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 |  |
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   t0{3,3,3,3,3} 6-simplex Heptapeton (hop) |
| 2 |  |
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t1{3,3,3,3,3} Rectified 6-simplex Rectified heptapeton (ril) |
| 3 |  |
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t0,1{3,3,3,3,3} Truncated 6-simplex Truncated heptapeton (til) |
| 4 |  |
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t2{3,3,3,3,3} Birectified 6-simplex Birectified heptapeton (bril) |
| 5 |  |
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 |
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t0,2{3,3,3,3,3} Cantellated 6-simplex Small rhombated heptapeton (sril) |
| 6 |  |
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t1,2{3,3,3,3,3} Bitruncated 6-simplex Bitruncated heptapeton (batal) |
| 7 |  |
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 |
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t0,1,2{3,3,3,3,3} Cantitruncated 6-simplex Great rhombated heptapeton (gril) |
| 8 |  |
 |
 |
 |
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t0,3{3,3,3,3,3} Runcinated 6-simplex Small prismated heptapeton (spil) |
| 9 |  |
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 |
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t1,3{3,3,3,3,3} Bicantellated 6-simplex Small birhombated heptapeton (sabril) |
| 10 |  |
 |
 |
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t0,1,3{3,3,3,3,3} Runcitruncated 6-simplex Prismatotruncated heptapeton (patal) |
| 11 |  |
 |
 |
 |
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t2,3{3,3,3,3,3} Tritruncated 6-simplex Tetradecapeton (fe) |
| 12 |  |
 |
 |
 |
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t0,2,3{3,3,3,3,3} Runcicantellated 6-simplex Prismatorhombated heptapeton (pril) |
| 13 |  |
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t1,2,3{3,3,3,3,3} Bicantitruncated 6-simplex Great birhombated heptapeton (gabril) |
| 14 |  |
 |
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 |
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t0,1,2,3{3,3,3,3,3} Runcicantitruncated 6-simplex Great prismated heptapeton (gapil) |
| 15 |  |
 |
 |
 |
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t0,4{3,3,3,3,3} Stericated 6-simplex Small cellated heptapeton (scal) |
| 16 |  |
 |
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 |
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t1,4{3,3,3,3,3} Biruncinated 6-simplex Small biprismato-tetradecapeton (sibpof) |
| 17 |  |
 |
 |
 |
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t0,1,4{3,3,3,3,3} Steritruncated 6-simplex cellitruncated heptapeton (catal) |
| 18 |  |
 |
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 |
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t0,2,4{3,3,3,3,3} Stericantellated 6-simplex Cellirhombated heptapeton (cral) |
| 19 |  |
 |
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t1,2,4{3,3,3,3,3} Biruncitruncated 6-simplex Biprismatorhombated heptapeton (bapril) |
| 20 |  |
 |
 |
 |
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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 Great biprismato-tetradecapeton (gibpof) |
| 25 |  |
 |
 |
 |
 |
t0,1,2,3,4{3,3,3,3,3} Steriruncicantitruncated 6-simplex Great cellated heptapeton (gacal) |
| 26 |  |
 |
 |
 |
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t0,5{3,3,3,3,3} Pentellated 6-simplex Small teri-tetradecapeton (staf) |
| 27 |  |
 |
 |
 |
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t0,1,5{3,3,3,3,3} Pentitruncated 6-simplex Tericellated heptapeton (tocal) |
| 28 |  |
 |
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t0,2,5{3,3,3,3,3} Penticantellated 6-simplex Teriprismated heptapeton (tapal) |
| 29 |  |
 |
 |
 |
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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 Teriprismatorhombi-tetradecapeton (taporf) |
| 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 tericellitrunki-tetradecapeton (tactaf) |
| 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 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
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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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