A6 polytope
# A6 polytope

Orthographic projections $A_6$ Coxeter plane :

* 6-simplex

In 6-dimensional geometry, there are 35 uniform polytopes with $A_6$ 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 $A_6$ Coxeter group, and other subgroups.

## Graphs

Symmetric orthographic projections of these 35 polytopes can be made in the $A_6,$ $A_5,$ $A_4,$ $A_3,$ $A_2$ Coxeter planes. $A_k$ 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.

| # | Coxeter-Dynkin diagram Schläfli symbol Name |
| - | - |
| 1 | $t_0{3,3,3,3,3}$ 6-simplex Heptapeton (hop) |
| 2 | $t_1{3,3,3,3,3}$ Rectified 6-simplex Rectified heptapeton (ril) |
| 3 | $t_{0,1}{3,3,3,3,3}$ Truncated 6-simplex Truncated heptapeton (til) |
| 4 | $t_2{3,3,3,3,3}$ Birectified 6-simplex Birectified heptapeton (bril) |
| 5 | $t_{0,2}{3,3,3,3,3}$ Cantellated 6-simplex Small rhombated heptapeton (sril) |
| 6 | $t_{1,2}{3,3,3,3,3}$ Bitruncated 6-simplex Bitruncated heptapeton (batal) |
| 7 | $t_{0,1,2}{3,3,3,3,3}$ Cantitruncated 6-simplex Great rhombated heptapeton (gril) |
| 8 | $t_{0,3}{3,3,3,3,3}$ Runcinated 6-simplex Small prismated heptapeton (spil) |
| 9 | $t_{1,3}{3,3,3,3,3}$ Bicantellated 6-simplex Small birhombated heptapeton (sabril) |
| 10 | $t_{0,1,3}{3,3,3,3,3}$ Runcitruncated 6-simplex Prismatotruncated heptapeton (patal) |
| 11 | $t_{2,3}{3,3,3,3,3}$ Tritruncated 6-simplex Tetradecapeton (fe) |
| 12 | $t_{0,2,3}{3,3,3,3,3}$ Runcicantellated 6-simplex Prismatorhombated heptapeton (pril) |
| 13 | $t_{1,2,3}{3,3,3,3,3}$ Bicantitruncated 6-simplex Great birhombated heptapeton (gabril) |
| 14 | $t_{0,1,2,3}{3,3,3,3,3}$ Runcicantitruncated 6-simplex Great prismated heptapeton (gapil) |
| 15 | $t_{0,4}{3,3,3,3,3}$ Stericated 6-simplex Small cellated heptapeton (scal) |
| 16 | $t_{1,4}{3,3,3,3,3}$ Biruncinated 6-simplex Small biprismato-tetradecapeton (sibpof) |
| 17 | $t_{0,1,4}{3,3,3,3,3}$ Steritruncated 6-simplex cellitruncated heptapeton (catal) |
| 18 | $t_{0,2,4}{3,3,3,3,3}$ Stericantellated 6-simplex Cellirhombated heptapeton (cral) |
| 19 | $t_{1,2,4}{3,3,3,3,3}$ Biruncitruncated 6-simplex Biprismatorhombated heptapeton (bapril) |
| 20 | $t_{0,1,2,4}{3,3,3,3,3}$ Stericantitruncated 6-simplex Celligreatorhombated heptapeton (cagral) |
| 21 | $t_{0,3,4}{3,3,3,3,3}$ Steriruncinated 6-simplex Celliprismated heptapeton (copal) |
| 22 | $t_{0,1,3,4}{3,3,3,3,3}$ Steriruncitruncated 6-simplex celliprismatotruncated heptapeton (captal) |
| 23 | $t_{0,2,3,4}{3,3,3,3,3}$ Steriruncicantellated 6-simplex celliprismatorhombated heptapeton (copril) |
| 24 | $t_{1,2,3,4}{3,3,3,3,3}$ Biruncicantitruncated 6-simplex Great biprismato-tetradecapeton (gibpof) |
| 25 | $t_{0,1,2,3,4}{3,3,3,3,3}$ Steriruncicantitruncated 6-simplex Great cellated heptapeton (gacal) |
| 26 | $t_{0,5}{3,3,3,3,3}$ Pentellated 6-simplex Small teri-tetradecapeton (staf) |
| 27 | $t_{0,1,5}{3,3,3,3,3}$ Pentitruncated 6-simplex Tericellated heptapeton (tocal) |
| 28 | $t_{0,2,5}{3,3,3,3,3}$ Penticantellated 6-simplex Teriprismated heptapeton (tapal) |
| 29 | $t_{0,1,2,5}{3,3,3,3,3}$ Penticantitruncated 6-simplex Terigreatorhombated heptapeton (togral) |
| 30 | $t_{0,1,3,5}{3,3,3,3,3}$ Pentiruncitruncated 6-simplex Tericellirhombated heptapeton (tocral) |
| 31 | $t_{0,2,3,5}{3,3,3,3,3}$ Pentiruncicantellated 6-simplex Teriprismatorhombi-tetradecapeton (taporf) |
| 32 | $t_{0,1,2,3,5}{3,3,3,3,3}$ Pentiruncicantitruncated 6-simplex Terigreatoprismated heptapeton (tagopal) |
| 33 | $t_{0,1,4,5}{3,3,3,3,3}$ Pentisteritruncated 6-simplex tericellitrunki-tetradecapeton (tactaf) |
| 34 | $t_{0,1,2,4,5}{3,3,3,3,3}$ Pentistericantitruncated 6-simplex tericelligreatorhombated heptapeton (tacogral) |
| 35 | $t_{0,1,2,3,4,5}{3,3,3,3,3}$ Omnitruncated 6-simplex Great teri-tetradecapeton (gotaf) |
