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[edit]

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[edit]

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[edit]

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, 1₆₁ - ; also as h{4,3⁸} .
  2. {3^6,1,1} - 9-orthoplex, 6₁₁ -

The A₉ family[edit]

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
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