Uniform 9-polytope

Graphs of three regular and related uniform polytopes

File:9-simplex t0.svg 9-simplex						File:9-simplex t1.svg Rectified 9-simplex
File:9-simplex t01.svg Truncated 9-simplex						File:9-simplex t02.svg Cantellated 9-simplex
File:9-simplex t03.svg Runcinated 9-simplex						File:9-simplex t04.svg Stericated 9-simplex
File:9-simplex t05.svg Pentellated 9-simplex						File:9-simplex t06.svg Hexicated 9-simplex
File:9-simplex t07.svg Heptellated 9-simplex						File:9-simplex t08.svg Octellated 9-simplex
File:9-orthoplex.svg 9-orthoplex						File:9-cube.svg 9-cube
File:Truncated 9-orthoplex.png Truncated 9-orthoplex						File:Truncated 9-cube.png Truncated 9-cube
File:Rectified enneacross.png Rectified 9-orthoplex						File:Rectified 9-cube.png Rectified 9-cube
File:9-demicube.svg 9-demicube						File:Truncated 9-demicube.png 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.

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.

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.

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]	Template:CDD
B9	[4,37]	Template:CDD
D9	[36,1,1]	Template:CDD

Selected regular and uniform 9-polytopes from each family include:

• Simplex family: A₉ [3⁸] - Template:CDD
  • 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 - Template:CDD
• Hypercube/orthoplex family: B₉ [4,3⁸] - Template:CDD
  • 511 uniform 9-polytopes as permutations of rings in the group diagram, including two regular ones:
    1. {4,3⁷} - 9-cube or enneract - Template:CDD
    2. {3⁷,4} - 9-orthoplex or enneacross - Template:CDD
• Demihypercube D₉ family: [3^6,1,1] - Template:CDD
  • 383 uniform 9-polytope as permutations of rings in the group diagram, including:
    1. {3^1,6,1} - 9-demicube or demienneract, 1₆₁ - Template:CDD; also as h{4,3⁸} Template:CDD.
    2. {3^6,1,1} - 9-orthoplex, 6₁₁ - Template:CDD

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	File:9-simplex t0.svg	Template:CDD t0{3,3,3,3,3,3,3,3} 9-simplex (day)	10	45	120	210	252	210	120	45	10
2	File:9-simplex t1.svg	Template:CDD t1{3,3,3,3,3,3,3,3} Rectified 9-simplex (reday)								360	45
3	File:9-simplex t2.svg	Template:CDD t2{3,3,3,3,3,3,3,3} Birectified 9-simplex (breday)								1260	120
4	File:9-simplex t3.svg	Template:CDD t3{3,3,3,3,3,3,3,3} Trirectified 9-simplex (treday)								2520	210
5	File:9-simplex t4.svg	Template:CDD t4{3,3,3,3,3,3,3,3} Quadrirectified 9-simplex (icoy)								3150	252
6	File:9-simplex t01.svg	Template:CDD t0,1{3,3,3,3,3,3,3,3} Truncated 9-simplex (teday)								405	90
7	File:9-simplex t02.svg	Template:CDD t0,2{3,3,3,3,3,3,3,3} Cantellated 9-simplex								2880	360
8	File:9-simplex t12.svg	Template:CDD t1,2{3,3,3,3,3,3,3,3} Bitruncated 9-simplex								1620	360
9	File:9-simplex t03.svg	Template:CDD t0,3{3,3,3,3,3,3,3,3} Runcinated 9-simplex								8820	840
10	File:9-simplex t13.svg	Template:CDD t1,3{3,3,3,3,3,3,3,3} Bicantellated 9-simplex								10080	1260
11	File:9-simplex t23.svg	Template:CDD t2,3{3,3,3,3,3,3,3,3} Tritruncated 9-simplex (treday)								3780	840
12	File:9-simplex t04.svg	Template:CDD t0,4{3,3,3,3,3,3,3,3} Stericated 9-simplex								15120	1260
13	File:9-simplex t14.svg	Template:CDD t1,4{3,3,3,3,3,3,3,3} Biruncinated 9-simplex								26460	2520
14	File:9-simplex t24.svg	Template:CDD t2,4{3,3,3,3,3,3,3,3} Tricantellated 9-simplex								20160	2520
15		Template:CDD t3,4{3,3,3,3,3,3,3,3} Quadritruncated 9-simplex								5670	1260
16	File:9-simplex t05.svg	Template:CDD t0,5{3,3,3,3,3,3,3,3} Pentellated 9-simplex								15750	1260
17		Template:CDD t1,5{3,3,3,3,3,3,3,3} Bistericated 9-simplex								37800	3150
18		Template:CDD t2,5{3,3,3,3,3,3,3,3} Triruncinated 9-simplex								44100	4200
19		Template:CDD t3,5{3,3,3,3,3,3,3,3} Quadricantellated 9-simplex								25200	3150
20	File:9-simplex t06.svg	Template:CDD t0,6{3,3,3,3,3,3,3,3} Hexicated 9-simplex								10080	840
21		Template:CDD t1,6{3,3,3,3,3,3,3,3} Bipentellated 9-simplex								31500	2520
22		Template:CDD t2,6{3,3,3,3,3,3,3,3} Tristericated 9-simplex								50400	4200
23	File:9-simplex t07.svg	Template:CDD t0,7{3,3,3,3,3,3,3,3} Heptellated 9-simplex								3780	360
24		Template:CDD t1,7{3,3,3,3,3,3,3,3} Bihexicated 9-simplex								15120	1260
25	File:9-simplex t08.svg	Template:CDD t0,8{3,3,3,3,3,3,3,3} Octellated 9-simplex								720	90
26	File:9-simplex t012.svg	Template:CDD t0,1,2{3,3,3,3,3,3,3,3} Cantitruncated 9-simplex								3240	720
27		Template:CDD t0,1,3{3,3,3,3,3,3,3,3} Runcitruncated 9-simplex								18900	2520
28		Template:CDD t0,2,3{3,3,3,3,3,3,3,3} Runcicantellated 9-simplex								12600	2520
29	File:9-simplex t123.svg	Template:CDD t1,2,3{3,3,3,3,3,3,3,3} Bicantitruncated 9-simplex								11340	2520
30		Template:CDD t0,1,4{3,3,3,3,3,3,3,3} Steritruncated 9-simplex								47880	5040
31		Template:CDD t0,2,4{3,3,3,3,3,3,3,3} Stericantellated 9-simplex								60480	7560
32		Template:CDD t1,2,4{3,3,3,3,3,3,3,3} Biruncitruncated 9-simplex								52920	7560
33		Template:CDD t0,3,4{3,3,3,3,3,3,3,3} Steriruncinated 9-simplex								27720	5040
34		Template:CDD t1,3,4{3,3,3,3,3,3,3,3} Biruncicantellated 9-simplex								41580	7560
35	File:9-simplex t234.svg	Template:CDD t2,3,4{3,3,3,3,3,3,3,3} Tricantitruncated 9-simplex								22680	5040
36		Template:CDD t0,1,5{3,3,3,3,3,3,3,3} Pentitruncated 9-simplex								66150	6300
37		Template:CDD t0,2,5{3,3,3,3,3,3,3,3} Penticantellated 9-simplex								126000	12600
38		Template:CDD t1,2,5{3,3,3,3,3,3,3,3} Bisteritruncated 9-simplex								107100	12600
39		Template:CDD t0,3,5{3,3,3,3,3,3,3,3} Pentiruncinated 9-simplex								107100	12600
40		Template:CDD t1,3,5{3,3,3,3,3,3,3,3} Bistericantellated 9-simplex								151200	18900
41		Template:CDD t2,3,5{3,3,3,3,3,3,3,3} Triruncitruncated 9-simplex								81900	12600
42		Template:CDD t0,4,5{3,3,3,3,3,3,3,3} Pentistericated 9-simplex								37800	6300
43		Template:CDD t1,4,5{3,3,3,3,3,3,3,3} Bisteriruncinated 9-simplex								81900	12600
44		Template:CDD t2,4,5{3,3,3,3,3,3,3,3} Triruncicantellated 9-simplex								75600	12600
45	File:9-simplex t345.svg	Template:CDD t3,4,5{3,3,3,3,3,3,3,3} Quadricantitruncated 9-simplex								28350	6300
46		Template:CDD t0,1,6{3,3,3,3,3,3,3,3} Hexitruncated 9-simplex								52920	5040
47		Template:CDD t0,2,6{3,3,3,3,3,3,3,3} Hexicantellated 9-simplex								138600	12600
48		Template:CDD t1,2,6{3,3,3,3,3,3,3,3} Bipentitruncated 9-simplex								113400	12600
49		Template:CDD t0,3,6{3,3,3,3,3,3,3,3} Hexiruncinated 9-simplex								176400	16800
50		Template:CDD t1,3,6{3,3,3,3,3,3,3,3} Bipenticantellated 9-simplex								239400	25200
51		Template:CDD t2,3,6{3,3,3,3,3,3,3,3} Tristeritruncated 9-simplex								126000	16800
52		Template:CDD t0,4,6{3,3,3,3,3,3,3,3} Hexistericated 9-simplex								113400	12600
53		Template:CDD t1,4,6{3,3,3,3,3,3,3,3} Bipentiruncinated 9-simplex								226800	25200
54		Template:CDD t2,4,6{3,3,3,3,3,3,3,3} Tristericantellated 9-simplex								201600	25200
55		Template:CDD t0,5,6{3,3,3,3,3,3,3,3} Hexipentellated 9-simplex								32760	5040
56		Template:CDD t1,5,6{3,3,3,3,3,3,3,3} Bipentistericated 9-simplex								94500	12600
57		Template:CDD t0,1,7{3,3,3,3,3,3,3,3} Heptitruncated 9-simplex								23940	2520
58		Template:CDD t0,2,7{3,3,3,3,3,3,3,3} Hepticantellated 9-simplex								83160	7560
59		Template:CDD t1,2,7{3,3,3,3,3,3,3,3} Bihexitruncated 9-simplex								64260	7560
60		Template:CDD t0,3,7{3,3,3,3,3,3,3,3} Heptiruncinated 9-simplex								144900	12600
61		Template:CDD t1,3,7{3,3,3,3,3,3,3,3} Bihexicantellated 9-simplex								189000	18900
62		Template:CDD t0,4,7{3,3,3,3,3,3,3,3} Heptistericated 9-simplex								138600	12600
63		Template:CDD t1,4,7{3,3,3,3,3,3,3,3} Bihexiruncinated 9-simplex								264600	25200
64		Template:CDD t0,5,7{3,3,3,3,3,3,3,3} Heptipentellated 9-simplex								71820	7560
65		Template:CDD t0,6,7{3,3,3,3,3,3,3,3} Heptihexicated 9-simplex								17640	2520
66		Template:CDD t0,1,8{3,3,3,3,3,3,3,3} Octitruncated 9-simplex								5400	720
67		Template:CDD t0,2,8{3,3,3,3,3,3,3,3} Octicantellated 9-simplex								25200	2520
68		Template:CDD t0,3,8{3,3,3,3,3,3,3,3} Octiruncinated 9-simplex								57960	5040
69		Template:CDD t0,4,8{3,3,3,3,3,3,3,3} Octistericated 9-simplex								75600	6300
70		Template:CDD t0,1,2,3{3,3,3,3,3,3,3,3} Runcicantitruncated 9-simplex								22680	5040
71		Template:CDD t0,1,2,4{3,3,3,3,3,3,3,3} Stericantitruncated 9-simplex								105840	15120
72		Template:CDD t0,1,3,4{3,3,3,3,3,3,3,3} Steriruncitruncated 9-simplex								75600	15120
73		Template:CDD t0,2,3,4{3,3,3,3,3,3,3,3} Steriruncicantellated 9-simplex								75600	15120
74		Template:CDD t1,2,3,4{3,3,3,3,3,3,3,3} Biruncicantitruncated 9-simplex								68040	15120
75		Template:CDD t0,1,2,5{3,3,3,3,3,3,3,3} Penticantitruncated 9-simplex								214200	25200
76		Template:CDD t0,1,3,5{3,3,3,3,3,3,3,3} Pentiruncitruncated 9-simplex								283500	37800
77		Template:CDD t0,2,3,5{3,3,3,3,3,3,3,3} Pentiruncicantellated 9-simplex								264600	37800
78		Template:CDD t1,2,3,5{3,3,3,3,3,3,3,3} Bistericantitruncated 9-simplex								245700	37800
79		Template:CDD t0,1,4,5{3,3,3,3,3,3,3,3} Pentisteritruncated 9-simplex								138600	25200
80		Template:CDD t0,2,4,5{3,3,3,3,3,3,3,3} Pentistericantellated 9-simplex								226800	37800
81		Template:CDD t1,2,4,5{3,3,3,3,3,3,3,3} Bisteriruncitruncated 9-simplex								189000	37800
82		Template:CDD t0,3,4,5{3,3,3,3,3,3,3,3} Pentisteriruncinated 9-simplex								138600	25200
83		Template:CDD t1,3,4,5{3,3,3,3,3,3,3,3} Bisteriruncicantellated 9-simplex								207900	37800
84		Template:CDD t2,3,4,5{3,3,3,3,3,3,3,3} Triruncicantitruncated 9-simplex								113400	25200
85		Template:CDD t0,1,2,6{3,3,3,3,3,3,3,3} Hexicantitruncated 9-simplex								226800	25200
86		Template:CDD t0,1,3,6{3,3,3,3,3,3,3,3} Hexiruncitruncated 9-simplex								453600	50400
87		Template:CDD t0,2,3,6{3,3,3,3,3,3,3,3} Hexiruncicantellated 9-simplex								403200	50400
88		Template:CDD t1,2,3,6{3,3,3,3,3,3,3,3} Bipenticantitruncated 9-simplex								378000	50400
89		Template:CDD t0,1,4,6{3,3,3,3,3,3,3,3} Hexisteritruncated 9-simplex								403200	50400
90		Template:CDD t0,2,4,6{3,3,3,3,3,3,3,3} Hexistericantellated 9-simplex								604800	75600
91		Template:CDD t1,2,4,6{3,3,3,3,3,3,3,3} Bipentiruncitruncated 9-simplex								529200	75600
92		Template:CDD t0,3,4,6{3,3,3,3,3,3,3,3} Hexisteriruncinated 9-simplex								352800	50400
93		Template:CDD t1,3,4,6{3,3,3,3,3,3,3,3} Bipentiruncicantellated 9-simplex								529200	75600
94		Template:CDD t2,3,4,6{3,3,3,3,3,3,3,3} Tristericantitruncated 9-simplex								302400	50400
95		Template:CDD t0,1,5,6{3,3,3,3,3,3,3,3} Hexipentitruncated 9-simplex								151200	25200
96		Template:CDD t0,2,5,6{3,3,3,3,3,3,3,3} Hexipenticantellated 9-simplex								352800	50400
97		Template:CDD t1,2,5,6{3,3,3,3,3,3,3,3} Bipentisteritruncated 9-simplex								277200	50400
98		Template:CDD t0,3,5,6{3,3,3,3,3,3,3,3} Hexipentiruncinated 9-simplex								352800	50400
99		Template:CDD t1,3,5,6{3,3,3,3,3,3,3,3} Bipentistericantellated 9-simplex								491400	75600
100		Template:CDD t2,3,5,6{3,3,3,3,3,3,3,3} Tristeriruncitruncated 9-simplex								252000	50400
101		Template:CDD t0,4,5,6{3,3,3,3,3,3,3,3} Hexipentistericated 9-simplex								151200	25200
102		Template:CDD t1,4,5,6{3,3,3,3,3,3,3,3} Bipentisteriruncinated 9-simplex								327600	50400
103		Template:CDD t0,1,2,7{3,3,3,3,3,3,3,3} Hepticantitruncated 9-simplex								128520	15120
104		Template:CDD t0,1,3,7{3,3,3,3,3,3,3,3} Heptiruncitruncated 9-simplex								359100	37800
105		Template:CDD t0,2,3,7{3,3,3,3,3,3,3,3} Heptiruncicantellated 9-simplex								302400	37800
106		Template:CDD t1,2,3,7{3,3,3,3,3,3,3,3} Bihexicantitruncated 9-simplex								283500	37800
107		Template:CDD t0,1,4,7{3,3,3,3,3,3,3,3} Heptisteritruncated 9-simplex								478800	50400
108		Template:CDD t0,2,4,7{3,3,3,3,3,3,3,3} Heptistericantellated 9-simplex								680400	75600
109		Template:CDD t1,2,4,7{3,3,3,3,3,3,3,3} Bihexiruncitruncated 9-simplex								604800	75600
110		Template:CDD t0,3,4,7{3,3,3,3,3,3,3,3} Heptisteriruncinated 9-simplex								378000	50400
111		Template:CDD t1,3,4,7{3,3,3,3,3,3,3,3} Bihexiruncicantellated 9-simplex								567000	75600
112		Template:CDD t0,1,5,7{3,3,3,3,3,3,3,3} Heptipentitruncated 9-simplex								321300	37800
113		Template:CDD t0,2,5,7{3,3,3,3,3,3,3,3} Heptipenticantellated 9-simplex								680400	75600
114		Template:CDD t1,2,5,7{3,3,3,3,3,3,3,3} Bihexisteritruncated 9-simplex								567000	75600
115		Template:CDD t0,3,5,7{3,3,3,3,3,3,3,3} Heptipentiruncinated 9-simplex								642600	75600
116		Template:CDD t1,3,5,7{3,3,3,3,3,3,3,3} Bihexistericantellated 9-simplex								907200	113400
117		Template:CDD t0,4,5,7{3,3,3,3,3,3,3,3} Heptipentistericated 9-simplex								264600	37800
118		Template:CDD t0,1,6,7{3,3,3,3,3,3,3,3} Heptihexitruncated 9-simplex								98280	15120
119		Template:CDD t0,2,6,7{3,3,3,3,3,3,3,3} Heptihexicantellated 9-simplex								302400	37800
120		Template:CDD t1,2,6,7{3,3,3,3,3,3,3,3} Bihexipentitruncated 9-simplex								226800	37800
121		Template:CDD t0,3,6,7{3,3,3,3,3,3,3,3} Heptihexiruncinated 9-simplex								428400	50400
122		Template:CDD t0,4,6,7{3,3,3,3,3,3,3,3} Heptihexistericated 9-simplex								302400	37800
123		Template:CDD t0,5,6,7{3,3,3,3,3,3,3,3} Heptihexipentellated 9-simplex								98280	15120
124		Template:CDD t0,1,2,8{3,3,3,3,3,3,3,3} Octicantitruncated 9-simplex								35280	5040
125		Template:CDD t0,1,3,8{3,3,3,3,3,3,3,3} Octiruncitruncated 9-simplex								136080	15120
126		Template:CDD t0,2,3,8{3,3,3,3,3,3,3,3} Octiruncicantellated 9-simplex								105840	15120
127		Template:CDD t0,1,4,8{3,3,3,3,3,3,3,3} Octisteritruncated 9-simplex								252000	25200
128		Template:CDD t0,2,4,8{3,3,3,3,3,3,3,3} Octistericantellated 9-simplex								340200	37800
129		Template:CDD t0,3,4,8{3,3,3,3,3,3,3,3} Octisteriruncinated 9-simplex								176400	25200
130		Template:CDD t0,1,5,8{3,3,3,3,3,3,3,3} Octipentitruncated 9-simplex								252000	25200
131		Template:CDD t0,2,5,8{3,3,3,3,3,3,3,3} Octipenticantellated 9-simplex								504000	50400
132		Template:CDD t0,3,5,8{3,3,3,3,3,3,3,3} Octipentiruncinated 9-simplex								453600	50400
133		Template:CDD t0,1,6,8{3,3,3,3,3,3,3,3} Octihexitruncated 9-simplex								136080	15120
134		Template:CDD t0,2,6,8{3,3,3,3,3,3,3,3} Octihexicantellated 9-simplex								378000	37800
135		Template:CDD t0,1,7,8{3,3,3,3,3,3,3,3} Octiheptitruncated 9-simplex								35280	5040
136		Template:CDD t0,1,2,3,4{3,3,3,3,3,3,3,3} Steriruncicantitruncated 9-simplex								136080	30240
137		Template:CDD t0,1,2,3,5{3,3,3,3,3,3,3,3} Pentiruncicantitruncated 9-simplex								491400	75600
138		Template:CDD t0,1,2,4,5{3,3,3,3,3,3,3,3} Pentistericantitruncated 9-simplex								378000	75600
139		Template:CDD t0,1,3,4,5{3,3,3,3,3,3,3,3} Pentisteriruncitruncated 9-simplex								378000	75600
140		Template:CDD t0,2,3,4,5{3,3,3,3,3,3,3,3} Pentisteriruncicantellated 9-simplex								378000	75600
141		Template:CDD t1,2,3,4,5{3,3,3,3,3,3,3,3} Bisteriruncicantitruncated 9-simplex								340200	75600
142		Template:CDD t0,1,2,3,6{3,3,3,3,3,3,3,3} Hexiruncicantitruncated 9-simplex								756000	100800
143		Template:CDD t0,1,2,4,6{3,3,3,3,3,3,3,3} Hexistericantitruncated 9-simplex								1058400	151200
144		Template:CDD t0,1,3,4,6{3,3,3,3,3,3,3,3} Hexisteriruncitruncated 9-simplex								982800	151200
145		Template:CDD t0,2,3,4,6{3,3,3,3,3,3,3,3} Hexisteriruncicantellated 9-simplex								982800	151200
146		Template:CDD t1,2,3,4,6{3,3,3,3,3,3,3,3} Bipentiruncicantitruncated 9-simplex								907200	151200
147		Template:CDD t0,1,2,5,6{3,3,3,3,3,3,3,3} Hexipenticantitruncated 9-simplex								554400	100800
148		Template:CDD t0,1,3,5,6{3,3,3,3,3,3,3,3} Hexipentiruncitruncated 9-simplex								907200	151200
149		Template:CDD t0,2,3,5,6{3,3,3,3,3,3,3,3} Hexipentiruncicantellated 9-simplex								831600	151200
150		Template:CDD t1,2,3,5,6{3,3,3,3,3,3,3,3} Bipentistericantitruncated 9-simplex								756000	151200
151		Template:CDD t0,1,4,5,6{3,3,3,3,3,3,3,3} Hexipentisteritruncated 9-simplex								554400	100800
152		Template:CDD t0,2,4,5,6{3,3,3,3,3,3,3,3} Hexipentistericantellated 9-simplex								907200	151200
153		Template:CDD t1,2,4,5,6{3,3,3,3,3,3,3,3} Bipentisteriruncitruncated 9-simplex								756000	151200
154		Template:CDD t0,3,4,5,6{3,3,3,3,3,3,3,3} Hexipentisteriruncinated 9-simplex								554400	100800
155		Template:CDD t1,3,4,5,6{3,3,3,3,3,3,3,3} Bipentisteriruncicantellated 9-simplex								831600	151200
156		Template:CDD t2,3,4,5,6{3,3,3,3,3,3,3,3} Tristeriruncicantitruncated 9-simplex								453600	100800
157		Template:CDD t0,1,2,3,7{3,3,3,3,3,3,3,3} Heptiruncicantitruncated 9-simplex								567000	75600
158		Template:CDD t0,1,2,4,7{3,3,3,3,3,3,3,3} Heptistericantitruncated 9-simplex								1209600	151200
159		Template:CDD t0,1,3,4,7{3,3,3,3,3,3,3,3} Heptisteriruncitruncated 9-simplex								1058400	151200
160		Template:CDD t0,2,3,4,7{3,3,3,3,3,3,3,3} Heptisteriruncicantellated 9-simplex								1058400	151200
161		Template:CDD t1,2,3,4,7{3,3,3,3,3,3,3,3} Bihexiruncicantitruncated 9-simplex								982800	151200
162		Template:CDD t0,1,2,5,7{3,3,3,3,3,3,3,3} Heptipenticantitruncated 9-simplex								1134000	151200
163		Template:CDD t0,1,3,5,7{3,3,3,3,3,3,3,3} Heptipentiruncitruncated 9-simplex								1701000	226800
164		Template:CDD t0,2,3,5,7{3,3,3,3,3,3,3,3} Heptipentiruncicantellated 9-simplex								1587600	226800
165		Template:CDD t1,2,3,5,7{3,3,3,3,3,3,3,3} Bihexistericantitruncated 9-simplex								1474200	226800
166		Template:CDD t0,1,4,5,7{3,3,3,3,3,3,3,3} Heptipentisteritruncated 9-simplex								982800	151200
167		Template:CDD t0,2,4,5,7{3,3,3,3,3,3,3,3} Heptipentistericantellated 9-simplex								1587600	226800
168		Template:CDD t1,2,4,5,7{3,3,3,3,3,3,3,3} Bihexisteriruncitruncated 9-simplex								1360800	226800
169		Template:CDD t0,3,4,5,7{3,3,3,3,3,3,3,3} Heptipentisteriruncinated 9-simplex								982800	151200
170		Template:CDD t1,3,4,5,7{3,3,3,3,3,3,3,3} Bihexisteriruncicantellated 9-simplex								1474200	226800
171		Template:CDD t0,1,2,6,7{3,3,3,3,3,3,3,3} Heptihexicantitruncated 9-simplex								453600	75600
172		Template:CDD t0,1,3,6,7{3,3,3,3,3,3,3,3} Heptihexiruncitruncated 9-simplex								1058400	151200
173		Template:CDD t0,2,3,6,7{3,3,3,3,3,3,3,3} Heptihexiruncicantellated 9-simplex								907200	151200
174		Template:CDD t1,2,3,6,7{3,3,3,3,3,3,3,3} Bihexipenticantitruncated 9-simplex								831600	151200
175		Template:CDD t0,1,4,6,7{3,3,3,3,3,3,3,3} Heptihexisteritruncated 9-simplex								1058400	151200
176		Template:CDD t0,2,4,6,7{3,3,3,3,3,3,3,3} Heptihexistericantellated 9-simplex								1587600	226800
177		Template:CDD t1,2,4,6,7{3,3,3,3,3,3,3,3} Bihexipentiruncitruncated 9-simplex								1360800	226800
178		Template:CDD t0,3,4,6,7{3,3,3,3,3,3,3,3} Heptihexisteriruncinated 9-simplex								907200	151200
179		Template:CDD t0,1,5,6,7{3,3,3,3,3,3,3,3} Heptihexipentitruncated 9-simplex								453600	75600
180		Template:CDD t0,2,5,6,7{3,3,3,3,3,3,3,3} Heptihexipenticantellated 9-simplex								1058400	151200
181		Template:CDD t0,3,5,6,7{3,3,3,3,3,3,3,3} Heptihexipentiruncinated 9-simplex								1058400	151200
182		Template:CDD t0,4,5,6,7{3,3,3,3,3,3,3,3} Heptihexipentistericated 9-simplex								453600	75600
183		Template:CDD t0,1,2,3,8{3,3,3,3,3,3,3,3} Octiruncicantitruncated 9-simplex								196560	30240
184		Template:CDD t0,1,2,4,8{3,3,3,3,3,3,3,3} Octistericantitruncated 9-simplex								604800	75600
185		Template:CDD t0,1,3,4,8{3,3,3,3,3,3,3,3} Octisteriruncitruncated 9-simplex								491400	75600
186		Template:CDD t0,2,3,4,8{3,3,3,3,3,3,3,3} Octisteriruncicantellated 9-simplex								491400	75600
187		Template:CDD t0,1,2,5,8{3,3,3,3,3,3,3,3} Octipenticantitruncated 9-simplex								856800	100800
188		Template:CDD t0,1,3,5,8{3,3,3,3,3,3,3,3} Octipentiruncitruncated 9-simplex								1209600	151200
189		Template:CDD t0,2,3,5,8{3,3,3,3,3,3,3,3} Octipentiruncicantellated 9-simplex								1134000	151200
190		Template:CDD t0,1,4,5,8{3,3,3,3,3,3,3,3} Octipentisteritruncated 9-simplex								655200	100800
191		Template:CDD t0,2,4,5,8{3,3,3,3,3,3,3,3} Octipentistericantellated 9-simplex								1058400	151200
192		Template:CDD t0,3,4,5,8{3,3,3,3,3,3,3,3} Octipentisteriruncinated 9-simplex								655200	100800
193		Template:CDD t0,1,2,6,8{3,3,3,3,3,3,3,3} Octihexicantitruncated 9-simplex								604800	75600
194		Template:CDD t0,1,3,6,8{3,3,3,3,3,3,3,3} Octihexiruncitruncated 9-simplex								1285200	151200
195		Template:CDD t0,2,3,6,8{3,3,3,3,3,3,3,3} Octihexiruncicantellated 9-simplex								1134000	151200
196		Template:CDD t0,1,4,6,8{3,3,3,3,3,3,3,3} Octihexisteritruncated 9-simplex								1209600	151200
197		Template:CDD t0,2,4,6,8{3,3,3,3,3,3,3,3} Octihexistericantellated 9-simplex								1814400	226800
198		Template:CDD t0,1,5,6,8{3,3,3,3,3,3,3,3} Octihexipentitruncated 9-simplex								491400	75600
199		Template:CDD t0,1,2,7,8{3,3,3,3,3,3,3,3} Octihepticantitruncated 9-simplex								196560	30240
200		Template:CDD t0,1,3,7,8{3,3,3,3,3,3,3,3} Octiheptiruncitruncated 9-simplex								604800	75600
201		Template:CDD t0,1,4,7,8{3,3,3,3,3,3,3,3} Octiheptisteritruncated 9-simplex								856800	100800
202		Template:CDD t0,1,2,3,4,5{3,3,3,3,3,3,3,3} Pentisteriruncicantitruncated 9-simplex								680400	151200
203		Template:CDD t0,1,2,3,4,6{3,3,3,3,3,3,3,3} Hexisteriruncicantitruncated 9-simplex								1814400	302400
204		Template:CDD t0,1,2,3,5,6{3,3,3,3,3,3,3,3} Hexipentiruncicantitruncated 9-simplex								1512000	302400
205		Template:CDD t0,1,2,4,5,6{3,3,3,3,3,3,3,3} Hexipentistericantitruncated 9-simplex								1512000	302400
206		Template:CDD t0,1,3,4,5,6{3,3,3,3,3,3,3,3} Hexipentisteriruncitruncated 9-simplex								1512000	302400
207		Template:CDD t0,2,3,4,5,6{3,3,3,3,3,3,3,3} Hexipentisteriruncicantellated 9-simplex								1512000	302400
208		Template:CDD t1,2,3,4,5,6{3,3,3,3,3,3,3,3} Bipentisteriruncicantitruncated 9-simplex								1360800	302400
209		Template:CDD t0,1,2,3,4,7{3,3,3,3,3,3,3,3} Heptisteriruncicantitruncated 9-simplex								1965600	302400
210		Template:CDD t0,1,2,3,5,7{3,3,3,3,3,3,3,3} Heptipentiruncicantitruncated 9-simplex								2948400	453600
211		Template:CDD t0,1,2,4,5,7{3,3,3,3,3,3,3,3} Heptipentistericantitruncated 9-simplex								2721600	453600
212		Template:CDD t0,1,3,4,5,7{3,3,3,3,3,3,3,3} Heptipentisteriruncitruncated 9-simplex								2721600	453600
213		Template:CDD t0,2,3,4,5,7{3,3,3,3,3,3,3,3} Heptipentisteriruncicantellated 9-simplex								2721600	453600
214		Template:CDD t1,2,3,4,5,7{3,3,3,3,3,3,3,3} Bihexisteriruncicantitruncated 9-simplex								2494800	453600
215		Template:CDD t0,1,2,3,6,7{3,3,3,3,3,3,3,3} Heptihexiruncicantitruncated 9-simplex								1663200	302400
216		Template:CDD t0,1,2,4,6,7{3,3,3,3,3,3,3,3} Heptihexistericantitruncated 9-simplex								2721600	453600
217		Template:CDD t0,1,3,4,6,7{3,3,3,3,3,3,3,3} Heptihexisteriruncitruncated 9-simplex								2494800	453600
218		Template:CDD t0,2,3,4,6,7{3,3,3,3,3,3,3,3} Heptihexisteriruncicantellated 9-simplex								2494800	453600
219		Template:CDD t1,2,3,4,6,7{3,3,3,3,3,3,3,3} Bihexipentiruncicantitruncated 9-simplex								2268000	453600
220		Template:CDD t0,1,2,5,6,7{3,3,3,3,3,3,3,3} Heptihexipenticantitruncated 9-simplex								1663200	302400
221		Template:CDD t0,1,3,5,6,7{3,3,3,3,3,3,3,3} Heptihexipentiruncitruncated 9-simplex								2721600	453600
222		Template:CDD t0,2,3,5,6,7{3,3,3,3,3,3,3,3} Heptihexipentiruncicantellated 9-simplex								2494800	453600
223		Template:CDD t1,2,3,5,6,7{3,3,3,3,3,3,3,3} Bihexipentistericantitruncated 9-simplex								2268000	453600
224		Template:CDD t0,1,4,5,6,7{3,3,3,3,3,3,3,3} Heptihexipentisteritruncated 9-simplex								1663200	302400
225		Template:CDD t0,2,4,5,6,7{3,3,3,3,3,3,3,3} Heptihexipentistericantellated 9-simplex								2721600	453600
226		Template:CDD t0,3,4,5,6,7{3,3,3,3,3,3,3,3} Heptihexipentisteriruncinated 9-simplex								1663200	302400
227		Template:CDD t0,1,2,3,4,8{3,3,3,3,3,3,3,3} Octisteriruncicantitruncated 9-simplex								907200	151200
228		Template:CDD t0,1,2,3,5,8{3,3,3,3,3,3,3,3} Octipentiruncicantitruncated 9-simplex								2116800	302400
229		Template:CDD t0,1,2,4,5,8{3,3,3,3,3,3,3,3} Octipentistericantitruncated 9-simplex								1814400	302400
230		Template:CDD t0,1,3,4,5,8{3,3,3,3,3,3,3,3} Octipentisteriruncitruncated 9-simplex								1814400	302400
231		Template:CDD t0,2,3,4,5,8{3,3,3,3,3,3,3,3} Octipentisteriruncicantellated 9-simplex								1814400	302400
232		Template:CDD t0,1,2,3,6,8{3,3,3,3,3,3,3,3} Octihexiruncicantitruncated 9-simplex								2116800	302400
233		Template:CDD t0,1,2,4,6,8{3,3,3,3,3,3,3,3} Octihexistericantitruncated 9-simplex								3175200	453600
234		Template:CDD t0,1,3,4,6,8{3,3,3,3,3,3,3,3} Octihexisteriruncitruncated 9-simplex								2948400	453600
235		Template:CDD t0,2,3,4,6,8{3,3,3,3,3,3,3,3} Octihexisteriruncicantellated 9-simplex								2948400	453600
236		Template:CDD t0,1,2,5,6,8{3,3,3,3,3,3,3,3} Octihexipenticantitruncated 9-simplex								1814400	302400
237		Template:CDD t0,1,3,5,6,8{3,3,3,3,3,3,3,3} Octihexipentiruncitruncated 9-simplex								2948400	453600
238		Template:CDD t0,2,3,5,6,8{3,3,3,3,3,3,3,3} Octihexipentiruncicantellated 9-simplex								2721600	453600
239		Template:CDD t0,1,4,5,6,8{3,3,3,3,3,3,3,3} Octihexipentisteritruncated 9-simplex								1814400	302400
240		Template:CDD t0,1,2,3,7,8{3,3,3,3,3,3,3,3} Octiheptiruncicantitruncated 9-simplex								907200	151200
241		Template:CDD t0,1,2,4,7,8{3,3,3,3,3,3,3,3} Octiheptistericantitruncated 9-simplex								2116800	302400
242		Template:CDD t0,1,3,4,7,8{3,3,3,3,3,3,3,3} Octiheptisteriruncitruncated 9-simplex								1814400	302400
243		Template:CDD t0,1,2,5,7,8{3,3,3,3,3,3,3,3} Octiheptipenticantitruncated 9-simplex								2116800	302400
244		Template:CDD t0,1,3,5,7,8{3,3,3,3,3,3,3,3} Octiheptipentiruncitruncated 9-simplex								3175200	453600
245		Template:CDD t0,1,2,6,7,8{3,3,3,3,3,3,3,3} Octiheptihexicantitruncated 9-simplex								907200	151200
246		Template:CDD t0,1,2,3,4,5,6{3,3,3,3,3,3,3,3} Hexipentisteriruncicantitruncated 9-simplex								2721600	604800
247		Template:CDD t0,1,2,3,4,5,7{3,3,3,3,3,3,3,3} Heptipentisteriruncicantitruncated 9-simplex								4989600	907200
248		Template:CDD t0,1,2,3,4,6,7{3,3,3,3,3,3,3,3} Heptihexisteriruncicantitruncated 9-simplex								4536000	907200
249		Template:CDD t0,1,2,3,5,6,7{3,3,3,3,3,3,3,3} Heptihexipentiruncicantitruncated 9-simplex								4536000	907200
250		Template:CDD t0,1,2,4,5,6,7{3,3,3,3,3,3,3,3} Heptihexipentistericantitruncated 9-simplex								4536000	907200
251		Template:CDD t0,1,3,4,5,6,7{3,3,3,3,3,3,3,3} Heptihexipentisteriruncitruncated 9-simplex								4536000	907200
252		Template:CDD t0,2,3,4,5,6,7{3,3,3,3,3,3,3,3} Heptihexipentisteriruncicantellated 9-simplex								4536000	907200
253		Template:CDD t1,2,3,4,5,6,7{3,3,3,3,3,3,3,3} Bihexipentisteriruncicantitruncated 9-simplex								4082400	907200
254		Template:CDD t0,1,2,3,4,5,8{3,3,3,3,3,3,3,3} Octipentisteriruncicantitruncated 9-simplex								3326400	604800
255		Template:CDD t0,1,2,3,4,6,8{3,3,3,3,3,3,3,3} Octihexisteriruncicantitruncated 9-simplex								5443200	907200
256		Template:CDD t0,1,2,3,5,6,8{3,3,3,3,3,3,3,3} Octihexipentiruncicantitruncated 9-simplex								4989600	907200
257		Template:CDD t0,1,2,4,5,6,8{3,3,3,3,3,3,3,3} Octihexipentistericantitruncated 9-simplex								4989600	907200
258		Template:CDD t0,1,3,4,5,6,8{3,3,3,3,3,3,3,3} Octihexipentisteriruncitruncated 9-simplex								4989600	907200
259		Template:CDD t0,2,3,4,5,6,8{3,3,3,3,3,3,3,3} Octihexipentisteriruncicantellated 9-simplex								4989600	907200
260		Template:CDD t0,1,2,3,4,7,8{3,3,3,3,3,3,3,3} Octiheptisteriruncicantitruncated 9-simplex								3326400	604800
261		Template:CDD t0,1,2,3,5,7,8{3,3,3,3,3,3,3,3} Octiheptipentiruncicantitruncated 9-simplex								5443200	907200
262		Template:CDD t0,1,2,4,5,7,8{3,3,3,3,3,3,3,3} Octiheptipentistericantitruncated 9-simplex								4989600	907200
263		Template:CDD t0,1,3,4,5,7,8{3,3,3,3,3,3,3,3} Octiheptipentisteriruncitruncated 9-simplex								4989600	907200
264		Template:CDD t0,1,2,3,6,7,8{3,3,3,3,3,3,3,3} Octiheptihexiruncicantitruncated 9-simplex								3326400	604800
265		Template:CDD t0,1,2,4,6,7,8{3,3,3,3,3,3,3,3} Octiheptihexistericantitruncated 9-simplex								5443200	907200
266		Template:CDD t0,1,2,3,4,5,6,7{3,3,3,3,3,3,3,3} Heptihexipentisteriruncicantitruncated 9-simplex								8164800	1814400
267		Template:CDD t0,1,2,3,4,5,6,8{3,3,3,3,3,3,3,3} Octihexipentisteriruncicantitruncated 9-simplex								9072000	1814400
268		Template:CDD t0,1,2,3,4,5,7,8{3,3,3,3,3,3,3,3} Octiheptipentisteriruncicantitruncated 9-simplex								9072000	1814400
269		Template:CDD t0,1,2,3,4,6,7,8{3,3,3,3,3,3,3,3} Octiheptihexisteriruncicantitruncated 9-simplex								9072000	1814400
270		Template:CDD t0,1,2,3,5,6,7,8{3,3,3,3,3,3,3,3} Octiheptihexipentiruncicantitruncated 9-simplex								9072000	1814400
271		Template:CDD 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	File:9-cube.svg	Template:CDD t0{4,3,3,3,3,3,3,3} 9-cube (enne)	18	144	672	2016	4032	5376	4608	2304	512
2	File:Truncated 9-cube.png	Template:CDD t0,1{4,3,3,3,3,3,3,3} Truncated 9-cube (ten)								2304	4608
3	File:Rectified 9-cube.png	Template:CDD t1{4,3,3,3,3,3,3,3} Rectified 9-cube (ren)								18432	2304
4	File:Birectified 9-cube.png	Template:CDD t2{4,3,3,3,3,3,3,3} Birectified 9-cube (barn)								64512	4608
5	File:Quintirectified 9-orthoplex.png	Template:CDD t3{4,3,3,3,3,3,3,3} Trirectified 9-cube (tarn)								96768	5376
6	File:Quadrirectified 9-orthoplex.png	Template:CDD t4{4,3,3,3,3,3,3,3} Quadrirectified 9-cube (nav) (Quadrirectified 9-orthoplex)								80640	4032
7	File:Trirectified 9-orthoplex.png	Template:CDD t3{3,3,3,3,3,3,3,4} Trirectified 9-orthoplex (tarv)								40320	2016
8	File:Birectified 9-orthoplex.png	Template:CDD t2{3,3,3,3,3,3,3,4} Birectified 9-orthoplex (brav)								12096	672
9	File:Rectified heptacross.png	Template:CDD t1{3,3,3,3,3,3,3,4} Rectified 9-orthoplex (riv)								2016	144
10	File:Truncated 9-orthoplex.png	Template:CDD t0,1{3,3,3,3,3,3,3,4} Truncated 9-orthoplex (tiv)								2160	288
11	File:Cross graph 9.png	Template:CDD 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	File:9-demicube t0 B9.svg	File:9-demicube t0 D9.svg	File:9-demicube t0 D8.svg	File:9-demicube t0 D7.svg	File:9-demicube t0 D6.svg	File:9-demicube t0 D5.svg	File:9-demicube t0 D4.svg	File:9-demicube t0 D3.svg	File:9-demicube t0 A7.svg	File:9-demicube t0 A5.svg	File:9-demicube t0 A3.svg	Template:CDD 9-demicube (henne)	(1,1,1,1,1,1,1,1,1)	274	2448	9888	23520	36288	37632	21404	4608	256	1.0606601
2	File:9-demicube t01 B9.svg	File:9-demicube t01 D9.svg	File:9-demicube t01 D8.svg	File:9-demicube t01 D7.svg	File:9-demicube t01 D6.svg	File:9-demicube t01 D5.svg	File:9-demicube t01 D4.svg	File:9-demicube t01 D3.svg	File:9-demicube t01 A7.svg	File:9-demicube t01 A5.svg	File:9-demicube t01 A3.svg	Template:CDD Truncated 9-demicube (thenne)	(1,1,3,3,3,3,3,3,3)								69120	9216	2.8504384
3	File:9-demicube t02 B9.svg	File:9-demicube t02 D9.svg	File:9-demicube t02 D8.svg	File:9-demicube t02 D7.svg	File:9-demicube t02 D6.svg	File:9-demicube t02 D5.svg	File:9-demicube t02 D4.svg	File:9-demicube t02 D3.svg	File:9-demicube t02 A7.svg	File:9-demicube t02 A5.svg	File:9-demicube t02 A3.svg	Template:CDD Cantellated 9-demicube	(1,1,1,3,3,3,3,3,3)								225792	21504	2.6692696
4	File:9-demicube t03 B9.svg	File:9-demicube t03 D9.svg	File:9-demicube t03 D8.svg	File:9-demicube t03 D7.svg	File:9-demicube t03 D6.svg	File:9-demicube t03 D5.svg	File:9-demicube t03 D4.svg	File:9-demicube t03 D3.svg	File:9-demicube t03 A7.svg	File:9-demicube t03 A5.svg	File:9-demicube t03 A3.svg	Template:CDD Runcinated 9-demicube	(1,1,1,1,3,3,3,3,3)								419328	32256	2.4748735
5	File:9-demicube t04 B9.svg	File:9-demicube t04 D9.svg	File:9-demicube t04 D8.svg	File:9-demicube t04 D7.svg	File:9-demicube t04 D6.svg	File:9-demicube t04 D5.svg	File:9-demicube t04 D4.svg	File:9-demicube t04 D3.svg	File:9-demicube t04 A7.svg	File:9-demicube t04 A5.svg	File:9-demicube t04 A3.svg	Template:CDD Stericated 9-demicube	(1,1,1,1,1,3,3,3,3)								483840	32256	2.2638462
6	File:9-demicube t05 B9.svg	File:9-demicube t05 D9.svg	File:9-demicube t05 D8.svg	File:9-demicube t05 D7.svg	File:9-demicube t05 D6.svg	File:9-demicube t05 D5.svg	File:9-demicube t05 D4.svg	File:9-demicube t05 D3.svg	File:9-demicube t05 A7.svg	File:9-demicube t05 A5.svg	File:9-demicube t05 A3.svg	Template:CDD Pentellated 9-demicube	(1,1,1,1,1,1,3,3,3)								354816	21504	2.0310094
7	File:9-demicube t06 B9.svg	File:9-demicube t06 D9.svg	File:9-demicube t06 D8.svg	File:9-demicube t06 D7.svg	File:9-demicube t06 D6.svg	File:9-demicube t06 D5.svg	File:9-demicube t06 D4.svg	File:9-demicube t06 D3.svg	File:9-demicube t06 A7.svg	File:9-demicube t06 A5.svg	File:9-demicube t06 A3.svg	Template:CDD Hexicated 9-demicube	(1,1,1,1,1,1,1,3,3)								161280	9216	1.7677668
8	File:9-demicube t07 B9.svg	File:9-demicube t07 D9.svg	File:9-demicube t07 D8.svg	File:9-demicube t07 D7.svg	File:9-demicube t07 D6.svg	File:9-demicube t07 D5.svg	File:9-demicube t07 D4.svg	File:9-demicube t07 D3.svg	File:9-demicube t07 A7.svg	File:9-demicube t07 A5.svg	File:9-demicube t07 A3.svg	Template:CDD Heptellated 9-demicube	(1,1,1,1,1,1,1,1,3)								41472	2304	1.4577379

Regular and uniform honeycombs

File:Coxeter diagram affine rank9 correspondence.png

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	${\tilde{A}}_8$	[3[9]]	Template:CDD	45
2	${\tilde{C}}_8$	[4,36,4]	Template:CDD	271
3	${\tilde{B}}_8$	h[4,36,4] [4,35,31,1]	Template:CDD	383 (128 new)
4	${\tilde{D}}_8$	q[4,36,4] [31,1,34,31,1]	Template:CDD	155 (15 new)
5	${\tilde{E}}_8$	[35,2,1]	Template:CDD	511

Regular and uniform tessellations include:

• ${\tilde{A}}_8$ 45 uniquely ringed forms
  • 8-simplex honeycomb: {3^[9]} Template:CDD
• ${\tilde{C}}_8$ 271 uniquely ringed forms
  • Regular 8-cube honeycomb: {4,3⁶,4}, Template:CDD
• ${\tilde{B}}_8$: 383 uniquely ringed forms, 255 shared with ${\tilde{C}}_8$, 128 new
  • 8-demicube honeycomb: h{4,3⁶,4} or {3^1,1,3⁵,4}, Template:CDD or Template:CDD
• ${\tilde{D}}_8$, [3^1,1,3⁴,3^1,1]: 155 unique ring permutations, and 15 are new, the first, Template:CDD, Coxeter called a quarter 8-cubic honeycomb, representing as q{4,3⁶,4}, or qδ₉.
• ${\tilde{E}}_8$ 511 forms
  • 5₂₁ honeycomb:Template:CDD
  • 2₅₁ honeycomb: Template:CDD
  • 1₅₂ honeycomb: Template:CDD

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.

${\overline{P}}_8$ = [3,3[8]]: Template:CDD

${\overline{Q}}_8$ = [31,1,33,32,1]: Template:CDD

${\overline{S}}_8$ = [4,34,32,1]: Template:CDD

${\overline{T}}_8$ = [34,3,1]: Template:CDD

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, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, Template:Isbn [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
• Template:KlitzingPolytopes

External links

• Polytope names
• Polytopes of Various Dimensions, Jonathan Bowers
• Multi-dimensional Glossary
• Template:PolyCell