# Uniform 9-polytope

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.

 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

A uniform 9-polytope is one which is vertex-transitive, and constructed from uniform 8-polytope facets.

## 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 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 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, 161 - ; also as h{4,38} .
2. {36,1,1} - 9-orthoplex, 611 -

## 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-faces7-faces6-faces5-faces4-facesCellsFacesEdgesVertices
1

t0{3,3,3,3,3,3,3,3}
9-simplex (day)

10451202102522101204510
2

t1{3,3,3,3,3,3,3,3}
Rectified 9-simplex (reday)

36045
3

t2{3,3,3,3,3,3,3,3}
Birectified 9-simplex (breday)

1260120
4

t3{3,3,3,3,3,3,3,3}
Trirectified 9-simplex (treday)

2520210
5

t4{3,3,3,3,3,3,3,3}

3150252
6

t0,1{3,3,3,3,3,3,3,3}
Truncated 9-simplex (teday)

40590
7

t0,2{3,3,3,3,3,3,3,3}
Cantellated 9-simplex

2880360
8

t1,2{3,3,3,3,3,3,3,3}
Bitruncated 9-simplex

1620360
9

t0,3{3,3,3,3,3,3,3,3}
Runcinated 9-simplex

8820840
10

t1,3{3,3,3,3,3,3,3,3}
Bicantellated 9-simplex

100801260
11

t2,3{3,3,3,3,3,3,3,3}
Tritruncated 9-simplex (treday)

3780840
12

t0,4{3,3,3,3,3,3,3,3}
Stericated 9-simplex

151201260
13

t1,4{3,3,3,3,3,3,3,3}
Biruncinated 9-simplex

264602520
14

t2,4{3,3,3,3,3,3,3,3}
Tricantellated 9-simplex

201602520
15

t3,4{3,3,3,3,3,3,3,3}

56701260
16

t0,5{3,3,3,3,3,3,3,3}
Pentellated 9-simplex

157501260
17

t1,5{3,3,3,3,3,3,3,3}
Bistericated 9-simplex

378003150
18

t2,5{3,3,3,3,3,3,3,3}
Triruncinated 9-simplex

441004200
19

t3,5{3,3,3,3,3,3,3,3}

252003150
20

t0,6{3,3,3,3,3,3,3,3}
Hexicated 9-simplex

10080840
21

t1,6{3,3,3,3,3,3,3,3}
Bipentellated 9-simplex

315002520
22

t2,6{3,3,3,3,3,3,3,3}
Tristericated 9-simplex

504004200
23

t0,7{3,3,3,3,3,3,3,3}
Heptellated 9-simplex

3780360
24

t1,7{3,3,3,3,3,3,3,3}
Bihexicated 9-simplex

151201260
25

t0,8{3,3,3,3,3,3,3,3}
Octellated 9-simplex

72090
26

t0,1,2{3,3,3,3,3,3,3,3}
Cantitruncated 9-simplex

3240720
27

t0,1,3{3,3,3,3,3,3,3,3}
Runcitruncated 9-simplex

189002520
28

t0,2,3{3,3,3,3,3,3,3,3}
Runcicantellated 9-simplex

126002520
29

t1,2,3{3,3,3,3,3,3,3,3}
Bicantitruncated 9-simplex

113402520
30

t0,1,4{3,3,3,3,3,3,3,3}
Steritruncated 9-simplex

478805040
31

t0,2,4{3,3,3,3,3,3,3,3}
Stericantellated 9-simplex

604807560
32

t1,2,4{3,3,3,3,3,3,3,3}
Biruncitruncated 9-simplex

529207560
33

t0,3,4{3,3,3,3,3,3,3,3}
Steriruncinated 9-simplex

277205040
34

t1,3,4{3,3,3,3,3,3,3,3}
Biruncicantellated 9-simplex

415807560
35

t2,3,4{3,3,3,3,3,3,3,3}
Tricantitruncated 9-simplex

226805040
36

t0,1,5{3,3,3,3,3,3,3,3}
Pentitruncated 9-simplex

661506300
37

t0,2,5{3,3,3,3,3,3,3,3}
Penticantellated 9-simplex

12600012600
38

t1,2,5{3,3,3,3,3,3,3,3}
Bisteritruncated 9-simplex

10710012600
39

t0,3,5{3,3,3,3,3,3,3,3}
Pentiruncinated 9-simplex

10710012600
40

t1,3,5{3,3,3,3,3,3,3,3}
Bistericantellated 9-simplex

15120018900
41

t2,3,5{3,3,3,3,3,3,3,3}
Triruncitruncated 9-simplex

8190012600
42

t0,4,5{3,3,3,3,3,3,3,3}
Pentistericated 9-simplex

378006300
43

t1,4,5{3,3,3,3,3,3,3,3}
Bisteriruncinated 9-simplex

8190012600
44

t2,4,5{3,3,3,3,3,3,3,3}
Triruncicantellated 9-simplex

7560012600
45

t3,4,5{3,3,3,3,3,3,3,3}

283506300
46

t0,1,6{3,3,3,3,3,3,3,3}
Hexitruncated 9-simplex

529205040
47

t0,2,6{3,3,3,3,3,3,3,3}
Hexicantellated 9-simplex

13860012600
48

t1,2,6{3,3,3,3,3,3,3,3}
Bipentitruncated 9-simplex

11340012600
49

t0,3,6{3,3,3,3,3,3,3,3}
Hexiruncinated 9-simplex

17640016800
50

t1,3,6{3,3,3,3,3,3,3,3}
Bipenticantellated 9-simplex

23940025200
51

t2,3,6{3,3,3,3,3,3,3,3}
Tristeritruncated 9-simplex

12600016800
52

t0,4,6{3,3,3,3,3,3,3,3}
Hexistericated 9-simplex

11340012600
53

t1,4,6{3,3,3,3,3,3,3,3}
Bipentiruncinated 9-simplex

22680025200
54

t2,4,6{3,3,3,3,3,3,3,3}
Tristericantellated 9-simplex

20160025200
55

t0,5,6{3,3,3,3,3,3,3,3}
Hexipentellated 9-simplex

327605040
56

t1,5,6{3,3,3,3,3,3,3,3}
Bipentistericated 9-simplex

9450012600
57

t0,1,7{3,3,3,3,3,3,3,3}
Heptitruncated 9-simplex

239402520
58

t0,2,7{3,3,3,3,3,3,3,3}
Hepticantellated 9-simplex

831607560
59

t1,2,7{3,3,3,3,3,3,3,3}
Bihexitruncated 9-simplex

642607560
60

t0,3,7{3,3,3,3,3,3,3,3}
Heptiruncinated 9-simplex

14490012600
61

t1,3,7{3,3,3,3,3,3,3,3}
Bihexicantellated 9-simplex

18900018900
62

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

13860012600
63

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

26460025200
64

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

718207560
65

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

176402520
66

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

5400720
67

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

252002520
68

t0,3,8{3,3,3,3,3,3,3,3}
Octiruncinated 9-simplex

579605040
69

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

756006300
70

t0,1,2,3{3,3,3,3,3,3,3,3}
Runcicantitruncated 9-simplex

226805040
71

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

10584015120
72

t0,1,3,4{3,3,3,3,3,3,3,3}
Steriruncitruncated 9-simplex

7560015120
73

t0,2,3,4{3,3,3,3,3,3,3,3}
Steriruncicantellated 9-simplex

7560015120
74

t1,2,3,4{3,3,3,3,3,3,3,3}
Biruncicantitruncated 9-simplex

6804015120
75

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

21420025200
76

t0,1,3,5{3,3,3,3,3,3,3,3}
Pentiruncitruncated 9-simplex

28350037800
77

t0,2,3,5{3,3,3,3,3,3,3,3}
Pentiruncicantellated 9-simplex

26460037800
78

t1,2,3,5{3,3,3,3,3,3,3,3}
Bistericantitruncated 9-simplex

24570037800
79

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

13860025200
80

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

22680037800
81

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

18900037800
82

t0,3,4,5{3,3,3,3,3,3,3,3}
Pentisteriruncinated 9-simplex

13860025200
83

t1,3,4,5{3,3,3,3,3,3,3,3}
Bisteriruncicantellated 9-simplex

20790037800
84

t2,3,4,5{3,3,3,3,3,3,3,3}
Triruncicantitruncated 9-simplex

11340025200
85

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

22680025200
86

t0,1,3,6{3,3,3,3,3,3,3,3}
Hexiruncitruncated 9-simplex

45360050400
87

t0,2,3,6{3,3,3,3,3,3,3,3}
Hexiruncicantellated 9-simplex

40320050400
88

t1,2,3,6{3,3,3,3,3,3,3,3}
Bipenticantitruncated 9-simplex

37800050400
89

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

40320050400
90

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

60480075600
91

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

52920075600
92

t0,3,4,6{3,3,3,3,3,3,3,3}
Hexisteriruncinated 9-simplex

35280050400
93

t1,3,4,6{3,3,3,3,3,3,3,3}
Bipentiruncicantellated 9-simplex

52920075600
94

t2,3,4,6{3,3,3,3,3,3,3,3}
Tristericantitruncated 9-simplex

30240050400
95

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

15120025200
96

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

35280050400
97

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

27720050400
98

t0,3,5,6{3,3,3,3,3,3,3,3}
Hexipentiruncinated 9-simplex

35280050400
99

t1,3,5,6{3,3,3,3,3,3,3,3}
Bipentistericantellated 9-simplex

49140075600
100

t2,3,5,6{3,3,3,3,3,3,3,3}
Tristeriruncitruncated 9-simplex

25200050400
101

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

15120025200
102

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

32760050400
103

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

12852015120
104

t0,1,3,7{3,3,3,3,3,3,3,3}
Heptiruncitruncated 9-simplex

35910037800
105

t0,2,3,7{3,3,3,3,3,3,3,3}
Heptiruncicantellated 9-simplex

30240037800
106

t1,2,3,7{3,3,3,3,3,3,3,3}
Bihexicantitruncated 9-simplex

28350037800
107

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

47880050400
108

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

68040075600
109

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

60480075600
110

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

37800050400
111

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

56700075600
112

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

32130037800
113

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

68040075600
114

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

56700075600
115

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

64260075600
116

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

907200113400
117

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

26460037800
118

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

9828015120
119

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

30240037800
120

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

22680037800
121

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

42840050400
122

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

30240037800
123

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

9828015120
124

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

352805040
125

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

13608015120
126

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

10584015120
127

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

25200025200
128

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

34020037800
129

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

17640025200
130

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

25200025200
131

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

50400050400
132

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

45360050400
133

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

13608015120
134

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

37800037800
135

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

352805040
136

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

13608030240
137

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

49140075600
138

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

37800075600
139

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

37800075600
140

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

37800075600
141

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

34020075600
142

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

756000100800
143

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

1058400151200
144

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

982800151200
145

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

982800151200
146

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

907200151200
147

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

554400100800
148

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

907200151200
149

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

831600151200
150

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

756000151200
151

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

554400100800
152

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

907200151200
153

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

756000151200
154

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

554400100800
155

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

831600151200
156

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

453600100800
157

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

56700075600
158

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

1209600151200
159

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

1058400151200
160

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

1058400151200
161

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

982800151200
162

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

1134000151200
163

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

1701000226800
164

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

1587600226800
165

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

1474200226800
166

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

982800151200
167

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

1587600226800
168

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

1360800226800
169

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

982800151200
170

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

1474200226800
171

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

45360075600
172

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

1058400151200
173

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

907200151200
174

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

831600151200
175

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

1058400151200
176

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

1587600226800
177

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

1360800226800
178

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

907200151200
179

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

45360075600
180

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

1058400151200
181

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

1058400151200
182

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

45360075600
183

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

19656030240
184

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

60480075600
185

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

49140075600
186

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

49140075600
187

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

856800100800
188

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

1209600151200
189

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

1134000151200
190

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

655200100800
191

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

1058400151200
192

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

655200100800
193

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

60480075600
194

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

1285200151200
195

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

1134000151200
196

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

1209600151200
197

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

1814400226800
198

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

49140075600
199

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

19656030240
200

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

60480075600
201

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

856800100800
202

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

680400151200
203

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

1814400302400
204

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

1512000302400
205

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

1512000302400
206

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

1512000302400
207

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

1512000302400
208

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

1360800302400
209

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

1965600302400
210

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

2948400453600
211

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

2721600453600
212

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

2721600453600
213

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

2721600453600
214

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

2494800453600
215

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

1663200302400
216

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

2721600453600
217

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

2494800453600
218

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

2494800453600
219

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

2268000453600
220

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

1663200302400
221

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

2721600453600
222

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

2494800453600
223

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

2268000453600
224

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

1663200302400
225

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

2721600453600
226

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

1663200302400
227

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

907200151200
228

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

2116800302400
229

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

1814400302400
230

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

1814400302400
231

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

1814400302400
232

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

2116800302400
233

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

3175200453600
234

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

2948400453600
235

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

2948400453600
236

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

1814400302400
237

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

2948400453600
238

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

2721600453600
239

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

1814400302400
240

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

907200151200
241

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

2116800302400
242

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

1814400302400
243

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

2116800302400
244

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

3175200453600
245

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

907200151200
246

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

2721600604800
247

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

4989600907200
248

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

4536000907200
249

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

4536000907200
250

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

4536000907200
251

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

4536000907200
252

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

4536000907200
253

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

4082400907200
254

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

3326400604800
255

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

5443200907200
256

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

4989600907200
257

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

4989600907200
258

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

4989600907200
259

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

4989600907200
260

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

3326400604800
261

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

5443200907200
262

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

4989600907200
263

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

4989600907200
264

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

3326400604800
265

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

5443200907200
266

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

81648001814400
267

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

90720001814400
268

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

90720001814400
269

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

90720001814400
270

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

90720001814400
271

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

163296003628800

## 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)
1814467220164032537646082304512
2
t0,1{4,3,3,3,3,3,3,3}
Truncated 9-cube (ten)
23044608
3
t1{4,3,3,3,3,3,3,3}
Rectified 9-cube (ren)
184322304
4
t2{4,3,3,3,3,3,3,3}
Birectified 9-cube (barn)
645124608
5
t3{4,3,3,3,3,3,3,3}
Trirectified 9-cube (tarn)
967685376
6
t4{4,3,3,3,3,3,3,3}
806404032
7
t3{3,3,3,3,3,3,3,4}
Trirectified 9-orthoplex (tarv)
403202016
8
t2{3,3,3,3,3,3,3,4}
Birectified 9-orthoplex (brav)
12096672
9
t1{3,3,3,3,3,3,3,4}
Rectified 9-orthoplex (riv)
2016144
10
t0,1{3,3,3,3,3,3,3,4}
Truncated 9-orthoplex (tiv)
2160288
11
t0{3,3,3,3,3,3,3,4}
9-orthoplex (vee)
5122304460853764032201667214418

## 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)
B9D9D8D7D6D5D4D3A7A5A3876543210
1
9-demicube (henne)
(1,1,1,1,1,1,1,1,1)274244898882352036288376322140446082561.0606601
2
Truncated 9-demicube (thenne)
(1,1,3,3,3,3,3,3,3)6912092162.8504384
3
Cantellated 9-demicube
(1,1,1,3,3,3,3,3,3)225792215042.6692696
4
Runcinated 9-demicube
(1,1,1,1,3,3,3,3,3)419328322562.4748735
5
Stericated 9-demicube
(1,1,1,1,1,3,3,3,3)483840322562.2638462
6
Pentellated 9-demicube
(1,1,1,1,1,1,3,3,3)354816215042.0310094
7
Hexicated 9-demicube
(1,1,1,1,1,1,1,3,3)16128092161.7677668
8
Heptellated 9-demicube
(1,1,1,1,1,1,1,1,3)4147223041.4577379

## Regular and uniform honeycombs

There are five fundamental affine Coxeter groups that generate regular and uniform tessellations in 8-space:

# Coxeter group Coxeter diagram Forms
1${\displaystyle {\tilde {A}}_{8}}$[3[9]]45
2${\displaystyle {\tilde {C}}_{8}}$[4,36,4]271
3${\displaystyle {\tilde {B}}_{8}}$h[4,36,4]
[4,35,31,1]
383 (128 new)
4${\displaystyle {\tilde {D}}_{8}}$q[4,36,4]
[31,1,34,31,1]
155 (15 new)
5${\displaystyle {\tilde {E}}_{8}}$[35,2,1]511

Regular and uniform tessellations include:

• ${\displaystyle {\tilde {A}}_{8}}$ 45 uniquely ringed forms
• ${\displaystyle {\tilde {C}}_{8}}$ 271 uniquely ringed forms
• ${\displaystyle {\tilde {B}}_{8}}$: 383 uniquely ringed forms, 255 shared with ${\displaystyle {\tilde {C}}_{8}}$, 128 new
• ${\displaystyle {\tilde {D}}_{8}}$, [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.
• ${\displaystyle {\tilde {E}}_{8}}$ 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.

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

## References

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