In 6dimensional geometry, there are 35 uniform polytopes with A6 symmetry. There is one selfdual regular form, the 6simplex 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.
#  A_{6} [7] 
A_{5} [6] 
A_{4} [5] 
A_{3} [4] 
A_{2} [3] 
CoxeterDynkin diagram Schläfli symbol Name 

1  t_{0}{3,3,3,3,3} 6simplex Heptapeton (hop) 

2  t_{1}{3,3,3,3,3} Rectified 6simplex Rectified heptapeton (ril) 

3  t_{0,1}{3,3,3,3,3} Truncated 6simplex Truncated heptapeton (til) 

4  t_{2}{3,3,3,3,3} Birectified 6simplex Birectified heptapeton (bril) 

5  t_{0,2}{3,3,3,3,3} Cantellated 6simplex Small rhombated heptapeton (sril) 

6  t_{1,2}{3,3,3,3,3} Bitruncated 6simplex Bitruncated heptapeton (batal) 

7  t_{0,1,2}{3,3,3,3,3} Cantitruncated 6simplex Great rhombated heptapeton (gril) 

8  t_{0,3}{3,3,3,3,3} Runcinated 6simplex Small prismated heptapeton (spil) 

9  t_{1,3}{3,3,3,3,3} Bicantellated 6simplex Small birhombated heptapeton (sabril) 

10  t_{0,1,3}{3,3,3,3,3} Runcitruncated 6simplex Prismatotruncated heptapeton (patal) 

11  t_{2,3}{3,3,3,3,3} Tritruncated 6simplex Tetradecapeton (fe) 

12  t_{0,2,3}{3,3,3,3,3} Runcicantellated 6simplex Prismatorhombated heptapeton (pril) 

13  t_{1,2,3}{3,3,3,3,3} Bicantitruncated 6simplex Great birhombated heptapeton (gabril) 

14  t_{0,1,2,3}{3,3,3,3,3} Runcicantitruncated 6simplex Great prismated heptapeton (gapil) 

15  t_{0,4}{3,3,3,3,3} Stericated 6simplex Small cellated heptapeton (scal) 

16  t_{1,4}{3,3,3,3,3} Biruncinated 6simplex Small biprismatotetradecapeton (sibpof) 

17  t_{0,1,4}{3,3,3,3,3} Steritruncated 6simplex cellitruncated heptapeton (catal) 

18  t_{0,2,4}{3,3,3,3,3} Stericantellated 6simplex Cellirhombated heptapeton (cral) 

19  t_{1,2,4}{3,3,3,3,3} Biruncitruncated 6simplex Biprismatorhombated heptapeton (bapril) 

20  t_{0,1,2,4}{3,3,3,3,3} Stericantitruncated 6simplex Celligreatorhombated heptapeton (cagral) 

21  t_{0,3,4}{3,3,3,3,3} Steriruncinated 6simplex Celliprismated heptapeton (copal) 

22  t_{0,1,3,4}{3,3,3,3,3} Steriruncitruncated 6simplex celliprismatotruncated heptapeton (captal) 

23  t_{0,2,3,4}{3,3,3,3,3} Steriruncicantellated 6simplex celliprismatorhombated heptapeton (copril) 

24  t_{1,2,3,4}{3,3,3,3,3} Biruncicantitruncated 6simplex Great biprismatotetradecapeton (gibpof) 

25  t_{0,1,2,3,4}{3,3,3,3,3} Steriruncicantitruncated 6simplex Great cellated heptapeton (gacal) 

26  t_{0,5}{3,3,3,3,3} Pentellated 6simplex Small teritetradecapeton (staf) 

27  t_{0,1,5}{3,3,3,3,3} Pentitruncated 6simplex Tericellated heptapeton (tocal) 

28  t_{0,2,5}{3,3,3,3,3} Penticantellated 6simplex Teriprismated heptapeton (tapal) 

29  t_{0,1,2,5}{3,3,3,3,3} Penticantitruncated 6simplex Terigreatorhombated heptapeton (togral) 

30  t_{0,1,3,5}{3,3,3,3,3} Pentiruncitruncated 6simplex Tericellirhombated heptapeton (tocral) 

31  t_{0,2,3,5}{3,3,3,3,3} Pentiruncicantellated 6simplex Teriprismatorhombitetradecapeton (taporf) 

32  t_{0,1,2,3,5}{3,3,3,3,3} Pentiruncicantitruncated 6simplex Terigreatoprismated heptapeton (tagopal) 

33  t_{0,1,4,5}{3,3,3,3,3} Pentisteritruncated 6simplex tericellitrunkitetradecapeton (tactaf) 

34  t_{0,1,2,4,5}{3,3,3,3,3} Pentistericantitruncated 6simplex tericelligreatorhombated heptapeton (tacogral) 

35  t_{0,1,2,3,4,5}{3,3,3,3,3} Omnitruncated 6simplex Great teritetradecapeton (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, WileyInterscience Publication, 1995, ISBN 9780471010036 [1]
(Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380407, MR 2,10]
(Paper 23) H.S.M. Coxeter, Regular and SemiRegular Polytopes II, [Math. Zeit. 188 (1985) 559591]
(Paper 24) H.S.M. Coxeter, Regular and SemiRegular Polytopes III, [Math. Zeit. 200 (1988) 345]
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  A_{n}  B_{n}  I_{2}(p) / D_{n}  E_{6} / E_{7} / E_{8} / F_{4} / G_{2}  H_{n}  
Regular polygon  Triangle  Square  pgon  Hexagon  Pentagon  
Uniform polyhedron  Tetrahedron  Octahedron • Cube  Demicube  Dodecahedron • Icosahedron  
Uniform 4polytope  5cell  16cell • Tesseract  Demitesseract  24cell  120cell • 600cell  
Uniform 5polytope  5simplex  5orthoplex • 5cube  5demicube  
Uniform 6polytope  6simplex  6orthoplex • 6cube  6demicube  1_{22} • 2_{21}  
Uniform 7polytope  7simplex  7orthoplex • 7cube  7demicube  1_{32} • 2_{31} • 3_{21}  
Uniform 8polytope  8simplex  8orthoplex • 8cube  8demicube  1_{42} • 2_{41} • 4_{21}  
Uniform 9polytope  9simplex  9orthoplex • 9cube  9demicube  
Uniform 10polytope  10simplex  10orthoplex • 10cube  10demicube  
Uniform npolytope  nsimplex  northoplex • ncube  ndemicube  1_{k2} • 2_{k1} • k_{21}  npentagonal polytope  
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds 
Hellenica World  Scientific Library
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