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.

- Regular 9-polytopes
- Euler characteristic
- Uniform 9-polytopes by fundamental Coxeter groups
- The A9 family
- The B9 family
- The D9 family
- Regular and uniform honeycombs
- Regular and uniform hyperbolic honeycombs
- References
- External links

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 {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.

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 | |
---|---|---|

A_{9} | [3^{8}] | |

B_{9} | [4,3^{7}] | |

D_{9} | [3^{6,1,1}] |

Selected regular and uniform 9-polytopes from each family include:

- Simplex family: A
_{9}[3^{8}] -- 271 uniform 9-polytopes as permutations of rings in the group diagram, including one regular:
- {3
^{8}} - 9-simplex or**deca-9-tope**or**decayotton**-

- {3

- 271 uniform 9-polytopes as permutations of rings in the group diagram, including one regular:
- Hypercube/orthoplex family: B
_{9}[4,3^{8}] -- 511 uniform 9-polytopes as permutations of rings in the group diagram, including two regular ones:
- {4,3
^{7}} - 9-cube or**enneract**- - {3
^{7},4} - 9-orthoplex or**enneacross**-

- {4,3

- 511 uniform 9-polytopes as permutations of rings in the group diagram, including two regular ones:
- Demihypercube D
_{9}family: [3^{6,1,1}] -- 383 uniform 9-polytope as permutations of rings in the group diagram, including:
- {3
^{1,6,1}} - 9-demicube or**demienneract**,**1**- ; also as h{4,3_{61}^{8}} . - {3
^{6,1,1}} - 9-orthoplex,**6**-_{11}

- {3

- 383 uniform 9-polytope as permutations of rings in the group diagram, including:

The A_{9} 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 |
| 10 | 45 | 120 | 210 | 252 | 210 | 120 | 45 | 10 | |

2 |
| 360 | 45 | ||||||||

3 |
| 1260 | 120 | ||||||||

4 |
| 2520 | 210 | ||||||||

5 |
| 3150 | 252 | ||||||||

6 |
| 405 | 90 | ||||||||

7 |
| 2880 | 360 | ||||||||

8 |
| 1620 | 360 | ||||||||

9 |
| 8820 | 840 | ||||||||

10 |
| 10080 | 1260 | ||||||||

11 |
| 3780 | 840 | ||||||||

12 |
| 15120 | 1260 | ||||||||

13 |
| 26460 | 2520 | ||||||||

14 |
| 20160 | 2520 | ||||||||

15 |
| 5670 | 1260 | ||||||||

16 |
| 15750 | 1260 | ||||||||

17 |
| 37800 | 3150 | ||||||||

18 |
| 44100 | 4200 | ||||||||

19 |
| 25200 | 3150 | ||||||||

20 |
| 10080 | 840 | ||||||||

21 |
| 31500 | 2520 | ||||||||

22 |
| 50400 | 4200 | ||||||||

23 |
| 3780 | 360 | ||||||||

24 |
| 15120 | 1260 | ||||||||

25 |
| 720 | 90 | ||||||||

26 |
| 3240 | 720 | ||||||||

27 |
| 18900 | 2520 | ||||||||

28 |
| 12600 | 2520 | ||||||||

29 |
| 11340 | 2520 | ||||||||

30 |
| 47880 | 5040 | ||||||||

31 |
| 60480 | 7560 | ||||||||

32 |
| 52920 | 7560 | ||||||||

33 |
| 27720 | 5040 | ||||||||

34 |
| 41580 | 7560 | ||||||||

35 |
| 22680 | 5040 | ||||||||

36 |
| 66150 | 6300 | ||||||||

37 |
| 126000 | 12600 | ||||||||

38 |
| 107100 | 12600 | ||||||||

39 |
| 107100 | 12600 | ||||||||

40 |
| 151200 | 18900 | ||||||||

41 |
| 81900 | 12600 | ||||||||

42 |
| 37800 | 6300 | ||||||||

43 |
| 81900 | 12600 | ||||||||

44 |
| 75600 | 12600 | ||||||||

45 |
| 28350 | 6300 | ||||||||

46 |
| 52920 | 5040 | ||||||||

47 |
| 138600 | 12600 | ||||||||

48 |
| 113400 | 12600 | ||||||||

49 |
| 176400 | 16800 | ||||||||

50 |
| 239400 | 25200 | ||||||||

51 |
| 126000 | 16800 | ||||||||

52 |
| 113400 | 12600 | ||||||||

53 |
| 226800 | 25200 | ||||||||

54 |
| 201600 | 25200 | ||||||||

55 |
| 32760 | 5040 | ||||||||

56 |
| 94500 | 12600 | ||||||||

57 |
| 23940 | 2520 | ||||||||

58 |
| 83160 | 7560 | ||||||||

59 |
| 64260 | 7560 | ||||||||

60 |
| 144900 | 12600 | ||||||||

61 |
| 189000 | 18900 | ||||||||

62 |
| 138600 | 12600 | ||||||||

63 |
| 264600 | 25200 | ||||||||

64 |
| 71820 | 7560 | ||||||||

65 |
| 17640 | 2520 | ||||||||

66 |
| 5400 | 720 | ||||||||

67 |
| 25200 | 2520 | ||||||||

68 |
| 57960 | 5040 | ||||||||

69 |
| 75600 | 6300 | ||||||||

70 |
| 22680 | 5040 | ||||||||

71 |
| 105840 | 15120 | ||||||||

72 |
| 75600 | 15120 | ||||||||

73 |
| 75600 | 15120 | ||||||||

74 |
| 68040 | 15120 | ||||||||

75 |
| 214200 | 25200 | ||||||||

76 |
| 283500 | 37800 | ||||||||

77 |
| 264600 | 37800 | ||||||||

78 |
| 245700 | 37800 | ||||||||

79 |
| 138600 | 25200 | ||||||||

80 |
| 226800 | 37800 | ||||||||

81 |
| 189000 | 37800 | ||||||||

82 |
| 138600 | 25200 | ||||||||

83 |
| 207900 | 37800 | ||||||||

84 |
| 113400 | 25200 | ||||||||

85 |
| 226800 | 25200 | ||||||||

86 |
| 453600 | 50400 | ||||||||

87 |
| 403200 | 50400 | ||||||||

88 |
| 378000 | 50400 | ||||||||

89 |
| 403200 | 50400 | ||||||||

90 |
| 604800 | 75600 | ||||||||

91 |
| 529200 | 75600 | ||||||||

92 |
| 352800 | 50400 | ||||||||

93 |
| 529200 | 75600 | ||||||||

94 |
| 302400 | 50400 | ||||||||

95 |
| 151200 | 25200 | ||||||||

96 |
| 352800 | 50400 | ||||||||

97 |
| 277200 | 50400 | ||||||||

98 |
| 352800 | 50400 | ||||||||

99 |
| 491400 | 75600 | ||||||||

100 |
| 252000 | 50400 | ||||||||

101 |
| 151200 | 25200 | ||||||||

102 |
| 327600 | 50400 | ||||||||

103 |
| 128520 | 15120 | ||||||||

104 |
| 359100 | 37800 | ||||||||

105 |
| 302400 | 37800 | ||||||||

106 |
| 283500 | 37800 | ||||||||

107 |
| 478800 | 50400 | ||||||||

108 |
| 680400 | 75600 | ||||||||

109 |
| 604800 | 75600 | ||||||||

110 |
| 378000 | 50400 | ||||||||

111 |
| 567000 | 75600 | ||||||||

112 |
| 321300 | 37800 | ||||||||

113 |
| 680400 | 75600 | ||||||||

114 |
| 567000 | 75600 | ||||||||

115 |
| 642600 | 75600 | ||||||||

116 |
| 907200 | 113400 | ||||||||

117 |
| 264600 | 37800 | ||||||||

118 |
| 98280 | 15120 | ||||||||

119 |
| 302400 | 37800 | ||||||||

120 |
| 226800 | 37800 | ||||||||

121 |
| 428400 | 50400 | ||||||||

122 |
| 302400 | 37800 | ||||||||

123 |
| 98280 | 15120 | ||||||||

124 |
| 35280 | 5040 | ||||||||

125 |
| 136080 | 15120 | ||||||||

126 |
| 105840 | 15120 | ||||||||

127 |
| 252000 | 25200 | ||||||||

128 |
| 340200 | 37800 | ||||||||

129 |
| 176400 | 25200 | ||||||||

130 |
| 252000 | 25200 | ||||||||

131 |
| 504000 | 50400 | ||||||||

132 |
| 453600 | 50400 | ||||||||

133 |
| 136080 | 15120 | ||||||||

134 |
| 378000 | 37800 | ||||||||

135 |
| 35280 | 5040 | ||||||||

136 |
| 136080 | 30240 | ||||||||

137 |
| 491400 | 75600 | ||||||||

138 |
| 378000 | 75600 | ||||||||

139 |
| 378000 | 75600 | ||||||||

140 |
| 378000 | 75600 | ||||||||

141 |
| 340200 | 75600 | ||||||||

142 |
| 756000 | 100800 | ||||||||

143 |
| 1058400 | 151200 | ||||||||

144 |