Uniform 9-polytope

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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.

Contents

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