# A7 polytope

Orthographic projections A$_7$ Coxeter plane :
* 7-simplex
In 7-dimensional geometry, there are 71 uniform polytopes with A$_7$ symmetry. There is one self-dual regular form, the 7-simplex with 8 vertices.
Each can be visualized as symmetric orthographic projections in Coxeter planes of the A$_7$ Coxeter group, and other subgroups.

## Graphs

Symmetric orthographic projections of these 71 polytopes can be made in the A$_7$, A$_6$, A$_5$, A$_4$, A$_3$, A$_2$ Coxeter planes. A$_k$ has [k+1] symmetry. For even k and symmetrically ringed-diagrams, symmetry doubles to [2(k+1)].
These 71 polytopes are each shown in these 6 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 Johnson name | A$_k$ orthogonal projection graphs | A$_k$ orthogonal projection graphs | A$_k$ orthogonal projection graphs | A$_k$ orthogonal projection graphs | A$_k$ orthogonal projection graphs | A$_k$ orthogonal projection graphs |
| # | Coxeter-Dynkin diagram Schläfli symbol Johnson name | A$_7$ [8] | A$_6$ [7] | A$_5$ [6] | A$_4$ [5] | A$_3$ [4] | A$_2$ [3] |
| 1 | t$_0${3,3,3,3,3,3} 7-simplex | | | | | | |
| 2 | t$_1${3,3,3,3,3,3} Rectified 7-simplex | | | | | | |
| 3 | t$_2${3,3,3,3,3,3} Birectified 7-simplex | | | | | | |
| 4 | t$_3${3,3,3,3,3,3} Trirectified 7-simplex | | | | | | |
| 5 | t$_{0,1}${3,3,3,3,3,3} Truncated 7-simplex | | | | | | |
| 6 | t$_{0,2}${3,3,3,3,3,3} Cantellated 7-simplex | | | | | | |
| 7 | t$_{1,2}${3,3,3,3,3,3} Bitruncated 7-simplex | | | | | | |
| 8 | t$_{0,3}${3,3,3,3,3,3} Runcinated 7-simplex | | | | | | |
| 9 | t$_{1,3}${3,3,3,3,3,3} Bicantellated 7-simplex | | | | | | |
| 10 | t$_{2,3}${3,3,3,3,3,3} Tritruncated 7-simplex | | | | | | |
| 11 | t$_{0,4}${3,3,3,3,3,3} Stericated 7-simplex | | | | | | |
| 12 | t$_{1,4}${3,3,3,3,3,3} Biruncinated 7-simplex | | | | | | |
| 13 | t$_{2,4}${3,3,3,3,3,3} Tricantellated 7-simplex | | | | | | |
| 14 | t$_{0,5}${3,3,3,3,3,3} Pentellated 7-simplex | | | | | | |
| 15 | t$_{1,5}${3,3,3,3,3,3} Bistericated 7-simplex | | | | | | |
| 16 | t$_{0,6}${3,3,3,3,3,3} Hexicated 7-simplex | | | | | | |
| 17 | t$_{0,1,2}${3,3,3,3,3,3} Cantitruncated 7-simplex | | | | | | |
| 18 | t$_{0,1,3}${3,3,3,3,3,3} Runcitruncated 7-simplex | | | | | | |
| 19 | t$_{0,2,3}${3,3,3,3,3,3} Runcicantellated 7-simplex | | | | | | |
| 20 | t$_{1,2,3}${3,3,3,3,3,3} Bicantitruncated 7-simplex | | | | | | |
| 21 | t$_{0,1,4}${3,3,3,3,3,3} Steritruncated 7-simplex | | | | | | |
| 22 | t$_{0,2,4}${3,3,3,3,3,3} Stericantellated 7-simplex | | | | | | |
| 23 | t$_{1,2,4}${3,3,3,3,3,3} Biruncitruncated 7-simplex | | | | | | |
| 24 | t$_{0,3,4}${3,3,3,3,3,3} Steriruncinated 7-simplex | | | | | | |
| 25 | t$_{1,3,4}${3,3,3,3,3,3} Biruncicantellated 7-simplex | | | | | | |
| 26 | t$_{2,3,4}${3,3,3,3,3,3} Tricantitruncated 7-simplex | | | | | | |
| 27 | t$_{0,1,5}${3,3,3,3,3,3} Pentitruncated 7-simplex | | | | | | |
| 28 | t$_{0,2,5}${3,3,3,3,3,3} Penticantellated 7-simplex | | | | | | |
| 29 | t$_{1,2,5}${3,3,3,3,3,3} Bisteritruncated 7-simplex | | | | | | |
| 30 | t$_{0,3,5}${3,3,3,3,3,3} Pentiruncinated 7-simplex | | | | | | |
| 31 | t$_{1,3,5}${3,3,3,3,3,3} Bistericantellated 7-simplex | | | | | | |
| 32 | t$_{0,4,5}${3,3,3,3,3,3} Pentistericated 7-simplex | | | | | | |
| 33 | t$_{0,1,6}${3,3,3,3,3,3} Hexitruncated 7-simplex | | | | | | |
| 34 | t$_{0,2,6}${3,3,3,3,3,3} Hexicantellated 7-simplex | | | | | | |
| 35 | t$_{0,3,6}${3,3,3,3,3,3} Hexiruncinated 7-simplex | | | | | | |
| 36 | t$_{0,1,2,3}${3,3,3,3,3,3} Runcicantitruncated 7-simplex | | | | | | |
| 37 | t$_{0,1,2,4}${3,3,3,3,3,3} Stericantitruncated 7-simplex | | | | | | |
| 38 | t$_{0,1,3,4}${3,3,3,3,3,3} Steriruncitruncated 7-simplex | | | | | | |
| 39 | t$_{0,2,3,4}${3,3,3,3,3,3} Steriruncicantellated 7-simplex | | | | | | |
| 40 | t$_{1,2,3,4}${3,3,3,3,3,3} Biruncicantitruncated 7-simplex | | | | | | |
| 41 | t$_{0,1,2,5}${3,3,3,3,3,3} Penticantitruncated 7-simplex | | | | | | |
| 42 | t$_{0,1,3,5}${3,3,3,3,3,3} Pentiruncitruncated 7-simplex | | | | | | |
| 43 | t$_{0,2,3,5}${3,3,3,3,3,3} Pentiruncicantellated 7-simplex | | | | | | |
| 44 | t$_{1,2,3,5}${3,3,3,3,3,3} Bistericantitruncated 7-simplex | | | | | | |
| 45 | t$_{0,1,4,5}${3,3,3,3,3,3} Pentisteritruncated 7-simplex | | | | | | |
| 46 | t$_{0,2,4,5}${3,3,3,3,3,3} Pentistericantellated 7-simplex | | | | | | |
| 47 | t$_{1,2,4,5}${3,3,3,3,3,3} Bisteriruncitruncated 7-simplex | | | | | | |
| 48 | t$_{0,3,4,5}${3,3,3,3,3,3} Pentisteriruncinated 7-simplex | | | | | | |
| 49 | t$_{0,1,2,6}${3,3,3,3,3,3} Hexicantitruncated 7-simplex | | | | | | |
| 50 | t$_{0,1,3,6}${3,3,3,3,3,3} Hexiruncitruncated 7-simplex | | | | | | |
| 51 | t$_{0,2,3,6}${3,3,3,3,3,3} Hexiruncicantellated 7-simplex | | | | | | |
| 52 | t$_{0,1,4,6}${3,3,3,3,3,3} Hexisteritruncated 7-simplex | | | | | | |
| 53 | t$_{0,2,4,6}${3,3,3,3,3,3} Hexistericantellated 7-simplex | | | | | | |
| 54 | t$_{0,1,5,6}${3,3,3,3,3,3} Hexipentitruncated 7-simplex | | | | | | |
| 55 | t$_{0,1,2,3,4}${3,3,3,3,3,3} Steriruncicantitruncated 7-simplex | | | | | | |
| 56 | t$_{0,1,2,3,5}${3,3,3,3,3,3} Pentiruncicantitruncated 7-simplex | | | | | | |
| 57 | t$_{0,1,2,4,5}${3,3,3,3,3,3} Pentistericantitruncated 7-simplex | | | | | | |
| 58 | t$_{0,1,3,4,5}${3,3,3,3,3,3} Pentisteriruncitruncated 7-simplex | | | | | | |
| 59 | t$_{0,2,3,4,5}${3,3,3,3,3,3} Pentisteriruncicantellated 7-simplex | | | | | | |
| 60 | t$_{1,2,3,4,5}${3,3,3,3,3,3} Bisteriruncicantitruncated 7-simplex | | | | | | |
| 61 | t$_{0,1,2,3,6}${3,3,3,3,3,3} Hexiruncicantitruncated 7-simplex | | | | | | |
| 62 | t$_{0,1,2,4,6}${3,3,3,3,3,3} Hexistericantitruncated 7-simplex | | | | | | |
| 63 | t$_{0,1,3,4,6}${3,3,3,3,3,3} Hexisteriruncitruncated 7-simplex | | | | | | |
| 64 | t$_{0,2,3,4,6}${3,3,3,3,3,3} Hexisteriruncicantellated 7-simplex | | | | | | |
| 65 | t$_{0,1,2,5,6}${3,3,3,3,3,3} Hexipenticantitruncated 7-simplex | | | | | | |
| 66 | t$_{0,1,3,5,6}${3,3,3,3,3,3} Hexipentiruncitruncated 7-simplex | | | | | | |
| 67 | t$_{0,1,2,3,4,5}${3,3,3,3,3,3} Pentisteriruncicantitruncated 7-simplex | | | | | | |
| 68 | t$_{0,1,2,3,4,6}${3,3,3,3,3,3} Hexisteriruncicantitruncated 7-simplex | | | | | | |
| 69 | t$_{0,1,2,3,5,6}${3,3,3,3,3,3} Hexipentiruncicantitruncated 7-simplex | | | | | | |
| 70 | t$_{0,1,2,4,5,6}${3,3,3,3,3,3} Hexipentistericantitruncated 7-simplex | | | | | | |
| 71 | t$_{0,1,2,3,4,5,6}${3,3,3,3,3,3} Omnitruncated 7-simplex | | | | | | |