List of B7 polytopes
Encyclopedia
7-cube |
7-orthoplex |
7-demicube |
In 7-dimensional geometry
Geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....
, 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 projection
Orthographic projection
Orthographic projection is a means of representing a three-dimensional object in two dimensions. It is a form of parallel projection, where all the projection lines are orthogonal to the projection plane, resulting in every plane of the scene appearing in affine transformation on the viewing surface...
s in Coxeter planes of the B7 Coxeter group, and other subgroups.
Graphs
Symmetric orthographic projectionOrthographic projection
Orthographic projection is a means of representing a three-dimensional object in two dimensions. It is a form of parallel projection, where all the projection lines are orthogonal to the projection plane, resulting in every plane of the scene appearing in affine transformation on the viewing surface...
s 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 Coxeter-Dynkin diagram In geometry, a Coxeter–Dynkin diagram is a graph with numerically labeled edges representing the spatial relations between a collection of mirrors... Schläfli symbol Johnson and Bowers Name |
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B7 [14] |
B6 [12] |
B5 [10] |
B4 [8] |
B3 [6] |
B2 [4] |
A5 [6] |
A3 [4] |
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1 | t0{3,3,3,3,3,4} 7-orthoplex Hecatonicosoctaexon (zee) |
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2 | t1{3,3,3,3,3,4} Rectified 7-orthoplex Rectified hecatonicosoctaexon (rez) |
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3 | t2{3,3,3,3,3,4} Birectified 7-orthoplex Birectified hecatonicosoctaexon (barz) |
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4 | t3{4,3,3,3,3,3} Trirectified 7-cube Hepteractihecatonicosoctaexon (sez) |
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5 | t2{4,3,3,3,3,3} Birectified 7-cube Birectified hepteract (bersa) |
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6 | t1{4,3,3,3,3,3} Rectified 7-cube Rectified 7-cube In seven-dimensional geometry, a rectified 7-cube is a convex uniform 7-polytope, being a rectification of the regular 7-cube.There are unique 7 degrees of rectifications, the zeroth being the 7-cube, and the 6th and last being the 7-cube. Vertices of the rectified 7-cube are located at the... Rectified hepteract (rasa) |
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7 | t0{4,3,3,3,3,3} 7-cube Hepteract (hept) |
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8 | t0,1{3,3,3,3,3,4} Truncated 7-orthoplex Truncated 7-orthoplex In seven-dimensional geometry, a truncated 7-orthoplex is a convex uniform 7-polytope, being a truncation of the regular 7-orthoplex.There are 6 truncations of the 7-orthoplex. Vertices of the truncation 7-orthoplex are located as pairs on the edge of the 7-orthoplex. Vertices of the bitruncated... Truncated hecatonicosoctaexon (Taz) |
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9 | t0,2{3,3,3,3,3,4} Cantellated 7-orthoplex Cantellated 7-orthoplex In seven-dimensional geometry, a cantellated 7-orthoplex is a convex uniform 7-polytope, being a cantellation of the regular 7-orthoplex.There are ten degrees of cantellation for the 7-orthoplex, including truncations... Small rhombated hecatonicosoctaexon (Sarz) |
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10 | t1,2{3,3,3,3,3,4} Bitruncated 7-orthoplex Bitruncated hecatonicosoctaexon (Botaz) |
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11 | t0,3{3,3,3,3,3,4} Runcinated 7-orthoplex Runcinated 7-orthoplex In seven-dimensional geometry, a runcinated 7-orthoplex is a convex uniform 7-polytope with 3rd order truncations of the regular 7-orthoplex.There are 16 unique runcinations of the 7-orthoplex with permutations of truncations, and cantellations... Small prismated hecatonicosoctaexon (Spaz) |
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12 | t1,3{3,3,3,3,3,4} Bicantellated 7-orthoplex Small birhombated hecatonicosoctaexon (Sebraz) |
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13 | t2,3{3,3,3,3,3,4} Tritruncated 7-orthoplex Tritruncated hecatonicosoctaexon (Totaz) |
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14 | t0,4{3,3,3,3,3,4} Stericated 7-orthoplex Stericated 7-orthoplex In seven-dimensional geometry, a stericated 7-orthoplex is a convex uniform 7-polytope with 4th order truncations of the regular 7-orthoplex.... Small cellated hecatonicosoctaexon (Scaz) |
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15 | t1,4{3,3,3,3,3,4} Biruncinated 7-orthoplex Small biprismated hecatonicosoctaexon (Sibpaz) |
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16 | t2,4{4,3,3,3,3,3} Tricantellated 7-cube Small trirhombi-hepteractihecatonicosoctaexon (Strasaz) |
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17 | t2,3{4,3,3,3,3,3} Tritruncated 7-cube Tritruncated hepteract (Tatsa) |
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18 | t0,5{3,3,3,3,3,4} Pentellated 7-orthoplex Pentellated 7-orthoplex In seven-dimensional geometry, a pentellated 7-orthoplex is a convex uniform 7-polytope with 5th order truncations of the regular 7-orthoplex.... Small terated hecatonicosoctaexon (Staz) |
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19 | t1,5{4,3,3,3,3,3} Bistericated 7-cube Small bicelli-hepteractihecatonicosoctaexon (Sabcosaz) |
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20 | t1,4{4,3,3,3,3,3} Biruncinated 7-cube Small biprismated hepteract (Sibposa) |
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21 | t1,3{4,3,3,3,3,3} Bicantellated 7-cube Small birhombated hepteract (Sibrosa) |
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22 | t1,2{4,3,3,3,3,3} Bitruncated 7-cube Bitruncated hepteract (Betsa) |
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23 | t0,6{4,3,3,3,3,3} Hexicated 7-cube Hexicated 7-cube In seven-dimensional geometry, a hexicated 7-cube is a convex uniform 7-polytope, including 6th-order truncations from the regular 7-cube.... Small peti-hepteractihecatonicosoctaexon (Suposaz) |
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24 | t0,5{4,3,3,3,3,3} Pentellated 7-cube Pentellated 7-cube In seven-dimensional geometry, a pentellated 7-cube is a convex uniform 7-polytope with 5th order truncations of the regular 7-cube. There are 32 unique pentellations of the 7-cube with permutations of truncations, cantellations, runcinations, and sterications... Small terated hepteract (Stesa) |
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25 | t0,4{4,3,3,3,3,3} Stericated 7-cube Stericated 7-cube In seven-dimensional geometry, a stericated 7-cube is a convex uniform 7-polytope with 4th order truncations of the regular 7-cube.There are 24 unique sterication for the 7-cube with permutations of truncations, cantellations, and runcinations... Small cellated hepteract (Scosa) |
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26 | t0,3{4,3,3,3,3,3} Runcinated 7-cube Runcinated 7-cube In seven-dimensional geometry, a runcinated 7-cube is a convex uniform 7-polytope with 3rd order truncations of the regular 7-cube.There are 16 unique runcinations of the 7-cube with permutations of truncations, and cantellations... Small prismated hepteract (Spesa) |
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27 | t0,2{4,3,3,3,3,3} Cantellated 7-cube Cantellated 7-cube In seven-dimensional geometry, a cantellated 7-cube is a convex uniform 7-polytope, being a cantellation of the regular 7-cube.There are 10 degrees of cantellation for the 7-cube, including truncations... Small rhombated hepteract (Sersa) |
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28 | t0,1{4,3,3,3,3,3} Truncated 7-cube Truncated 7-cube In seven-dimensional geometry, a truncated 7-cube is a convex uniform 7-polytope, being a truncation of the regular 7-cube.There are 6 truncations for the 7-cube. Vertices of the truncated 7-cube are located as pairs on the edge of the 7-cube. Vertices of the bitruncated 7-cube are located on the... Truncated hepteract (Tasa) |
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29 | t0,1,2{3,3,3,3,3,4} Cantitruncated 7-orthoplex Great rhombated hecatonicosoctaexon (Garz) |
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30 | t0,1,3{3,3,3,3,3,4} Runcitruncated 7-orthoplex Prismatotruncated hecatonicosoctaexon (Potaz) |
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31 | t0,2,3{3,3,3,3,3,4} Runcicantellated 7-orthoplex Prismatorhombated hecatonicosoctaexon (Parz) |
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32 | t1,2,3{3,3,3,3,3,4} Bicantitruncated 7-orthoplex Great birhombated hecatonicosoctaexon (Gebraz) |
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33 | t0,1,4{3,3,3,3,3,4} Steritruncated 7-orthoplex Cellitruncated hecatonicosoctaexon (Catz) |
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34 | t0,2,4{3,3,3,3,3,4} Stericantellated 7-orthoplex Cellirhombated hecatonicosoctaexon (Craze) |
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35 | t1,2,4{3,3,3,3,3,4} Biruncitruncated 7-orthoplex Biprismatotruncated hecatonicosoctaexon (Baptize) |
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36 | t0,3,4{3,3,3,3,3,4} Steriruncinated 7-orthoplex Celliprismated hecatonicosoctaexon (Copaz) |
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37 | t1,3,4{3,3,3,3,3,4} Biruncicantellated 7-orthoplex Biprismatorhombated hecatonicosoctaexon (Boparz) |
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38 | t2,3,4{4,3,3,3,3,3} Tricantitruncated 7-cube Great trirhombi-hepteractihecatonicosoctaexon (Gotrasaz) |
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39 | t0,1,5{3,3,3,3,3,4} Pentitruncated 7-orthoplex Teritruncated hecatonicosoctaexon (Tetaz) |
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40 | t0,2,5{3,3,3,3,3,4} Penticantellated 7-orthoplex Terirhombated hecatonicosoctaexon (Teroz) |
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41 | t1,2,5{3,3,3,3,3,4} Bisteritruncated 7-orthoplex Bicellitruncated hecatonicosoctaexon (Boctaz) |
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42 | t0,3,5{3,3,3,3,3,4} Pentiruncinated 7-orthoplex Teriprismated hecatonicosoctaexon (Topaz) |
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43 | t1,3,5{4,3,3,3,3,3} Bistericantellated 7-cube Bicellirhombi-hepteractihecatonicosoctaexon (Bacresaz) |
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44 | t1,3,4{4,3,3,3,3,3} Biruncicantellated 7-cube Biprismatorhombated hepteract (Bopresa) |
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45 | t0,4,5{3,3,3,3,3,4} Pentistericated 7-orthoplex Tericellated hecatonicosoctaexon (Tocaz) |
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46 | t1,2,5{4,3,3,3,3,3} Bisteritruncated 7-cube Bicellitruncated hepteract (Bactasa) |
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47 | t1,2,4{4,3,3,3,3,3} Biruncitruncated 7-cube Biprismatotruncated hepteract (Biptesa) |
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48 | t1,2,3{4,3,3,3,3,3} Bicantitruncated 7-cube Great birhombated hepteract (Gibrosa) |
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49 | t0,1,6{3,3,3,3,3,4} Hexitruncated 7-orthoplex Petitruncated hecatonicosoctaexon (Putaz) |
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50 | t0,2,6{3,3,3,3,3,4} Hexicantellated 7-orthoplex Petirhombated hecatonicosoctaexon (Puraz) |
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51 | t0,4,5{4,3,3,3,3,3} Pentistericated 7-cube Tericellated hepteract (Tacosa) |
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52 | | | t0,3,6{4,3,3,3,3,3} Hexiruncinated 7-cube Petiprismato-hepteractihecatonicosoctaexon (Pupsez) |
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53 | t0,3,5{4,3,3,3,3,3} Pentiruncinated 7-cube Teriprismated hepteract (Tapsa) |
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54 | t0,3,4{4,3,3,3,3,3} Steriruncinated 7-cube Celliprismated hepteract (Capsa) |
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55 | t0,2,6{4,3,3,3,3,3} Hexicantellated 7-cube Petirhombated hepteract (Purosa) |
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56 | t0,2,5{4,3,3,3,3,3} Penticantellated 7-cube Terirhombated hepteract (Tersa) |
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57 | t0,2,4{4,3,3,3,3,3} Stericantellated 7-cube Cellirhombated hepteract (Carsa) |
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58 | t0,2,3{4,3,3,3,3,3} Runcicantellated 7-cube Prismatorhombated hepteract (Parsa) |
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59 | t0,1,6{4,3,3,3,3,3} Hexitruncated 7-cube Petitruncated hepteract (Putsa) |
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60 | t0,1,5{4,3,3,3,3,3} Pentitruncated 7-cube Teritruncated hepteract (Tetsa) |
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61 | t0,1,4{4,3,3,3,3,3} Steritruncated 7-cube Cellitruncated hepteract (Catsa) |
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62 | t0,1,3{4,3,3,3,3,3} Runcitruncated 7-cube Prismatotruncated hepteract (Petsa) |
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63 | t0,1,2{4,3,3,3,3,3} Cantitruncated 7-cube Great rhombated hepteract (Gersa) |
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64 | t0,1,2,3{3,3,3,3,3,4} Runcicantitruncated 7-orthoplex Great prismated hecatonicosoctaexon (Gopaz) |
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65 | t0,1,2,4{3,3,3,3,3,4} Stericantitruncated 7-orthoplex Celligreatorhombated hecatonicosoctaexon (Cogarz) |
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66 | t0,1,3,4{3,3,3,3,3,4} Steriruncitruncated 7-orthoplex Celliprismatotruncated hecatonicosoctaexon (Captaz) |
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67 | t0,2,3,4{3,3,3,3,3,4} Steriruncicantellated 7-orthoplex Celliprismatorhombated hecatonicosoctaexon (Caparz) |
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68 | t1,2,3,4{3,3,3,3,3,4} Biruncicantitruncated 7-orthoplex Great biprismated hecatonicosoctaexon (Gibpaz) |
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69 | t0,1,2,5{3,3,3,3,3,4} Penticantitruncated 7-orthoplex Terigreatorhombated hecatonicosoctaexon (Tograz) |
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70 | t0,1,3,5{3,3,3,3,3,4} Pentiruncitruncated 7-orthoplex Teriprismatotruncated hecatonicosoctaexon (Toptaz) |
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71 | t0,2,3,5{3,3,3,3,3,4} Pentiruncicantellated 7-orthoplex Teriprismatorhombated hecatonicosoctaexon (Toparz) |
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72 | t1,2,3,5{3,3,3,3,3,4} Bistericantitruncated 7-orthoplex Bicelligreatorhombated hecatonicosoctaexon (Becogarz) |
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73 | t0,1,4,5{3,3,3,3,3,4} Pentisteritruncated 7-orthoplex Tericellitruncated hecatonicosoctaexon (Tacotaz) |
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74 | t0,2,4,5{3,3,3,3,3,4} Pentistericantellated 7-orthoplex Tericellirhombated hecatonicosoctaexon (Tocarz) |
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75 | t1,2,4,5{4,3,3,3,3,3} Bisteriruncitruncated 7-cube Bicelliprismatotrunki-hepteractihecatonicosoctaexon (Bocaptosaz) |
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76 | t0,3,4,5{3,3,3,3,3,4} Pentisteriruncinated 7-orthoplex Tericelliprismated hecatonicosoctaexon (Tecpaz) |
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77 | t1,2,3,5{4,3,3,3,3,3} Bistericantitruncated 7-cube Bicelligreatorhombated hepteract (Becgresa) |
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78 | t1,2,3,4{4,3,3,3,3,3} Biruncicantitruncated 7-cube Great biprismated hepteract (Gibposa) |
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79 | t0,1,2,6{3,3,3,3,3,4} Hexicantitruncated 7-orthoplex Petigreatorhombated hecatonicosoctaexon (Pugarez) |
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80 | t0,1,3,6{3,3,3,3,3,4} Hexiruncitruncated 7-orthoplex Petiprismatotruncated hecatonicosoctaexon (Papataz) |
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81 | t0,2,3,6{3,3,3,3,3,4} Hexiruncicantellated 7-orthoplex Petiprismatorhombated hecatonicosoctaexon (Puparez) |
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82 | t0,3,4,5{4,3,3,3,3,3} Pentisteriruncinated 7-cube Tericelliprismated hepteract (Tecpasa) |
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83 | t0,1,4,6{3,3,3,3,3,4} Hexisteritruncated 7-orthoplex Peticellitruncated hecatonicosoctaexon (Pucotaz) |
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84 | t0,2,4,6{4,3,3,3,3,3} Hexistericantellated 7-cube Peticellirhombi-hepteractihecatonicosoctaexon (Pucrosaz) |
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85 | t0,2,4,5{4,3,3,3,3,3} Pentistericantellated 7-cube Tericellirhombated hepteract (Tecresa) |
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86 | t0,2,3,6{4,3,3,3,3,3} Hexiruncicantellated 7-cube Petiprismatorhombated hepteract (Pupresa) |
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87 | t0,2,3,5{4,3,3,3,3,3} Pentiruncicantellated 7-cube Teriprismatorhombated hepteract (Topresa) |
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88 | t0,2,3,4{4,3,3,3,3,3} Steriruncicantellated 7-cube Celliprismatorhombated hepteract (Copresa) |
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89 | t0,1,5,6{4,3,3,3,3,3} Hexipentitruncated 7-cube Petiteritrunki-hepteractihecatonicosoctaexon (Putatosez) |
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90 | t0,1,4,6{4,3,3,3,3,3} Hexisteritruncated 7-cube Peticellitruncated hepteract (Pacutsa) |
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91 | t0,1,4,5{4,3,3,3,3,3} Pentisteritruncated 7-cube Tericellitruncated hepteract (Tecatsa) |
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92 | t0,1,3,6{4,3,3,3,3,3} Hexiruncitruncated 7-cube Petiprismatotruncated hepteract (Pupetsa) |
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93 | t0,1,3,5{4,3,3,3,3,3} Pentiruncitruncated 7-cube Teriprismatotruncated hepteract (Toptosa) |
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94 | t0,1,3,4{4,3,3,3,3,3} Steriruncitruncated 7-cube Celliprismatotruncated hepteract (Captesa) |
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95 | t0,1,2,6{4,3,3,3,3,3} Hexicantitruncated 7-cube Petigreatorhombated hepteract (Pugrosa) |
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96 | t0,1,2,5{4,3,3,3,3,3} Penticantitruncated 7-cube Terigreatorhombated hepteract (Togresa) |
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97 | t0,1,2,4{4,3,3,3,3,3} Stericantitruncated 7-cube Celligreatorhombated hepteract (Cogarsa) |
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98 | t0,1,2,3{4,3,3,3,3,3} Runcicantitruncated 7-cube Great prismated hepteract (Gapsa) |
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99 | t0,1,2,3,4{3,3,3,3,3,4} Steriruncicantitruncated 7-orthoplex Great cellated hecatonicosoctaexon (Gocaz) |
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100 | | | t0,1,2,3,5{3,3,3,3,3,4} Pentiruncicantitruncated 7-orthoplex Terigreatoprismated hecatonicosoctaexon (Tegopaz) |
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101 | | | t0,1,2,4,5{3,3,3,3,3,4} Pentistericantitruncated 7-orthoplex Tericelligreatorhombated hecatonicosoctaexon (Tecagraz) |
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102 | | | t0,1,3,4,5{3,3,3,3,3,4} Pentisteriruncitruncated 7-orthoplex Tericelliprismatotruncated hecatonicosoctaexon (Tecpotaz) |
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103 | | | t0,2,3,4,5{3,3,3,3,3,4} Pentisteriruncicantellated 7-orthoplex Tericelliprismatorhombated hecatonicosoctaexon (Tacparez) |
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104 | | | t1,2,3,4,5{4,3,3,3,3,3} Bisteriruncicantitruncated 7-cube Great bicelli-hepteractihecatonicosoctaexon (Gabcosaz) |
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105 | t0,1,2,3,6{3,3,3,3,3,4} Hexiruncicantitruncated 7-orthoplex Petigreatoprismated hecatonicosoctaexon (Pugopaz) |
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106 | | | t0,1,2,4,6{3,3,3,3,3,4} Hexistericantitruncated 7-orthoplex Peticelligreatorhombated hecatonicosoctaexon (Pucagraz) |
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107 | | | t0,1,3,4,6{3,3,3,3,3,4} Hexisteriruncitruncated 7-orthoplex Peticelliprismatotruncated hecatonicosoctaexon (Pucpotaz) |
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108 | | | t0,2,3,4,6{4,3,3,3,3,3} Hexisteriruncicantellated 7-cube Peticelliprismatorhombi-hepteractihecatonicosoctaexon (Pucprosaz) |
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109 | | | t0,2,3,4,5{4,3,3,3,3,3} Pentisteriruncicantellated 7-cube Tericelliprismatorhombated hepteract (Tocpresa) |
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110 | t0,1,2,5,6{3,3,3,3,3,4} Hexipenticantitruncated 7-orthoplex Petiterigreatorhombated hecatonicosoctaexon (Putegraz) |
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111 | | | t0,1,3,5,6{4,3,3,3,3,3} Hexipentiruncitruncated 7-cube Petiteriprismatotrunki-hepteractihecatonicosoctaexon (Putpetsaz) |
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112 | | | t0,1,3,4,6{4,3,3,3,3,3} Hexisteriruncitruncated 7-cube Peticelliprismatotruncated hepteract (Pucpetsa) |
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113 | | | t0,1,3,4,5{4,3,3,3,3,3} Pentisteriruncitruncated 7-cube Tericelliprismatotruncated hepteract (Tecpetsa) |
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114 | t0,1,2,5,6{4,3,3,3,3,3} Hexipenticantitruncated 7-cube Petiterigreatorhombated hepteract (Putgresa) |
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115 | | | t0,1,2,4,6{4,3,3,3,3,3} Hexistericantitruncated 7-cube Peticelligreatorhombated hepteract (Pucagrosa) |
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116 | | | t0,1,2,4,5{4,3,3,3,3,3} Pentistericantitruncated 7-cube Tericelligreatorhombated hepteract (Tecgresa) |
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117 | t0,1,2,3,6{4,3,3,3,3,3} Hexiruncicantitruncated 7-cube Petigreatoprismated hepteract (Pugopsa) |
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118 | | | t0,1,2,3,5{4,3,3,3,3,3} Pentiruncicantitruncated 7-cube Terigreatoprismated hepteract (Togapsa) |
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119 | t0,1,2,3,4{4,3,3,3,3,3} Steriruncicantitruncated 7-cube Great cellated hepteract (Gacosa) |
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120 | | | t0,1,2,3,4,5{3,3,3,3,3,4} Pentisteriruncicantitruncated 7-orthoplex Great terated hecatonicosoctaexon (Gotaz) |
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121 | | | t0,1,2,3,4,6{3,3,3,3,3,4} Hexisteriruncicantitruncated 7-orthoplex Petigreatocellated hecatonicosoctaexon (Pugacaz) |
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122 | | | t0,1,2,3,5,6{3,3,3,3,3,4} Hexipentiruncicantitruncated 7-orthoplex Petiterigreatoprismated hecatonicosoctaexon (Putgapaz) |
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123 | | | t0,1,2,4,5,6{4,3,3,3,3,3} Hexipentistericantitruncated 7-cube Petitericelligreatorhombi-hepteractihecatonicosoctaexon (Putcagrasaz) |
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124 | | | t0,1,2,3,5,6{4,3,3,3,3,3} Hexipentiruncicantitruncated 7-cube Petiterigreatoprismated hepteract (Putgapsa) |
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125 | | | t0,1,2,3,4,6{4,3,3,3,3,3} Hexisteriruncicantitruncated 7-cube Petigreatocellated hepteract (Pugacasa) |
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126 | | | t0,1,2,3,4,5{4,3,3,3,3,3} Pentisteriruncicantitruncated 7-cube Great terated hepteract (Gotesa) |
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127 | | | t0,1,2,3,4,5,6{4,3,3,3,3,3} Omnitruncated 7-cube Great peti-hepteractihecatonicosoctaexon (Guposaz) |
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128 | h0{4,3,3,3,3,3} 7-demicube Hemihepteract |