# 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) |