B7 polytope
 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
[edit]Symmetric 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