In 8-dimensional geometry, there are 256 uniform polytopes with B8 symmetry. There are two regular forms, the 8-orthoplex, and 8-cube with 16 and 256 vertices respectively. The 8-demicube is added with half the symmetry.

They can be visualized as symmetric orthographic projections in Coxeter planes of the B8 Coxeter group, and other subgroups.

Graphs

Symmetric orthographic projections of these 256 polytopes can be made in the B8, B7, B6, B5, B4, B3, B2, A7, A5, A3, Coxeter planes. Ak has [k+1] symmetry, and Bk has [2k] symmetry.

These 256 polytopes are each shown in these 10 symmetry planes, with vertices and edges drawn, and vertices colored by the number of overlapping vertices in each projective position.

Element counts | Coxeter-Dynkin diagram SchlĂ¤fli symbol Name |
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B_{8}[16] |
B_{7}[14] |
B_{6}[12] |
B_{5}[10] |
B_{4}[8] |
B_{3}[6] |
B_{2}[4] |
A_{7}[8] |
A_{5}[6] |
A_{3}[4] |
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1 | t _{0}{3,3,3,3,3,3,4}8-orthoplex Diacosipentacontahexazetton (ek) |
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2 | t _{1}{3,3,3,3,3,3,4}Rectified 8-orthoplex Rectified diacosipentacontahexazetton (rek) |
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3 | t _{2}{3,3,3,3,3,3,4}Birectified 8-orthoplex Birectified diacosipentacontahexazetton (bark) |
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4 | t _{3}{3,3,3,3,3,3,4}Trirectified 8-orthoplex Trirectified diacosipentacontahexazetton (tark) |
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5 | t _{3}{4,3,3,3,3,3,3}Trirectified 8-cube Trirectified octeract (tro) |
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6 | t _{2}{4,3,3,3,3,3,3}Birectified 8-cube Birectified octeract (bro) |
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7 | t _{1}{4,3,3,3,3,3,3}Rectified 8-cube Rectified octeract (recto) |
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8 | t _{0}{4,3,3,3,3,3,3}8-cube Octeract (octo) |
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9 | t _{0,1}{3,3,3,3,3,3,4}Truncated 8-orthoplex Truncated diacosipentacontahexazetton (tek) |
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10 | t _{0,2}{3,3,3,3,3,3,4}Cantellated 8-orthoplex Small rhombated diacosipentacontahexazetton (srek) |
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11 | t _{1,2}{3,3,3,3,3,3,4}Bitruncated 8-orthoplex Bitruncated diacosipentacontahexazetton (batek) |
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12 | t _{0,3}{3,3,3,3,3,3,4}Runcinated 8-orthoplex Small prismated diacosipentacontahexazetton (spek) |
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13 | t _{1,3}{3,3,3,3,3,3,4}Bicantellated 8-orthoplex Small birhombated diacosipentacontahexazetton (sabork) |
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14 | t _{2,3}{3,3,3,3,3,3,4}Tritruncated 8-orthoplex Tritruncated diacosipentacontahexazetton (tatek) |
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15 | t _{0,4}{3,3,3,3,3,3,4}Stericated 8-orthoplex Small cellated diacosipentacontahexazetton (scak) |
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16 | t _{1,4}{3,3,3,3,3,3,4}Biruncinated 8-orthoplex Small biprismated diacosipentacontahexazetton (sabpek) |
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17 | t _{2,4}{3,3,3,3,3,3,4}Tricantellated 8-orthoplex Small trirhombated diacosipentacontahexazetton (satrek) |
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18 | t _{3,4}{4,3,3,3,3,3,3}Quadritruncated 8-cube Octeractidiacosipentacontahexazetton (oke) |
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19 | t _{0,5}{3,3,3,3,3,3,4}Pentellated 8-orthoplex Small terated diacosipentacontahexazetton (setek) |
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20 | t _{1,5}{3,3,3,3,3,3,4}Bistericated 8-orthoplex Small bicellated diacosipentacontahexazetton (sibcak) |
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21 | t _{2,5}{4,3,3,3,3,3,3}Triruncinated 8-cube Small triprismato-octeractidiacosipentacontahexazetton (sitpoke) |
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22 | t _{2,4}{4,3,3,3,3,3,3}Tricantellated 8-cube Small trirhombated octeract (satro) |
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23 | t _{2,3}{4,3,3,3,3,3,3}Tritruncated 8-cube Tritruncated octeract (tato) |
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24 | t _{0,6}{3,3,3,3,3,3,4}Hexicated 8-orthoplex Small petated diacosipentacontahexazetton (supek) |
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25 | t _{1,6}{4,3,3,3,3,3,3}Bipentellated 8-cube Small biteri-octeractidiacosipentacontahexazetton (sabtoke) |
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26 | t _{1,5}{4,3,3,3,3,3,3}Bistericated 8-cube Small bicellated octeract (sobco) |
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27 | t _{1,4}{4,3,3,3,3,3,3}Biruncinated 8-cube Small biprismated octeract (sabepo) |
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28 | t _{1,3}{4,3,3,3,3,3,3}Bicantellated 8-cube Small birhombated octeract (subro) |
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29 | t _{1,2}{4,3,3,3,3,3,3}Bitruncated 8-cube Bitruncated octeract (bato) |
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30 | t _{0,7}{4,3,3,3,3,3,3}Heptellated 8-cube Small exi-octeractidiacosipentacontahexazetton (saxoke) |
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31 | t _{0,6}{4,3,3,3,3,3,3}Hexicated 8-cube Small petated octeract (supo) |
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32 | t _{0,5}{4,3,3,3,3,3,3}Pentellated 8-cube Small terated octeract (soto) |
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33 | t _{0,4}{4,3,3,3,3,3,3}Stericated 8-cube Small cellated octeract (soco) |
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34 | t _{0,3}{4,3,3,3,3,3,3}Runcinated 8-cube Small prismated octeract (sopo) |
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35 | t _{0,2}{4,3,3,3,3,3,3}Cantellated 8-cube Small rhombated octeract (soro) |
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36 | t _{0,1}{4,3,3,3,3,3,3}Truncated 8-cube Truncated octeract (tocto) |
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37 | t _{0,1,2}{3,3,3,3,3,3,4}Cantitruncated 8-orthoplex Great rhombated diacosipentacontahexazetton |
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38 | t _{0,1,3}{3,3,3,3,3,3,4}Runcitruncated 8-orthoplex Prismatotruncated diacosipentacontahexazetton |
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39 | t _{0,2,3}{3,3,3,3,3,3,4}Runcicantellated 8-orthoplex Prismatorhombated diacosipentacontahexazetton |
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40 | t _{1,2,3}{3,3,3,3,3,3,4}Bicantitruncated 8-orthoplex Great birhombated diacosipentacontahexazetton |
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41 | t _{0,1,4}{3,3,3,3,3,3,4}Steritruncated 8-orthoplex Cellitruncated diacosipentacontahexazetton |
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42 | t _{0,2,4}{3,3,3,3,3,3,4}Stericantellated 8-orthoplex Cellirhombated diacosipentacontahexazetton |
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43 | t _{1,2,4}{3,3,3,3,3,3,4}Biruncitruncated 8-orthoplex Biprismatotruncated diacosipentacontahexazetton |
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44 | t _{0,3,4}{3,3,3,3,3,3,4}Steriruncinated 8-orthoplex Celliprismated diacosipentacontahexazetton |
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45 | t _{1,3,4}{3,3,3,3,3,3,4}Biruncicantellated 8-orthoplex Biprismatorhombated diacosipentacontahexazetton |
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46 | t _{2,3,4}{3,3,3,3,3,3,4}Tricantitruncated 8-orthoplex Great trirhombated diacosipentacontahexazetton |
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47 | t _{0,1,5}{3,3,3,3,3,3,4}Pentitruncated 8-orthoplex Teritruncated diacosipentacontahexazetton |
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48 | t _{0,2,5}{3,3,3,3,3,3,4}Penticantellated 8-orthoplex Terirhombated diacosipentacontahexazetton |
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49 | t _{1,2,5}{3,3,3,3,3,3,4}Bisteritruncated 8-orthoplex Bicellitruncated diacosipentacontahexazetton |
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50 | t _{0,3,5}{3,3,3,3,3,3,4}Pentiruncinated 8-orthoplex Teriprismated diacosipentacontahexazetton |
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51 | t _{1,3,5}{3,3,3,3,3,3,4}Bistericantellated 8-orthoplex Bicellirhombated diacosipentacontahexazetton |
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52 | t _{2,3,5}{3,3,3,3,3,3,4}Triruncitruncated 8-orthoplex Triprismatotruncated diacosipentacontahexazetton |
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53 | t _{0,4,5}{3,3,3,3,3,3,4}Pentistericated 8-orthoplex Tericellated diacosipentacontahexazetton |
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54 | t _{1,4,5}{3,3,3,3,3,3,4}Bisteriruncinated 8-orthoplex Bicelliprismated diacosipentacontahexazetton |
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55 | t _{2,3,5}{4,3,3,3,3,3,3}Triruncitruncated 8-cube Triprismatotruncated octeract |
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56 | t _{2,3,4}{4,3,3,3,3,3,3}Tricantitruncated 8-cube Great trirhombated octeract |
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57 | t _{0,1,6}{3,3,3,3,3,3,4}Hexitruncated 8-orthoplex Petitruncated diacosipentacontahexazetton |
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58 | t _{0,2,6}{3,3,3,3,3,3,4}Hexicantellated 8-orthoplex Petirhombated diacosipentacontahexazetton |
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59 | t _{1,2,6}{3,3,3,3,3,3,4}Bipentitruncated 8-orthoplex Biteritruncated diacosipentacontahexazetton |
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60 | t _{0,3,6}{3,3,3,3,3,3,4}Hexiruncinated 8-orthoplex Petiprismated diacosipentacontahexazetton |
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61 | t _{1,3,6}{3,3,3,3,3,3,4}Bipenticantellated 8-orthoplex Biterirhombated diacosipentacontahexazetton |
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62 | t _{1,4,5}{4,3,3,3,3,3,3}Bisteriruncinated 8-cube Bicelliprismated octeract |
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63 | t _{0,4,6}{3,3,3,3,3,3,4}Hexistericated 8-orthoplex Peticellated diacosipentacontahexazetton |
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64 | t _{1,3,6}{4,3,3,3,3,3,3}Bipenticantellated 8-cube Biterirhombated octeract |
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65 | t _{1,3,5}{4,3,3,3,3,3,3}Bistericantellated 8-cube Bicellirhombated octeract |
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66 | t _{1,3,4}{4,3,3,3,3,3,3}Biruncicantellated 8-cube Biprismatorhombated octeract |
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67 | t _{0,5,6}{3,3,3,3,3,3,4}Hexipentellated 8-orthoplex Petiterated diacosipentacontahexazetton |
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68 | t _{1,2,6}{4,3,3,3,3,3,3}Bipentitruncated 8-cube Biteritruncated octeract |
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69 | t _{1,2,5}{4,3,3,3,3,3,3}Bisteritruncated 8-cube Bicellitruncated octeract |
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70 | t _{1,2,4}{4,3,3,3,3,3,3}Biruncitruncated 8-cube Biprismatotruncated octeract |
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71 | t _{1,2,3}{4,3,3,3,3,3,3}Bicantitruncated 8-cube Great birhombated octeract |
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72 | t _{0,1,7}{3,3,3,3,3,3,4}Heptitruncated 8-orthoplex Exitruncated diacosipentacontahexazetton |
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73 | t _{0,2,7}{3,3,3,3,3,3,4}Hepticantellated 8-orthoplex Exirhombated diacosipentacontahexazetton |
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74 | t _{0,5,6}{4,3,3,3,3,3,3}Hexipentellated 8-cube Petiterated octeract |
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75 | t _{0,3,7}{3,3,3,3,3,3,4}Heptiruncinated 8-orthoplex Exiprismated diacosipentacontahexazetton |
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76 | t _{0,4,6}{4,3,3,3,3,3,3}Hexistericated 8-cube Peticellated octeract |
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77 | t _{0,4,5}{4,3,3,3,3,3,3}Pentistericated 8-cube Tericellated octeract |
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78 | t _{0,3,7}{4,3,3,3,3,3,3}Heptiruncinated 8-cube Exiprismated octeract |
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79 | t _{0,3,6}{4,3,3,3,3,3,3}Hexiruncinated 8-cube Petiprismated octeract |
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80 | t _{0,3,5}{4,3,3,3,3,3,3}Pentiruncinated 8-cube Teriprismated octeract |
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81 | t _{0,3,4}{4,3,3,3,3,3,3}Steriruncinated 8-cube Celliprismated octeract |
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82 | t _{0,2,7}{4,3,3,3,3,3,3}Hepticantellated 8-cube Exirhombated octeract |
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83 | t _{0,2,6}{4,3,3,3,3,3,3}Hexicantellated 8-cube Petirhombated octeract |
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84 | t _{0,2,5}{4,3,3,3,3,3,3}Penticantellated 8-cube Terirhombated octeract |
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85 | t _{0,2,4}{4,3,3,3,3,3,3}Stericantellated 8-cube Cellirhombated octeract |
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86 | t _{0,2,3}{4,3,3,3,3,3,3}Runcicantellated 8-cube Prismatorhombated octeract |
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87 | t _{0,1,7}{4,3,3,3,3,3,3}Heptitruncated 8-cube Exitruncated octeract |
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88 | t _{0,1,6}{4,3,3,3,3,3,3}Hexitruncated 8-cube Petitruncated octeract |
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89 | t _{0,1,5}{4,3,3,3,3,3,3}Pentitruncated 8-cube Teritruncated octeract |
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90 | t _{0,1,4}{4,3,3,3,3,3,3}Steritruncated 8-cube Cellitruncated octeract |
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91 | t _{0,1,3}{4,3,3,3,3,3,3}Runcitruncated 8-cube Prismatotruncated octeract |
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92 | t _{0,1,2}{4,3,3,3,3,3,3}Cantitruncated 8-cube Great rhombated octeract |
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93 | t _{0,1,2,3}{3,3,3,3,3,3,4}Runcicantitruncated 8-orthoplex Great prismated diacosipentacontahexazetton |
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94 | t _{0,1,2,4}{3,3,3,3,3,3,4}Stericantitruncated 8-orthoplex Celligreatorhombated diacosipentacontahexazetton |
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95 | t _{0,1,3,4}{3,3,3,3,3,3,4}Steriruncitruncated 8-orthoplex Celliprismatotruncated diacosipentacontahexazetton |
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96 | t _{0,2,3,4}{3,3,3,3,3,3,4}Steriruncicantellated 8-orthoplex Celliprismatorhombated diacosipentacontahexazetton |
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97 | t _{1,2,3,4}{3,3,3,3,3,3,4}Biruncicantitruncated 8-orthoplex Great biprismated diacosipentacontahexazetton |
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98 | t _{0,1,2,5}{3,3,3,3,3,3,4}Penticantitruncated 8-orthoplex Terigreatorhombated diacosipentacontahexazetton |
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99 | t _{0,1,3,5}{3,3,3,3,3,3,4}Pentiruncitruncated 8-orthoplex Teriprismatotruncated diacosipentacontahexazetton |
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100 | t _{0,2,3,5}{3,3,3,3,3,3,4}Pentiruncicantellated 8-orthoplex Teriprismatorhombated diacosipentacontahexazetton |
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101 | t _{1,2,3,5}{3,3,3,3,3,3,4}Bistericantitruncated 8-orthoplex Bicelligreatorhombated diacosipentacontahexazetton |
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102 | t _{0,1,4,5}{3,3,3,3,3,3,4}Pentisteritruncated 8-orthoplex Tericellitruncated diacosipentacontahexazetton |
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103 | t _{0,2,4,5}{3,3,3,3,3,3,4}Pentistericantellated 8-orthoplex Tericellirhombated diacosipentacontahexazetton |
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104 | t _{1,2,4,5}{3,3,3,3,3,3,4}Bisteriruncitruncated 8-orthoplex Bicelliprismatotruncated diacosipentacontahexazetton |
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105 | t _{0,3,4,5}{3,3,3,3,3,3,4}Pentisteriruncinated 8-orthoplex Tericelliprismated diacosipentacontahexazetton |
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106 | t _{1,3,4,5}{3,3,3,3,3,3,4}Bisteriruncicantellated 8-orthoplex Bicelliprismatorhombated diacosipentacontahexazetton |
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107 | t _{2,3,4,5}{4,3,3,3,3,3,3}Triruncicantitruncated 8-cube Great triprismato-octeractidiacosipentacontahexazetton |
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108 | t _{0,1,2,6}{3,3,3,3,3,3,4}Hexicantitruncated 8-orthoplex Petigreatorhombated diacosipentacontahexazetton |
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109 | t _{0,1,3,6}{3,3,3,3,3,3,4}Hexiruncitruncated 8-orthoplex Petiprismatotruncated diacosipentacontahexazetton |
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110 | t _{0,2,3,6}{3,3,3,3,3,3,4}Hexiruncicantellated 8-orthoplex Petiprismatorhombated diacosipentacontahexazetton |
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111 | t _{1,2,3,6}{3,3,3,3,3,3,4}Bipenticantitruncated 8-orthoplex Biterigreatorhombated diacosipentacontahexazetton |
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112 | t _{0,1,4,6}{3,3,3,3,3,3,4}Hexisteritruncated 8-orthoplex Peticellitruncated diacosipentacontahexazetton |
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113 | t _{0,2,4,6}{3,3,3,3,3,3,4}Hexistericantellated 8-orthoplex Peticellirhombated diacosipentacontahexazetton |
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114 | t _{1,2,4,6}{3,3,3,3,3,3,4}Bipentiruncitruncated 8-orthoplex Biteriprismatotruncated diacosipentacontahexazetton |
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115 | t _{0,3,4,6}{3,3,3,3,3,3,4}Hexisteriruncinated 8-orthoplex Peticelliprismated diacosipentacontahexazetton |
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116 | t _{1,3,4,6}{4,3,3,3,3,3,3}Bipentiruncicantellated 8-cube Biteriprismatorhombi-octeractidiacosipentacontahexazetton |
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117 | t _{1,3,4,5}{4,3,3,3,3,3,3}Bisteriruncicantellated 8-cube Bicelliprismatorhombated octeract |
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118 | t _{0,1,5,6}{3,3,3,3,3,3,4}Hexipentitruncated 8-orthoplex Petiteritruncated diacosipentacontahexazetton |
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119 | t _{0,2,5,6}{3,3,3,3,3,3,4}Hexipenticantellated 8-orthoplex Petiterirhombated diacosipentacontahexazetton |
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120 | t _{1,2,5,6}{4,3,3,3,3,3,3}Bipentisteritruncated 8-cube Bitericellitrunki-octeractidiacosipentacontahexazetton |
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121 | t _{0,3,5,6}{3,3,3,3,3,3,4}Hexipentiruncinated 8-orthoplex Petiteriprismated diacosipentacontahexazetton |
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122 | t _{1,2,4,6}{4,3,3,3,3,3,3}Bipentiruncitruncated 8-cube Biteriprismatotruncated octeract |
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123 | t _{1,2,4,5}{4,3,3,3,3,3,3}Bisteriruncitruncated 8-cube Bicelliprismatotruncated octeract |
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124 | t _{0,4,5,6}{3,3,3,3,3,3,4}Hexipentistericated 8-orthoplex Petitericellated diacosipentacontahexazetton |
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125 | t _{1,2,3,6}{4,3,3,3,3,3,3}Bipenticantitruncated 8-cube Biterigreatorhombated octeract |
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126 | t _{1,2,3,5}{4,3,3,3,3,3,3}Bistericantitruncated 8-cube Bicelligreatorhombated octeract |
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127 | t _{1,2,3,4}{4,3,3,3,3,3,3}Biruncicantitruncated 8-cube Great biprismated octeract |
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128 | t _{0,1,2,7}{3,3,3,3,3,3,4}Hepticantitruncated 8-orthoplex Exigreatorhombated diacosipentacontahexazetton |
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129 | t _{0,1,3,7}{3,3,3,3,3,3,4}Heptiruncitruncated 8-orthoplex Exiprismatotruncated diacosipentacontahexazetton |
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130 | t _{0,2,3,7}{3,3,3,3,3,3,4}Heptiruncicantellated 8-orthoplex Exiprismatorhombated diacosipentacontahexazetton |
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131 | t _{0,4,5,6}{4,3,3,3,3,3,3}Hexipentistericated 8-cube Petitericellated octeract |
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132 | t _{0,1,4,7}{3,3,3,3,3,3,4}Heptisteritruncated 8-orthoplex Exicellitruncated diacosipentacontahexazetton |
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133 | t _{0,2,4,7}{3,3,3,3,3,3,4}Heptistericantellated 8-orthoplex Exicellirhombated diacosipentacontahexazetton |
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134 | t _{0,3,5,6}{4,3,3,3,3,3,3}Hexipentiruncinated 8-cube Petiteriprismated octeract |
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135 | t _{0,3,4,7}{4,3,3,3,3,3,3}Heptisteriruncinated 8-cube Exicelliprismato-octeractidiacosipentacontahexazetton |
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136 | t _{0,3,4,6}{4,3,3,3,3,3,3}Hexisteriruncinated 8-cube Peticelliprismated octeract |
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137 | t _{0,3,4,5}{4,3,3,3,3,3,3}Pentisteriruncinated 8-cube Tericelliprismated octeract |
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138 | t _{0,1,5,7}{3,3,3,3,3,3,4}Heptipentitruncated 8-orthoplex Exiteritruncated diacosipentacontahexazetton |
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139 | t _{0,2,5,7}{4,3,3,3,3,3,3}Heptipenticantellated 8-cube Exiterirhombi-octeractidiacosipentacontahexazetton |
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140 | t _{0,2,5,6}{4,3,3,3,3,3,3}Hexipenticantellated 8-cube Petiterirhombated octeract |
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141 | t _{0,2,4,7}{4,3,3,3,3,3,3}Heptistericantellated 8-cube Exicellirhombated octeract |
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142 | t _{0,2,4,6}{4,3,3,3,3,3,3}Hexistericantellated 8-cube Peticellirhombated octeract |
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143 | t _{0,2,4,5}{4,3,3,3,3,3,3}Pentistericantellated 8-cube Tericellirhombated octeract |
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144 | t _{0,2,3,7}{4,3,3,3,3,3,3}Heptiruncicantellated 8-cube Exiprismatorhombated octeract |
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145 | t _{0,2,3,6}{4,3,3,3,3,3,3}Hexiruncicantellated 8-cube Petiprismatorhombated octeract |
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146 |