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.

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

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}
Quadrirectified 9-simplex (icoy)

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}
Quadritruncated 9-simplex

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}
Quadricantellated 9-simplex

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}
Quadricantitruncated 9-simplex

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}
Quadrirectified 9-cube (nav)
(Quadrirectified 9-orthoplex)
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)
Element counts Circumrad
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[3[9]]45
2[4,36,4]271
3h[4,36,4]
[4,35,31,1]
383 (128 new)
4q[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

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.

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

References

  1. 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
    • (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)".
Fundamental convex regular and uniform polytopes in dimensions 2–10
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 4-polytope 5-cell 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
Fundamental convex regular and uniform honeycombs in dimensions 2-9
Space Family / /
E2 Uniform tiling {3[3]} δ3 hδ3 qδ3 Hexagonal
E3 Uniform convex honeycomb {3[4]} δ4 hδ4 qδ4
E4 Uniform 4-honeycomb {3[5]} δ5 hδ5 qδ5 24-cell honeycomb
E5 Uniform 5-honeycomb {3[6]} δ6 hδ6 qδ6
E6 Uniform 6-honeycomb {3[7]} δ7 hδ7 qδ7 222
E7 Uniform 7-honeycomb {3[8]} δ8 hδ8 qδ8 133331
E8 Uniform 8-honeycomb {3[9]} δ9 hδ9 qδ9 152251521
E9 Uniform 9-honeycomb {3[10]} δ10 hδ10 qδ10
En-1 Uniform (n-1)-honeycomb {3[n]} δn hδn qδn 1k22k1k21
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