simplex
Teriprismatorhombi-tetradecapeton (taporf)
32 |  | 80px | 80px | 80px |  |
t0,1,2,3,5{3,3,3,3,3}
Pentiruncicantitruncated 6-simplex
Terigreatoprismated heptapeton (tagopal)
33 |  | 80px | 80px | 80px |  |
t0,1,4,5{3,3,3,3,3}
Pentisteritruncated 6-simplex
tericellitrunki-tetradecapeton (tactaf)
34 |  | 80px | 80px | 80px |  |
t0,1,2,4,5{3,3,3,3,3}
Pentistericantitruncated 6-simplex
tericelligreatorhombated heptapeton (tacogral)
35 |  | 80px | 80px | 80px |  |
t0,1,2,3,4,5{3,3,3,3,3}
Omnitruncated 6-simplex
Great teri-tetradecapeton (gotaf)

## 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

## External links

* Klitzing, Richard. "6D uniform polytopes (polypeta)". https://bendwavy.org/klitzing/dimensions/polypeta.htm.

## Notes

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

* v
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* e

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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