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.

Graphs

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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
Tetradecapeton (fe)
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
Small biprismato-tetradecapeton (sibpof)
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
Great biprismato-tetradecapeton (gibpof)
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
Small teri-tetradecapeton (staf)
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
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

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  • 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
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Notes

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Family An Bn I2(p) / Dn E6 / E7 / E8 / F4 / G2 Hn
Regular polygon Triangle Square p-gon Hexagon Pentagon
Uniform polyhedron Tetrahedron OctahedronCube Demicube DodecahedronIcosahedron
Uniform polychoron Pentachoron 16-cellTesseract Demitesseract 24-cell 120-cell600-cell
Uniform 5-polytope 5-simplex 5-orthoplex5-cube 5-demicube
Uniform 6-polytope 6-simplex 6-orthoplex6-cube 6-demicube 122221
Uniform 7-polytope 7-simplex 7-orthoplex7-cube 7-demicube 132231321
Uniform 8-polytope 8-simplex 8-orthoplex8-cube 8-demicube 142241421
Uniform 9-polytope 9-simplex 9-orthoplex9-cube 9-demicube
Uniform 10-polytope 10-simplex 10-orthoplex10-cube 10-demicube
Uniform n-polytope n-simplex n-orthoplexn-cube n-demicube 1k22k1k21 n-pentagonal polytope
Topics: Polytope familiesRegular polytopeList of regular polytopes and compounds