# Uniform 9-polytope

In nine-dimensional geometry, a polyyotton (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 polyyotton is one which is vertex-transitive, and constructed from uniform facets.

A proposed name for 9-polytope is polyyotton (plural: polyyotta), created from poly-, yotta- (a variation on octa, meaning eight) and -on.

## Regular 9-polytopes

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:

1. {3,3,3,3,3,3,3,3} - 9-simplex
2. {4,3,3,3,3,3,3,3} - 9-cube
3. {3,3,3,3,3,3,3,4} - 9-orthoplex

There are no nonconvex regular 9-polytopes.

## Euler characteristic

The Euler characteristic for 9-polytopes that are topological 8-spheres (including all convex 9-polytopes) is zero. χ=V-E+F-C+f4-f5+f6-f7+f8=2.

## Uniform 9-polytopes by fundamental Coxeter groups

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:
1. {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:
1. {4,37} - 9-cube or enneract -
2. {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:
1. {31,6,1} - 9-demicube or demienneract, 16,1 - ; also as h{4,38} .
2. {36,1,1} - 9-orthoplex, 61,1 -

## The A9 family

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{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}

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}

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}

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}

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

t0,2,4,7{3,3,3,3,3,3,3,3}
Heptistericantellated 9-simplex

680400 75600
109

t1,2,4,7{3,3,3,3,3,3,3,3}
Bihexiruncitruncated 9-simplex

604800 75600
110

t0,3,4,7{3,3,3,3,3,3,3,3}
Heptisteriruncinated 9-simplex

378000 50400
111

t1,3,4,7{3,3,3,3,3,3,3,3}
Bihexiruncicantellated 9-simplex

567000 75600
112

t0,1,5,7{3,3,3,3,3,3,3,3}
Heptipentitruncated 9-simplex

321300 37800
113

t0,2,5,7{3,3,3,3,3,3,3,3}
Heptipenticantellated 9-simplex

680400 75600
114

t1,2,5,7{3,3,3,3,3,3,3,3}
Bihexisteritruncated 9-simplex

567000 75600
115

t0,3,5,7{3,3,3,3,3,3,3,3}
Heptipentiruncinated 9-simplex

642600 75600
116

t1,3,5,7{3,3,3,3,3,3,3,3}
Bihexistericantellated 9-simplex

907200 113400
117

t0,4,5,7{3,3,3,3,3,3,3,3}
Heptipentistericated 9-simplex

264600 37800
118

t0,1,6,7{3,3,3,3,3,3,3,3}
Heptihexitruncated 9-simplex

98280 15120
119

t0,2,6,7{3,3,3,3,3,3,3,3}
Heptihexicantellated 9-simplex

302400 37800
120

t1,2,6,7{3,3,3,3,3,3,3,3}
Bihexipentitruncated 9-simplex

226800 37800
121

t0,3,6,7{3,3,3,3,3,3,3,3}
Heptihexiruncinated 9-simplex

428400 50400
122

t0,4,6,7{3,3,3,3,3,3,3,3}
Heptihexistericated 9-simplex

302400 37800
123

t0,5,6,7{3,3,3,3,3,3,3,3}
Heptihexipentellated 9-simplex

98280 15120
124

t0,1,2,8{3,3,3,3,3,3,3,3}
Octicantitruncated 9-simplex

35280 5040
125

t0,1,3,8{3,3,3,3,3,3,3,3}
Octiruncitruncated 9-simplex

136080 15120
126

t0,2,3,8{3,3,3,3,3,3,3,3}
Octiruncicantellated 9-simplex

105840 15120
127

t0,1,4,8{3,3,3,3,3,3,3,3}
Octisteritruncated 9-simplex

252000 25200
128

t0,2,4,8{3,3,3,3,3,3,3,3}
Octistericantellated 9-simplex

340200 37800
129

t0,3,4,8{3,3,3,3,3,3,3,3}
Octisteriruncinated 9-simplex

176400 25200
130

t0,1,5,8{3,3,3,3,3,3,3,3}
Octipentitruncated 9-simplex

252000 25200
131

t0,2,5,8{3,3,3,3,3,3,3,3}
Octipenticantellated 9-simplex

504000 50400
132

t0,3,5,8{3,3,3,3,3,3,3,3}
Octipentiruncinated 9-simplex

453600 50400
133

t0,1,6,8{3,3,3,3,3,3,3,3}
Octihexitruncated 9-simplex

136080 15120
134

t0,2,6,8{3,3,3,3,3,3,3,3}
Octihexicantellated 9-simplex

378000 37800
135

t0,1,7,8{3,3,3,3,3,3,3,3}
Octiheptitruncated 9-simplex

35280 5040
136

t0,1,2,3,4{3,3,3,3,3,3,3,3}
Steriruncicantitruncated 9-simplex

136080 30240
137

t0,1,2,3,5{3,3,3,3,3,3,3,3}
Pentiruncicantitruncated 9-simplex

491400 75600
138

t0,1,2,4,5{3,3,3,3,3,3,3,3}
Pentistericantitruncated 9-simplex

378000 75600
139

t0,1,3,4,5{3,3,3,3,3,3,3,3}
Pentisteriruncitruncated 9-simplex

378000 75600
140

t0,2,3,4,5{3,3,3,3,3,3,3,3}
Pentisteriruncicantellated 9-simplex

378000 75600
141

t1,2,3,4,5{3,3,3,3,3,3,3,3}
Bisteriruncicantitruncated 9-simplex

340200 75600
142

t0,1,2,3,6{3,3,3,3,3,3,3,3}
Hexiruncicantitruncated 9-simplex

756000 100800
143

t0,1,2,4,6{3,3,3,3,3,3,3,3}
Hexistericantitruncated 9-simplex

1058400 151200
144

t0,1,3,4,6{3,3,3,3,3,3,3,3}
Hexisteriruncitruncated 9-simplex

982800 151200
145

t0,2,3,4,6{3,3,3,3,3,3,3,3}
Hexisteriruncicantellated 9-simplex

982800 151200
146

t1,2,3,4,6{3,3,3,3,3,3,3,3}
Bipentiruncicantitruncated 9-simplex

907200 151200
147

t0,1,2,5,6{3,3,3,3,3,3,3,3}
Hexipenticantitruncated 9-simplex

554400 100800
148

t0,1,3,5,6{3,3,3,3,3,3,3,3}
Hexipentiruncitruncated 9-simplex

907200 151200
149

t0,2,3,5,6{3,3,3,3,3,3,3,3}
Hexipentiruncicantellated 9-simplex

831600 151200
150

t1,2,3,5,6{3,3,3,3,3,3,3,3}
Bipentistericantitruncated 9-simplex

756000 151200
151

t0,1,4,5,6{3,3,3,3,3,3,3,3}
Hexipentisteritruncated 9-simplex

554400 100800
152

t0,2,4,5,6{3,3,3,3,3,3,3,3}
Hexipentistericantellated 9-simplex

907200 151200
153

t1,2,4,5,6{3,3,3,3,3,3,3,3}
Bipentisteriruncitruncated 9-simplex

756000 151200
154

t0,3,4,5,6{3,3,3,3,3,3,3,3}
Hexipentisteriruncinated 9-simplex

554400 100800
155

t1,3,4,5,6{3,3,3,3,3,3,3,3}
Bipentisteriruncicantellated 9-simplex

831600 151200
156

t2,3,4,5,6{3,3,3,3,3,3,3,3}
Tristeriruncicantitruncated 9-simplex

453600 100800
157

t0,1,2,3,7{3,3,3,3,3,3,3,3}
Heptiruncicantitruncated 9-simplex

567000 75600
158

t0,1,2,4,7{3,3,3,3,3,3,3,3}
Heptistericantitruncated 9-simplex

1209600 151200
159

t0,1,3,4,7{3,3,3,3,3,3,3,3}
Heptisteriruncitruncated 9-simplex

1058400 151200
160

t0,2,3,4,7{3,3,3,3,3,3,3,3}
Heptisteriruncicantellated 9-simplex

1058400 151200
161

t1,2,3,4,7{3,3,3,3,3,3,3,3}
Bihexiruncicantitruncated 9-simplex

982800 151200
162

t0,1,2,5,7{3,3,3,3,3,3,3,3}
Heptipenticantitruncated 9-simplex

1134000 151200
163

t0,1,3,5,7{3,3,3,3,3,3,3,3}
Heptipentiruncitruncated 9-simplex

1701000 226800
164

t0,2,3,5,7{3,3,3,3,3,3,3,3}
Heptipentiruncicantellated 9-simplex

1587600 226800
165

t1,2,3,5,7{3,3,3,3,3,3,3,3}
Bihexistericantitruncated 9-simplex

1474200 226800
166

t0,1,4,5,7{3,3,3,3,3,3,3,3}
Heptipentisteritruncated 9-simplex

982800 151200
167

t0,2,4,5,7{3,3,3,3,3,3,3,3}
Heptipentistericantellated 9-simplex

1587600 226800
168

t1,2,4,5,7{3,3,3,3,3,3,3,3}
Bihexisteriruncitruncated 9-simplex

1360800 226800
169

t0,3,4,5,7{3,3,3,3,3,3,3,3}
Heptipentisteriruncinated 9-simplex

982800 151200
170

t1,3,4,5,7{3,3,3,3,3,3,3,3}
Bihexisteriruncicantellated 9-simplex

1474200 226800
171

t0,1,2,6,7{3,3,3,3,3,3,3,3}
Heptihexicantitruncated 9-simplex

453600 75600
172

t0,1,3,6,7{3,3,3,3,3,3,3,3}
Heptihexiruncitruncated 9-simplex

1058400 151200
173

t0,2,3,6,7{3,3,3,3,3,3,3,3}
Heptihexiruncicantellated 9-simplex

907200 151200
174

t1,2,3,6,7{3,3,3,3,3,3,3,3}
Bihexipenticantitruncated 9-simplex

831600 151200
175

t0,1,4,6,7{3,3,3,3,3,3,3,3}
Heptihexisteritruncated 9-simplex

1058400 151200
176

t0,2,4,6,7{3,3,3,3,3,3,3,3}
Heptihexistericantellated 9-simplex

1587600 226800
177

t1,2,4,6,7{3,3,3,3,3,3,3,3}
Bihexipentiruncitruncated 9-simplex

1360800 226800
178

t0,3,4,6,7{3,3,3,3,3,3,3,3}
Heptihexisteriruncinated 9-simplex

907200 151200
179

t0,1,5,6,7{3,3,3,3,3,3,3,3}
Heptihexipentitruncated 9-simplex

453600 75600
180

t0,2,5,6,7{3,3,3,3,3,3,3,3}
Heptihexipenticantellated 9-simplex

1058400 151200
181

t0,3,5,6,7{3,3,3,3,3,3,3,3}
Heptihexipentiruncinated 9-simplex

1058400 151200
182

t0,4,5,6,7{3,3,3,3,3,3,3,3}
Heptihexipentistericated 9-simplex

453600 75600
183

t0,1,2,3,8{3,3,3,3,3,3,3,3}
Octiruncicantitruncated 9-simplex

196560 30240
184

t0,1,2,4,8{3,3,3,3,3,3,3,3}
Octistericantitruncated 9-simplex

604800 75600
185

t0,1,3,4,8{3,3,3,3,3,3,3,3}
Octisteriruncitruncated 9-simplex

491400 75600
186

t0,2,3,4,8{3,3,3,3,3,3,3,3}
Octisteriruncicantellated 9-simplex

491400 75600
187

t0,1,2,5,8{3,3,3,3,3,3,3,3}
Octipenticantitruncated 9-simplex

856800 100800
188

t0,1,3,5,8{3,3,3,3,3,3,3,3}
Octipentiruncitruncated 9-simplex

1209600 151200
189

t0,2,3,5,8{3,3,3,3,3,3,3,3}
Octipentiruncicantellated 9-simplex

1134000 151200
190

t0,1,4,5,8{3,3,3,3,3,3,3,3}
Octipentisteritruncated 9-simplex

655200 100800
191

t0,2,4,5,8{3,3,3,3,3,3,3,3}
Octipentistericantellated 9-simplex

1058400 151200
192

t0,3,4,5,8{3,3,3,3,3,3,3,3}
Octipentisteriruncinated 9-simplex

655200 100800
193

t0,1,2,6,8{3,3,3,3,3,3,3,3}
Octihexicantitruncated 9-simplex

604800 75600
194

t0,1,3,6,8{3,3,3,3,3,3,3,3}
Octihexiruncitruncated 9-simplex

1285200 151200
195

t0,2,3,6,8{3,3,3,3,3,3,3,3}
Octihexiruncicantellated 9-simplex

1134000 151200
196

t0,1,4,6,8{3,3,3,3,3,3,3,3}
Octihexisteritruncated 9-simplex

1209600 151200
197

t0,2,4,6,8{3,3,3,3,3,3,3,3}
Octihexistericantellated 9-simplex

1814400 226800
198

t0,1,5,6,8{3,3,3,3,3,3,3,3}
Octihexipentitruncated 9-simplex

491400 75600
199

t0,1,2,7,8{3,3,3,3,3,3,3,3}
Octihepticantitruncated 9-simplex

196560 30240
200

t0,1,3,7,8{3,3,3,3,3,3,3,3}
Octiheptiruncitruncated 9-simplex

604800 75600
201

t0,1,4,7,8{3,3,3,3,3,3,3,3}
Octiheptisteritruncated 9-simplex

856800 100800
202

t0,1,2,3,4,5{3,3,3,3,3,3,3,3}
Pentisteriruncicantitruncated 9-simplex

680400 151200
203

t0,1,2,3,4,6{3,3,3,3,3,3,3,3}
Hexisteriruncicantitruncated 9-simplex

1814400 302400
204

t0,1,2,3,5,6{3,3,3,3,3,3,3,3}
Hexipentiruncicantitruncated 9-simplex

1512000 302400
205

t0,1,2,4,5,6{3,3,3,3,3,3,3,3}
Hexipentistericantitruncated 9-simplex

1512000 302400
206

t0,1,3,4,5,6{3,3,3,3,3,3,3,3}
Hexipentisteriruncitruncated 9-simplex

1512000 302400
207

t0,2,3,4,5,6{3,3,3,3,3,3,3,3}
Hexipentisteriruncicantellated 9-simplex

1512000 302400
208

t1,2,3,4,5,6{3,3,3,3,3,3,3,3}
Bipentisteriruncicantitruncated 9-simplex

1360800 302400
209

t0,1,2,3,4,7{3,3,3,3,3,3,3,3}
Heptisteriruncicantitruncated 9-simplex

1965600 302400
210

t0,1,2,3,5,7{3,3,3,3,3,3,3,3}
Heptipentiruncicantitruncated 9-simplex

2948400 453600
211

t0,1,2,4,5,7{3,3,3,3,3,3,3,3}
Heptipentistericantitruncated 9-simplex

2721600 453600
212

t0,1,3,4,5,7{3,3,3,3,3,3,3,3}
Heptipentisteriruncitruncated 9-simplex

2721600 453600
213

t0,2,3,4,5,7{3,3,3,3,3,3,3,3}
Heptipentisteriruncicantellated 9-simplex

2721600 453600
214

t1,2,3,4,5,7{3,3,3,3,3,3,3,3}
Bihexisteriruncicantitruncated 9-simplex

2494800 453600
215

t0,1,2,3,6,7{3,3,3,3,3,3,3,3}
Heptihexiruncicantitruncated 9-simplex

1663200 302400
216

t0,1,2,4,6,7{3,3,3,3,3,3,3,3}
Heptihexistericantitruncated 9-simplex

2721600 453600
217

t0,1,3,4,6,7{3,3,3,3,3,3,3,3}
Heptihexisteriruncitruncated 9-simplex

2494800 453600
218

t0,2,3,4,6,7{3,3,3,3,3,3,3,3}
Heptihexisteriruncicantellated 9-simplex

2494800 453600
219

t1,2,3,4,6,7{3,3,3,3,3,3,3,3}
Bihexipentiruncicantitruncated 9-simplex

2268000 453600
220

t0,1,2,5,6,7{3,3,3,3,3,3,3,3}
Heptihexipenticantitruncated 9-simplex

1663200 302400
221

t0,1,3,5,6,7{3,3,3,3,3,3,3,3}
Heptihexipentiruncitruncated 9-simplex

2721600 453600
222

t0,2,3,5,6,7{3,3,3,3,3,3,3,3}
Heptihexipentiruncicantellated 9-simplex

2494800 453600
223

t1,2,3,5,6,7{3,3,3,3,3,3,3,3}
Bihexipentistericantitruncated 9-simplex

2268000 453600
224

t0,1,4,5,6,7{3,3,3,3,3,3,3,3}
Heptihexipentisteritruncated 9-simplex

1663200 302400
225

t0,2,4,5,6,7{3,3,3,3,3,3,3,3}
Heptihexipentistericantellated 9-simplex

2721600 453600
226

t0,3,4,5,6,7{3,3,3,3,3,3,3,3}
Heptihexipentisteriruncinated 9-simplex

1663200 302400
227

t0,1,2,3,4,8{3,3,3,3,3,3,3,3}
Octisteriruncicantitruncated 9-simplex

907200 151200
228

t0,1,2,3,5,8{3,3,3,3,3,3,3,3}
Octipentiruncicantitruncated 9-simplex

2116800 302400
229

t0,1,2,4,5,8{3,3,3,3,3,3,3,3}
Octipentistericantitruncated 9-simplex

1814400 302400
230

t0,1,3,4,5,8{3,3,3,3,3,3,3,3}
Octipentisteriruncitruncated 9-simplex

1814400 302400
231

t0,2,3,4,5,8{3,3,3,3,3,3,3,3}
Octipentisteriruncicantellated 9-simplex

1814400 302400
232

t0,1,2,3,6,8{3,3,3,3,3,3,3,3}
Octihexiruncicantitruncated 9-simplex

2116800 302400
233

t0,1,2,4,6,8{3,3,3,3,3,3,3,3}
Octihexistericantitruncated 9-simplex

3175200 453600
234

t0,1,3,4,6,8{3,3,3,3,3,3,3,3}
Octihexisteriruncitruncated 9-simplex

2948400 453600
235

t0,2,3,4,6,8{3,3,3,3,3,3,3,3}
Octihexisteriruncicantellated 9-simplex

2948400 453600
236

t0,1,2,5,6,8{3,3,3,3,3,3,3,3}
Octihexipenticantitruncated 9-simplex

1814400 302400
237

t0,1,3,5,6,8{3,3,3,3,3,3,3,3}
Octihexipentiruncitruncated 9-simplex

2948400 453600
238

t0,2,3,5,6,8{3,3,3,3,3,3,3,3}
Octihexipentiruncicantellated 9-simplex

2721600 453600
239

t0,1,4,5,6,8{3,3,3,3,3,3,3,3}
Octihexipentisteritruncated 9-simplex

1814400 302400
240

t0,1,2,3,7,8{3,3,3,3,3,3,3,3}
Octiheptiruncicantitruncated 9-simplex

907200 151200
241

t0,1,2,4,7,8{3,3,3,3,3,3,3,3}
Octiheptistericantitruncated 9-simplex

2116800 302400
242

t0,1,3,4,7,8{3,3,3,3,3,3,3,3}
Octiheptisteriruncitruncated 9-simplex

1814400 302400
243

t0,1,2,5,7,8{3,3,3,3,3,3,3,3}
Octiheptipenticantitruncated 9-simplex

2116800 302400
244

t0,1,3,5,7,8{3,3,3,3,3,3,3,3}
Octiheptipentiruncitruncated 9-simplex

3175200 453600
245

t0,1,2,6,7,8{3,3,3,3,3,3,3,3}
Octiheptihexicantitruncated 9-simplex

907200 151200
246

t0,1,2,3,4,5,6{3,3,3,3,3,3,3,3}
Hexipentisteriruncicantitruncated 9-simplex

2721600 604800
247

t0,1,2,3,4,5,7{3,3,3,3,3,3,3,3}
Heptipentisteriruncicantitruncated 9-simplex

4989600 907200
248

t0,1,2,3,4,6,7{3,3,3,3,3,3,3,3}
Heptihexisteriruncicantitruncated 9-simplex

4536000 907200
249

t0,1,2,3,5,6,7{3,3,3,3,3,3,3,3}
Heptihexipentiruncicantitruncated 9-simplex

4536000 907200
250

t0,1,2,4,5,6,7{3,3,3,3,3,3,3,3}
Heptihexipentistericantitruncated 9-simplex

4536000 907200
251

t0,1,3,4,5,6,7{3,3,3,3,3,3,3,3}
Heptihexipentisteriruncitruncated 9-simplex

4536000 907200
252

t0,2,3,4,5,6,7{3,3,3,3,3,3,3,3}
Heptihexipentisteriruncicantellated 9-simplex

4536000 907200
253

t1,2,3,4,5,6,7{3,3,3,3,3,3,3,3}
Bihexipentisteriruncicantitruncated 9-simplex

4082400 907200
254

t0,1,2,3,4,5,8{3,3,3,3,3,3,3,3}
Octipentisteriruncicantitruncated 9-simplex

3326400 604800
255

t0,1,2,3,4,6,8{3,3,3,3,3,3,3,3}
Octihexisteriruncicantitruncated 9-simplex

5443200 907200
256

t0,1,2,3,5,6,8{3,3,3,3,3,3,3,3}
Octihexipentiruncicantitruncated 9-simplex

4989600 907200
257

t0,1,2,4,5,6,8{3,3,3,3,3,3,3,3}
Octihexipentistericantitruncated 9-simplex

4989600 907200
258

t0,1,3,4,5,6,8{3,3,3,3,3,3,3,3}
Octihexipentisteriruncitruncated 9-simplex

4989600 907200
259

t0,2,3,4,5,6,8{3,3,3,3,3,3,3,3}
Octihexipentisteriruncicantellated 9-simplex

4989600 907200
260

t0,1,2,3,4,7,8{3,3,3,3,3,3,3,3}
Octiheptisteriruncicantitruncated 9-simplex

3326400 604800
261

t0,1,2,3,5,7,8{3,3,3,3,3,3,3,3}
Octiheptipentiruncicantitruncated 9-simplex

5443200 907200
262

t0,1,2,4,5,7,8{3,3,3,3,3,3,3,3}
Octiheptipentistericantitruncated 9-simplex

4989600 907200
263

t0,1,3,4,5,7,8{3,3,3,3,3,3,3,3}
Octiheptipentisteriruncitruncated 9-simplex

4989600 907200
264

t0,1,2,3,6,7,8{3,3,3,3,3,3,3,3}
Octiheptihexiruncicantitruncated 9-simplex

3326400 604800
265

t0,1,2,4,6,7,8{3,3,3,3,3,3,3,3}
Octiheptihexistericantitruncated 9-simplex

5443200 907200
266

t0,1,2,3,4,5,6,7{3,3,3,3,3,3,3,3}
Heptihexipentisteriruncicantitruncated 9-simplex

8164800 1814400
267

t0,1,2,3,4,5,6,8{3,3,3,3,3,3,3,3}
Octihexipentisteriruncicantitruncated 9-simplex

9072000 1814400
268

t0,1,2,3,4,5,7,8{3,3,3,3,3,3,3,3}
Octiheptipentisteriruncicantitruncated 9-simplex

9072000 1814400
269

t0,1,2,3,4,6,7,8{3,3,3,3,3,3,3,3}
Octiheptihexisteriruncicantitruncated 9-simplex

9072000 1814400
270

t0,1,2,3,5,6,7,8{3,3,3,3,3,3,3,3}
Octiheptihexipentiruncicantitruncated 9-simplex

9072000 1814400
271

t0,1,2,3,4,5,6,7,8{3,3,3,3,3,3,3,3}
Omnitruncated 9-simplex

16329600 3628800

## The B9 family

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{4,3,3,3,3,3,3,3}
9-cube (enne)
18 144 672 2016 4032 5376 4608 2304 512
2
t0,1{4,3,3,3,3,3,3,3}
Truncated 9-cube (ten)
2304 4608
3
t1{4,3,3,3,3,3,3,3}
Rectified 9-cube (ren)
18432 2304
4
t2{4,3,3,3,3,3,3,3}
Birectified 9-cube (barn)
64512 4608
5
t3{4,3,3,3,3,3,3,3}
Trirectified 9-cube (tarn)
96768 5376
6
t4{4,3,3,3,3,3,3,3}
80640 4032
7
t3{3,3,3,3,3,3,3,4}
Trirectified 9-orthoplex (tarv)
40320 2016
8
t2{3,3,3,3,3,3,3,4}
Birectified 9-orthoplex (brav)
12096 672
9
t1{3,3,3,3,3,3,3,4}
Rectified 9-orthoplex (riv)
2016 144
10
t0,1{3,3,3,3,3,3,3,4}
Truncated 9-orthoplex (tiv)
2160 288
11
t0{3,3,3,3,3,3,3,4}
9-orthoplex (vee)
512 2304 4608 5376 4032 2016 672 144 18

## The D9 family

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.

# Coxeter plane graphs Coxeter-Dynkin diagram
Schläfli symbol
Base point
(Alternately signed)
B9 D9 D8 D7 D6 D5 D4 D3 A7 A5 A3 8 7 6 5 4 3 2 1 0
1
9-demicube (henne)
(1,1,1,1,1,1,1,1,1) 274 2448 9888 23520 36288 37632 21404 4608 256 1.0606601
2
Truncated 9-demicube (thenne)
(1,1,3,3,3,3,3,3,3) 69120 9216 2.8504384
3
Cantellated 9-demicube
(1,1,1,3,3,3,3,3,3) 225792 21504 2.6692696
4
Runcinated 9-demicube
(1,1,1,1,3,3,3,3,3) 419328 32256 2.4748735
5
Stericated 9-demicube
(1,1,1,1,1,3,3,3,3) 483840 32256 2.2638462
6
Pentellated 9-demicube
(1,1,1,1,1,1,3,3,3) 354816 21504 2.0310094
7
Hexicated 9-demicube
(1,1,1,1,1,1,1,3,3) 161280 9216 1.7677668
8
Heptellated 9-demicube
(1,1,1,1,1,1,1,1,3) 41472 2304 1.4577379

## Regular and uniform honeycombs

Coxeter-Dynkin diagram correspondences between families and higher symmetry within diagrams. Nodes of the same color in each row represent identical mirrors. Black nodes are not active in the correspondence.

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[9]] 45
2 ${\tilde{C}}_8$ [4,36,4] 271
3 ${\tilde{B}}_8$ h[4,36,4]
[4,35,31,1]
383 (128 new)
4 ${\tilde{D}}_8$ q[4,36,4]
[31,1,34,31,1]
155 (15 new)
5 ${\tilde{E}}_8$ [35,2,1] 511

Regular and uniform tessellations include:

• ${\tilde{A}}_8$ 45 uniquely ringed forms
• ${\tilde{C}}_8$ 271 uniquely ringed forms
• ${\tilde{B}}_8$: 383 uniquely ringed forms, 255 shared with ${\tilde{C}}_8$, 128 new
• ${\tilde{D}}_8$, [31,1,34,31,1]: 155 unique ring permutations, and 15 are new, the first, , Coxeter called a quarter 8-cubic honeycomb, representing as q{4,36,4}, or qδ9.
• ${\tilde{E}}_8$ 511 forms

### Regular and uniform hyperbolic honeycombs

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 noncompact hyperbolic Coxeter groups of rank 9, each generating uniform honeycombs in 8-space as permutations of rings of the Coxeter diagrams.

 ${\bar{P}}_8$ = [3,3[8]]: ${\bar{Q}}_8$ = [31,1,33,32,1]: ${\bar{S}}_8$ = [4,34,32,1]: ${\bar{T}}_8$ = [34,3,1]:

## References

• T. Gosset: On the Regular and Semi-Regular Figures in Space of n Dimensions, Messenger of Mathematics, Macmillan, 1900
• A. Boole Stott: Geometrical deduction of semiregular from regular polytopes and space fillings, Verhandelingen of the Koninklijke academy van Wetenschappen width unit Amsterdam, Eerste Sectie 11,1, Amsterdam, 1910
• H.S.M. Coxeter:
• H.S.M. Coxeter, M.S. Longuet-Higgins und J.C.P. Miller: Uniform Polyhedra, Philosophical Transactions of the Royal Society of London, Londne, 1954
• H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973
• Kaleidoscopes: Selected Writings of H.S.M. Coxeter, editied by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6 [1]
• (Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380-407, MR 2,10]
• (Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559-591]
• (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
• N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. Dissertation, University of Toronto, 1966
• Richard Klitzing, 9D, uniform polytopes (polyyotta)