# A8 polytope

Orthographic projections A$_8$ Coxeter plane :
* 8-simplex
In 8-dimensional geometry, there are 135 uniform polytopes with A$_8$ symmetry. There is one self-dual regular form, the 8-simplex with 9 vertices.
Each can be visualized as symmetric orthographic projections in Coxeter planes of the A$_8$ Coxeter group, and other subgroups.

## Graphs

Symmetric orthographic projections of these 135 polytopes can be made in the A$_8$, A$_7$, A$_6$, A$_5$, A$_4$, A$_3$, A$_2$ Coxeter planes. A$_k$ has [k+1] symmetry.
These 135 polytopes are each shown in these 7 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 | A$_k$ orthogonal projection graphs |
| # | Coxeter-Dynkin diagram Schläfli symbol Johnson name | A$_8$ [9] | A$_7$ [8] | A$_6$ [7] | A$_5$ [6] | A$_4$ [5] | A$_3$ [4] | A$_2$ [3] |
| 1 | t$_0${3,3,3,3,3,3,3} 8-simplex | | | | | | | |
| 2 | t$_1${3,3,3,3,3,3,3} Rectified 8-simplex | | | | | | | |
| 3 | t$_2${3,3,3,3,3,3,3} Birectified 8-simplex | | | | | | | |
| 4 | t$_3${3,3,3,3,3,3,3} Trirectified 8-simplex | | | | | | | |
| 5 | t$_{0,1}${3,3,3,3,3,3,3} Truncated 8-simplex | | | | | | | |
| 6 | t$_{0,2}${3,3,3,3,3,3,3} Cantellated 8-simplex | | | | | | | |
| 7 | t$_{1,2}${3,3,3,3,3,3,3} Bitruncated 8-simplex | | | | | | | |
| 8 | t$_{0,3}${3,3,3,3,3,3,3} Runcinated 8-simplex | | | | | | | |
| 9 | t$_{1,3}${3,3,3,3,3,3,3} Bicantellated 8-simplex | | | | | | | |
| 10 | t$_{2,3}${3,3,3,3,3,3,3} Tritruncated 8-simplex | | | | | | | |
| 11 | t$_{0,4}${3,3,3,3,3,3,3} Stericated 8-simplex | | | | | | | |
| 12 | t$_{1,4}${3,3,3,3,3,3,3} Biruncinated 8-simplex | | | | | | | |
| 13 | t$_{2,4}${3,3,3,3,3,3,3} Tricantellated 8-simplex | | | | | | | |
| 14 | t$_{3,4}${3,3,3,3,3,3,3} Quadritruncated 8-simplex | | | | | | | |
| 15 | t$_{0,5}${3,3,3,3,3,3,3} Pentellated 8-simplex | | | | | | | |
| 16 | t$_{1,5}${3,3,3,3,3,3,3} Bistericated 8-simplex | | | | | | | |
| 17 | t$_{2,5}${3,3,3,3,3,3,3} Triruncinated 8-simplex | | | | | | | |
| 18 | t$_{0,6}${3,3,3,3,3,3,3} Hexicated 8-simplex | | | | | | | |
| 19 | t$_{1,6}${3,3,3,3,3,3,3} Bipentellated 8-simplex | | | | | | | |
| 20 | t$_{0,7}${3,3,3,3,3,3,3} Heptellated 8-simplex | | | | | | | |
| 21 | t$_{0,1,2}${3,3,3,3,3,3,3} Cantitruncated 8-simplex | | | | | | | |
| 22 | t$_{0,1,3}${3,3,3,3,3,3,3} Runcitruncated 8-simplex | | | | | | | |
| 23 | t$_{0,2,3}${3,3,3,3,3,3,3} Runcicantellated 8-simplex | | | | | | | |
| 24 | t$_{1,2,3}${3,3,3,3,3,3,3} Bicantitruncated 8-simplex | | | | | | | |
| 25 | t$_{0,1,4}${3,3,3,3,3,3,3} Steritruncated 8-simplex | | | | | | | |
| 26 | t$_{0,2,4}${3,3,3,3,3,3,3} Stericantellated 8-simplex | | | | | | | |
| 27 | t$_{1,2,4}${3,3,3,3,3,3,3} Biruncitruncated 8-simplex | | | | | | | |
| 28 | t$_{0,3,4}${3,3,3,3,3,3,3} Steriruncinated 8-simplex | | | | | | | |
| 29 | t$_{1,3,4}${3,3,3,3,3,3,3} Biruncicantellated 8-simplex | | | | | | | |
| 30 | t$_{2,3,4}${3,3,3,3,3,3,3} Tricantitruncated 8-simplex | | | | | | | |
| 31 | t$_{0,1,5}${3,3,3,3,3,3,3} Pentitruncated 8-simplex | | | | | | | |
| 32 | t$_{0,2,5}${3,3,3,3,3,3,3} Penticantellated 8-simplex | | | | | | | |
| 33 | t$_{1,2,5}${3,3,3,3,3,3,3} Bisteritruncated 8-simplex | | | | | | | |
| 34 | t$_{0,3,5}${3,3,3,3,3,3,3} Pentiruncinated 8-simplex | | | | | | | |
| 35 | t$_{1,3,5}${3,3,3,3,3,3,3} Bistericantellated 8-simplex | | | | | | | |
| 36 | t$_{2,3,5}${3,3,3,3,3,3,3} Triruncitruncated 8-simplex | | | | | | | |
| 37 | t$_{0,4,5}${3,3,3,3,3,3,3} Pentistericated 8-simplex | | | | | | | |
| 38 | t$_{1,4,5}${3,3,3,3,3,3,3} Bisteriruncinated 8-simplex | | | | | | | |
| 39 | t$_{0,1,6}${3,3,3,3,3,3,3} Hexitruncated 8-simplex | | | | | | | |
| 40 | t$_{0,2,6}${3,3,3,3,3,3,3} Hexicantellated 8-simplex | | | | | | | |
| 41 | t$_{1,2,6}${3,3,3,3,3,3,3} Bipentitruncated 8-simplex | | | | | | | |
| 42 | t$_{0,3,6}${3,3,3,3,3,3,3} Hexiruncinated 8-simplex | | | | | | | |
| 43 | t$_{1,3,6}${3,3,3,3,3,3,3} Bipenticantellated 8-simplex | | | | | | | |
| 44 | t$_{0,4,6}${3,3,3,3,3,3,3} Hexistericated 8-simplex | | | | | | | |
| 45 | t$_{0,5,6}${3,3,3,3,3,3,3} Hexipentellated 8-simplex | | | | | | | |
| 46 | t$_{0,1,7}${3,3,3,3,3,3,3} Heptitruncated 8-simplex | | | | | | | |
| 47 | t$_{0,2,7}${3,3,3,3,3,3,3} Hepticantellated 8-simplex | | | | | | | |
| 48 | t$_{0,3,7}${3,3,3,3,3,3,3} Heptiruncinated 8-simplex | | | | | | | |
| 49 | t$_{0,1,2,3}${3,3,3,3,3,3,3} Runcicantitruncated 8-simplex | | | | | | | |
| 50 | t$_{0,1,2,4}${3,3,3,3,3,3,3} Stericantitruncated 8-simplex | | | | | | | |
| 51 | t$_{0,1,3,4}${3,3,3,3,3,3,3} Steriruncitruncated 8-simplex | | | | | | | |
| 52 | t$_{0,2,3,4}${3,3,3,3,3,3,3} Steriruncicantellated 8-simplex | | | | | | | |
| 53 | t$_{1,2,3,4}${3,3,3,3,3,3,3} Biruncicantitruncated 8-simplex | | | | | | | |
| 54 | t$_{0,1,2,5}${3,3,3,3,3,3,3} Penticantitruncated 8-simplex | | | | | | | |
| 55 | t$_{0,1,3,5}${3,3,3,3,3,3,3} Pentiruncitruncated 8-simplex | | | | | | | |
| 56 | t$_{0,2,3,5}${3,3,3,3,3,3,3} Pentiruncicantellated 8-simplex | | | | | | | |
| 57 | t$_{1,2,3,5}${3,3,3,3,3,3,3} Bistericantitruncated 8-simplex | | | | | | | |
| 58 | t$_{0,1,4,5}${3,3,3,3,3,3,3} Pentisteritruncated 8-simplex | | | | | | | |
| 59 | t$_{0,2,4,5}${3,3,3,3,3,3,3} Pentistericantellated 8-simplex | | | | | | | |
| 60 | t$_{1,2,4,5}${3,3,3,3,3,3,3} Bisteriruncitruncated 8-simplex | | | | | | | |
| 61 | t$_{0,3,4,5}${3,3,3,3,3,3,3} Pentisteriruncinated 8-simplex | | | | | | | |
| 62 | t$_{1,3,4,5}${3,3,3,3,3,3,3} Bisteriruncicantellated 8-simplex | | | | | | | |
| 63 | t$_{2,3,4,5}${3,3,3,3,3,3,3} Triruncicantitruncated 8-simplex | | | | | | | |
| 64 | t$_{0,1,2,6}${3,3,3,3,3,3,3} Hexicantitruncated 8-simplex | | | | | | | |
| 65 | t$_{0,1,3,6}${3,3,3,3,3,3,3} Hexiruncitruncated 8-simplex | | | | | | | |
| 66 | t$_{0,2,3,6}${3,3,3,3,3,3,3} Hexiruncicantellated 8-simplex | | | | | | | |
| 67 | t$_{1,2,3,6}${3,3,3,3,3,3,3} Bipenticantitruncated 8-simplex | | | | | | | |
| 68 | t$_{0,1,4,6}${3,3,3,3,3,3,3} Hexisteritruncated 8-simplex | | | | | | | |
| 69 | t$_{0,2,4,6}${3,3,3,3,3,3,3} Hexistericantellated 8-simplex | | | | | | | |
| 70 | t$_{1,2,4,6}${3,3,3,3,3,3,3} Bipentiruncitruncated 8-simplex | | | | | | | |
| 71 | t$_{0,3,4,6}${3,3,3,3,3,3,3} Hexisteriruncinated 8-simplex | | | | | | | |
| 72 | t$_{1,3,4,6}${3,3,3,3,3,3,3} Bipentiruncicantellated 8-simplex | | | | | | | |
| 73 | t$_{0,1,5,6}${3,3,3,3,3,3,3} Hexipentitruncated 8-simplex | | | | | | | |
| 74 | t$_{0,2,5,6}${3,3,3,3,3,3,3} Hexipenticantellated 8-simplex | | | | | | | |
| 75 | t$_{1,2,5,6}${3,3,3,3,3,3,3} Bipentisteritruncated 8-simplex | | | | | | | |
| 76 | t$_{0,3,5,6}${3,3,3,3,3,3,3} Hexipentiruncinated 8-simplex | | | | | | | |
| 77 | t$_{0,4,5,6}${3,3,3,3,3,3,3} Hexipentistericated 8-simplex | | | | | | | |
| 78 | t$_{0,1,2,7}${3,3,3,3,3,3,3} Hepticantitruncated 8-simplex | | | | | | | |
| 79 | t$_{0,1,3,7}${3,3,3,3,3,3,3} Heptiruncitruncated 8-simplex | | | | | | | |
| 80 | t$_{0,2,3,7}${3,3,3,3,3,3,3} Heptiruncicantellated 8-simplex | | | | | | | |
| 81 | t$_{0,1,4,7}${3,3,3,3,3,3,3} Heptisteritruncated 8-simplex | | | | | | | |
| 82 | t$_{0,2,4,7}${3,3,3,3,3,3,3} Heptistericantellated 8-simplex | | | | | | | |
| 83 | t$_{0,3,4,7}${3,3,3,3,3,3,3} Heptisteriruncinated 8-simplex | | | | | | | |
| 84 | t$_{0,1,5,7}${3,3,3,3,3,3,3} Heptipentitruncated 8-simplex | | | | | | | |
| 85 | t$_{0,2,5,7}${3,3,3,3,3,3,3} Heptipenticantellated 8-simplex | | | | | | | |
| 86 | t$_{0,1,6,7}${3,3,3,3,3,3,3} Heptihexitruncated 8-simplex | | | | | | | |
| 87 | t$_{0,1,2,3,4}${3,3,3,3,3,3,3} Steriruncicantitruncated 8-simplex | | | | | | | |
| 88 | t$_{0,1,2,3,5}${3,3,3,3,3,3,3} Pentiruncicantitruncated 8-simplex | | | | | | | |
| 89 | t$_{0,1,2,4,5}${3,3,3,3,3,3,3} Pentistericantitruncated 8-simplex | | | | | | | |
| 90 | t$_{0,1,3,4,5}${3,3,3,3,3,3,3} Pentisteriruncitruncated 8-simplex | | | | | | | |
| 91 | t$_{0,2,3,4,5}${3,3,3,3,3,3,3} Pentisteriruncicantellated 8-simplex | | | | | | | |
| 92 | t$_{1,2,3,4,5}${3,3,3,3,3,3,3} Bisteriruncicantitruncated 8-simplex | | | | | | | |
| 93 | t$_{0,1,2,3,6}${3,3,3,3,3,3,3} Hexiruncicantitruncated 8-simplex | | | | | | | |
| 94 | t$_{0,1,2,4,6}${3,3,3,3,3,3,3} Hexistericantitruncated 8-simplex | | | | | | | |
| 95 | t$_{0,1,3,4,6}${3,3,3,3,3,3,3} Hexisteriruncitruncated 8-simplex | | | | | | | |
| 96 | t$_{0,2,3,4,6}${3,3,3,3,3,3,3} Hexisteriruncicantellated 8-simplex | | | | | | | |
| 97 | t$_{1,2,3,4,6}${3,3,3,3,3,3,3} Bipentiruncicantitruncated 8-simplex | | | | | | | |
| 98 | t$_{0,1,2,5,6}${3,3,3,3,3,3,3} Hexipenticantitruncated 8-simplex | | | | | | | |
| 99 | t$_{0,1,3,5,6}${3,3,3,3,3,3,3} Hexipentiruncitruncated 8-simplex | | | | | | | |
| 100 | t$_{0,2,3,5,6}${3,3,3,3,3,3,3} Hexipentiruncicantellated 8-simplex | | | | | | | |
| 101 | t$_{1,2,3,5,6}${3,3,3,3,3,3,3} Bipentistericantitruncated 8-simplex | | | | | | | |
| 102 | t$_{0,1,4,5,6}${3,3,3,3,3,3,3} Hexipentisteritruncated 8-simplex | | | | | | | |
| 103 | t$_{0,2,4,5,6}${3,3,3,3,3,3,3} Hexipentistericantellated 8-simplex | | | | | | | |
| 104 | t$_{0,3,4,5,6}${3,3,3,3,3,3,3} Hexipentisteriruncinated 8-simplex | | | | | | | |
| 105 | t$_{0,1,2,3,7}${3,3,3,3,3,3,3} Heptiruncicantitruncated 8-simplex | | | | | | | |
| 106 | t$_{0,1,2,4,7}${3,3,3,3,3,3,3} Heptistericantitruncated 8-simplex | | | | | | | |
| 107 | t$_{0,1,3,4,7}${3,3,3,3,3,3,3} Heptisteriruncitruncated 8-simplex | | | | | | | |
| 108 | t$_{0,2,3,4,7}${3,3,3,3,3,3,3} Heptisteriruncicantellated 8-simplex | | | | | | | |
| 109 | t$_{0,1,2,5,7}${3,3,3,3,3,3,3} Heptipenticantitruncated 8-simplex | | | | | | | |
| 110 | t$_{0,1,3,5,7}${3,3,3,3,3,3,3} Heptipentiruncitruncated 8-simplex | | | | | | | |
| 111 | t$_{0,2,3,5,7}${3,3,3,3,3,3,3} Heptipentiruncicantellated 8-simplex | | | | | | | |
| 112 | t$_{0,1,4,5,7}${3,3,3,3,3,3,3} Heptipentisteritruncated 8-simplex | | | | | | | |
| 113 | t$_{0,1,2,6,7}${3,3,3,3,3,3,3} Heptihexicantitruncated 8-simplex | | | | | | | |
| 114 | t$_{0,1,3,6,7}${3,3,3,3,3,3,3} Heptihexiruncitruncated 8-simplex | | | | | | | |
| 115 | t$_{0,1,2,3,4,5}${3,3,3,3,3,3,3} Pentisteriruncicantitruncated 8-simplex | | | | | | | |
| 116 | t$_{0,1,2,3,4,6}${3,3,3,3,3,3,3} Hexisteriruncicantitruncated 8-simplex | | | | | | | |
| 117 | t$_{0,1,2,3,5,6}${3,3,3,3,3,3,3} Hexipentiruncicantitruncated 8-simplex | | | | | | | |
| 118 | t$_{0,1,2,4,5,6}${3,3,3,3,3,3,3} Hexipentistericantitruncated 8-simplex | | | | | | | |
| 119 | t$_{0,1,3,4,5,6}${3,3,3,3,3,3,3} Hexipentisteriruncitruncated 8-simplex | | | | | | | |
| 120 | t$_{0,2,3,4,5,6}${3,3,3,3,3,3,3} Hexipentisteriruncicantellated 8-simplex | | | | | | | |
| 121 | t$_{1,2,3,4,5,6}${3,3,3,3,3,3,3} Bipentisteriruncicantitruncated 8-simplex | | | | | | | |
| 122 | t$_{0,1,2,3,4,7}${3,3,3,3,3,3,3} Heptisteriruncicantitruncated 8-simplex | | | | | | | |
| 123 | t$_{0,1,2,3,5,7}${3,3,3,3,3,3,3} Heptipentiruncicantitruncated 8-simplex | | | | | | | |
| 124 | t$_{0,1,2,4,5,7}${3,3,3,3,3,3,3} Heptipentistericantitruncated 8-simplex | | | | | | | |
| 125 | t$_{0,1,3,4,5,7}${3,3,3,3,3,3,3} Heptipentisteriruncitruncated 8-simplex | | | | | | | |
| 126 | t$_{0,2,3,4,5,7}${3,3,3,3,3,3,3} Heptipentisteriruncicantellated 8-simplex | | | | | | | |
| 127 | t$_{0,1,2,3,6,7}${3,3,3,3,3,3,3} Heptihexiruncicantitruncated 8-simplex | | | | | | | |
| 128 | t$_{0,1,2,4,6,7}${3,3,3,3,3,3,3} Heptihexistericantitruncated 8-simplex | | | | | | | |
| 129 | t$_{0,1,3,4,6,7}${3,3,3,3,3,3,3} Heptihexisteriruncitruncated 8-simplex | | | | | | | |
| 130 | t$_{0,1,2,5,6,7}${3,3,3,3,3,3,3} Heptihexipenticantitruncated 8-simplex | | | | | | | |
| 131 | t$_{0,1,2,3,4,5,6}${3,3,3,3,3,3,3} Hexipentisteriruncicantitruncated 8-simplex | | | | | | | |
| 132 | t$_{0,1,2,3,4,5,7}${3,3,3,3,3,3,3} Heptipentisteriruncicantitruncated 8-simplex | | | | | | | |
| 133 | t$_{0,1,2,3,4,6,7}${3,3,3,3,3,3,3} Heptihexisteriruncicantitruncated 8-simplex | | | | | | | |
| 134 | t$_{0,1,2,3,5,6,7}${3,3,3,3,3,3,3} Heptihexipentiruncicantitruncated 8-simplex | | | | | | | |
| 135 | t$_{0,1,2,3,4,5,6,7}${3,3,3,3,3,3,3} Omnitruncated 8-simplex | | | | | | | |