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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 6-simplex t0.svg 6-simplex t0 A5.svg 6-simplex t0 A4.svg 6-simplex t0 A3.svg 6-simplex t0 A2.svg CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
t0{3,3,3,3,3}
6-simplex
Heptapeton (hop)
2 6-simplex t1.svg 6-simplex t1 A5.svg 6-simplex t1 A4.svg 6-simplex t1 A3.svg 6-simplex t1 A2.svg CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
t1{3,3,3,3,3}
Rectified 6-simplex
Rectified heptapeton (ril)
3 6-simplex t01.svg 6-simplex t01 A5.svg 6-simplex t01 A4.svg 6-simplex t01 A3.svg 6-simplex t01 A2.svg CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
t0,1{3,3,3,3,3}
Truncated 6-simplex
Truncated heptapeton (til)
4 6-simplex t2.svg 6-simplex t2 A5.svg 6-simplex t2 A4.svg 6-simplex t2 A3.svg 6-simplex t2 A2.svg CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
t2{3,3,3,3,3}
Birectified 6-simplex
Birectified heptapeton (bril)
5 6-simplex t02.svg 6-simplex t02 A5.svg 6-simplex t02 A4.svg 6-simplex t02 A3.svg 6-simplex t02 A2.svg CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
t0,2{3,3,3,3,3}
Cantellated 6-simplex
Small rhombated heptapeton (sril)
6 6-simplex t12.svg 6-simplex t12 A5.svg 6-simplex t12 A4.svg 6-simplex t12 A3.svg 6-simplex t12 A2.svg CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
t1,2{3,3,3,3,3}
Bitruncated 6-simplex
Bitruncated heptapeton (batal)
7 6-simplex t012.svg 6-simplex t012 A5.svg 6-simplex t012 A4.svg 6-simplex t012 A3.svg 6-simplex t012 A2.svg CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
t0,1,2{3,3,3,3,3}
Cantitruncated 6-simplex
Great rhombated heptapeton (gril)
8 6-simplex t03.svg 6-simplex t03 A5.svg 6-simplex t03 A4.svg 6-simplex t03 A3.svg 6-simplex t03 A2.svg CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
t0,3{3,3,3,3,3}
Runcinated 6-simplex
Small prismated heptapeton (spil)
9 6-simplex t13.svg 6-simplex t13 A5.svg 6-simplex t13 A4.svg 6-simplex t13 A3.svg 6-simplex t13 A2.svg CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
t1,3{3,3,3,3,3}
Bicantellated 6-simplex
Small birhombated heptapeton (sabril)
10 6-simplex t013.svg 6-simplex t013 A5.svg 6-simplex t013 A4.svg 6-simplex t013 A3.svg 6-simplex t013 A2.svg CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
t0,1,3{3,3,3,3,3}
Runcitruncated 6-simplex
Prismatotruncated heptapeton (patal)
11 6-simplex t23.svg 6-simplex t23 A5.svg 6-simplex t23 A4.svg 6-simplex t23 A3.svg 6-simplex t23 A2.svg CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
t2,3{3,3,3,3,3}
Tritruncated 6-simplex
Tetradecapeton (fe)
12 6-simplex t023.svg 6-simplex t023 A5.svg 6-simplex t023 A4.svg 6-simplex t023 A3.svg 6-simplex t023 A2.svg CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
t0,2,3{3,3,3,3,3}
Runcicantellated 6-simplex
Prismatorhombated heptapeton (pril)
13 6-simplex t123.svg 6-simplex t123 A5.svg 6-simplex t123 A4.svg 6-simplex t123 A3.svg 6-simplex t123 A2.svg CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
t1,2,3{3,3,3,3,3}
Bicantitruncated 6-simplex
Great birhombated heptapeton (gabril)
14 6-simplex t0123.svg 6-simplex t0123 A5.svg 6-simplex t0123 A4.svg 6-simplex t0123 A3.svg 6-simplex t0123 A2.svg CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
t0,1,2,3{3,3,3,3,3}
Runcicantitruncated 6-simplex
Great prismated heptapeton (gapil)
15 6-simplex t04.svg 6-simplex t04 A5.svg 6-simplex t04 A4.svg 6-simplex t04 A3.svg 6-simplex t04 A2.svg CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
t0,4{3,3,3,3,3}
Stericated 6-simplex
Small cellated heptapeton (scal)
16 6-simplex t14.svg 6-simplex t14 A5.svg 6-simplex t14 A4.svg 6-simplex t14 A3.svg 6-simplex t14 A2.svg CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
t1,4{3,3,3,3,3}
Biruncinated 6-simplex
Small biprismato-tetradecapeton (sibpof)
17 6-simplex t014.svg 6-simplex t014 A5.svg 6-simplex t014 A4.svg 6-simplex t014 A3.svg 6-simplex t014 A2.svg CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
t0,1,4{3,3,3,3,3}
Steritruncated 6-simplex
cellitruncated heptapeton (catal)
18 6-simplex t024.svg 6-simplex t024 A5.svg 6-simplex t024 A4.svg 6-simplex t024 A3.svg 6-simplex t024 A2.svg CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
t0,2,4{3,3,3,3,3}
Stericantellated 6-simplex
Cellirhombated heptapeton (cral)
19 6-simplex t124.svg 6-simplex t124 A5.svg 6-simplex t124 A4.svg 6-simplex t124 A3.svg 6-simplex t124 A2.svg CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
t1,2,4{3,3,3,3,3}
Biruncitruncated 6-simplex
Biprismatorhombated heptapeton (bapril)
20 6-simplex t0124.svg 6-simplex t0124 A5.svg 6-simplex t0124 A4.svg 6-simplex t0124 A3.svg 6-simplex t0124 A2.svg CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
t0,1,2,4{3,3,3,3,3}
Stericantitruncated 6-simplex
Celligreatorhombated heptapeton (cagral)
21 6-simplex t034.svg 6-simplex t034 A5.svg 6-simplex t034 A4.svg 6-simplex t034 A3.svg 6-simplex t034 A2.svg CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
t0,3,4{3,3,3,3,3}
Steriruncinated 6-simplex
Celliprismated heptapeton (copal)
22 6-simplex t0134.svg 6-simplex t0134 A5.svg 6-simplex t0134 A4.svg 6-simplex t0134 A3.svg 6-simplex t0134 A2.svg CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
t0,1,3,4{3,3,3,3,3}
Steriruncitruncated 6-simplex
celliprismatotruncated heptapeton (captal)
23 6-simplex t0234.svg 6-simplex t0234 A5.svg 6-simplex t0234 A4.svg 6-simplex t0234 A3.svg 6-simplex t0234 A2.svg CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
t0,2,3,4{3,3,3,3,3}
Steriruncicantellated 6-simplex
celliprismatorhombated heptapeton (copril)
24 6-simplex t1234.svg 6-simplex t1234 A5.svg 6-simplex t1234 A4.svg 6-simplex t1234 A3.svg 6-simplex t1234 A2.svg CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
t1,2,3,4{3,3,3,3,3}
Biruncicantitruncated 6-simplex
Great biprismato-tetradecapeton (gibpof)
25 6-simplex t01234.svg 6-simplex t01234 A5.svg 6-simplex t01234 A4.svg 6-simplex t01234 A3.svg 6-simplex t01234 A2.svg CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
t0,1,2,3,4{3,3,3,3,3}
Steriruncicantitruncated 6-simplex
Great cellated heptapeton (gacal)
26 6-simplex t05.svg 6-simplex t05 A5.svg 6-simplex t05 A4.svg 6-simplex t05 A3.svg 6-simplex t05 A2.svg CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
t0,5{3,3,3,3,3}
Pentellated 6-simplex
Small teri-tetradecapeton (staf)
27 6-simplex t015.svg 6-simplex t015 A5.svg 6-simplex t015 A4.svg 6-simplex t015 A3.svg 6-simplex t015 A2.svg CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
t0,1,5{3,3,3,3,3}
Pentitruncated 6-simplex
Tericellated heptapeton (tocal)
28 6-simplex t025.svg 6-simplex t025 A5.svg 6-simplex t025 A4.svg 6-simplex t025 A3.svg 6-simplex t025 A2.svg CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
t0,2,5{3,3,3,3,3}
Penticantellated 6-simplex
Teriprismated heptapeton (tapal)
29 6-simplex t0125.svg 6-simplex t0125 A5.svg 6-simplex t0125 A4.svg 6-simplex t0125 A3.svg 6-simplex t0125 A2.svg CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
t0,1,2,5{3,3,3,3,3}
Penticantitruncated 6-simplex
Terigreatorhombated heptapeton (togral)
30 6-simplex t0135.svg 6-simplex t0135 A5.svg 6-simplex t0135 A4.svg 6-simplex t0135 A3.svg 6-simplex t0135 A2.svg CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
t0,1,3,5{3,3,3,3,3}
Pentiruncitruncated 6-simplex
Tericellirhombated heptapeton (tocral)
31 6-simplex t0235.svg 6-simplex t0235 A5.svg 6-simplex t0235 A4.svg 6-simplex t0235 A3.svg 6-simplex t0235 A2.svg CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
t0,2,3,5{3,3,3,3,3}
Pentiruncicantellated 6-simplex
Teriprismatorhombi-tetradecapeton (taporf)
32 6-simplex t01235.svg 6-simplex t01235 A5.svg 6-simplex t01235 A4.svg 6-simplex t01235 A3.svg 6-simplex t01235 A2.svg CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
t0,1,2,3,5{3,3,3,3,3}
Pentiruncicantitruncated 6-simplex
Terigreatoprismated heptapeton (tagopal)
33 6-simplex t0145.svg 6-simplex t0145 A5.svg 6-simplex t0145 A4.svg 6-simplex t0145 A3.svg 6-simplex t0145 A2.svg CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
t0,1,4,5{3,3,3,3,3}
Pentisteritruncated 6-simplex
tericellitrunki-tetradecapeton (tactaf)
34 6-simplex t01245.svg 6-simplex t01245 A5.svg 6-simplex t01245 A4.svg 6-simplex t01245 A3.svg 6-simplex t01245 A2.svg CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
t0,1,2,4,5{3,3,3,3,3}
Pentistericantitruncated 6-simplex
tericelligreatorhombated heptapeton (tacogral)
35 6-simplex t012345.svg 6-simplex t012345 A5.svg 6-simplex t012345 A4.svg 6-simplex t012345 A3.svg 6-simplex t012345 A2.svg CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
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)".

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

Hellenica World - Scientific Library

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