A8 polytope
 8-simplex    |
In 8-dimensional geometry, there are 135 uniform polytopes with A8 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 A8 Coxeter group, and other subgroups.
Graphs
Symmetric orthographic projections of these 135 polytopes can be made in the A8, A7, A6, A5, A4, A3, A2 Coxeter planes. Ak 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 |
Ak orthogonal projection graphs | ||||||
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| A8 [9] |
A7 [8] |
A6 [7] |
A5 [6] |
A4 [5] |
A3 [4] |
A2 [3] | ||
| 1 | t0{3,3,3,3,3,3,3} 8-simplex |
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| 2 | t1{3,3,3,3,3,3,3} Rectified 8-simplex |
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| 3 | t2{3,3,3,3,3,3,3} Birectified 8-simplex |
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| 4 | t3{3,3,3,3,3,3,3} Trirectified 8-simplex |
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| 5 | t0,1{3,3,3,3,3,3,3} Truncated 8-simplex |
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| 6 | t0,2{3,3,3,3,3,3,3} Cantellated 8-simplex |
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| 7 | t1,2{3,3,3,3,3,3,3} Bitruncated 8-simplex |
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| 8 | t0,3{3,3,3,3,3,3,3} Runcinated 8-simplex |
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| 9 | t1,3{3,3,3,3,3,3,3} Bicantellated 8-simplex |
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| 10 | t2,3{3,3,3,3,3,3,3} Tritruncated 8-simplex |
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| 11 | t0,4{3,3,3,3,3,3,3} Stericated 8-simplex |
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| 12 | t1,4{3,3,3,3,3,3,3} Biruncinated 8-simplex |
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| 13 | t2,4{3,3,3,3,3,3,3} Tricantellated 8-simplex |
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| 14 | t3,4{3,3,3,3,3,3,3} Quadritruncated 8-simplex |
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| 15 | t0,5{3,3,3,3,3,3,3} Pentellated 8-simplex |
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| 16 | t1,5{3,3,3,3,3,3,3} Bistericated 8-simplex |
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| 17 | t2,5{3,3,3,3,3,3,3} Triruncinated 8-simplex |
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| 18 | t0,6{3,3,3,3,3,3,3} Hexicated 8-simplex |
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| 19 | t1,6{3,3,3,3,3,3,3} Bipentellated 8-simplex |
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| 20 | t0,7{3,3,3,3,3,3,3} Heptellated 8-simplex |
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| 21 | t0,1,2{3,3,3,3,3,3,3} Cantitruncated 8-simplex |
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| 22 | t0,1,3{3,3,3,3,3,3,3} Runcitruncated 8-simplex |
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| 23 | t0,2,3{3,3,3,3,3,3,3} Runcicantellated 8-simplex |
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| 24 | t1,2,3{3,3,3,3,3,3,3} Bicantitruncated 8-simplex |
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| 25 | t0,1,4{3,3,3,3,3,3,3} Steritruncated 8-simplex |
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| 26 | t0,2,4{3,3,3,3,3,3,3} Stericantellated 8-simplex |
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| 27 | t1,2,4{3,3,3,3,3,3,3} Biruncitruncated 8-simplex |
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| 28 | t0,3,4{3,3,3,3,3,3,3} Steriruncinated 8-simplex |
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| 29 | t1,3,4{3,3,3,3,3,3,3} Biruncicantellated 8-simplex |
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| 30 | t2,3,4{3,3,3,3,3,3,3} Tricantitruncated 8-simplex |
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| 31 | t0,1,5{3,3,3,3,3,3,3} Pentitruncated 8-simplex |
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| 32 | t0,2,5{3,3,3,3,3,3,3} Penticantellated 8-simplex |
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| 33 | t1,2,5{3,3,3,3,3,3,3} Bisteritruncated 8-simplex |
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| 34 | t0,3,5{3,3,3,3,3,3,3} Pentiruncinated 8-simplex |
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| 35 | t1,3,5{3,3,3,3,3,3,3} Bistericantellated 8-simplex |
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| 36 | t2,3,5{3,3,3,3,3,3,3} Triruncitruncated 8-simplex |
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| 37 | t0,4,5{3,3,3,3,3,3,3} Pentistericated 8-simplex |
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| 38 | t1,4,5{3,3,3,3,3,3,3} Bisteriruncinated 8-simplex |
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| 39 | t0,1,6{3,3,3,3,3,3,3} Hexitruncated 8-simplex |
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| 40 | t0,2,6{3,3,3,3,3,3,3} Hexicantellated 8-simplex |
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| 41 | t1,2,6{3,3,3,3,3,3,3} Bipentitruncated 8-simplex |
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| 42 | t0,3,6{3,3,3,3,3,3,3} Hexiruncinated 8-simplex |
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| 43 | t1,3,6{3,3,3,3,3,3,3} Bipenticantellated 8-simplex |
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| 44 | t0,4,6{3,3,3,3,3,3,3} Hexistericated 8-simplex |
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| 45 | t0,5,6{3,3,3,3,3,3,3} Hexipentellated 8-simplex |
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| 46 | t0,1,7{3,3,3,3,3,3,3} Heptitruncated 8-simplex |
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| 47 | t0,2,7{3,3,3,3,3,3,3} Hepticantellated 8-simplex |
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| 48 | t0,3,7{3,3,3,3,3,3,3} Heptiruncinated 8-simplex |
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| 49 | t0,1,2,3{3,3,3,3,3,3,3} Runcicantitruncated 8-simplex |
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| 50 | t0,1,2,4{3,3,3,3,3,3,3} Stericantitruncated 8-simplex |
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| 51 | t0,1,3,4{3,3,3,3,3,3,3} Steriruncitruncated 8-simplex |
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| 52 | t0,2,3,4{3,3,3,3,3,3,3} Steriruncicantellated 8-simplex |
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| 53 | t1,2,3,4{3,3,3,3,3,3,3} Biruncicantitruncated 8-simplex |
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| 54 | t0,1,2,5{3,3,3,3,3,3,3} Penticantitruncated 8-simplex |
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| 55 | t0,1,3,5{3,3,3,3,3,3,3} Pentiruncitruncated 8-simplex |
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| 56 | t0,2,3,5{3,3,3,3,3,3,3} Pentiruncicantellated 8-simplex |
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| 57 | t1,2,3,5{3,3,3,3,3,3,3} Bistericantitruncated 8-simplex |
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| 58 | t0,1,4,5{3,3,3,3,3,3,3} Pentisteritruncated 8-simplex |
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| 59 | t0,2,4,5{3,3,3,3,3,3,3} Pentistericantellated 8-simplex |
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| 60 | t1,2,4,5{3,3,3,3,3,3,3} Bisteriruncitruncated 8-simplex |
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| 61 | t0,3,4,5{3,3,3,3,3,3,3} Pentisteriruncinated 8-simplex |
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| 62 | t1,3,4,5{3,3,3,3,3,3,3} Bisteriruncicantellated 8-simplex |
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| 63 | t2,3,4,5{3,3,3,3,3,3,3} Triruncicantitruncated 8-simplex |
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| 64 | t0,1,2,6{3,3,3,3,3,3,3} Hexicantitruncated 8-simplex |
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| 65 | t0,1,3,6{3,3,3,3,3,3,3} Hexiruncitruncated 8-simplex |
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| 66 | t0,2,3,6{3,3,3,3,3,3,3} Hexiruncicantellated 8-simplex |
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| 67 | t1,2,3,6{3,3,3,3,3,3,3} Bipenticantitruncated 8-simplex |
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| 68 | t0,1,4,6{3,3,3,3,3,3,3} Hexisteritruncated 8-simplex |
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| 69 | t0,2,4,6{3,3,3,3,3,3,3} Hexistericantellated 8-simplex |
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| 70 | t1,2,4,6{3,3,3,3,3,3,3} Bipentiruncitruncated 8-simplex |
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| 71 | t0,3,4,6{3,3,3,3,3,3,3} Hexisteriruncinated 8-simplex |
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| 72 | t1,3,4,6{3,3,3,3,3,3,3} Bipentiruncicantellated 8-simplex |
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| 73 | t0,1,5,6{3,3,3,3,3,3,3} Hexipentitruncated 8-simplex |
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| 74 | t0,2,5,6{3,3,3,3,3,3,3} Hexipenticantellated 8-simplex |
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| 75 | t1,2,5,6{3,3,3,3,3,3,3} Bipentisteritruncated 8-simplex |
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| 76 | t0,3,5,6{3,3,3,3,3,3,3} Hexipentiruncinated 8-simplex |
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| 77 | t0,4,5,6{3,3,3,3,3,3,3} Hexipentistericated 8-simplex |
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| 78 | t0,1,2,7{3,3,3,3,3,3,3} Hepticantitruncated 8-simplex |
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| 79 | t0,1,3,7{3,3,3,3,3,3,3} Heptiruncitruncated 8-simplex |
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| 80 | t0,2,3,7{3,3,3,3,3,3,3} Heptiruncicantellated 8-simplex |
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| 81 | t0,1,4,7{3,3,3,3,3,3,3} Heptisteritruncated 8-simplex |
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| 82 | t0,2,4,7{3,3,3,3,3,3,3} Heptistericantellated 8-simplex |
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| 83 | t0,3,4,7{3,3,3,3,3,3,3} Heptisteriruncinated 8-simplex |
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| 84 | t0,1,5,7{3,3,3,3,3,3,3} Heptipentitruncated 8-simplex |
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| 85 | t0,2,5,7{3,3,3,3,3,3,3} Heptipenticantellated 8-simplex |
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| 86 | t0,1,6,7{3,3,3,3,3,3,3} Heptihexitruncated 8-simplex |
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| 87 | t0,1,2,3,4{3,3,3,3,3,3,3} Steriruncicantitruncated 8-simplex |
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| 88 | t0,1,2,3,5{3,3,3,3,3,3,3} Pentiruncicantitruncated 8-simplex |
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| 89 | t0,1,2,4,5{3,3,3,3,3,3,3} Pentistericantitruncated 8-simplex |
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| 90 | t0,1,3,4,5{3,3,3,3,3,3,3} Pentisteriruncitruncated 8-simplex |
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| 91 | t0,2,3,4,5{3,3,3,3,3,3,3} Pentisteriruncicantellated 8-simplex |
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| 92 | t1,2,3,4,5{3,3,3,3,3,3,3} Bisteriruncicantitruncated 8-simplex |
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| 93 | t0,1,2,3,6{3,3,3,3,3,3,3} Hexiruncicantitruncated 8-simplex |
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| 94 | t0,1,2,4,6{3,3,3,3,3,3,3} Hexistericantitruncated 8-simplex |
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| 95 | t0,1,3,4,6{3,3,3,3,3,3,3} Hexisteriruncitruncated 8-simplex |
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| 96 | t0,2,3,4,6{3,3,3,3,3,3,3} Hexisteriruncicantellated 8-simplex |
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| 97 | t1,2,3,4,6{3,3,3,3,3,3,3} Bipentiruncicantitruncated 8-simplex |
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| 98 | t0,1,2,5,6{3,3,3,3,3,3,3} Hexipenticantitruncated 8-simplex |
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| 99 | t0,1,3,5,6{3,3,3,3,3,3,3} Hexipentiruncitruncated 8-simplex |
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| 100 | t0,2,3,5,6{3,3,3,3,3,3,3} Hexipentiruncicantellated 8-simplex |
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| 101 | t1,2,3,5,6{3,3,3,3,3,3,3} Bipentistericantitruncated 8-simplex |
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| 102 | t0,1,4,5,6{3,3,3,3,3,3,3} Hexipentisteritruncated 8-simplex |
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| 103 | t0,2,4,5,6{3,3,3,3,3,3,3} Hexipentistericantellated 8-simplex |
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| 104 | t0,3,4,5,6{3,3,3,3,3,3,3} Hexipentisteriruncinated 8-simplex |
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| 105 | t0,1,2,3,7{3,3,3,3,3,3,3} Heptiruncicantitruncated 8-simplex |
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| 106 | t0,1,2,4,7{3,3,3,3,3,3,3} Heptistericantitruncated 8-simplex |
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| 107 | t0,1,3,4,7{3,3,3,3,3,3,3} Heptisteriruncitruncated 8-simplex |
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| 108 | t0,2,3,4,7{3,3,3,3,3,3,3} Heptisteriruncicantellated 8-simplex |
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| 109 | t0,1,2,5,7{3,3,3,3,3,3,3} Heptipenticantitruncated 8-simplex |
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| 110 | t0,1,3,5,7{3,3,3,3,3,3,3} Heptipentiruncitruncated 8-simplex |
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| 111 | t0,2,3,5,7{3,3,3,3,3,3,3} Heptipentiruncicantellated 8-simplex |
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| 112 | t0,1,4,5,7{3,3,3,3,3,3,3} Heptipentisteritruncated 8-simplex |
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| 113 | t0,1,2,6,7{3,3,3,3,3,3,3} Heptihexicantitruncated 8-simplex |
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| 114 | t0,1,3,6,7{3,3,3,3,3,3,3} Heptihexiruncitruncated 8-simplex |
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| 115 | t0,1,2,3,4,5{3,3,3,3,3,3,3} Pentisteriruncicantitruncated 8-simplex |
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| 116 | t0,1,2,3,4,6{3,3,3,3,3,3,3} Hexisteriruncicantitruncated 8-simplex |
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| 117 | t0,1,2,3,5,6{3,3,3,3,3,3,3} Hexipentiruncicantitruncated 8-simplex |
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| 118 | t0,1,2,4,5,6{3,3,3,3,3,3,3} Hexipentistericantitruncated 8-simplex |
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| 119 | t0,1,3,4,5,6{3,3,3,3,3,3,3} Hexipentisteriruncitruncated 8-simplex |
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| 120 | t0,2,3,4,5,6{3,3,3,3,3,3,3} Hexipentisteriruncicantellated 8-simplex |
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| 121 | t1,2,3,4,5,6{3,3,3,3,3,3,3} Bipentisteriruncicantitruncated 8-simplex |
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| 122 | t0,1,2,3,4,7{3,3,3,3,3,3,3} Heptisteriruncicantitruncated 8-simplex |
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| 123 | t0,1,2,3,5,7{3,3,3,3,3,3,3} Heptipentiruncicantitruncated 8-simplex |
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| 124 | t0,1,2,4,5,7{3,3,3,3,3,3,3} Heptipentistericantitruncated 8-simplex |
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| 125 | t0,1,3,4,5,7{3,3,3,3,3,3,3} Heptipentisteriruncitruncated 8-simplex |
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| 126 | t0,2,3,4,5,7{3,3,3,3,3,3,3} Heptipentisteriruncicantellated 8-simplex |
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| 127 | t0,1,2,3,6,7{3,3,3,3,3,3,3} Heptihexiruncicantitruncated 8-simplex |
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| 128 | t0,1,2,4,6,7{3,3,3,3,3,3,3} Heptihexistericantitruncated 8-simplex |
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| 129 | t0,1,3,4,6,7{3,3,3,3,3,3,3} Heptihexisteriruncitruncated 8-simplex |
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| 130 | t0,1,2,5,6,7{3,3,3,3,3,3,3} Heptihexipenticantitruncated 8-simplex |
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| 131 | t0,1,2,3,4,5,6{3,3,3,3,3,3,3} Hexipentisteriruncicantitruncated 8-simplex |
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| 132 | t0,1,2,3,4,5,7{3,3,3,3,3,3,3} Heptipentisteriruncicantitruncated 8-simplex |
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| 133 | t0,1,2,3,4,6,7{3,3,3,3,3,3,3} Heptihexisteriruncicantitruncated 8-simplex |
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| 134 | t0,1,2,3,5,6,7{3,3,3,3,3,3,3} Heptihexipentiruncicantitruncated 8-simplex |
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| 135 | t0,1,2,3,4,5,6,7{3,3,3,3,3,3,3} Omnitruncated 8-simplex |
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References
- H.S.M. Coxeter:
- H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973
- Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6 [1]
- (Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380-407, MR 2,10]
- (Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559-591]
- (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
- N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. Dissertation, University of Toronto, 1966
External links
- Klitzing, Richard. "8D uniform polytopes (polyzetta)".
Notes
| Fundamental convex regular and uniform polytopes in dimensions 2–10 | ||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Family | An | Bn | I2(p) / Dn | E6 / E7 / E8 / E9 / E10 / F4 / G2 | Hn | |||||||
| Regular polygon | Triangle | Square | p-gon | Hexagon | Pentagon | |||||||
| Uniform polyhedron | Tetrahedron | Octahedron • Cube | Demicube | Dodecahedron • Icosahedron | ||||||||
| Uniform 4-polytope | 5-cell | 16-cell • Tesseract | Demitesseract | 24-cell | 120-cell • 600-cell | |||||||
| Uniform 5-polytope | 5-simplex | 5-orthoplex • 5-cube | 5-demicube | |||||||||
| Uniform 6-polytope | 6-simplex | 6-orthoplex • 6-cube | 6-demicube | 122 • 221 | ||||||||
| Uniform 7-polytope | 7-simplex | 7-orthoplex • 7-cube | 7-demicube | 132 • 231 • 321 | ||||||||
| Uniform 8-polytope | 8-simplex | 8-orthoplex • 8-cube | 8-demicube | 142 • 241 • 421 | ||||||||
| Uniform 9-polytope | 9-simplex | 9-orthoplex • 9-cube | 9-demicube | |||||||||
| Uniform 10-polytope | 10-simplex | 10-orthoplex • 10-cube | 10-demicube | |||||||||
| Uniform n-polytope | n-simplex | n-orthoplex • n-cube | n-demicube | 1k2 • 2k1 • k21 | n-pentagonal polytope | |||||||
| Topics: Polytope families • Regular polytope • List of regular polytopes and compounds | ||||||||||||
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