Uniform 9-polytope

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 polyyotton (or 9-polytope) is a polytope contained by 8-polytope facets. Each 7-polytope ridge being shared by exactly two 8-polytope facets.

A uniform polyyotton is one which is vertex-transitive, and constructed from uniform facets.

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

1 Regular 9-polytopes
2 Euler characteristic
3 Uniform 9-polytopes by fundamental Coxeter groups
4 The A₉ family
5 The B₉ family
6 The D₉ family
7 Regular and uniform honeycombs
- 7.1 Regular and uniform hyperbolic honeycombs
8 References
9 External links

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:

{3,3,3,3,3,3,3,3} - 9-simplex
{4,3,3,3,3,3,3,3} - 9-cube
{3,3,3,3,3,3,3,4} - 9-orthoplex

There are no nonconvex regular 9-polytopes.

Euler characteristic

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

Uniform 9-polytopes by fundamental Coxeter groups

Uniform 9-polytopes with reflective symmetry can be generated by these three Coxeter groups, represented by permutations of rings of the Coxeter-Dynkin diagrams:

Coxeter group		Coxeter-Dynkin diagram
A₉	[3⁸]
B₉	[4,3⁷]
D₉	[3^6,1,1]

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: [3^6,1,1] -
- 383 uniform 9-polytope as permutations of rings in the group diagram, including:
  1. {3^1,6,1} - 9-demicube or demienneract, 1_6,1 - ; also as h{4,3⁸} .
  2. {3^6,1,1} - 9-orthoplex, 6_1,1 -

The A₉ family

The A₉ family has symmetry of order 3628800 (10 factorial).

There are 256+16-1=271 forms based on all permutations of the Coxeter-Dynkin diagrams with one or more rings. These are all enumerated below. Bowers-style acronym names are given in parentheses for cross-referencing.

#	Graph	Coxeter-Dynkin diagram Schläfli symbol Name	Element counts
#	Graph	Coxeter-Dynkin diagram Schläfli symbol Name	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		t_0,1{3,3,3,3,3,3,3,3} Truncated 9-simplex (teday)								405	90
7		t_0,2{3,3,3,3,3,3,3,3} Cantellated 9-simplex								2880	360
8		t_1,2{3,3,3,3,3,3,3,3} Bitruncated 9-simplex								1620	360
9		t_0,3{3,3,3,3,3,3,3,3} Runcinated 9-simplex								8820	840
10		t_1,3{3,3,3,3,3,3,3,3} Bicantellated 9-simplex								10080	1260
11		t_2,3{3,3,3,3,3,3,3,3} Tritruncated 9-simplex (treday)								3780	840
12		t_0,4{3,3,3,3,3,3,3,3} Stericated 9-simplex								15120	1260
13		t_1,4{3,3,3,3,3,3,3,3} Biruncinated 9-simplex								26460	2520
14		t_2,4{3,3,3,3,3,3,3,3} Tricantellated 9-simplex								20160	2520
15		t_3,4{3,3,3,3,3,3,3,3} Quadritruncated 9-simplex								5670	1260
16		t_0,5{3,3,3,3,3,3,3,3} Pentellated 9-simplex								15750	1260
17		t_1,5{3,3,3,3,3,3,3,3} Bistericated 9-simplex								37800	3150
18		t_2,5{3,3,3,3,3,3,3,3} Triruncinated 9-simplex								44100	4200
19		t_3,5{3,3,3,3,3,3,3,3} Quadricantellated 9-simplex								25200	3150
20		t_0,6{3,3,3,3,3,3,3,3} Hexicated 9-simplex								10080	840
21		t_1,6{3,3,3,3,3,3,3,3} Bipentellated 9-simplex								31500	2520
22		t_2,6{3,3,3,3,3,3,3,3} Tristericated 9-simplex								50400	4200
23		t_0,7{3,3,3,3,3,3,3,3} Heptellated 9-simplex								3780	360
24		t_1,7{3,3,3,3,3,3,3,3} Bihexicated 9-simplex								15120	1260
25		t_0,8{3,3,3,3,3,3,3,3} Octellated 9-simplex								720	90
26		t_0,1,2{3,3,3,3,3,3,3,3} Cantitruncated 9-simplex								3240	720
27		t_0,1,3{3,3,3,3,3,3,3,3} Runcitruncated 9-simplex								18900	2520
28		t_0,2,3{3,3,3,3,3,3,3,3} Runcicantellated 9-simplex								12600	2520
29		t_1,2,3{3,3,3,3,3,3,3,3} Bicantitruncated 9-simplex								11340	2520
30		t_0,1,4{3,3,3,3,3,3,3,3} Steritruncated 9-simplex								47880	5040
31		t_0,2,4{3,3,3,3,3,3,3,3} Stericantellated 9-simplex								60480	7560
32		t_1,2,4{3,3,3,3,3,3,3,3} Biruncitruncated 9-simplex								52920	7560
33		t_0,3,4{3,3,3,3,3,3,3,3} Steriruncinated 9-simplex								27720	5040
34		t_1,3,4{3,3,3,3,3,3,3,3} Biruncicantellated 9-simplex								41580	7560
35		t_2,3,4{3,3,3,3,3,3,3,3} Tricantitruncated 9-simplex								22680	5040
36		t_0,1,5{3,3,3,3,3,3,3,3} Pentitruncated 9-simplex								66150	6300
37		t_0,2,5{3,3,3,3,3,3,3,3} Penticantellated 9-simplex								126000	12600
38		t_1,2,5{3,3,3,3,3,3,3,3} Bisteritruncated 9-simplex								107100	12600
39		t_0,3,5{3,3,3,3,3,3,3,3} Pentiruncinated 9-simplex								107100	12600
40		t_1,3,5{3,3,3,3,3,3,3,3} Bistericantellated 9-simplex								151200	18900
41		t_2,3,5{3,3,3,3,3,3,3,3} Triruncitruncated 9-simplex								81900	12600
42		t_0,4,5{3,3,3,3,3,3,3,3} Pentistericated 9-simplex								37800	6300
43		t_1,4,5{3,3,3,3,3,3,3,3} Bisteriruncinated 9-simplex								81900	12600
44		t_2,4,5{3,3,3,3,3,3,3,3} Triruncicantellated 9-simplex								75600	12600
45		t_3,4,5{3,3,3,3,3,3,3,3} Quadricantitruncated 9-simplex								28350	6300
46		t_0,1,6{3,3,3,3,3,3,3,3} Hexitruncated 9-simplex								52920	5040
47		t_0,2,6{3,3,3,3,3,3,3,3} Hexicantellated 9-simplex								138600	12600
48		t_1,2,6{3,3,3,3,3,3,3,3} Bipentitruncated 9-simplex								113400	12600
49		t_0,3,6{3,3,3,3,3,3,3,3} Hexiruncinated 9-simplex								176400	16800
50		t_1,3,6{3,3,3,3,3,3,3,3} Bipenticantellated 9-simplex								239400	25200
51		t_2,3,6{3,3,3,3,3,3,3,3} Tristeritruncated 9-simplex								126000	16800
52		t_0,4,6{3,3,3,3,3,3,3,3} Hexistericated 9-simplex								113400	12600
53		t_1,4,6{3,3,3,3,3,3,3,3} Bipentiruncinated 9-simplex								226800	25200
54		t_2,4,6{3,3,3,3,3,3,3,3} Tristericantellated 9-simplex								201600	25200
55		t_0,5,6{3,3,3,3,3,3,3,3} Hexipentellated 9-simplex								32760	5040
56		t_1,5,6{3,3,3,3,3,3,3,3} Bipentistericated 9-simplex								94500	12600
57		t_0,1,7{3,3,3,3,3,3,3,3} Heptitruncated 9-simplex								23940	2520
58		t_0,2,7{3,3,3,3,3,3,3,3} Hepticantellated 9-simplex								83160	7560
59		t_1,2,7{3,3,3,3,3,3,3,3} Bihexitruncated 9-simplex								64260	7560
60		t_0,3,7{3,3,3,3,3,3,3,3} Heptiruncinated 9-simplex								144900	12600
61		t_1,3,7{3,3,3,3,3,3,3,3} Bihexicantellated 9-simplex								189000	18900
62		t_0,4,7{3,3,3,3,3,3,3,3} Heptistericated 9-simplex								138600	12600
63		t_1,4,7{3,3,3,3,3,3,3,3} Bihexiruncinated 9-simplex								264600	25200
64		t_0,5,7{3,3,3,3,3,3,3,3} Heptipentellated 9-simplex								71820	7560
65		t_0,6,7{3,3,3,3,3,3,3,3} Heptihexicated 9-simplex								17640	2520
66		t_0,1,8{3,3,3,3,3,3,3,3} Octitruncated 9-simplex								5400	720
67		t_0,2,8{3,3,3,3,3,3,3,3} Octicantellated 9-simplex								25200	2520
68		t_0,3,8{3,3,3,3,3,3,3,3} Octiruncinated 9-simplex								57960	5040
69		t_0,4,8{3,3,3,3,3,3,3,3} Octistericated 9-simplex								75600	6300
70		t_0,1,2,3{3,3,3,3,3,3,3,3} Runcicantitruncated 9-simplex								22680	5040
71		t_0,1,2,4{3,3,3,3,3,3,3,3} Stericantitruncated 9-simplex								105840	15120
72		t_0,1,3,4{3,3,3,3,3,3,3,3} Steriruncitruncated 9-simplex								75600	15120
73		t_0,2,3,4{3,3,3,3,3,3,3,3} Steriruncicantellated 9-simplex								75600	15120
74		t_1,2,3,4{3,3,3,3,3,3,3,3} Biruncicantitruncated 9-simplex								68040	15120
75		t_0,1,2,5{3,3,3,3,3,3,3,3} Penticantitruncated 9-simplex								214200	25200
76		t_0,1,3,5{3,3,3,3,3,3,3,3} Pentiruncitruncated 9-simplex								283500	37800
77		t_0,2,3,5{3,3,3,3,3,3,3,3} Pentiruncicantellated 9-simplex								264600	37800
78		t_1,2,3,5{3,3,3,3,3,3,3,3} Bistericantitruncated 9-simplex								245700	37800
79		t_0,1,4,5{3,3,3,3,3,3,3,3} Pentisteritruncated 9-simplex								138600	25200
80		t_0,2,4,5{3,3,3,3,3,3,3,3} Pentistericantellated 9-simplex								226800	37800
81		t_1,2,4,5{3,3,3,3,3,3,3,3} Bisteriruncitruncated 9-simplex								189000	37800
82		t_0,3,4,5{3,3,3,3,3,3,3,3} Pentisteriruncinated 9-simplex								138600	25200
83		t_1,3,4,5{3,3,3,3,3,3,3,3} Bisteriruncicantellated 9-simplex								207900	37800
84		t_2,3,4,5{3,3,3,3,3,3,3,3} Triruncicantitruncated 9-simplex								113400	25200
85		t_0,1,2,6{3,3,3,3,3,3,3,3} Hexicantitruncated 9-simplex								226800	25200
86		t_0,1,3,6{3,3,3,3,3,3,3,3} Hexiruncitruncated 9-simplex								453600	50400
87		t_0,2,3,6{3,3,3,3,3,3,3,3} Hexiruncicantellated 9-simplex								403200	50400
88		t_1,2,3,6{3,3,3,3,3,3,3,3} Bipenticantitruncated 9-simplex								378000	50400
89		t_0,1,4,6{3,3,3,3,3,3,3,3} Hexisteritruncated 9-simplex								403200	50400
90		t_0,2,4,6{3,3,3,3,3,3,3,3} Hexistericantellated 9-simplex								604800	75600
91		t_1,2,4,6{3,3,3,3,3,3,3,3} Bipentiruncitruncated 9-simplex								529200	75600
92		t_0,3,4,6{3,3,3,3,3,3,3,3} Hexisteriruncinated 9-simplex								352800	50400
93		t_1,3,4,6{3,3,3,3,3,3,3,3} Bipentiruncicantellated 9-simplex								529200	75600
94		t_2,3,4,6{3,3,3,3,3,3,3,3} Tristericantitruncated 9-simplex								302400	50400
95		t_0,1,5,6{3,3,3,3,3,3,3,3} Hexipentitruncated 9-simplex								151200	25200
96		t_0,2,5,6{3,3,3,3,3,3,3,3} Hexipenticantellated 9-simplex								352800	50400
97		t_1,2,5,6{3,3,3,3,3,3,3,3} Bipentisteritruncated 9-simplex								277200	50400
98		t_0,3,5,6{3,3,3,3,3,3,3,3} Hexipentiruncinated 9-simplex								352800	50400
99		t_1,3,5,6{3,3,3,3,3,3,3,3} Bipentistericantellated 9-simplex								491400	75600
100		t_2,3,5,6{3,3,3,3,3,3,3,3} Tristeriruncitruncated 9-simplex								252000	50400
101		t_0,4,5,6{3,3,3,3,3,3,3,3} Hexipentistericated 9-simplex								151200	25200
102		t_1,4,5,6{3,3,3,3,3,3,3,3} Bipentisteriruncinated 9-simplex								327600	50400
103		t_0,1,2,7{3,3,3,3,3,3,3,3} Hepticantitruncated 9-simplex								128520	15120
104		t_0,1,3,7{3,3,3,3,3,3,3,3} Heptiruncitruncated 9-simplex								359100	37800
105		t_0,2,3,7{3,3,3,3,3,3,3,3} Heptiruncicantellated 9-simplex								302400	37800
106		t_1,2,3,7{3,3,3,3,3,3,3,3} Bihexicantitruncated 9-simplex								283500	37800
107		t_0,1,4,7{3,3,3,3,3,3,3,3} Heptisteritruncated 9-simplex								478800	50400
108		t_0,2,4,7{3,3,3,3,3,3,3,3} Heptistericantellated 9-simplex								680400	75600
109		t_1,2,4,7{3,3,3,3,3,3,3,3} Bihexiruncitruncated 9-simplex								604800	75600
110		t_0,3,4,7{3,3,3,3,3,3,3,3} Heptisteriruncinated 9-simplex								378000	50400
111		t_1,3,4,7{3,3,3,3,3,3,3,3} Bihexiruncicantellated 9-simplex								567000	75600
112		t_0,1,5,7{3,3,3,3,3,3,3,3} Heptipentitruncated 9-simplex								321300	37800
113		t_0,2,5,7{3,3,3,3,3,3,3,3} Heptipenticantellated 9-simplex								680400	75600
114		t_1,2,5,7{3,3,3,3,3,3,3,3} Bihexisteritruncated 9-simplex								567000	75600
115		t_0,3,5,7{3,3,3,3,3,3,3,3} Heptipentiruncinated 9-simplex								642600	75600
116		t_1,3,5,7{3,3,3,3,3,3,3,3} Bihexistericantellated 9-simplex								907200	113400
117		t_0,4,5,7{3,3,3,3,3,3,3,3} Heptipentistericated 9-simplex								264600	37800
118		t_0,1,6,7{3,3,3,3,3,3,3,3} Heptihexitruncated 9-simplex								98280	15120
119		t_0,2,6,7{3,3,3,3,3,3,3,3} Heptihexicantellated 9-simplex								302400	37800
120		t_1,2,6,7{3,3,3,3,3,3,3,3} Bihexipentitruncated 9-simplex								226800	37800
121		t_0,3,6,7{3,3,3,3,3,3,3,3} Heptihexiruncinated 9-simplex								428400	50400
122		t_0,4,6,7{3,3,3,3,3,3,3,3} Heptihexistericated 9-simplex								302400	37800
123		t_0,5,6,7{3,3,3,3,3,3,3,3} Heptihexipentellated 9-simplex								98280	15120
124		t_0,1,2,8{3,3,3,3,3,3,3,3} Octicantitruncated 9-simplex								35280	5040
125		t_0,1,3,8{3,3,3,3,3,3,3,3} Octiruncitruncated 9-simplex								136080	15120
126		t_0,2,3,8{3,3,3,3,3,3,3,3} Octiruncicantellated 9-simplex								105840	15120
127		t_0,1,4,8{3,3,3,3,3,3,3,3} Octisteritruncated 9-simplex								252000	25200
128		t_0,2,4,8{3,3,3,3,3,3,3,3} Octistericantellated 9-simplex								340200	37800
129		t_0,3,4,8{3,3,3,3,3,3,3,3} Octisteriruncinated 9-simplex								176400	25200
130		t_0,1,5,8{3,3,3,3,3,3,3,3} Octipentitruncated 9-simplex								252000	25200
131		t_0,2,5,8{3,3,3,3,3,3,3,3} Octipenticantellated 9-simplex								504000	50400
132		t_0,3,5,8{3,3,3,3,3,3,3,3} Octipentiruncinated 9-simplex								453600	50400
133		t_0,1,6,8{3,3,3,3,3,3,3,3} Octihexitruncated 9-simplex								136080	15120
134		t_0,2,6,8{3,3,3,3,3,3,3,3} Octihexicantellated 9-simplex								378000	37800
135		t_0,1,7,8{3,3,3,3,3,3,3,3} Octiheptitruncated 9-simplex								35280	5040
136		t_0,1,2,3,4{3,3,3,3,3,3,3,3} Steriruncicantitruncated 9-simplex								136080	30240
137		t_0,1,2,3,5{3,3,3,3,3,3,3,3} Pentiruncicantitruncated 9-simplex								491400	75600
138		t_0,1,2,4,5{3,3,3,3,3,3,3,3} Pentistericantitruncated 9-simplex								378000	75600
139		t_0,1,3,4,5{3,3,3,3,3,3,3,3} Pentisteriruncitruncated 9-simplex								378000	75600
140		t_0,2,3,4,5{3,3,3,3,3,3,3,3} Pentisteriruncicantellated 9-simplex								378000	75600
141		t_1,2,3,4,5{3,3,3,3,3,3,3,3} Bisteriruncicantitruncated 9-simplex								340200	75600
142		t_0,1,2,3,6{3,3,3,3,3,3,3,3} Hexiruncicantitruncated 9-simplex								756000	100800
143		t_0,1,2,4,6{3,3,3,3,3,3,3,3} Hexistericantitruncated 9-simplex								1058400	151200
144		t_0,1,3,4,6{3,3,3,3,3,3,3,3} Hexisteriruncitruncated 9-simplex								982800	151200
145		t_0,2,3,4,6{3,3,3,3,3,3,3,3} Hexisteriruncicantellated 9-simplex								982800	151200
146		t_1,2,3,4,6{3,3,3,3,3,3,3,3} Bipentiruncicantitruncated 9-simplex								907200	151200
147		t_0,1,2,5,6{3,3,3,3,3,3,3,3} Hexipenticantitruncated 9-simplex								554400	100800
148		t_0,1,3,5,6{3,3,3,3,3,3,3,3} Hexipentiruncitruncated 9-simplex								907200	151200
149		t_0,2,3,5,6{3,3,3,3,3,3,3,3} Hexipentiruncicantellated 9-simplex								831600	151200
150		t_1,2,3,5,6{3,3,3,3,3,3,3,3} Bipentistericantitruncated 9-simplex								756000	151200
151		t_0,1,4,5,6{3,3,3,3,3,3,3,3} Hexipentisteritruncated 9-simplex								554400	100800
152		t_0,2,4,5,6{3,3,3,3,3,3,3,3} Hexipentistericantellated 9-simplex								907200	151200
153		t_1,2,4,5,6{3,3,3,3,3,3,3,3} Bipentisteriruncitruncated 9-simplex								756000	151200
154		t_0,3,4,5,6{3,3,3,3,3,3,3,3} Hexipentisteriruncinated 9-simplex								554400	100800
155		t_1,3,4,5,6{3,3,3,3,3,3,3,3} Bipentisteriruncicantellated 9-simplex								831600	151200
156		t_2,3,4,5,6{3,3,3,3,3,3,3,3} Tristeriruncicantitruncated 9-simplex								453600	100800
157		t_0,1,2,3,7{3,3,3,3,3,3,3,3} Heptiruncicantitruncated 9-simplex								567000	75600
158		t_0,1,2,4,7{3,3,3,3,3,3,3,3} Heptistericantitruncated 9-simplex								1209600	151200
159		t_0,1,3,4,7{3,3,3,3,3,3,3,3} Heptisteriruncitruncated 9-simplex								1058400	151200
160		t_0,2,3,4,7{3,3,3,3,3,3,3,3} Heptisteriruncicantellated 9-simplex								1058400	151200
161		t_1,2,3,4,7{3,3,3,3,3,3,3,3} Bihexiruncicantitruncated 9-simplex								982800	151200
162		t_0,1,2,5,7{3,3,3,3,3,3,3,3} Heptipenticantitruncated 9-simplex								1134000	151200
163		t_0,1,3,5,7{3,3,3,3,3,3,3,3} Heptipentiruncitruncated 9-simplex								1701000	226800
164		t_0,2,3,5,7{3,3,3,3,3,3,3,3} Heptipentiruncicantellated 9-simplex								1587600	226800
165		t_1,2,3,5,7{3,3,3,3,3,3,3,3} Bihexistericantitruncated 9-simplex								1474200	226800
166		t_0,1,4,5,7{3,3,3,3,3,3,3,3} Heptipentisteritruncated 9-simplex								982800	151200
167		t_0,2,4,5,7{3,3,3,3,3,3,3,3} Heptipentistericantellated 9-simplex								1587600	226800
168		t_1,2,4,5,7{3,3,3,3,3,3,3,3} Bihexisteriruncitruncated 9-simplex								1360800	226800
169		t_0,3,4,5,7{3,3,3,3,3,3,3,3} Heptipentisteriruncinated 9-simplex								982800	151200
170		t_1,3,4,5,7{3,3,3,3,3,3,3,3} Bihexisteriruncicantellated 9-simplex								1474200	226800
171		t_0,1,2,6,7{3,3,3,3,3,3,3,3} Heptihexicantitruncated 9-simplex								453600	75600
172		t_0,1,3,6,7{3,3,3,3,3,3,3,3} Heptihexiruncitruncated 9-simplex								1058400	151200
173		t_0,2,3,6,7{3,3,3,3,3,3,3,3} Heptihexiruncicantellated 9-simplex								907200	151200
174		t_1,2,3,6,7{3,3,3,3,3,3,3,3} Bihexipenticantitruncated 9-simplex								831600	151200
175		t_0,1,4,6,7{3,3,3,3,3,3,3,3} Heptihexisteritruncated 9-simplex								1058400	151200
176		t_0,2,4,6,7{3,3,3,3,3,3,3,3} Heptihexistericantellated 9-simplex								1587600	226800
177		t_1,2,4,6,7{3,3,3,3,3,3,3,3} Bihexipentiruncitruncated 9-simplex								1360800	226800
178		t_0,3,4,6,7{3,3,3,3,3,3,3,3} Heptihexisteriruncinated 9-simplex								907200	151200
179		t_0,1,5,6,7{3,3,3,3,3,3,3,3} Heptihexipentitruncated 9-simplex								453600	75600
180		t_0,2,5,6,7{3,3,3,3,3,3,3,3} Heptihexipenticantellated 9-simplex								1058400	151200
181		t_0,3,5,6,7{3,3,3,3,3,3,3,3} Heptihexipentiruncinated 9-simplex								1058400	151200
182		t_0,4,5,6,7{3,3,3,3,3,3,3,3} Heptihexipentistericated 9-simplex								453600	75600
183		t_0,1,2,3,8{3,3,3,3,3,3,3,3} Octiruncicantitruncated 9-simplex								196560	30240
184		t_0,1,2,4,8{3,3,3,3,3,3,3,3} Octistericantitruncated 9-simplex								604800	75600
185		t_0,1,3,4,8{3,3,3,3,3,3,3,3} Octisteriruncitruncated 9-simplex								491400	75600
186		t_0,2,3,4,8{3,3,3,3,3,3,3,3} Octisteriruncicantellated 9-simplex								491400	75600
187		t_0,1,2,5,8{3,3,3,3,3,3,3,3} Octipenticantitruncated 9-simplex								856800	100800
188		t_0,1,3,5,8{3,3,3,3,3,3,3,3} Octipentiruncitruncated 9-simplex								1209600	151200
189		t_0,2,3,5,8{3,3,3,3,3,3,3,3} Octipentiruncicantellated 9-simplex								1134000	151200
190		t_0,1,4,5,8{3,3,3,3,3,3,3,3} Octipentisteritruncated 9-simplex								655200	100800
191		t_0,2,4,5,8{3,3,3,3,3,3,3,3} Octipentistericantellated 9-simplex								1058400	151200
192		t_0,3,4,5,8{3,3,3,3,3,3,3,3} Octipentisteriruncinated 9-simplex								655200	100800
193		t_0,1,2,6,8{3,3,3,3,3,3,3,3} Octihexicantitruncated 9-simplex								604800	75600
194		t_0,1,3,6,8{3,3,3,3,3,3,3,3} Octihexiruncitruncated 9-simplex								1285200	151200
195		t_0,2,3,6,8{3,3,3,3,3,3,3,3} Octihexiruncicantellated 9-simplex								1134000	151200
196		t_0,1,4,6,8{3,3,3,3,3,3,3,3} Octihexisteritruncated 9-simplex								1209600	151200
197		t_0,2,4,6,8{3,3,3,3,3,3,3,3} Octihexistericantellated 9-simplex								1814400	226800
198		t_0,1,5,6,8{3,3,3,3,3,3,3,3} Octihexipentitruncated 9-simplex								491400	75600
199		t_0,1,2,7,8{3,3,3,3,3,3,3,3} Octihepticantitruncated 9-simplex								196560	30240
200		t_0,1,3,7,8{3,3,3,3,3,3,3,3} Octiheptiruncitruncated 9-simplex								604800	75600
201		t_0,1,4,7,8{3,3,3,3,3,3,3,3} Octiheptisteritruncated 9-simplex								856800	100800
202		t_0,1,2,3,4,5{3,3,3,3,3,3,3,3} Pentisteriruncicantitruncated 9-simplex								680400	151200
203		t_0,1,2,3,4,6{3,3,3,3,3,3,3,3} Hexisteriruncicantitruncated 9-simplex								1814400	302400
204		t_0,1,2,3,5,6{3,3,3,3,3,3,3,3} Hexipentiruncicantitruncated 9-simplex								1512000	302400
205		t_0,1,2,4,5,6{3,3,3,3,3,3,3,3} Hexipentistericantitruncated 9-simplex								1512000	302400
206		t_0,1,3,4,5,6{3,3,3,3,3,3,3,3} Hexipentisteriruncitruncated 9-simplex								1512000	302400
207		t_0,2,3,4,5,6{3,3,3,3,3,3,3,3} Hexipentisteriruncicantellated 9-simplex								1512000	302400
208		t_1,2,3,4,5,6{3,3,3,3,3,3,3,3} Bipentisteriruncicantitruncated 9-simplex								1360800	302400
209		t_0,1,2,3,4,7{3,3,3,3,3,3,3,3} Heptisteriruncicantitruncated 9-simplex								1965600	302400
210		t_0,1,2,3,5,7{3,3,3,3,3,3,3,3} Heptipentiruncicantitruncated 9-simplex								2948400	453600
211		t_0,1,2,4,5,7{3,3,3,3,3,3,3,3} Heptipentistericantitruncated 9-simplex								2721600	453600
212		t_0,1,3,4,5,7{3,3,3,3,3,3,3,3} Heptipentisteriruncitruncated 9-simplex								2721600	453600
213		t_0,2,3,4,5,7{3,3,3,3,3,3,3,3} Heptipentisteriruncicantellated 9-simplex								2721600	453600
214		t_1,2,3,4,5,7{3,3,3,3,3,3,3,3} Bihexisteriruncicantitruncated 9-simplex								2494800	453600
215		t_0,1,2,3,6,7{3,3,3,3,3,3,3,3} Heptihexiruncicantitruncated 9-simplex								1663200	302400
216		t_0,1,2,4,6,7{3,3,3,3,3,3,3,3} Heptihexistericantitruncated 9-simplex								2721600	453600
217		t_0,1,3,4,6,7{3,3,3,3,3,3,3,3} Heptihexisteriruncitruncated 9-simplex								2494800	453600
218		t_0,2,3,4,6,7{3,3,3,3,3,3,3,3} Heptihexisteriruncicantellated 9-simplex								2494800	453600
219		t_1,2,3,4,6,7{3,3,3,3,3,3,3,3} Bihexipentiruncicantitruncated 9-simplex								2268000	453600
220		t_0,1,2,5,6,7{3,3,3,3,3,3,3,3} Heptihexipenticantitruncated 9-simplex								1663200	302400
221		t_0,1,3,5,6,7{3,3,3,3,3,3,3,3} Heptihexipentiruncitruncated 9-simplex								2721600	453600
222		t_0,2,3,5,6,7{3,3,3,3,3,3,3,3} Heptihexipentiruncicantellated 9-simplex								2494800	453600
223		t_1,2,3,5,6,7{3,3,3,3,3,3,3,3} Bihexipentistericantitruncated 9-simplex								2268000	453600
224		t_0,1,4,5,6,7{3,3,3,3,3,3,3,3} Heptihexipentisteritruncated 9-simplex								1663200	302400
225		t_0,2,4,5,6,7{3,3,3,3,3,3,3,3} Heptihexipentistericantellated 9-simplex								2721600	453600
226		t_0,3,4,5,6,7{3,3,3,3,3,3,3,3} Heptihexipentisteriruncinated 9-simplex								1663200	302400
227		t_0,1,2,3,4,8{3,3,3,3,3,3,3,3} Octisteriruncicantitruncated 9-simplex								907200	151200
228		t_0,1,2,3,5,8{3,3,3,3,3,3,3,3} Octipentiruncicantitruncated 9-simplex								2116800	302400
229		t_0,1,2,4,5,8{3,3,3,3,3,3,3,3} Octipentistericantitruncated 9-simplex								1814400	302400
230		t_0,1,3,4,5,8{3,3,3,3,3,3,3,3} Octipentisteriruncitruncated 9-simplex								1814400	302400
231		t_0,2,3,4,5,8{3,3,3,3,3,3,3,3} Octipentisteriruncicantellated 9-simplex								1814400	302400
232		t_0,1,2,3,6,8{3,3,3,3,3,3,3,3} Octihexiruncicantitruncated 9-simplex								2116800	302400
233		t_0,1,2,4,6,8{3,3,3,3,3,3,3,3} Octihexistericantitruncated 9-simplex								3175200	453600
234		t_0,1,3,4,6,8{3,3,3,3,3,3,3,3} Octihexisteriruncitruncated 9-simplex								2948400	453600
235		t_0,2,3,4,6,8{3,3,3,3,3,3,3,3} Octihexisteriruncicantellated 9-simplex								2948400	453600
236		t_0,1,2,5,6,8{3,3,3,3,3,3,3,3} Octihexipenticantitruncated 9-simplex								1814400	302400
237		t_0,1,3,5,6,8{3,3,3,3,3,3,3,3} Octihexipentiruncitruncated 9-simplex								2948400	453600
238		t_0,2,3,5,6,8{3,3,3,3,3,3,3,3} Octihexipentiruncicantellated 9-simplex								2721600	453600
239		t_0,1,4,5,6,8{3,3,3,3,3,3,3,3} Octihexipentisteritruncated 9-simplex								1814400	302400
240		t_0,1,2,3,7,8{3,3,3,3,3,3,3,3} Octiheptiruncicantitruncated 9-simplex								907200	151200
241		t_0,1,2,4,7,8{3,3,3,3,3,3,3,3} Octiheptistericantitruncated 9-simplex								2116800	302400
242		t_0,1,3,4,7,8{3,3,3,3,3,3,3,3} Octiheptisteriruncitruncated 9-simplex								1814400	302400
243		t_0,1,2,5,7,8{3,3,3,3,3,3,3,3} Octiheptipenticantitruncated 9-simplex								2116800	302400
244		t_0,1,3,5,7,8{3,3,3,3,3,3,3,3} Octiheptipentiruncitruncated 9-simplex								3175200	453600
245		t_0,1,2,6,7,8{3,3,3,3,3,3,3,3} Octiheptihexicantitruncated 9-simplex								907200	151200
246		t_{0,1,2,3,4,5,6}{3,3,3,3,3,3,3,3} Hexipentisteriruncicantitruncated 9-simplex								2721600	604800
247		t_{0,1,2,3,4,5,7}{3,3,3,3,3,3,3,3} Heptipentisteriruncicantitruncated 9-simplex								4989600	907200
248		t_{0,1,2,3,4,6,7}{3,3,3,3,3,3,3,3} Heptihexisteriruncicantitruncated 9-simplex								4536000	907200
249		t_{0,1,2,3,5,6,7}{3,3,3,3,3,3,3,3} Heptihexipentiruncicantitruncated 9-simplex								4536000	907200
250		t_{0,1,2,4,5,6,7}{3,3,3,3,3,3,3,3} Heptihexipentistericantitruncated 9-simplex								4536000	907200
251		t_{0,1,3,4,5,6,7}{3,3,3,3,3,3,3,3} Heptihexipentisteriruncitruncated 9-simplex								4536000	907200
252		t_{0,2,3,4,5,6,7}{3,3,3,3,3,3,3,3} Heptihexipentisteriruncicantellated 9-simplex								4536000	907200
253		t_{1,2,3,4,5,6,7}{3,3,3,3,3,3,3,3} Bihexipentisteriruncicantitruncated 9-simplex								4082400	907200
254		t_{0,1,2,3,4,5,8}{3,3,3,3,3,3,3,3} Octipentisteriruncicantitruncated 9-simplex								3326400	604800
255		t_{0,1,2,3,4,6,8}{3,3,3,3,3,3,3,3} Octihexisteriruncicantitruncated 9-simplex								5443200	907200
256		t_{0,1,2,3,5,6,8}{3,3,3,3,3,3,3,3} Octihexipentiruncicantitruncated 9-simplex								4989600	907200
257		t_{0,1,2,4,5,6,8}{3,3,3,3,3,3,3,3} Octihexipentistericantitruncated 9-simplex								4989600	907200
258		t_{0,1,3,4,5,6,8}{3,3,3,3,3,3,3,3} Octihexipentisteriruncitruncated 9-simplex								4989600	907200
259		t_{0,2,3,4,5,6,8}{3,3,3,3,3,3,3,3} Octihexipentisteriruncicantellated 9-simplex								4989600	907200
260		t_{0,1,2,3,4,7,8}{3,3,3,3,3,3,3,3} Octiheptisteriruncicantitruncated 9-simplex								3326400	604800
261		t_{0,1,2,3,5,7,8}{3,3,3,3,3,3,3,3} Octiheptipentiruncicantitruncated 9-simplex								5443200	907200
262		t_{0,1,2,4,5,7,8}{3,3,3,3,3,3,3,3} Octiheptipentistericantitruncated 9-simplex								4989600	907200
263		t_{0,1,3,4,5,7,8}{3,3,3,3,3,3,3,3} Octiheptipentisteriruncitruncated 9-simplex								4989600	907200
264		t_{0,1,2,3,6,7,8}{3,3,3,3,3,3,3,3} Octiheptihexiruncicantitruncated 9-simplex								3326400	604800
265		t_{0,1,2,4,6,7,8}{3,3,3,3,3,3,3,3} Octiheptihexistericantitruncated 9-simplex								5443200	907200
266		t_{0,1,2,3,4,5,6,7}{3,3,3,3,3,3,3,3} Heptihexipentisteriruncicantitruncated 9-simplex								8164800	1814400
267		t_{0,1,2,3,4,5,6,8}{3,3,3,3,3,3,3,3} Octihexipentisteriruncicantitruncated 9-simplex								9072000	1814400
268		t_{0,1,2,3,4,5,7,8}{3,3,3,3,3,3,3,3} Octiheptipentisteriruncicantitruncated 9-simplex								9072000	1814400
269		t_{0,1,2,3,4,6,7,8}{3,3,3,3,3,3,3,3} Octiheptihexisteriruncicantitruncated 9-simplex								9072000	1814400
270		t_{0,1,2,3,5,6,7,8}{3,3,3,3,3,3,3,3} Octiheptihexipentiruncicantitruncated 9-simplex								9072000	1814400
271		t_{0,1,2,3,4,5,6,7,8}{3,3,3,3,3,3,3,3} Omnitruncated 9-simplex								16329600	3628800

The B₉ family

There are 511 forms based on all permutations of the Coxeter-Dynkin diagrams with one or more rings.

Eleven cases are shown below: Nine rectified forms and 2 truncations. Bowers-style acronym names are given in parentheses for cross-referencing. Bowers-style acronym names are given in parentheses for cross-referencing.

#	Graph	Coxeter-Dynkin diagram Schläfli symbol Name	Element counts
#	Graph	Coxeter-Dynkin diagram Schläfli symbol Name	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		t_0,1{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		t_0,1{3,3,3,3,3,3,3,4} Truncated 9-orthoplex (tiv)								2160	288
11		t₀{3,3,3,3,3,3,3,4} 9-orthoplex (vee)	512	2304	4608	5376	4032	2016	672	144	18

The D₉ family

The D₉ family has symmetry of order 92,897,280 (9 factorial × 2⁸).

This family has 3×128−1=383 Wythoffian uniform polytopes, generated by marking one or more nodes of the D₉ Coxeter-Dynkin diagram. Of these, 255 (2×128−1) are repeated from the B₉ family and 128 are unique to this family, with the eight 1 or 2 ringed forms listed below. Bowers-style acronym names are given in parentheses for cross-referencing.

#	Coxeter plane graphs											Coxeter-Dynkin diagram Schläfli symbol	Base point (Alternately signed)	Element counts									Circumrad
#	B₉	D₉	D₈	D₇	D₆	D₅	D₄	D₃	A₇	A₅	A₃	Coxeter-Dynkin diagram Schläfli symbol	Base point (Alternately signed)	8	7	6	5	4	3	2	1	0	Circumrad
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

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

#	Coxeter group
1	${\tilde{A}}_8$	[3^[8]]
2	${\tilde{B}}_8$	h[4,3⁶,4] [4,3⁵,3^1,1]
3	${\tilde{C}}_8$	[4,3⁶,4]
4	${\tilde{D}}_8$	q[4,3⁶,4] [3^1,1,3⁴,3^1,1]
5	${\tilde{E}}_8$	[3^5,2,1]

Regular and uniform tessellations include:

:
- Regular octeractic honeycomb: {4,3⁶,4},
:
- Uniform demiocteractic honeycomb: h{4,3⁶,4} or {3^1,1,3⁵,4}, or
:
- Uniform 5₂₁ honeycomb:
- Uniform 2₅₁ honeycomb:
- Uniform 1₅₂ honeycomb:

Regular and uniform hyperbolic honeycombs

There are no compact hyperbolic Coxeter groups of rank 9, groups that can generate honeycombs with all finite facets, and a finite vertex figure. However there are 4 noncompact hyperbolic Coxeter groups of rank 9, each generating uniform honeycombs in 8-space as permutations of rings of the Coxeter diagrams.

${\bar{P}}_8$ = [3,3^[8]]:	${\bar{Q}}_8$ = [3^1,1,3³,3^2,1]:	${\bar{S}}_8$ = [4,3⁴,3^2,1]:	${\bar{T}}_8$ = [3^4,3,1]:

References

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

External links

Polytope names
Polytopes of Various Dimensions, Jonathan Bowers
Multi-dimensional Glossary
Glossary for hyperspace, George Olshevsky.


Family	A_n	BC_n	D_n	E₆ / E₇ / E₈ / F₄ / G₂	H_n
Regular polygon	Triangle	Square		Hexagon	Pentagon
Uniform polyhedron	Tetrahedron	Octahedron • Cube	Demicube		Dodecahedron • Icosahedron
Uniform polychoron	5-cell	16-cell • Tesseract	Demitesseract	24-cell	120-cell • 600-cell
Uniform 5-polytope	5-simplex	5-orthoplex • 5-cube	5-demicube
Uniform 6-polytope	6-simplex	6-orthoplex • 6-cube	6-demicube	1₂₂ • 2₂₁
Uniform 7-polytope	7-simplex	7-orthoplex • 7-cube	7-demicube	1₃₂ • 2₃₁ • 3₂₁
Uniform 8-polytope	8-simplex	8-orthoplex • 8-cube	8-demicube	1₄₂ • 2₄₁ • 4₂₁
Uniform 9-polytope	9-simplex	9-orthoplex • 9-cube	9-demicube
Uniform 10-polytope	10-simplex	10-orthoplex • 10-cube	10-demicube
n-polytopes	n-simplex	n-orthoplex • n-cube	n-demicube	1_k2 • 2_k1 • k₂₁	pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes