A8 polytope
# 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 | | | | | | | |
