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.

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

Regular 9-polytopes

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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

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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

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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

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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}
Quadrirectified 9-simplex (icoy)

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}
Quadritruncated 9-simplex

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}
Quadricantellated 9-simplex

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}
Quadricantitruncated 9-simplex

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

edit

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}
Quadrirectified 9-cube (nav)
(Quadrirectified 9-orthoplex)
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

edit

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)
Element counts Circumrad
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

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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 [3[9]] 45
2 [4,36,4] 271
3 h[4,36,4]
[4,35,31,1]
383 (128 new)
4 q[4,36,4]
[31,1,34,31,1]
155 (15 new)
5 [35,2,1] 511

Regular and uniform tessellations include:

  • 45 uniquely ringed forms
  • 271 uniquely ringed forms
  • : 383 uniquely ringed forms, 255 shared with , 128 new
  • , [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.
  • 511 forms

Regular and uniform hyperbolic honeycombs

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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.

= [3,3[8]]:
= [31,1,33,32,1]:
= [4,34,32,1]:
= [34,3,1]:

References

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  1. ^ a b c Richeson, D.; Euler's Gem: The Polyhedron Formula and the Birth of Topoplogy, Princeton, 2008.
  • 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, edited 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
  • Klitzing, Richard. "9D uniform polytopes (polyyotta)".
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Family An Bn I2(p) / Dn E6 / E7 / E8 / F4 / G2 Hn
Regular polygon Triangle Square p-gon Hexagon Pentagon
Uniform polyhedron Tetrahedron OctahedronCube Demicube DodecahedronIcosahedron
Uniform polychoron Pentachoron 16-cellTesseract Demitesseract 24-cell 120-cell600-cell
Uniform 5-polytope 5-simplex 5-orthoplex5-cube 5-demicube
Uniform 6-polytope 6-simplex 6-orthoplex6-cube 6-demicube 122221
Uniform 7-polytope 7-simplex 7-orthoplex7-cube 7-demicube 132231321
Uniform 8-polytope 8-simplex 8-orthoplex8-cube 8-demicube 142241421
Uniform 9-polytope 9-simplex 9-orthoplex9-cube 9-demicube
Uniform 10-polytope 10-simplex 10-orthoplex10-cube 10-demicube
Uniform n-polytope n-simplex n-orthoplexn-cube n-demicube 1k22k1k21 n-pentagonal polytope
Topics: Polytope familiesRegular polytopeList of regular polytopes and compounds
Space Family / /
E2 Uniform tiling 0[3] δ3 3 3 Hexagonal
E3 Uniform convex honeycomb 0[4] δ4 4 4
E4 Uniform 4-honeycomb 0[5] δ5 5 5 24-cell honeycomb
E5 Uniform 5-honeycomb 0[6] δ6 6 6
E6 Uniform 6-honeycomb 0[7] δ7 7 7 222
E7 Uniform 7-honeycomb 0[8] δ8 8 8 133331
E8 Uniform 8-honeycomb 0[9] δ9 9 9 152251521
E9 Uniform 9-honeycomb 0[10] δ10 10 10
E10 Uniform 10-honeycomb 0[11] δ11 11 11
En-1 Uniform (n-1)-honeycomb 0[n] δn n n 1k22k1k21