A uniform 9-polytope is one which is vertex-transitive, and constructed from uniform 8-polytope facets.
Regular 9-polytopes can be represented by the Schläfli symbol, with w 8-polytope facets around each peak.
There are exactly three such convex regular 9-polytopes:
There are no nonconvex regular 9-polytopes.
The topology of any given 9-polytope is defined by its Betti numbers and torsion coefficients.[1]
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.[1]
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.[1]
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 | [3<sup>8</sup>] | ||
| B9 | [4,3<sup>7</sup>] | ||
| D9 | [3<sup>6,1,1</sup>] | ||
Selected regular and uniform 9-polytopes from each family include:
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 diagram Schläfli symbol Name | Element counts | ||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 8-faces | 7-faces | 6-faces | 5-faces | 4-faces | Cells | Faces | Edges | Vertices | ||||
| 1 | t0 9-simplex (day) | 10 | 45 | 120 | 210 | 252 | 210 | 120 | 45 | 10 | ||
| 2 | t1 Rectified 9-simplex (reday) | 360 | 45 | |||||||||
| 3 | t2 Birectified 9-simplex (breday) | 1260 | 120 | |||||||||
| 4 | t3 Trirectified 9-simplex (treday) | 2520 | 210 | |||||||||
| 5 | t4 Quadrirectified 9-simplex (icoy) | 3150 | 252 | |||||||||
| 6 | t0,1 Truncated 9-simplex (teday) | 405 | 90 | |||||||||
| 7 | t0,2 Cantellated 9-simplex | 2880 | 360 | |||||||||
| 8 | t1,2 Bitruncated 9-simplex | 1620 | 360 | |||||||||
| 9 | t0,3 Runcinated 9-simplex | 8820 | 840 | |||||||||
| 10 | t1,3 Bicantellated 9-simplex | 10080 | 1260 | |||||||||
| 11 | t2,3 Tritruncated 9-simplex (treday) | 3780 | 840 | |||||||||
| 12 | t0,4 Stericated 9-simplex | 15120 | 1260 | |||||||||
| 13 | t1,4 Biruncinated 9-simplex | 26460 | 2520 | |||||||||
| 14 | t2,4 Tricantellated 9-simplex | 20160 | 2520 | |||||||||
| 15 | t3,4 Quadritruncated 9-simplex | 5670 | 1260 | |||||||||
| 16 | t0,5 Pentellated 9-simplex | 15750 | 1260 | |||||||||
| 17 | t1,5 Bistericated 9-simplex | 37800 | 3150 | |||||||||
| 18 | t2,5 Triruncinated 9-simplex | 44100 | 4200 | |||||||||
| 19 | t3,5 Quadricantellated 9-simplex | 25200 | 3150 | |||||||||
| 20 | t0,6 Hexicated 9-simplex | 10080 | 840 | |||||||||
| 21 | t1,6 Bipentellated 9-simplex | 31500 | 2520 | |||||||||
| 22 | t2,6 Tristericated 9-simplex | 50400 | 4200 | |||||||||
| 23 | t0,7 Heptellated 9-simplex | 3780 | 360 | |||||||||
| 24 | t1,7 Bihexicated 9-simplex | 15120 | 1260 | |||||||||
| 25 | t0,8 Octellated 9-simplex | 720 | 90 | |||||||||
| 26 | t0,1,2 Cantitruncated 9-simplex | 3240 | 720 | |||||||||
| 27 | t0,1,3 Runcitruncated 9-simplex | 18900 | 2520 | |||||||||
| 28 | t0,2,3 Runcicantellated 9-simplex | 12600 | 2520 | |||||||||
| 29 | t1,2,3 Bicantitruncated 9-simplex | 11340 | 2520 | |||||||||
| 30 | t0,1,4 Steritruncated 9-simplex | 47880 | 5040 | |||||||||
| 31 | t0,2,4 Stericantellated 9-simplex | 60480 | 7560 | |||||||||
| 32 | t1,2,4 Biruncitruncated 9-simplex | 52920 | 7560 | |||||||||
| 33 | t0,3,4 Steriruncinated 9-simplex | 27720 | 5040 | |||||||||
| 34 | t1,3,4 Biruncicantellated 9-simplex | 41580 | 7560 | |||||||||
| 35 | t2,3,4 Tricantitruncated 9-simplex | 22680 | 5040 | |||||||||
| 36 | t0,1,5 Pentitruncated 9-simplex | 66150 | 6300 | |||||||||
| 37 | t0,2,5 Penticantellated 9-simplex | 126000 | 12600 | |||||||||
| 38 | t1,2,5 Bisteritruncated 9-simplex | 107100 | 12600 | |||||||||
| 39 | t0,3,5 Pentiruncinated 9-simplex | 107100 | 12600 | |||||||||
| 40 | t1,3,5 Bistericantellated 9-simplex | 151200 | 18900 | |||||||||
| 41 | t2,3,5 Triruncitruncated 9-simplex | 81900 | 12600 | |||||||||
| 42 | t0,4,5 Pentistericated 9-simplex | 37800 | 6300 | |||||||||
| 43 | t1,4,5 Bisteriruncinated 9-simplex | 81900 | 12600 | |||||||||
| 44 | t2,4,5 Triruncicantellated 9-simplex | 75600 | 12600 | |||||||||
| 45 | t3,4,5 Quadricantitruncated 9-simplex | 28350 | 6300 | |||||||||
| 46 | t0,1,6 Hexitruncated 9-simplex | 52920 | 5040 | |||||||||
| 47 | t0,2,6 Hexicantellated 9-simplex | 138600 | 12600 | |||||||||
| 48 | t1,2,6 Bipentitruncated 9-simplex | 113400 | 12600 | |||||||||
| 49 | t0,3,6 Hexiruncinated 9-simplex | 176400 | 16800 | |||||||||
| 50 | t1,3,6 Bipenticantellated 9-simplex | 239400 | 25200 | |||||||||
| 51 | t2,3,6 Tristeritruncated 9-simplex | 126000 | 16800 | |||||||||
| 52 | t0,4,6 Hexistericated 9-simplex | 113400 | 12600 | |||||||||
| 53 | t1,4,6 Bipentiruncinated 9-simplex | 226800 | 25200 | |||||||||
| 54 | t2,4,6 Tristericantellated 9-simplex | 201600 | 25200 | |||||||||
| 55 | t0,5,6 Hexipentellated 9-simplex | 32760 | 5040 | |||||||||
| 56 | t1,5,6 Bipentistericated 9-simplex | 94500 | 12600 | |||||||||
| 57 | t0,1,7 Heptitruncated 9-simplex | 23940 | 2520 | |||||||||
| 58 | t0,2,7 Hepticantellated 9-simplex | 83160 | 7560 | |||||||||
| 59 | t1,2,7 Bihexitruncated 9-simplex | 64260 | 7560 | |||||||||
| 60 | t0,3,7 Heptiruncinated 9-simplex | 144900 | 12600 | |||||||||
| 61 | t1,3,7 Bihexicantellated 9-simplex | 189000 | 18900 | |||||||||
| 62 | t0,4,7 Heptistericated 9-simplex | 138600 | 12600 | |||||||||
| 63 | t1,4,7 Bihexiruncinated 9-simplex | 264600 | 25200 | |||||||||
| 64 | t0,5,7 Heptipentellated 9-simplex | 71820 | 7560 | |||||||||
| 65 | t0,6,7 Heptihexicated 9-simplex | 17640 | 2520 | |||||||||
| 66 | t0,1,8 Octitruncated 9-simplex | 5400 | 720 | |||||||||
| 67 | t0,2,8 Octicantellated 9-simplex | 25200 | 2520 | |||||||||
| 68 | t0,3,8 Octiruncinated 9-simplex | 57960 | 5040 | |||||||||
| 69 | t0,4,8 Octistericated 9-simplex | 75600 | 6300 | |||||||||
| 70 | t0,1,2,3 Runcicantitruncated 9-simplex | 22680 | 5040 | |||||||||
| 71 | t0,1,2,4 Stericantitruncated 9-simplex | 105840 | 15120 | |||||||||
| 72 | t0,1,3,4 Steriruncitruncated 9-simplex | 75600 | 15120 | |||||||||
| 73 | t0,2,3,4 Steriruncicantellated 9-simplex | 75600 | 15120 | |||||||||
| 74 | t1,2,3,4 Biruncicantitruncated 9-simplex | 68040 | 15120 | |||||||||
| 75 | t0,1,2,5 Penticantitruncated 9-simplex | 214200 | 25200 | |||||||||
| 76 | t0,1,3,5 Pentiruncitruncated 9-simplex | 283500 | 37800 | |||||||||
| 77 | t0,2,3,5 Pentiruncicantellated 9-simplex | 264600 | 37800 | |||||||||
| 78 | t1,2,3,5 Bistericantitruncated 9-simplex | 245700 | 37800 | |||||||||
| 79 | t0,1,4,5 Pentisteritruncated 9-simplex | 138600 | 25200 | |||||||||
| 80 | t0,2,4,5 Pentistericantellated 9-simplex | 226800 | 37800 | |||||||||
| 81 | t1,2,4,5 Bisteriruncitruncated 9-simplex | 189000 | 37800 | |||||||||
| 82 | t0,3,4,5 Pentisteriruncinated 9-simplex | 138600 | 25200 | |||||||||
| 83 | t1,3,4,5 Bisteriruncicantellated 9-simplex | 207900 | 37800 | |||||||||
| 84 | t2,3,4,5 Triruncicantitruncated 9-simplex | 113400 | 25200 | |||||||||
| 85 | t0,1,2,6 Hexicantitruncated 9-simplex | 226800 | 25200 | |||||||||
| 86 | t0,1,3,6 Hexiruncitruncated 9-simplex | 453600 | 50400 | |||||||||
| 87 | t0,2,3,6 Hexiruncicantellated 9-simplex | 403200 | 50400 | |||||||||
| 88 | t1,2,3,6 Bipenticantitruncated 9-simplex | 378000 | 50400 | |||||||||
| 89 | t0,1,4,6 Hexisteritruncated 9-simplex | 403200 | 50400 | |||||||||
| 90 | t0,2,4,6 Hexistericantellated 9-simplex | 604800 | 75600 | |||||||||
| 91 | t1,2,4,6 Bipentiruncitruncated 9-simplex | 529200 | 75600 | |||||||||
| 92 | t0,3,4,6 Hexisteriruncinated 9-simplex | 352800 | 50400 | |||||||||
| 93 | t1,3,4,6 Bipentiruncicantellated 9-simplex | 529200 | 75600 | |||||||||
| 94 | t2,3,4,6 Tristericantitruncated 9-simplex | 302400 | 50400 | |||||||||
| 95 | t0,1,5,6 Hexipentitruncated 9-simplex | 151200 | 25200 | |||||||||
| 96 | t0,2,5,6 Hexipenticantellated 9-simplex | 352800 | 50400 | |||||||||
| 97 | t1,2,5,6 Bipentisteritruncated 9-simplex | 277200 | 50400 | |||||||||
| 98 | t0,3,5,6 Hexipentiruncinated 9-simplex | 352800 | 50400 | |||||||||
| 99 | t1,3,5,6 Bipentistericantellated 9-simplex | 491400 | 75600 | |||||||||
| 100 | t2,3,5,6 Tristeriruncitruncated 9-simplex | 252000 | 50400 | |||||||||
| 101 | t0,4,5,6 Hexipentistericated 9-simplex | 151200 | 25200 | |||||||||
| 102 | t1,4,5,6 Bipentisteriruncinated 9-simplex | 327600 | 50400 | |||||||||
| 103 | t0,1,2,7 Hepticantitruncated 9-simplex | 128520 | 15120 | |||||||||
| 104 | t0,1,3,7 Heptiruncitruncated 9-simplex | 359100 | 37800 | |||||||||
| 105 | t0,2,3,7 Heptiruncicantellated 9-simplex | 302400 | 37800 | |||||||||
| 106 | t1,2,3,7 Bihexicantitruncated 9-simplex | 283500 | 37800 | |||||||||
| 107 | t0,1,4,7 Heptisteritruncated 9-simplex | 478800 | 50400 | |||||||||
| 108 | t0,2,4,7 Heptistericantellated 9-simplex | 680400 | 75600 | |||||||||
| 109 | t1,2,4,7 Bihexiruncitruncated 9-simplex | 604800 | 75600 | |||||||||
| 110 | t0,3,4,7 Heptisteriruncinated 9-simplex | 378000 | 50400 | |||||||||
| 111 | t1,3,4,7 Bihexiruncicantellated 9-simplex | 567000 | 75600 | |||||||||
| 112 | t0,1,5,7 Heptipentitruncated 9-simplex | 321300 | 37800 | |||||||||
| 113 | t0,2,5,7 Heptipenticantellated 9-simplex | 680400 | 75600 | |||||||||
| 114 | t1,2,5,7 Bihexisteritruncated 9-simplex | 567000 | 75600 | |||||||||
| 115 | t0,3,5,7 Heptipentiruncinated 9-simplex | 642600 | 75600 | |||||||||
| 116 | t1,3,5,7 Bihexistericantellated 9-simplex | 907200 | 113400 | |||||||||
| 117 | t0,4,5,7 Heptipentistericated 9-simplex | 264600 | 37800 | |||||||||
| 118 | t0,1,6,7 Heptihexitruncated 9-simplex | 98280 | 15120 | |||||||||
| 119 | t0,2,6,7 Heptihexicantellated 9-simplex | 302400 | 37800 | |||||||||
| 120 | t1,2,6,7 Bihexipentitruncated 9-simplex | 226800 | 37800 | |||||||||
| 121 | t0,3,6,7 Heptihexiruncinated 9-simplex | 428400 | 50400 | |||||||||
| 122 | t0,4,6,7 Heptihexistericated 9-simplex | 302400 | 37800 | |||||||||
| 123 | t0,5,6,7 Heptihexipentellated 9-simplex | 98280 | 15120 | |||||||||
| 124 | t0,1,2,8 Octicantitruncated 9-simplex | 35280 | 5040 | |||||||||
| 125 | t0,1,3,8 Octiruncitruncated 9-simplex | 136080 | 15120 | |||||||||
| 126 | t0,2,3,8 Octiruncicantellated 9-simplex | 105840 | 15120 | |||||||||
| 127 | t0,1,4,8 Octisteritruncated 9-simplex | 252000 | 25200 | |||||||||
| 128 | t0,2,4,8 Octistericantellated 9-simplex | 340200 | 37800 | |||||||||
| 129 | t0,3,4,8 Octisteriruncinated 9-simplex | 176400 | 25200 | |||||||||
| 130 | t0,1,5,8 Octipentitruncated 9-simplex | 252000 | 25200 | |||||||||
| 131 | t0,2,5,8 Octipenticantellated 9-simplex | 504000 | 50400 | |||||||||
| 132 | t0,3,5,8 Octipentiruncinated 9-simplex | 453600 | 50400 | |||||||||
| 133 | t0,1,6,8 Octihexitruncated 9-simplex | 136080 | 15120 | |||||||||
| 134 | t0,2,6,8 Octihexicantellated 9-simplex | 378000 | 37800 | |||||||||
| 135 | t0,1,7,8 Octiheptitruncated 9-simplex | 35280 | 5040 | |||||||||
| 136 | t0,1,2,3,4 Steriruncicantitruncated 9-simplex | 136080 | 30240 | |||||||||
| 137 | t0,1,2,3,5 Pentiruncicantitruncated 9-simplex | 491400 | 75600 | |||||||||
| 138 | t0,1,2,4,5 Pentistericantitruncated 9-simplex | 378000 | 75600 | |||||||||
| 139 | t0,1,3,4,5 Pentisteriruncitruncated 9-simplex | 378000 | 75600 | |||||||||
| 140 | t0,2,3,4,5 Pentisteriruncicantellated 9-simplex | 378000 | 75600 | |||||||||
| 141 | t1,2,3,4,5 Bisteriruncicantitruncated 9-simplex | 340200 | 75600 | |||||||||
| 142 | t0,1,2,3,6 Hexiruncicantitruncated 9-simplex | 756000 | 100800 | |||||||||
| 143 | t0,1,2,4,6 Hexistericantitruncated 9-simplex | 1058400 | 151200 | |||||||||
| 144 | t0,1,3,4,6 Hexisteriruncitruncated 9-simplex | 982800 | 151200 | |||||||||
| 145 | t0,2,3,4,6 Hexisteriruncicantellated 9-simplex | 982800 | 151200 | |||||||||
| 146 | t1,2,3,4,6 Bipentiruncicantitruncated 9-simplex | 907200 | 151200 | |||||||||
| 147 | t0,1,2,5,6 Hexipenticantitruncated 9-simplex | 554400 | 100800 | |||||||||
| 148 | t0,1,3,5,6 Hexipentiruncitruncated 9-simplex | 907200 | 151200 | |||||||||
| 149 | t0,2,3,5,6 Hexipentiruncicantellated 9-simplex | 831600 | 151200 | |||||||||
| 150 | t1,2,3,5,6 Bipentistericantitruncated 9-simplex | 756000 | 151200 | |||||||||
| 151 | t0,1,4,5,6 Hexipentisteritruncated 9-simplex | 554400 | 100800 | |||||||||
| 152 | t0,2,4,5,6 Hexipentistericantellated 9-simplex | 907200 | 151200 | |||||||||
| 153 | t1,2,4,5,6 Bipentisteriruncitruncated 9-simplex | 756000 | 151200 | |||||||||
| 154 | t0,3,4,5,6 Hexipentisteriruncinated 9-simplex | 554400 | 100800 | |||||||||
| 155 | t1,3,4,5,6 Bipentisteriruncicantellated 9-simplex | 831600 | 151200 | |||||||||
| 156 | t2,3,4,5,6 Tristeriruncicantitruncated 9-simplex | 453600 | 100800 | |||||||||
| 157 | t0,1,2,3,7 Heptiruncicantitruncated 9-simplex | 567000 | 75600 | |||||||||
| 158 | t0,1,2,4,7 Heptistericantitruncated 9-simplex | 1209600 | 151200 | |||||||||
| 159 | t0,1,3,4,7 Heptisteriruncitruncated 9-simplex | 1058400 | 151200 | |||||||||
| 160 | t0,2,3,4,7 Heptisteriruncicantellated 9-simplex | 1058400 | 151200 | |||||||||
| 161 | t1,2,3,4,7 Bihexiruncicantitruncated 9-simplex | 982800 | 151200 | |||||||||
| 162 | t0,1,2,5,7 Heptipenticantitruncated 9-simplex | 1134000 | 151200 | |||||||||
| 163 | t0,1,3,5,7 Heptipentiruncitruncated 9-simplex | 1701000 | 226800 | |||||||||
| 164 | t0,2,3,5,7 Heptipentiruncicantellated 9-simplex | 1587600 | 226800 | |||||||||
| 165 | t1,2,3,5,7 Bihexistericantitruncated 9-simplex | 1474200 | 226800 | |||||||||
| 166 | t0,1,4,5,7 Heptipentisteritruncated 9-simplex | 982800 | 151200 | |||||||||
| 167 | t0,2,4,5,7 Heptipentistericantellated 9-simplex | 1587600 | 226800 | |||||||||
| 168 | t1,2,4,5,7 Bihexisteriruncitruncated 9-simplex | 1360800 | 226800 | |||||||||
| 169 | t0,3,4,5,7 Heptipentisteriruncinated 9-simplex | 982800 | 151200 | |||||||||
| 170 | t1,3,4,5,7 Bihexisteriruncicantellated 9-simplex | 1474200 | 226800 | |||||||||
| 171 | t0,1,2,6,7 Heptihexicantitruncated 9-simplex | 453600 | 75600 | |||||||||
| 172 | t0,1,3,6,7 Heptihexiruncitruncated 9-simplex | 1058400 | 151200 | |||||||||
| 173 | t0,2,3,6,7 Heptihexiruncicantellated 9-simplex | 907200 | 151200 | |||||||||
| 174 | t1,2,3,6,7 Bihexipenticantitruncated 9-simplex | 831600 | 151200 | |||||||||
| 175 | t0,1,4,6,7 Heptihexisteritruncated 9-simplex | 1058400 | 151200 | |||||||||
| 176 | t0,2,4,6,7 Heptihexistericantellated 9-simplex | 1587600 | 226800 | |||||||||
| 177 | t1,2,4,6,7 Bihexipentiruncitruncated 9-simplex | 1360800 | 226800 | |||||||||
| 178 | t0,3,4,6,7 Heptihexisteriruncinated 9-simplex | 907200 | 151200 | |||||||||
| 179 | t0,1,5,6,7 Heptihexipentitruncated 9-simplex | 453600 | 75600 | |||||||||
| 180 | t0,2,5,6,7 Heptihexipenticantellated 9-simplex | 1058400 | 151200 | |||||||||
| 181 | t0,3,5,6,7 Heptihexipentiruncinated 9-simplex | 1058400 | 151200 | |||||||||
| 182 | t0,4,5,6,7 Heptihexipentistericated 9-simplex | 453600 | 75600 | |||||||||
| 183 | t0,1,2,3,8 Octiruncicantitruncated 9-simplex | 196560 | 30240 | |||||||||
| 184 | t0,1,2,4,8 Octistericantitruncated 9-simplex | 604800 | 75600 | |||||||||
| 185 | t0,1,3,4,8 Octisteriruncitruncated 9-simplex | 491400 | 75600 | |||||||||
| 186 | t0,2,3,4,8 Octisteriruncicantellated 9-simplex | 491400 | 75600 | |||||||||
| 187 | t0,1,2,5,8 Octipenticantitruncated 9-simplex | 856800 | 100800 | |||||||||
| 188 | t0,1,3,5,8 Octipentiruncitruncated 9-simplex | 1209600 | 151200 | |||||||||
| 189 | t0,2,3,5,8 Octipentiruncicantellated 9-simplex | 1134000 | 151200 | |||||||||
| 190 | t0,1,4,5,8 Octipentisteritruncated 9-simplex | 655200 | 100800 | |||||||||
| 191 | t0,2,4,5,8 Octipentistericantellated 9-simplex | 1058400 | 151200 | |||||||||
| 192 | t0,3,4,5,8 Octipentisteriruncinated 9-simplex | 655200 | 100800 | |||||||||
| 193 | t0,1,2,6,8 Octihexicantitruncated 9-simplex | 604800 | 75600 | |||||||||
| 194 | t0,1,3,6,8 Octihexiruncitruncated 9-simplex | 1285200 | 151200 | |||||||||
| 195 | t0,2,3,6,8 Octihexiruncicantellated 9-simplex | 1134000 | 151200 | |||||||||
| 196 | t0,1,4,6,8 Octihexisteritruncated 9-simplex | 1209600 | 151200 | |||||||||
| 197 | t0,2,4,6,8 Octihexistericantellated 9-simplex | 1814400 | 226800 | |||||||||
| 198 | t0,1,5,6,8 Octihexipentitruncated 9-simplex | 491400 | 75600 | |||||||||
| 199 | t0,1,2,7,8 Octihepticantitruncated 9-simplex | 196560 | 30240 | |||||||||
| 200 | t0,1,3,7,8 Octiheptiruncitruncated 9-simplex | 604800 | 75600 | |||||||||
| 201 | t0,1,4,7,8 Octiheptisteritruncated 9-simplex | 856800 | 100800 | |||||||||
| 202 | t0,1,2,3,4,5 Pentisteriruncicantitruncated 9-simplex | 680400 | 151200 | |||||||||
| 203 | t0,1,2,3,4,6 Hexisteriruncicantitruncated 9-simplex | 1814400 | 302400 | |||||||||
| 204 | t0,1,2,3,5,6 Hexipentiruncicantitruncated 9-simplex | 1512000 | 302400 | |||||||||
| 205 | t0,1,2,4,5,6 Hexipentistericantitruncated 9-simplex | 1512000 | 302400 | |||||||||
| 206 | t0,1,3,4,5,6 Hexipentisteriruncitruncated 9-simplex | 1512000 | 302400 | |||||||||
| 207 | t0,2,3,4,5,6 Hexipentisteriruncicantellated 9-simplex | 1512000 | 302400 | |||||||||
| 208 | t1,2,3,4,5,6 Bipentisteriruncicantitruncated 9-simplex | 1360800 | 302400 | |||||||||
| 209 | t0,1,2,3,4,7 Heptisteriruncicantitruncated 9-simplex | 1965600 | 302400 | |||||||||
| 210 | t0,1,2,3,5,7 Heptipentiruncicantitruncated 9-simplex | 2948400 | 453600 | |||||||||
| 211 | t0,1,2,4,5,7 Heptipentistericantitruncated 9-simplex | 2721600 | 453600 | |||||||||
| 212 | t0,1,3,4,5,7 Heptipentisteriruncitruncated 9-simplex | 2721600 | 453600 | |||||||||
| 213 | t0,2,3,4,5,7 Heptipentisteriruncicantellated 9-simplex | 2721600 | 453600 | |||||||||
| 214 | t1,2,3,4,5,7 Bihexisteriruncicantitruncated 9-simplex | 2494800 | 453600 | |||||||||
| 215 | t0,1,2,3,6,7 Heptihexiruncicantitruncated 9-simplex | 1663200 | 302400 | |||||||||
| 216 | t0,1,2,4,6,7 Heptihexistericantitruncated 9-simplex | 2721600 | 453600 | |||||||||
| 217 | t0,1,3,4,6,7 Heptihexisteriruncitruncated 9-simplex | 2494800 | 453600 | |||||||||
| 218 | t0,2,3,4,6,7 Heptihexisteriruncicantellated 9-simplex | 2494800 | 453600 | |||||||||
| 219 | t1,2,3,4,6,7 Bihexipentiruncicantitruncated 9-simplex | 2268000 | 453600 | |||||||||
| 220 | t0,1,2,5,6,7 Heptihexipenticantitruncated 9-simplex | 1663200 | 302400 | |||||||||
| 221 | t0,1,3,5,6,7 Heptihexipentiruncitruncated 9-simplex | 2721600 | 453600 | |||||||||
| 222 | t0,2,3,5,6,7 Heptihexipentiruncicantellated 9-simplex | 2494800 | 453600 | |||||||||
| 223 | t1,2,3,5,6,7 Bihexipentistericantitruncated 9-simplex | 2268000 | 453600 | |||||||||
| 224 | t0,1,4,5,6,7 Heptihexipentisteritruncated 9-simplex | 1663200 | 302400 | |||||||||
| 225 | t0,2,4,5,6,7 Heptihexipentistericantellated 9-simplex | 2721600 | 453600 | |||||||||
| 226 | t0,3,4,5,6,7 Heptihexipentisteriruncinated 9-simplex | 1663200 | 302400 | |||||||||
| 227 | t0,1,2,3,4,8 Octisteriruncicantitruncated 9-simplex | 907200 | 151200 | |||||||||
| 228 | t0,1,2,3,5,8 Octipentiruncicantitruncated 9-simplex | 2116800 | 302400 | |||||||||
| 229 | t0,1,2,4,5,8 Octipentistericantitruncated 9-simplex | 1814400 | 302400 | |||||||||
| 230 | t0,1,3,4,5,8 Octipentisteriruncitruncated 9-simplex | 1814400 | 302400 | |||||||||
| 231 | t0,2,3,4,5,8 Octipentisteriruncicantellated 9-simplex | 1814400 | 302400 | |||||||||
| 232 | t0,1,2,3,6,8 Octihexiruncicantitruncated 9-simplex | 2116800 | 302400 | |||||||||
| 233 | t0,1,2,4,6,8 Octihexistericantitruncated 9-simplex | 3175200 | 453600 | |||||||||
| 234 | t0,1,3,4,6,8 Octihexisteriruncitruncated 9-simplex | 2948400 | 453600 | |||||||||
| 235 | t0,2,3,4,6,8 Octihexisteriruncicantellated 9-simplex | 2948400 | 453600 | |||||||||
| 236 | t0,1,2,5,6,8 Octihexipenticantitruncated 9-simplex | 1814400 | 302400 | |||||||||
| 237 | t0,1,3,5,6,8 Octihexipentiruncitruncated 9-simplex | 2948400 | 453600 | |||||||||
| 238 | t0,2,3,5,6,8 Octihexipentiruncicantellated 9-simplex | 2721600 | 453600 | |||||||||
| 239 | t0,1,4,5,6,8 Octihexipentisteritruncated 9-simplex | 1814400 | 302400 | |||||||||
| 240 | t0,1,2,3,7,8 Octiheptiruncicantitruncated 9-simplex | 907200 | 151200 | |||||||||
| 241 | t0,1,2,4,7,8 Octiheptistericantitruncated 9-simplex | 2116800 | 302400 | |||||||||
| 242 | t0,1,3,4,7,8 Octiheptisteriruncitruncated 9-simplex | 1814400 | 302400 | |||||||||
| 243 | t0,1,2,5,7,8 Octiheptipenticantitruncated 9-simplex | 2116800 | 302400 | |||||||||
| 244 | t0,1,3,5,7,8 Octiheptipentiruncitruncated 9-simplex | 3175200 | 453600 | |||||||||
| 245 | t0,1,2,6,7,8 Octiheptihexicantitruncated 9-simplex | 907200 | 151200 | |||||||||
| 246 | t0,1,2,3,4,5,6 Hexipentisteriruncicantitruncated 9-simplex | 2721600 | 604800 | |||||||||
| 247 | t0,1,2,3,4,5,7 Heptipentisteriruncicantitruncated 9-simplex | 4989600 | 907200 | |||||||||
| 248 | t0,1,2,3,4,6,7 Heptihexisteriruncicantitruncated 9-simplex | 4536000 | 907200 | |||||||||
| 249 | t0,1,2,3,5,6,7 Heptihexipentiruncicantitruncated 9-simplex | 4536000 | 907200 | |||||||||
| 250 | t0,1,2,4,5,6,7 Heptihexipentistericantitruncated 9-simplex | 4536000 | 907200 | |||||||||
| 251 | t0,1,3,4,5,6,7 Heptihexipentisteriruncitruncated 9-simplex | 4536000 | 907200 | |||||||||
| 252 | t0,2,3,4,5,6,7 Heptihexipentisteriruncicantellated 9-simplex | 4536000 | 907200 | |||||||||
| 253 | t1,2,3,4,5,6,7 Bihexipentisteriruncicantitruncated 9-simplex | 4082400 | 907200 | |||||||||
| 254 | t0,1,2,3,4,5,8 Octipentisteriruncicantitruncated 9-simplex | 3326400 | 604800 | |||||||||
| 255 | t0,1,2,3,4,6,8 Octihexisteriruncicantitruncated 9-simplex | 5443200 | 907200 | |||||||||
| 256 | t0,1,2,3,5,6,8 Octihexipentiruncicantitruncated 9-simplex | 4989600 | 907200 | |||||||||
| 257 | t0,1,2,4,5,6,8 Octihexipentistericantitruncated 9-simplex | 4989600 | 907200 | |||||||||
| 258 | t0,1,3,4,5,6,8 Octihexipentisteriruncitruncated 9-simplex | 4989600 | 907200 | |||||||||
| 259 | t0,2,3,4,5,6,8 Octihexipentisteriruncicantellated 9-simplex | 4989600 | 907200 | |||||||||
| 260 | t0,1,2,3,4,7,8 Octiheptisteriruncicantitruncated 9-simplex | 3326400 | 604800 | |||||||||
| 261 | t0,1,2,3,5,7,8 Octiheptipentiruncicantitruncated 9-simplex | 5443200 | 907200 | |||||||||
| 262 | t0,1,2,4,5,7,8 Octiheptipentistericantitruncated 9-simplex | 4989600 | 907200 | |||||||||
| 263 | t0,1,3,4,5,7,8 Octiheptipentisteriruncitruncated 9-simplex | 4989600 | 907200 | |||||||||
| 264 | t0,1,2,3,6,7,8 Octiheptihexiruncicantitruncated 9-simplex | 3326400 | 604800 | |||||||||
| 265 | t0,1,2,4,6,7,8 Octiheptihexistericantitruncated 9-simplex | 5443200 | 907200 | |||||||||
| 266 | t0,1,2,3,4,5,6,7 Heptihexipentisteriruncicantitruncated 9-simplex | 8164800 | 1814400 | |||||||||
| 267 | t0,1,2,3,4,5,6,8 Octihexipentisteriruncicantitruncated 9-simplex | 9072000 | 1814400 | |||||||||
| 268 | t0,1,2,3,4,5,7,8 Octiheptipentisteriruncicantitruncated 9-simplex | 9072000 | 1814400 | |||||||||
| 269 | t0,1,2,3,4,6,7,8 Octiheptihexisteriruncicantitruncated 9-simplex | 9072000 | 1814400 | |||||||||
| 270 | t0,1,2,3,5,6,7,8 Octiheptihexipentiruncicantitruncated 9-simplex | 9072000 | 1814400 | |||||||||
| 271 | t0,1,2,3,4,5,6,7,8 Omnitruncated 9-simplex | 16329600 | 3628800 | |||||||||
There are 511 forms based on all permutations of the Coxeter-Dynkin diagrams with one or more rings.
Eleven cases are shown below: Nine rectified forms and 2 truncations. Bowers-style acronym names are given in parentheses for cross-referencing. Bowers-style acronym names are given in parentheses for cross-referencing.
| Graph | Coxeter-Dynkin diagram Schläfli symbol Name | Element counts | ||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 8-faces | 7-faces | 6-faces | 5-faces | 4-faces | Cells | Faces | Edges | Vertices | ||||
| 1 | t0 9-cube (enne) | 18 | 144 | 672 | 2016 | 4032 | 5376 | 4608 | 2304 | 512 | ||
| 2 | t0,1 Truncated 9-cube (ten) | 2304 | 4608 | |||||||||
| 3 | t1 Rectified 9-cube (ren) | 18432 | 2304 | |||||||||
| 4 | t2 Birectified 9-cube (barn) | 64512 | 4608 | |||||||||
| 5 | t3 Trirectified 9-cube (tarn) | 96768 | 5376 | |||||||||
| 6 | t4 Quadrirectified 9-cube (nav) (Quadrirectified 9-orthoplex) | 80640 | 4032 | |||||||||
| 7 | t3 Trirectified 9-orthoplex (tarv) | 40320 | 2016 | |||||||||
| 8 | t2 Birectified 9-orthoplex (brav) | 12096 | 672 | |||||||||
| 9 | t1 Rectified 9-orthoplex (riv) | 2016 | 144 | |||||||||
| 10 | t0,1 Truncated 9-orthoplex (tiv) | 2160 | 288 | |||||||||
| 11 | t0 9-orthoplex (vee) | 512 | 2304 | 4608 | 5376 | 4032 | 2016 | 672 | 144 | 18 | ||
The D9 family has symmetry of order 92,897,280 (9 factorial × 28).
This family has 3×128−1=383 Wythoffian uniform polytopes, generated by marking one or more nodes of the D9 Coxeter-Dynkin diagram. Of these, 255 (2×128−1) are repeated from the B9 family and 128 are unique to this family, with the eight 1 or 2 ringed forms listed below. Bowers-style acronym names are given in parentheses for cross-referencing.
There are five fundamental affine Coxeter groups that generate regular and uniform tessellations in 8-space:
| Coxeter group | Coxeter diagram | Forms | |||
|---|---|---|---|---|---|
| 1 | {\tilde{A}}8 | [3<sup>[9]] | 45 | ||
| 2 | {\tilde{C}}8 | [4,3<sup>6</sup>,4] | 271 | ||
| 3 | {\tilde{B}}8 | h[4,3<sup>6</sup>,4] [4,3<sup>5</sup>,3<sup>1,1</sup>] | 383 (128 new) | ||
| 4 | {\tilde{D}}8 | q[4,3<sup>6</sup>,4] [3<sup>1,1</sup>,3<sup>4</sup>,3<sup>1,1</sup>] | 155 (15 new) | ||
| 5 | {\tilde{E}}8 | [3<sup>5,2,1</sup>] | 511 | ||
Regular and uniform tessellations include:
{\tilde{A}}8
{\tilde{C}}8
{\tilde{B}}8
{\tilde{C}}8
{\tilde{D}}8
{\tilde{E}}8
There are no compact hyperbolic Coxeter groups of rank 9, groups that can generate honeycombs with all finite facets, and a finite vertex figure. However, there are 4 paracompact hyperbolic Coxeter groups of rank 9, each generating uniform honeycombs in 8-space as permutations of rings of the Coxeter diagrams.
{\bar{P}}8 | {\bar{Q}}8 | {\bar{S}}8 | {\bar{T}}8 |