uniform, polytope, graphs, three, regular, related, uniform, polytopes, simplex, rectified, simplex, truncated, simplex, cantellated, simplex, runcinated, simplex, stericated, simplex, pentellated, simplex, hexicated, simplex, heptellated, simplex, octellated,. Graphs of three regular and related uniform polytopes 9 simplex Rectified 9 simplex Truncated 9 simplex Cantellated 9 simplex Runcinated 9 simplex Stericated 9 simplex Pentellated 9 simplex Hexicated 9 simplex Heptellated 9 simplex Octellated 9 simplex 9 orthoplex 9 cube Truncated 9 orthoplex Truncated 9 cube Rectified 9 orthoplex Rectified 9 cube 9 demicube Truncated 9 demicube In nine dimensional geometry a nine dimensional polytope or 9 polytope is a polytope contained by 8 polytope facets Each 7 polytope ridge being shared by exactly two 8 polytope facets A uniform 9 polytope is one which is vertex transitive and constructed from uniform 8 polytope facets Contents 1 Regular 9 polytopes 2 Euler characteristic 3 Uniform 9 polytopes by fundamental Coxeter groups 4 The A9 family 5 The B9 family 6 The D9 family 7 Regular and uniform honeycombs 7 1 Regular and uniform hyperbolic honeycombs 8 References 9 External links Regular 9 polytopes Edit Regular 9 polytopes can be represented by the Schlafli symbol p q r s t u v w with w p q r s t u v 8 polytope facets around each peak There are exactly three such convex regular 9 polytopes 3 3 3 3 3 3 3 3 9 simplex 4 3 3 3 3 3 3 3 9 cube 3 3 3 3 3 3 3 4 9 orthoplex There are no nonconvex regular 9 polytopes Euler characteristic Edit The topology of any given 9 polytope is defined by its Betti numbers and torsion coefficients 91 1 93 The value of the Euler characteristic used to characterise polyhedra does not generalize usefully to higher dimensions whatever their underlying topology This inadequacy of the Euler characteristic to reliably distinguish between different topologies in higher dimensions led to the discovery of the more sophisticated Betti numbers 91 1 93 Similarly the notion of orientability of a polyhedron is insufficient to characterise the surface twistings of toroidal polytopes and this led to the use of torsion coefficients 91 1 93 Uniform 9 polytopes by fundamental Coxeter groups Edit Uniform 9 polytopes with reflective symmetry can be generated by these three Coxeter groups represented by permutations of rings of the Coxeter Dynkin diagrams Coxeter group Coxeter Dynkin diagram A9 38 B9 4 37 D9 36 1 1 Selected regular and uniform 9 polytopes from each family include Simplex family A9 38 271 uniform 9 polytopes as permutations of rings in the group diagram including one regular 38 9 simplex or deca 9 tope or decayotton Hypercube orthoplex family B9 4 38 511 uniform 9 polytopes as permutations of rings in the group diagram including two regular ones 4 37 9 cube or enneract 37 4 9 orthoplex or enneacross Demihypercube D9 family 36 1 1 383 uniform 9 polytope as permutations of rings in the group diagram including 31 6 1 9 demicube or demienneract 161 also as h 4 38 36 1 1 9 orthoplex 611 The A9 family Edit The A9 family has symmetry of order 3628800 10 factorial There are 256 16 1 271 forms based on all permutations of the Coxeter Dynkin diagrams with one or more rings These are all enumerated below Bowers style acronym names are given in parentheses for cross referencing Graph Coxeter Dynkin diagramSchlafli symbolName Element counts 8 faces 7 faces 6 faces 5 faces 4 faces Cells Faces Edges Vertices 1 t0 3 3 3 3 3 3 3 3 9 simplex day 10 45 120 210 252 210 120 45 10 2 t1 3 3 3 3 3 3 3 3 Rectified 9 simplex reday 360 45 3 t2 3 3 3 3 3 3 3 3 Birectified 9 simplex breday 1260 120 4 t3 3 3 3 3 3 3 3 3 Trirectified 9 simplex treday 2520 210 5 t4 3 3 3 3 3 3 3 3 Quadrirectified 9 simplex icoy 3150 252 6 t0 1 3 3 3 3 3 3 3 3 Truncated 9 simplex teday 405 90 7 t0 2 3 3 3 3 3 3 3 3 Cantellated 9 simplex 2880 360 8 t1 2 3 3 3 3 3 3 3 3 Bitruncated 9 simplex 1620 360 9 t0 3 3 3 3 3 3 3 3 3 Runcinated 9 simplex 8820 840 10 t1 3 3 3 3 3 3 3 3 3 Bicantellated 9 simplex 10080 1260 11 t2 3 3 3 3 3 3 3 3 3 Tritruncated 9 simplex treday 3780 840 12 t0 4 3 3 3 3 3 3 3 3 Stericated 9 simplex 15120 1260 13 t1 4 3 3 3 3 3 3 3 3 Biruncinated 9 simplex 26460 2520 14 t2 4 3 3 3 3 3 3 3 3 Tricantellated 9 simplex 20160 2520 15 t3 4 3 3 3 3 3 3 3 3 Quadritruncated 9 simplex 5670 1260 16 t0 5 3 3 3 3 3 3 3 3 Pentellated 9 simplex 15750 1260 17 t1 5 3 3 3 3 3 3 3 3 Bistericated 9 simplex 37800 3150 18 t2 5 3 3 3 3 3 3 3 3 Triruncinated 9 simplex 44100 4200 19 t3 5 3 3 3 3 3 3 3 3 Quadricantellated 9 simplex 25200 3150 20 t0 6 3 3 3 3 3 3 3 3 Hexicated 9 simplex 10080 840 21 t1 6 3 3 3 3 3 3 3 3 Bipentellated 9 simplex 31500 2520 22 t2 6 3 3 3 3 3 3 3 3 Tristericated 9 simplex 50400 4200 23 t0 7 3 3 3 3 3 3 3 3 Heptellated 9 simplex 3780 360 24 t1 7 3 3 3 3 3 3 3 3 Bihexicated 9 simplex 15120 1260 25 t0 8 3 3 3 3 3 3 3 3 Octellated 9 simplex 720 90 26 t0 1 2 3 3 3 3 3 3 3 3 Cantitruncated 9 simplex 3240 720 27 t0 1 3 3 3 3 3 3 3 3 3 Runcitruncated 9 simplex 18900 2520 28 t0 2 3 3 3 3 3 3 3 3 3 Runcicantellated 9 simplex 12600 2520 29 t1 2 3 3 3 3 3 3 3 3 3 Bicantitruncated 9 simplex 11340 2520 30 t0 1 4 3 3 3 3 3 3 3 3 Steritruncated 9 simplex 47880 5040 31 t0 2 4 3 3 3 3 3 3 3 3 Stericantellated 9 simplex 60480 7560 32 t1 2 4 3 3 3 3 3 3 3 3 Biruncitruncated 9 simplex 52920 7560 33 t0 3 4 3 3 3 3 3 3 3 3 Steriruncinated 9 simplex 27720 5040 34 t1 3 4 3 3 3 3 3 3 3 3 Biruncicantellated 9 simplex 41580 7560 35 t2 3 4 3 3 3 3 3 3 3 3 Tricantitruncated 9 simplex 22680 5040 36 t0 1 5 3 3 3 3 3 3 3 3 Pentitruncated 9 simplex 66150 6300 37 t0 2 5 3 3 3 3 3 3 3 3 Penticantellated 9 simplex 126000 12600 38 t1 2 5 3 3 3 3 3 3 3 3 Bisteritruncated 9 simplex 107100 12600 39 t0 3 5 3 3 3 3 3 3 3 3 Pentiruncinated 9 simplex 107100 12600 40 t1 3 5 3 3 3 3 3 3 3 3 Bistericantellated 9 simplex 151200 18900 41 t2 3 5 3 3 3 3 3 3 3 3 Triruncitruncated 9 simplex 81900 12600 42 t0 4 5 3 3 3 3 3 3 3 3 Pentistericated 9 simplex 37800 6300 43 t1 4 5 3 3 3 3 3 3 3 3 Bisteriruncinated 9 simplex 81900 12600 44 t2 4 5 3 3 3 3 3 3 3 3 Triruncicantellated 9 simplex 75600 12600 45 t3 4 5 3 3 3 3 3 3 3 3 Quadricantitruncated 9 simplex 28350 6300 46 t0 1 6 3 3 3 3 3 3 3 3 Hexitruncated 9 simplex 52920 5040 47 t0 2 6 3 3 3 3 3 3 3 3 Hexicantellated 9 simplex 138600 12600 48 t1 2 6 3 3 3 3 3 3 3 3 Bipentitruncated 9 simplex 113400 12600 49 t0 3 6 3 3 3 3 3 3 3 3 Hexiruncinated 9 simplex 176400 16800 50 t1 3 6 3 3 3 3 3 3 3 3 Bipenticantellated 9 simplex 239400 25200 51 t2 3 6 3 3 3 3 3 3 3 3 Tristeritruncated 9 simplex 126000 16800 52 t0 4 6 3 3 3 3 3 3 3 3 Hexistericated 9 simplex 113400 12600 53 t1 4 6 3 3 3 3 3 3 3 3 Bipentiruncinated 9 simplex 226800 25200 54 t2 4 6 3 3 3 3 3 3 3 3 Tristericantellated 9 simplex 201600 25200 55 t0 5 6 3 3 3 3 3 3 3 3 Hexipentellated 9 simplex 32760 5040 56 t1 5 6 3 3 3 3 3 3 3 3 Bipentistericated 9 simplex 94500 12600 57 t0 1 7 3 3 3 3 3 3 3 3 Heptitruncated 9 simplex 23940 2520 58 t0 2 7 3 3 3 3 3 3 3 3 Hepticantellated 9 simplex 83160 7560 59 t1 2 7 3 3 3 3 3 3 3 3 Bihexitruncated 9 simplex 64260 7560 60 t0 3 7 3 3 3 3 3 3 3 3 Heptiruncinated 9 simplex 144900 12600 61 t1 3 7 3 3 3 3 3 3 3 3 Bihexicantellated 9 simplex 189000 18900 62 t0 4 7 3 3 3 3 3 3 3 3 Heptistericated 9 simplex 138600 12600 63 t1 4 7 3 3 3 3 3 3 3 3 Bihexiruncinated 9 simplex 264600 25200 64 t0 5 7 3 3 3 3 3 3 3 3 Heptipentellated 9 simplex 71820 7560 65 t0 6 7 3 3 3 3 3 3 3 3 Heptihexicated 9 simplex 17640 2520 66 t0 1 8 3 3 3 3 3 3 3 3 Octitruncated 9 simplex 5400 720 67 t0 2 8 3 3 3 3 3 3 3 3 Octicantellated 9 simplex 25200 2520 68 t0 3 8 3 3 3 3 3 3 3 3 Octiruncinated 9 simplex 57960 5040 69 t0 4 8 3 3 3 3 3 3 3 3 Octistericated 9 simplex 75600 6300 70 t0 1 2 3 3 3 3 3 3 3 3 3 Runcicantitruncated 9 simplex 22680 5040 71 t0 1 2 4 3 3 3 3 3 3 3 3 Stericantitruncated 9 simplex 105840 15120 72 t0 1 3 4 3 3 3 3 3 3 3 3 Steriruncitruncated 9 simplex 75600 15120 73 t0 2 3 4 3 3 3 3 3 3 3 3 Steriruncicantellated 9 simplex 75600 15120 74 t1 2 3 4 3 3 3 3 3 3 3 3 Biruncicantitruncated 9 simplex 68040 15120 75 t0 1 2 5 3 3 3 3 3 3 3 3 Penticantitruncated 9 simplex 214200 25200 76 t0 1 3 5 3 3 3 3 3 3 3 3 Pentiruncitruncated 9 simplex 283500 37800 77 t0 2 3 5 3 3 3 3 3 3 3 3 Pentiruncicantellated 9 simplex 264600 37800 78 t1 2 3 5 3 3 3 3 3 3 3 3 Bistericantitruncated 9 simplex 245700 37800 79 t0 1 4 5 3 3 3 3 3 3 3 3 Pentisteritruncated 9 simplex 138600 25200 80 t0 2 4 5 3 3 3 3 3 3 3 3 Pentistericantellated 9 simplex 226800 37800 81 t1 2 4 5 3 3 3 3 3 3 3 3 Bisteriruncitruncated 9 simplex 189000 37800 82 t0 3 4 5 3 3 3 3 3 3 3 3 Pentisteriruncinated 9 simplex 138600 25200 83 t1 3 4 5 3 3 3 3 3 3 3 3 Bisteriruncicantellated 9 simplex 207900 37800 84 t2 3 4 5 3 3 3 3 3 3 3 3 Triruncicantitruncated 9 simplex 113400 25200 85 t0 1 2 6 3 3 3 3 3 3 3 3 Hexicantitruncated 9 simplex 226800 25200 86 t0 1 3 6 3 3 3 3 3 3 3 3 Hexiruncitruncated 9 simplex 453600 50400 87 t0 2 3 6 3 3 3 3 3 3 3 3 Hexiruncicantellated 9 simplex 403200 50400 88 t1 2 3 6 3 3 3 3 3 3 3 3 Bipenticantitruncated 9 simplex 378000 50400 89 t0 1 4 6 3 3 3 3 3 3 3 3 Hexisteritruncated 9 simplex 403200 50400 90 t0 2 4 6 3 3 3 3 3 3 3 3 Hexistericantellated 9 simplex 604800 75600 91 t1 2 4 6 3 3 3 3 3 3 3 3 Bipentiruncitruncated 9 simplex 529200 75600 92 t0 3 4 6 3 3 3 3 3 3 3 3 Hexisteriruncinated 9 simplex 352800 50400 93 t1 3 4 6 3 3 3 3 3 3 3 3 Bipentiruncicantellated 9 simplex 529200 75600 94 t2 3 4 6 3 3 3 3 3 3 3 3 Tristericantitruncated 9 simplex 302400 50400 95 t0 1 5 6 3 3 3 3 3 3 3 3 Hexipentitruncated 9 simplex 151200 25200 96 t0 2 5 6 3 3 3 3 3 3 3 3 Hexipenticantellated 9 simplex 352800 50400 97 t1 2 5 6 3 3 3 3 3 3 3 3 Bipentisteritruncated 9 simplex 277200 50400 98 t0 3 5 6 3 3 3 3 3 3 3 3 Hexipentiruncinated 9 simplex 352800 50400 99 t1 3 5 6 3 3 3 3 3 3 3 3 Bipentistericantellated 9 simplex 491400 75600 100 t2 3 5 6 3 3 3 3 3 3 3 3 Tristeriruncitruncated 9 simplex 252000 50400 101 t0 4 5 6 3 3 3 3 3 3 3 3 Hexipentistericated 9 simplex 151200 25200 102 t1 4 5 6 3 3 3 3 3 3 3 3 Bipentisteriruncinated 9 simplex 327600 50400 103 t0 1 2 7 3 3 3 3 3 3 3 3 Hepticantitruncated 9 simplex 128520 15120 104 t0 1 3 7 3 3 3 3 3 3 3 3 Heptiruncitruncated 9 simplex 359100 37800 105 t0 2 3 7 3 3 3 3 3 3 3 3 Heptiruncicantellated 9 simplex 302400 37800 106 t1 2 3 7 3 3 3 3 3 3 3 3 Bihexicantitruncated 9 simplex 283500 37800 107 t0 1 4 7 3 3 3 3 3 3 3 3 Heptisteritruncated 9 simplex 478800 50400 108 span, wikipedia, wiki, book, books, library,

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