 7-cube    
|
 7-orthoplex
|
 7-demicube 
|
In 7-dimensional geometry, there are 128 uniform polytopes with B7 symmetry. There are two regular forms, the 7-orthoplex, and 8-cube with 14 and 128 vertices respectively. The 7-demicube is added with half of the symmetry.
They can be visualized as symmetric orthographic projections in Coxeter planes of the B7 Coxeter group, and other subgroups.
Graphs
editSymmetric orthographic projections of these 128 polytopes can be made in the B7, B6, B5, B4, B3, B2, A5, A3, Coxeter planes. Ak has [k+1] symmetry, and Bk has [2k] symmetry.
These 128 polytopes are each shown in these 8 symmetry planes, with vertices and edges drawn, and vertices colored by the number of overlapping vertices in each projective position.
| # | Coxeter plane graphs | Coxeter-Dynkin diagram Schläfli symbol Johnson and Bowers Name | |||||||
|---|---|---|---|---|---|---|---|---|---|
| B7 [14] |
B6 [12] |
B5 [10] |
B4 [8] |
B3 [6] |
B2 [4] |
A5 [6] |
A3 [4] | ||
| 1 | |
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{3,3,3,3,3,4} 7-orthoplex Hecatonicosoctaexon (zee) |
| 2 |  |
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t1{3,3,3,3,3,4} Rectified 7-orthoplex Rectified hecatonicosoctaexon (rez) |
| 3 |  |
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t2{3,3,3,3,3,4} Birectified 7-orthoplex Birectified hecatonicosoctaexon (barz) |
| 4 |  |
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t3{4,3,3,3,3,3} Trirectified 7-cube Hepteractihecatonicosoctaexon (sez) |
| 5 |  |
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t2{4,3,3,3,3,3} Birectified 7-cube Birectified hepteract (bersa) |
| 6 |  |
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t1{4,3,3,3,3,3} Rectified 7-cube Rectified hepteract (rasa) |
| 7 | |
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{4,3,3,3,3,3} 7-cube Hepteract (hept) |
| 128 | |
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h{4,3,3,3,3,3} 7-demicube Hemihepteract |
| 8 |  |
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t0,1{3,3,3,3,3,4} Truncated 7-orthoplex Truncated hecatonicosoctaexon (Taz) |
| 9 |  |
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t0,2{3,3,3,3,3,4} Cantellated 7-orthoplex Small rhombated hecatonicosoctaexon (Sarz) |
| 10 |  |
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t1,2{3,3,3,3,3,4} Bitruncated 7-orthoplex Bitruncated hecatonicosoctaexon (Botaz) |
| 11 |  |
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t0,3{3,3,3,3,3,4} Runcinated 7-orthoplex Small prismated hecatonicosoctaexon (Spaz) |
| 12 |  |
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t1,3{3,3,3,3,3,4} Bicantellated 7-orthoplex Small birhombated hecatonicosoctaexon (Sebraz) |
| 13 |  |
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t2,3{3,3,3,3,3,4} Tritruncated 7-orthoplex Tritruncated hecatonicosoctaexon (Totaz) |
| 14 |  |
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t0,4{3,3,3,3,3,4} Stericated 7-orthoplex Small cellated hecatonicosoctaexon (Scaz) |
| 15 |  |
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t1,4{3,3,3,3,3,4} Biruncinated 7-orthoplex Small biprismated hecatonicosoctaexon (Sibpaz) |
| 16 |  |
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t2,4{4,3,3,3,3,3} Tricantellated 7-cube Small trirhombi-hepteractihecatonicosoctaexon (Strasaz) |
| 17 |  |
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t2,3{4,3,3,3,3,3} Tritruncated 7-cube Tritruncated hepteract (Tatsa) |
| 18 |  |
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t0,5{3,3,3,3,3,4} Pentellated 7-orthoplex Small terated hecatonicosoctaexon (Staz) |
| 19 |  |
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t1,5{4,3,3,3,3,3} Bistericated 7-cube Small bicelli-hepteractihecatonicosoctaexon (Sabcosaz) |
| 20 |  |
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t1,4{4,3,3,3,3,3} Biruncinated 7-cube Small biprismated hepteract (Sibposa) |
| 21 |  |
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t1,3{4,3,3,3,3,3} Bicantellated 7-cube Small birhombated hepteract (Sibrosa) |
| 22 |  |
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t1,2{4,3,3,3,3,3} Bitruncated 7-cube Bitruncated hepteract (Betsa) |
| 23 |  |
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t0,6{4,3,3,3,3,3} Hexicated 7-cube Small peti-hepteractihecatonicosoctaexon (Suposaz) |
| 24 |  |
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t0,5{4,3,3,3,3,3} Pentellated 7-cube Small terated hepteract (Stesa) |
| 25 |  |
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t0,4{4,3,3,3,3,3} Stericated 7-cube Small cellated hepteract (Scosa) |
| 26 |  |
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t0,3{4,3,3,3,3,3} Runcinated 7-cube Small prismated hepteract (Spesa) |
| 27 |  |
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t0,2{4,3,3,3,3,3} Cantellated 7-cube Small rhombated hepteract (Sersa) |
| 28 |  |
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t0,1{4,3,3,3,3,3} Truncated 7-cube Truncated hepteract (Tasa) |
| 29 |  |
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t0,1,2{3,3,3,3,3,4} Cantitruncated 7-orthoplex Great rhombated hecatonicosoctaexon (Garz) | |
| 30 |  |
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t0,1,3{3,3,3,3,3,4} Runcitruncated 7-orthoplex Prismatotruncated hecatonicosoctaexon (Potaz) | |
| 31 |  |
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t0,2,3{3,3,3,3,3,4} Runcicantellated 7-orthoplex Prismatorhombated hecatonicosoctaexon (Parz) | |
| 32 |  |
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t1,2,3{3,3,3,3,3,4} Bicantitruncated 7-orthoplex Great birhombated hecatonicosoctaexon (Gebraz) | |
| 33 |  |
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t0,1,4{3,3,3,3,3,4} Steritruncated 7-orthoplex Cellitruncated hecatonicosoctaexon (Catz) |
| 34 |  |
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t0,2,4{3,3,3,3,3,4} Stericantellated 7-orthoplex Cellirhombated hecatonicosoctaexon (Craze) |
| 35 |  |
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t1,2,4{3,3,3,3,3,4} Biruncitruncated 7-orthoplex Biprismatotruncated hecatonicosoctaexon (Baptize) |
| 36 |  |
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t0,3,4{3,3,3,3,3,4} Steriruncinated 7-orthoplex Celliprismated hecatonicosoctaexon (Copaz) | |
| 37 |  |
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t1,3,4{3,3,3,3,3,4} Biruncicantellated 7-orthoplex Biprismatorhombated hecatonicosoctaexon (Boparz) |
| 38 |  |
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t2,3,4{4,3,3,3,3,3} Tricantitruncated 7-cube Great trirhombi-hepteractihecatonicosoctaexon (Gotrasaz) | |
| 39 |  |
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t0,1,5{3,3,3,3,3,4} Pentitruncated 7-orthoplex Teritruncated hecatonicosoctaexon (Tetaz) |
| 40 |  |
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t0,2,5{3,3,3,3,3,4} Penticantellated 7-orthoplex Terirhombated hecatonicosoctaexon (Teroz) |
| 41 |  |
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t1,2,5{3,3,3,3,3,4} Bisteritruncated 7-orthoplex Bicellitruncated hecatonicosoctaexon (Boctaz) |
| 42 |  |
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t0,3,5{3,3,3,3,3,4} Pentiruncinated 7-orthoplex Teriprismated hecatonicosoctaexon (Topaz) |
| 43 |  |
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t1,3,5{4,3,3,3,3,3} Bistericantellated 7-cube Bicellirhombi-hepteractihecatonicosoctaexon (Bacresaz) |
| 44 |  |
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t1,3,4{4,3,3,3,3,3} Biruncicantellated 7-cube Biprismatorhombated hepteract (Bopresa) |
| 45 |  |
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t0,4,5{3,3,3,3,3,4} Pentistericated 7-orthoplex Tericellated hecatonicosoctaexon (Tocaz) |
| 46 |  |
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t1,2,5{4,3,3,3,3,3} Bisteritruncated 7-cube Bicellitruncated hepteract (Bactasa) |
| 47 |  |
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t1,2,4{4,3,3,3,3,3} Biruncitruncated 7-cube Biprismatotruncated hepteract (Biptesa) |
| 48 |  |
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t1,2,3{4,3,3,3,3,3} Bicantitruncated 7-cube Great birhombated hepteract (Gibrosa) |
| 49 |  |
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t0,1,6{3,3,3,3,3,4} Hexitruncated 7-orthoplex Petitruncated hecatonicosoctaexon (Putaz) |
| 50 |  |
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t0,2,6{3,3,3,3,3,4} Hexicantellated 7-orthoplex Petirhombated hecatonicosoctaexon (Puraz) |
| 51 |  |
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t0,4,5{4,3,3,3,3,3} Pentistericated 7-cube Tericellated hepteract (Tacosa) |
| 52 |  |
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t0,3,6{4,3,3,3,3,3} Hexiruncinated 7-cube Petiprismato-hepteractihecatonicosoctaexon (Pupsez) | |
| 53 |  |
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t0,3,5{4,3,3,3,3,3} Pentiruncinated 7-cube Teriprismated hepteract (Tapsa) |
| 54 |  |
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t0,3,4{4,3,3,3,3,3} Steriruncinated 7-cube Celliprismated hepteract (Capsa) |
| 55 |  |
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t0,2,6{4,3,3,3,3,3} Hexicantellated 7-cube Petirhombated hepteract (Purosa) |
| 56 |  |
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t0,2,5{4,3,3,3,3,3} Penticantellated 7-cube Terirhombated hepteract (Tersa) |
| 57 |  |
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t0,2,4{4,3,3,3,3,3} Stericantellated 7-cube Cellirhombated hepteract (Carsa) |
| 58 |  |
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t0,2,3{4,3,3,3,3,3} Runcicantellated 7-cube Prismatorhombated hepteract (Parsa) |
| 59 |  |
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t0,1,6{4,3,3,3,3,3} Hexitruncated 7-cube Petitruncated hepteract (Putsa) |
| 60 |  |
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t0,1,5{4,3,3,3,3,3} Pentitruncated 7-cube Teritruncated hepteract (Tetsa) |
| 61 |  |
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t0,1,4{4,3,3,3,3,3} Steritruncated 7-cube Cellitruncated hepteract (Catsa) |
| 62 |  |
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t0,1,3{4,3,3,3,3,3} Runcitruncated 7-cube Prismatotruncated hepteract (Petsa) |
| 63 |  |
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t0,1,2{4,3,3,3,3,3} Cantitruncated 7-cube Great rhombated hepteract (Gersa) |
| 64 |  |
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t0,1,2,3{3,3,3,3,3,4} Runcicantitruncated 7-orthoplex Great prismated hecatonicosoctaexon (Gopaz) | |
| 65 |  |
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t0,1,2,4{3,3,3,3,3,4} Stericantitruncated 7-orthoplex Celligreatorhombated hecatonicosoctaexon (Cogarz) |
| 66 |  |
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t0,1,3,4{3,3,3,3,3,4} Steriruncitruncated 7-orthoplex Celliprismatotruncated hecatonicosoctaexon (Captaz) |
| 67 |  |
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t0,2,3,4{3,3,3,3,3,4} Steriruncicantellated 7-orthoplex Celliprismatorhombated hecatonicosoctaexon (Caparz) |
| 68 |  |
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t1,2,3,4{3,3,3,3,3,4} Biruncicantitruncated 7-orthoplex Great biprismated hecatonicosoctaexon (Gibpaz) |
| 69 |  |
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t0,1,2,5{3,3,3,3,3,4} Penticantitruncated 7-orthoplex Terigreatorhombated hecatonicosoctaexon (Tograz) |
| 70 |  |
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t0,1,3,5{3,3,3,3,3,4} Pentiruncitruncated 7-orthoplex Teriprismatotruncated hecatonicosoctaexon (Toptaz) |
| 71 |  |
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t0,2,3,5{3,3,3,3,3,4} Pentiruncicantellated 7-orthoplex Teriprismatorhombated hecatonicosoctaexon (Toparz) |
| 72 |  |
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t1,2,3,5{3,3,3,3,3,4} Bistericantitruncated 7-orthoplex Bicelligreatorhombated hecatonicosoctaexon (Becogarz) |
| 73 |  |
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t0,1,4,5{3,3,3,3,3,4} Pentisteritruncated 7-orthoplex Tericellitruncated hecatonicosoctaexon (Tacotaz) |
| 74 |  |
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t0,2,4,5{3,3,3,3,3,4} Pentistericantellated 7-orthoplex Tericellirhombated hecatonicosoctaexon (Tocarz) |
| 75 |  |
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t1,2,4,5{4,3,3,3,3,3} Bisteriruncitruncated 7-cube Bicelliprismatotrunki-hepteractihecatonicosoctaexon (Bocaptosaz) |
| 76 |  |
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t0,3,4,5{3,3,3,3,3,4} Pentisteriruncinated 7-orthoplex Tericelliprismated hecatonicosoctaexon (Tecpaz) |
| 77 |  |
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t1,2,3,5{4,3,3,3,3,3} Bistericantitruncated 7-cube Bicelligreatorhombated hepteract (Becgresa) |
| 78 |  |
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t1,2,3,4{4,3,3,3,3,3} Biruncicantitruncated 7-cube Great biprismated hepteract (Gibposa) |
| 79 |  |
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t0,1,2,6{3,3,3,3,3,4} Hexicantitruncated 7-orthoplex Petigreatorhombated hecatonicosoctaexon (Pugarez) |
| 80 |  |
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t0,1,3,6{3,3,3,3,3,4} Hexiruncitruncated 7-orthoplex Petiprismatotruncated hecatonicosoctaexon (Papataz) |
| 81 |  |
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t0,2,3,6{3,3,3,3,3,4} Hexiruncicantellated 7-orthoplex Petiprismatorhombated hecatonicosoctaexon (Puparez) |
| 82 |  |
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t0,3,4,5{4,3,3,3,3,3} Pentisteriruncinated 7-cube Tericelliprismated hepteract (Tecpasa) |
| 83 |  |
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t0,1,4,6{3,3,3,3,3,4} Hexisteritruncated 7-orthoplex Peticellitruncated hecatonicosoctaexon (Pucotaz) |
| 84 |  |
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t0,2,4,6{4,3,3,3,3,3} Hexistericantellated 7-cube Peticellirhombi-hepteractihecatonicosoctaexon (Pucrosaz) |
| 85 |  |
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t0,2,4,5{4,3,3,3,3,3} Pentistericantellated 7-cube Tericellirhombated hepteract (Tecresa) |
| 86 |  |
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t0,2,3,6{4,3,3,3,3,3} Hexiruncicantellated 7-cube Petiprismatorhombated hepteract (Pupresa) |
| 87 |  |
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t0,2,3,5{4,3,3,3,3,3} Pentiruncicantellated 7-cube Teriprismatorhombated hepteract (Topresa) |
| 88 |  |
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t0,2,3,4{4,3,3,3,3,3} Steriruncicantellated 7-cube Celliprismatorhombated hepteract (Copresa) |
| 89 |  |
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t0,1,5,6{4,3,3,3,3,3} Hexipentitruncated 7-cube Petiteritrunki-hepteractihecatonicosoctaexon (Putatosez) |
| 90 |  |
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t0,1,4,6{4,3,3,3,3,3} Hexisteritruncated 7-cube Peticellitruncated hepteract (Pacutsa) |
| 91 |  |
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t0,1,4,5{4,3,3,3,3,3} Pentisteritruncated 7-cube Tericellitruncated hepteract (Tecatsa) |
| 92 |  |
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t0,1,3,6{4,3,3,3,3,3} Hexiruncitruncated 7-cube Petiprismatotruncated hepteract (Pupetsa) |
| 93 |  |
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t0,1,3,5{4,3,3,3,3,3} Pentiruncitruncated 7-cube Teriprismatotruncated hepteract (Toptosa) |
| 94 |  |
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t0,1,3,4{4,3,3,3,3,3} Steriruncitruncated 7-cube Celliprismatotruncated hepteract (Captesa) |
| 95 |  |
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t0,1,2,6{4,3,3,3,3,3} Hexicantitruncated 7-cube Petigreatorhombated hepteract (Pugrosa) |
| 96 |  |
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t0,1,2,5{4,3,3,3,3,3} Penticantitruncated 7-cube Terigreatorhombated hepteract (Togresa) |
| 97 |  |
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t0,1,2,4{4,3,3,3,3,3} Stericantitruncated 7-cube Celligreatorhombated hepteract (Cogarsa) |
| 98 |  |
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t0,1,2,3{4,3,3,3,3,3} Runcicantitruncated 7-cube Great prismated hepteract (Gapsa) |
| 99 |  |
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t0,1,2,3,4{3,3,3,3,3,4} Steriruncicantitruncated 7-orthoplex Great cellated hecatonicosoctaexon (Gocaz) |
| 100 |  |
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t0,1,2,3,5{3,3,3,3,3,4} Pentiruncicantitruncated 7-orthoplex Terigreatoprismated hecatonicosoctaexon (Tegopaz) | |
| 101 |  |
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t0,1,2,4,5{3,3,3,3,3,4} Pentistericantitruncated 7-orthoplex Tericelligreatorhombated hecatonicosoctaexon (Tecagraz) | |
| 102 |  |
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 |
t0,1,3,4,5{3,3,3,3,3,4} Pentisteriruncitruncated 7-orthoplex Tericelliprismatotruncated hecatonicosoctaexon (Tecpotaz) | |
| 103 |  |
 |
 |
 |
 |
 |
 |
t0,2,3,4,5{3,3,3,3,3,4} Pentisteriruncicantellated 7-orthoplex Tericelliprismatorhombated hecatonicosoctaexon (Tacparez) | |
| 104 |  |
 |
 |
 |
 |
 |
 |
t1,2,3,4,5{4,3,3,3,3,3} Bisteriruncicantitruncated 7-cube Great bicelli-hepteractihecatonicosoctaexon (Gabcosaz) | |
| 105 |  |
 |
 |
 |
 |
 |
 |
 |
t0,1,2,3,6{3,3,3,3,3,4} Hexiruncicantitruncated 7-orthoplex Petigreatoprismated hecatonicosoctaexon (Pugopaz) |
| 106 |  |
 |
 |
 |
 |
 |
 |
t0,1,2,4,6{3,3,3,3,3,4} Hexistericantitruncated 7-orthoplex Peticelligreatorhombated hecatonicosoctaexon (Pucagraz) | |
| 107 |  |
 |
 |
 |
 |
 |
 |
t0,1,3,4,6{3,3,3,3,3,4} Hexisteriruncitruncated 7-orthoplex Peticelliprismatotruncated hecatonicosoctaexon (Pucpotaz) | |
| 108 |  |
 |
 |
 |
 |
 |
 |
t0,2,3,4,6{4,3,3,3,3,3} Hexisteriruncicantellated 7-cube Peticelliprismatorhombi-hepteractihecatonicosoctaexon (Pucprosaz) | |
| 109 |  |
 |
 |
 |
 |
 |
 |
t0,2,3,4,5{4,3,3,3,3,3} Pentisteriruncicantellated 7-cube Tericelliprismatorhombated hepteract (Tocpresa) | |
| 110 |  |
 |
 |
 |
 |
 |
 |
 |
t0,1,2,5,6{3,3,3,3,3,4} Hexipenticantitruncated 7-orthoplex Petiterigreatorhombated hecatonicosoctaexon (Putegraz) |
| 111 |  |
 |
 |
 |
 |
 |
 |
t0,1,3,5,6{4,3,3,3,3,3} Hexipentiruncitruncated 7-cube Petiteriprismatotrunki-hepteractihecatonicosoctaexon (Putpetsaz) | |
| 112 |  |
 |
 |
 |
 |
 |
 |
t0,1,3,4,6{4,3,3,3,3,3} Hexisteriruncitruncated 7-cube Peticelliprismatotruncated hepteract (Pucpetsa) | |
| 113 |  |
 |
 |
 |
 |
 |
 |
t0,1,3,4,5{4,3,3,3,3,3} Pentisteriruncitruncated 7-cube Tericelliprismatotruncated hepteract (Tecpetsa) | |
| 114 |  |
 |
 |
 |
 |
 |
 |
 |
t0,1,2,5,6{4,3,3,3,3,3} Hexipenticantitruncated 7-cube Petiterigreatorhombated hepteract (Putgresa) |
| 115 |  |
 |
 |
 |
 |
 |
 |
t0,1,2,4,6{4,3,3,3,3,3} Hexistericantitruncated 7-cube Peticelligreatorhombated hepteract (Pucagrosa) | |
| 116 |  |
 |
 |
 |
 |
 |
 |
t0,1,2,4,5{4,3,3,3,3,3} Pentistericantitruncated 7-cube Tericelligreatorhombated hepteract (Tecgresa) | |
| 117 |  |
 |
 |
 |
 |
 |
t0,1,2,3,6{4,3,3,3,3,3} Hexiruncicantitruncated 7-cube Petigreatoprismated hepteract (Pugopsa) | ||
| 118 |  |
 |
 |
 |
 |
t0,1,2,3,5{4,3,3,3,3,3} Pentiruncicantitruncated 7-cube Terigreatoprismated hepteract (Togapsa) | |||
| 119 |  |
 |
 |
 |
 |
 |
t0,1,2,3,4{4,3,3,3,3,3} Steriruncicantitruncated 7-cube Great cellated hepteract (Gacosa) | ||
| 120 |  |
 |
 |
 |
 |
 |
 |
t0,1,2,3,4,5{3,3,3,3,3,4} Pentisteriruncicantitruncated 7-orthoplex Great terated hecatonicosoctaexon (Gotaz) | |
| 121 |  |
 |
 |
 |
 |
 |
t0,1,2,3,4,6{3,3,3,3,3,4} Hexisteriruncicantitruncated 7-orthoplex Petigreatocellated hecatonicosoctaexon (Pugacaz) | ||
| 122 |  |
 |
 |
 |
 |
 |
 |
t0,1,2,3,5,6{3,3,3,3,3,4} Hexipentiruncicantitruncated 7-orthoplex Petiterigreatoprismated hecatonicosoctaexon (Putgapaz) | |
| 123 |  |
 |
 |
 |
 |
 |
 |
t0,1,2,4,5,6{4,3,3,3,3,3} Hexipentistericantitruncated 7-cube Petitericelligreatorhombi-hepteractihecatonicosoctaexon (Putcagrasaz) | |
| 124 |  |
 |
 |
 |
 |
 |
 |
t0,1,2,3,5,6{4,3,3,3,3,3} Hexipentiruncicantitruncated 7-cube Petiterigreatoprismated hepteract (Putgapsa) | |
| 125 |  |
 |
 |
 |
 |
 |
 |
t0,1,2,3,4,6{4,3,3,3,3,3} Hexisteriruncicantitruncated 7-cube Petigreatocellated hepteract (Pugacasa) | |
| 126 |  |
 |
 |
 |
 |
 |
 |
t0,1,2,3,4,5{4,3,3,3,3,3} Pentisteriruncicantitruncated 7-cube Great terated hepteract (Gotesa) | |
| 127 |  |
 |
 |
 |
 |
 |
 |
t0,1,2,3,4,5,6{4,3,3,3,3,3} Omnitruncated 7-cube Great peti-hepteractihecatonicosoctaexon (Guposaz) | |
References
edit- 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
- Klitzing, Richard. "7D uniform polytopes (polyexa)".
- Notes