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 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

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: [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, 1₆₁ - ; also as h{4,3⁸} .
    2. {3^6,1,1} - 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-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

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

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

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

Regular and uniform tessellations include:

• ${\displaystyle {\tilde {A}}_{8}}$ 45 uniquely ringed forms
  • 8-simplex honeycomb: {3^[9]}
• ${\displaystyle {\tilde {C}}_{8}}$ 271 uniquely ringed forms
  • Regular 8-cube honeycomb: {4,3⁶,4},
• ${\displaystyle {\tilde {B}}_{8}}$: 383 uniquely ringed forms, 255 shared with ${\displaystyle {\tilde {C}}_{8}}$, 128 new
  • 8-demicube honeycomb: h{4,3⁶,4} or {3^1,1,3⁵,4}, or
• ${\displaystyle {\tilde {D}}_{8}}$, [3^1,1,3⁴,3^1,1]: 155 unique ring permutations, and 15 are new, the first, , Coxeter called a quarter 8-cubic honeycomb, representing as q{4,3⁶,4}, or qδ₉.
• ${\displaystyle {\tilde {E}}_{8}}$ 511 forms
  • 5₂₁ honeycomb:
  • 2₅₁ honeycomb:
  • 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 paracompact hyperbolic Coxeter groups of rank 9, each generating uniform honeycombs in 8-space as permutations of rings of the Coxeter diagrams.

${\displaystyle {\bar {P}}_{8}}$ = [3,3[8]]:

${\displaystyle {\bar {Q}}_{8}}$ = [31,1,33,32,1]:

${\displaystyle {\bar {S}}_{8}}$ = [4,34,32,1]:

${\displaystyle {\bar {T}}_{8}}$ = [34,3,1]:

References

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)".

External links

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

v t e 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	Octahedron • Cube	Demicube		Dodecahedron • Icosahedron
Uniform polychoron	Pentachoron	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
Uniform n-polytope	n-simplex	n-orthoplex • n-cube	n-demicube	1k2 • 2k1 • k21	n-pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds

v t e Fundamental convex regular and uniform honeycombs in dimensions 2–9
Space	Family	${\displaystyle {\tilde {A}}_{n-1}}$	${\displaystyle {\tilde {C}}_{n-1}}$	${\displaystyle {\tilde {B}}_{n-1}}$	${\displaystyle {\tilde {D}}_{n-1}}$	${\displaystyle {\tilde {G}}_{2}}$ / ${\displaystyle {\tilde {F}}_{4}}$ / ${\displaystyle {\tilde {E}}_{n-1}}$
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	133 • 331
E8	Uniform 8-honeycomb	{3[9]}	δ9	hδ9	qδ9	152 • 251 • 521
E9	Uniform 9-honeycomb	{3[10]}	δ10	hδ10	qδ10
E10	Uniform 10-honeycomb	{3[11]}	δ11	hδ11	qδ11
En-1	Uniform (n-1)-honeycomb	{3[n]}	δn	hδn	qδn	1k2 • 2k1 • k21