Uniform 8-polytope

Graphs of three regular and related uniform polytopes.
8-simplex	Rectified 8-simplex		Truncated 8-simplex
Cantellated 8-simplex	Runcinated 8-simplex		Stericated 8-simplex
Pentellated 8-simplex	Hexicated 8-simplex		Heptellated 8-simplex
8-orthoplex	Rectified 8-orthoplex		Truncated 8-orthoplex
Cantellated 8-orthoplex		Runcinated 8-orthoplex
Hexicated 8-orthoplex		Cantellated 8-cube
Runcinated 8-cube	Stericated 8-cube		Pentellated 8-cube
Hexicated 8-cube		Heptellated 8-cube
8-cube	Rectified 8-cube		Truncated 8-cube
8-demicube	Truncated 8-demicube		Cantellated 8-demicube
Runcinated 8-demicube		Stericated 8-demicube
Pentellated 8-demicube		Hexicated 8-demicube
4₂₁	1₄₂		2₄₁

In eight-dimensional geometry, an eight-dimensional polytope or 8-polytope is a polytope contained by 7-polytope facets. Each 6-polytope ridge being shared by exactly two 7-polytope facets.

A uniform 8-polytope is one which is vertex-transitive, and constructed from uniform 7-polytope facets.

Regular 8-polytopes

Regular 8-polytopes can be represented by the Schläfli symbol {p,q,r,s,t,u,v}, with v {p,q,r,s,t,u} 7-polytope facets around each peak.

There are exactly three such convex regular 8-polytopes:

{3,3,3,3,3,3,3} - 8-simplex
{4,3,3,3,3,3,3} - 8-cube
{3,3,3,3,3,3,4} - 8-orthoplex

There are no nonconvex regular 8-polytopes.

Characteristics

The topology of any given 8-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, and is zero for all 8-polytopes, 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 8-polytopes by fundamental Coxeter groups

Uniform 8-polytopes with reflective symmetry can be generated by these four Coxeter groups, represented by permutations of rings of the Coxeter-Dynkin diagrams:

#	Coxeter group		Forms
1	A₈	[3⁷]	135
2	BC₈	[4,3⁶]	255
3	D₈	[3^5,1,1]	191 (64 unique)
4	E₈	[3^4,2,1]	255

Selected regular and uniform 8-polytopes from each family include:

Simplex family: A₈ [3⁷] -
- 135 uniform 8-polytopes as permutations of rings in the group diagram, including one regular:
  1. {3⁷} - 8-simplex or ennea-9-tope or enneazetton -
Hypercube/orthoplex family: B₈ [4,3⁶] -
- 255 uniform 8-polytopes as permutations of rings in the group diagram, including two regular ones:
  1. {4,3⁶} - 8-cube or octeract-
  2. {3⁶,4} - 8-orthoplex or octacross -
Demihypercube D₈ family: [3^5,1,1] -
- 191 uniform 8-polytopes as permutations of rings in the group diagram, including:
  1. {3,3^5,1} - 8-demicube or demiocteract, 1₅₁ - ; also as h{4,3⁶} .
  2. {3,3,3,3,3,3^1,1} - 8-orthoplex, 5₁₁ -
E-polytope family E₈ family: [3^4,1,1] -
- 255 uniform 8-polytopes as permutations of rings in the group diagram, including:
  1. {3,3,3,3,3^2,1} - Thorold Gosset's semiregular 4₂₁,
  2. {3,3^4,2} - the uniform 1₄₂, ,
  3. {3,3,3^4,1} - the uniform 2₄₁,

Uniform prismatic forms

There are many uniform prismatic families, including:

Uniform 8-polytope prism families
#	Coxeter group		Coxeter-Dynkin diagram
7+1
1	A₇A₁	[3,3,3,3,3,3]×[ ]
2	B₇A₁	[4,3,3,3,3,3]×[ ]
3	D₇A₁	[3^4,1,1]×[ ]
4	E₇A₁	[3^3,2,1]×[ ]
6+2
1	A₆I₂(p)	[3,3,3,3,3]×[p]
2	B₆I₂(p)	[4,3,3,3,3]×[p]
3	D₆I₂(p)	[3^3,1,1]×[p]
4	E₆I₂(p)	[3,3,3,3,3]×[p]
6+1+1
1	A₆A₁A₁	[3,3,3,3,3]×[ ]x[ ]
2	B₆A₁A₁	[4,3,3,3,3]×[ ]x[ ]
3	D₆A₁A₁	[3^3,1,1]×[ ]x[ ]
4	E₆A₁A₁	[3,3,3,3,3]×[ ]x[ ]
5+3
1	A₅A₃	[3⁴]×[3,3]
2	B₅A₃	[4,3³]×[3,3]
3	D₅A₃	[3^2,1,1]×[3,3]
4	A₅B₃	[3⁴]×[4,3]
5	B₅B₃	[4,3³]×[4,3]
6	D₅B₃	[3^2,1,1]×[4,3]
7	A₅H₃	[3⁴]×[5,3]
8	B₅H₃	[4,3³]×[5,3]
9	D₅H₃	[3^2,1,1]×[5,3]
5+2+1
1	A₅I₂(p)A₁	[3,3,3]×[p]×[ ]
2	B₅I₂(p)A₁	[4,3,3]×[p]×[ ]
3	D₅I₂(p)A₁	[3^2,1,1]×[p]×[ ]
5+1+1+1
1	A₅A₁A₁A₁	[3,3,3]×[ ]×[ ]×[ ]
2	B₅A₁A₁A₁	[4,3,3]×[ ]×[ ]×[ ]
3	D₅A₁A₁A₁	[3^2,1,1]×[ ]×[ ]×[ ]
4+4
1	A₄A₄	[3,3,3]×[3,3,3]
2	B₄A₄	[4,3,3]×[3,3,3]
3	D₄A₄	[3^1,1,1]×[3,3,3]
4	F₄A₄	[3,4,3]×[3,3,3]
5	H₄A₄	[5,3,3]×[3,3,3]
6	B₄B₄	[4,3,3]×[4,3,3]
7	D₄B₄	[3^1,1,1]×[4,3,3]
8	F₄B₄	[3,4,3]×[4,3,3]
9	H₄B₄	[5,3,3]×[4,3,3]
10	D₄D₄	[3^1,1,1]×[3^1,1,1]
11	F₄D₄	[3,4,3]×[3^1,1,1]
12	H₄D₄	[5,3,3]×[3^1,1,1]
13	F₄×F₄	[3,4,3]×[3,4,3]
14	H₄×F₄	[5,3,3]×[3,4,3]
15	H₄H₄	[5,3,3]×[5,3,3]
4+3+1
1	A₄A₃A₁	[3,3,3]×[3,3]×[ ]
2	A₄B₃A₁	[3,3,3]×[4,3]×[ ]
3	A₄H₃A₁	[3,3,3]×[5,3]×[ ]
4	B₄A₃A₁	[4,3,3]×[3,3]×[ ]
5	B₄B₃A₁	[4,3,3]×[4,3]×[ ]
6	B₄H₃A₁	[4,3,3]×[5,3]×[ ]
7	H₄A₃A₁	[5,3,3]×[3,3]×[ ]
8	H₄B₃A₁	[5,3,3]×[4,3]×[ ]
9	H₄H₃A₁	[5,3,3]×[5,3]×[ ]
10	F₄A₃A₁	[3,4,3]×[3,3]×[ ]
11	F₄B₃A₁	[3,4,3]×[4,3]×[ ]
12	F₄H₃A₁	[3,4,3]×[5,3]×[ ]
13	D₄A₃A₁	[3^1,1,1]×[3,3]×[ ]
14	D₄B₃A₁	[3^1,1,1]×[4,3]×[ ]
15	D₄H₃A₁	[3^1,1,1]×[5,3]×[ ]
4+2+2
...
4+2+1+1
...
4+1+1+1+1
...
3+3+2
1	A₃A₃I₂(p)	[3,3]×[3,3]×[p]
2	B₃A₃I₂(p)	[4,3]×[3,3]×[p]
3	H₃A₃I₂(p)	[5,3]×[3,3]×[p]
4	B₃B₃I₂(p)	[4,3]×[4,3]×[p]
5	H₃B₃I₂(p)	[5,3]×[4,3]×[p]
6	H₃H₃I₂(p)	[5,3]×[5,3]×[p]
3+3+1+1
1	A₃²A₁²	[3,3]×[3,3]×[ ]×[ ]
2	B₃A₃A₁²	[4,3]×[3,3]×[ ]×[ ]
3	H₃A₃A₁²	[5,3]×[3,3]×[ ]×[ ]
4	B₃B₃A₁²	[4,3]×[4,3]×[ ]×[ ]
5	H₃B₃A₁²	[5,3]×[4,3]×[ ]×[ ]
6	H₃H₃A₁²	[5,3]×[5,3]×[ ]×[ ]
3+2+2+1
1	A₃I₂(p)I₂(q)A₁	[3,3]×[p]×[q]×[ ]
2	B₃I₂(p)I₂(q)A₁	[4,3]×[p]×[q]×[ ]
3	H₃I₂(p)I₂(q)A₁	[5,3]×[p]×[q]×[ ]
3+2+1+1+1
1	A₃I₂(p)A₁³	[3,3]×[p]×[ ]x[ ]×[ ]
2	B₃I₂(p)A₁³	[4,3]×[p]×[ ]x[ ]×[ ]
3	H₃I₂(p)A₁³	[5,3]×[p]×[ ]x[ ]×[ ]
3+1+1+1+1+1
1	A₃A₁⁵	[3,3]×[ ]x[ ]×[ ]x[ ]×[ ]
2	B₃A₁⁵	[4,3]×[ ]x[ ]×[ ]x[ ]×[ ]
3	H₃A₁⁵	[5,3]×[ ]x[ ]×[ ]x[ ]×[ ]
2+2+2+2
1	I₂(p)I₂(q)I₂(r)I₂(s)	[p]×[q]×[r]×[s]
2+2+2+1+1
1	I₂(p)I₂(q)I₂(r)A₁²	[p]×[q]×[r]×[ ]×[ ]
2+2+1+1+1+1
2	I₂(p)I₂(q)A₁⁴	[p]×[q]×[ ]×[ ]×[ ]×[ ]
2+1+1+1+1+1+1
1	I₂(p)A₁⁶	[p]×[ ]×[ ]×[ ]×[ ]×[ ]×[ ]
1+1+1+1+1+1+1+1
1	A₁⁸	[ ]×[ ]×[ ]×[ ]×[ ]×[ ]×[ ]×[ ]

The A₈ family

The A₈ family has symmetry of order 362880 (9 factorial).

There are 135 forms based on all permutations of the Coxeter-Dynkin diagrams with one or more rings. (128+8-1 cases) These are all enumerated below. Bowers-style acronym names are given in parentheses for cross-referencing.

See also a list of 8-simplex polytopes for symmetric Coxeter plane graphs of these polytopes.

A₈ uniform polytopes
#	Coxeter-Dynkin diagram	Truncation indices	Johnson name	Basepoint	Element counts
#	Coxeter-Dynkin diagram	Truncation indices	Johnson name	Basepoint	7	6	5	4	3	2	1	0
1		t₀	8-simplex (ene)	(0,0,0,0,0,0,0,0,1)	9	36	84	126	126	84	36	9
2		t₁	Rectified 8-simplex (rene)	(0,0,0,0,0,0,0,1,1)	18	108	336	630	576	588	252	36
3		t₂	Birectified 8-simplex (bene)	(0,0,0,0,0,0,1,1,1)	18	144	588	1386	2016	1764	756	84
4		t₃	Trirectified 8-simplex (trene)	(0,0,0,0,0,1,1,1,1)							1260	126
5		t_0,1	Truncated 8-simplex (tene)	(0,0,0,0,0,0,0,1,2)							288	72
6		t_0,2	Cantellated 8-simplex	(0,0,0,0,0,0,1,1,2)							1764	252
7		t_1,2	Bitruncated 8-simplex	(0,0,0,0,0,0,1,2,2)							1008	252
8		t_0,3	Runcinated 8-simplex	(0,0,0,0,0,1,1,1,2)							4536	504
9		t_1,3	Bicantellated 8-simplex	(0,0,0,0,0,1,1,2,2)							5292	756
10		t_2,3	Tritruncated 8-simplex	(0,0,0,0,0,1,2,2,2)							2016	504
11		t_0,4	Stericated 8-simplex	(0,0,0,0,1,1,1,1,2)							6300	630
12		t_1,4	Biruncinated 8-simplex	(0,0,0,0,1,1,1,2,2)							11340	1260
13		t_2,4	Tricantellated 8-simplex	(0,0,0,0,1,1,2,2,2)							8820	1260
14		t_3,4	Quadritruncated 8-simplex	(0,0,0,0,1,2,2,2,2)							2520	630
15		t_0,5	Pentellated 8-simplex	(0,0,0,1,1,1,1,1,2)							5040	504
16		t_1,5	Bistericated 8-simplex	(0,0,0,1,1,1,1,2,2)							12600	1260
17		t_2,5	Triruncinated 8-simplex	(0,0,0,1,1,1,2,2,2)							15120	1680
18		t_0,6	Hexicated 8-simplex	(0,0,1,1,1,1,1,1,2)							2268	252
19		t_1,6	Bipentellated 8-simplex	(0,0,1,1,1,1,1,2,2)							7560	756
20		t_0,7	Heptellated 8-simplex	(0,1,1,1,1,1,1,1,2)							504	72
21		t_0,1,2	Cantitruncated 8-simplex	(0,0,0,0,0,0,1,2,3)							2016	504
22		t_0,1,3	Runcitruncated 8-simplex	(0,0,0,0,0,1,1,2,3)							9828	1512
23		t_0,2,3	Runcicantellated 8-simplex	(0,0,0,0,0,1,2,2,3)							6804	1512
24		t_1,2,3	Bicantitruncated 8-simplex	(0,0,0,0,0,1,2,3,3)							6048	1512
25		t_0,1,4	Steritruncated 8-simplex	(0,0,0,0,1,1,1,2,3)							20160	2520
26		t_0,2,4	Stericantellated 8-simplex	(0,0,0,0,1,1,2,2,3)							26460	3780
27		t_1,2,4	Biruncitruncated 8-simplex	(0,0,0,0,1,1,2,3,3)							22680	3780
28		t_0,3,4	Steriruncinated 8-simplex	(0,0,0,0,1,2,2,2,3)							12600	2520
29		t_1,3,4	Biruncicantellated 8-simplex	(0,0,0,0,1,2,2,3,3)							18900	3780
30		t_2,3,4	Tricantitruncated 8-simplex	(0,0,0,0,1,2,3,3,3)							10080	2520
31		t_0,1,5	Pentitruncated 8-simplex	(0,0,0,1,1,1,1,2,3)							21420	2520
32		t_0,2,5	Penticantellated 8-simplex	(0,0,0,1,1,1,2,2,3)							42840	5040
33		t_1,2,5	Bisteritruncated 8-simplex	(0,0,0,1,1,1,2,3,3)							35280	5040
34		t_0,3,5	Pentiruncinated 8-simplex	(0,0,0,1,1,2,2,2,3)							37800	5040
35		t_1,3,5	Bistericantellated 8-simplex	(0,0,0,1,1,2,2,3,3)							52920	7560
36		t_2,3,5	Triruncitruncated 8-simplex	(0,0,0,1,1,2,3,3,3)							27720	5040
37		t_0,4,5	Pentistericated 8-simplex	(0,0,0,1,2,2,2,2,3)							13860	2520
38		t_1,4,5	Bisteriruncinated 8-simplex	(0,0,0,1,2,2,2,3,3)							30240	5040
39		t_0,1,6	Hexitruncated 8-simplex	(0,0,1,1,1,1,1,2,3)							12096	1512
40		t_0,2,6	Hexicantellated 8-simplex	(0,0,1,1,1,1,2,2,3)							34020	3780
41		t_1,2,6	Bipentitruncated 8-simplex	(0,0,1,1,1,1,2,3,3)							26460	3780
42		t_0,3,6	Hexiruncinated 8-simplex	(0,0,1,1,1,2,2,2,3)							45360	5040
43		t_1,3,6	Bipenticantellated 8-simplex	(0,0,1,1,1,2,2,3,3)							60480	7560
44		t_0,4,6	Hexistericated 8-simplex	(0,0,1,1,2,2,2,2,3)							30240	3780
45		t_0,5,6	Hexipentellated 8-simplex	(0,0,1,2,2,2,2,2,3)							9072	1512
46		t_0,1,7	Heptitruncated 8-simplex	(0,1,1,1,1,1,1,2,3)							3276	504
47		t_0,2,7	Hepticantellated 8-simplex	(0,1,1,1,1,1,2,2,3)							12852	1512
48		t_0,3,7	Heptiruncinated 8-simplex	(0,1,1,1,1,2,2,2,3)							23940	2520
49		t_0,1,2,3	Runcicantitruncated 8-simplex	(0,0,0,0,0,1,2,3,4)							12096	3024
50		t_0,1,2,4	Stericantitruncated 8-simplex	(0,0,0,0,1,1,2,3,4)							45360	7560
51		t_0,1,3,4	Steriruncitruncated 8-simplex	(0,0,0,0,1,2,2,3,4)							34020	7560
52		t_0,2,3,4	Steriruncicantellated 8-simplex	(0,0,0,0,1,2,3,3,4)							34020	7560
53		t_1,2,3,4	Biruncicantitruncated 8-simplex	(0,0,0,0,1,2,3,4,4)							30240	7560
54		t_0,1,2,5	Penticantitruncated 8-simplex	(0,0,0,1,1,1,2,3,4)							70560	10080
55		t_0,1,3,5	Pentiruncitruncated 8-simplex	(0,0,0,1,1,2,2,3,4)							98280	15120
56		t_0,2,3,5	Pentiruncicantellated 8-simplex	(0,0,0,1,1,2,3,3,4)							90720	15120
57		t_1,2,3,5	Bistericantitruncated 8-simplex	(0,0,0,1,1,2,3,4,4)							83160	15120
58		t_0,1,4,5	Pentisteritruncated 8-simplex	(0,0,0,1,2,2,2,3,4)							50400	10080
59		t_0,2,4,5	Pentistericantellated 8-simplex	(0,0,0,1,2,2,3,3,4)							83160	15120
60		t_1,2,4,5	Bisteriruncitruncated 8-simplex	(0,0,0,1,2,2,3,4,4)							68040	15120
61		t_0,3,4,5	Pentisteriruncinated 8-simplex	(0,0,0,1,2,3,3,3,4)							50400	10080
62		t_1,3,4,5	Bisteriruncicantellated 8-simplex	(0,0,0,1,2,3,3,4,4)							75600	15120
63		t_2,3,4,5	Triruncicantitruncated 8-simplex	(0,0,0,1,2,3,4,4,4)							40320	10080
64		t_0,1,2,6	Hexicantitruncated 8-simplex	(0,0,1,1,1,1,2,3,4)							52920	7560
65		t_0,1,3,6	Hexiruncitruncated 8-simplex	(0,0,1,1,1,2,2,3,4)							113400	15120
66		t_0,2,3,6	Hexiruncicantellated 8-simplex	(0,0,1,1,1,2,3,3,4)							98280	15120
67		t_1,2,3,6	Bipenticantitruncated 8-simplex	(0,0,1,1,1,2,3,4,4)							90720	15120
68		t_0,1,4,6	Hexisteritruncated 8-simplex	(0,0,1,1,2,2,2,3,4)							105840	15120
69		t_0,2,4,6	Hexistericantellated 8-simplex	(0,0,1,1,2,2,3,3,4)							158760	22680
70		t_1,2,4,6	Bipentiruncitruncated 8-simplex	(0,0,1,1,2,2,3,4,4)							136080	22680
71		t_0,3,4,6	Hexisteriruncinated 8-simplex	(0,0,1,1,2,3,3,3,4)							90720	15120
72		t_1,3,4,6	Bipentiruncicantellated 8-simplex	(0,0,1,1,2,3,3,4,4)							136080	22680
73		t_0,1,5,6	Hexipentitruncated 8-simplex	(0,0,1,2,2,2,2,3,4)							41580	7560
74		t_0,2,5,6	Hexipenticantellated 8-simplex	(0,0,1,2,2,2,3,3,4)							98280	15120
75		t_1,2,5,6	Bipentisteritruncated 8-simplex	(0,0,1,2,2,2,3,4,4)							75600	15120
76		t_0,3,5,6	Hexipentiruncinated 8-simplex	(0,0,1,2,2,3,3,3,4)							98280	15120
77		t_0,4,5,6	Hexipentistericated 8-simplex	(0,0,1,2,3,3,3,3,4)							41580	7560
78		t_0,1,2,7	Hepticantitruncated 8-simplex	(0,1,1,1,1,1,2,3,4)							18144	3024
79		t_0,1,3,7	Heptiruncitruncated 8-simplex	(0,1,1,1,1,2,2,3,4)							56700	7560
80		t_0,2,3,7	Heptiruncicantellated 8-simplex	(0,1,1,1,1,2,3,3,4)							45360	7560
81		t_0,1,4,7	Heptisteritruncated 8-simplex	(0,1,1,1,2,2,2,3,4)							80640	10080
82		t_0,2,4,7	Heptistericantellated 8-simplex	(0,1,1,1,2,2,3,3,4)							113400	15120
83		t_0,3,4,7	Heptisteriruncinated 8-simplex	(0,1,1,1,2,3,3,3,4)							60480	10080
84		t_0,1,5,7	Heptipentitruncated 8-simplex	(0,1,1,2,2,2,2,3,4)							56700	7560
85		t_0,2,5,7	Heptipenticantellated 8-simplex	(0,1,1,2,2,2,3,3,4)							120960	15120
86		t_0,1,6,7	Heptihexitruncated 8-simplex	(0,1,2,2,2,2,2,3,4)							18144	3024
87		t_0,1,2,3,4	Steriruncicantitruncated 8-simplex	(0,0,0,0,1,2,3,4,5)							60480	15120
88		t_0,1,2,3,5	Pentiruncicantitruncated 8-simplex	(0,0,0,1,1,2,3,4,5)							166320	30240
89		t_0,1,2,4,5	Pentistericantitruncated 8-simplex	(0,0,0,1,2,2,3,4,5)							136080	30240
90		t_0,1,3,4,5	Pentisteriruncitruncated 8-simplex	(0,0,0,1,2,3,3,4,5)							136080	30240
91		t_0,2,3,4,5	Pentisteriruncicantellated 8-simplex	(0,0,0,1,2,3,4,4,5)							136080	30240
92		t_1,2,3,4,5	Bisteriruncicantitruncated 8-simplex	(0,0,0,1,2,3,4,5,5)							120960	30240
93		t_0,1,2,3,6	Hexiruncicantitruncated 8-simplex	(0,0,1,1,1,2,3,4,5)							181440	30240
94		t_0,1,2,4,6	Hexistericantitruncated 8-simplex	(0,0,1,1,2,2,3,4,5)							272160	45360
95		t_0,1,3,4,6	Hexisteriruncitruncated 8-simplex	(0,0,1,1,2,3,3,4,5)							249480	45360
96		t_0,2,3,4,6	Hexisteriruncicantellated 8-simplex	(0,0,1,1,2,3,4,4,5)							249480	45360
97		t_1,2,3,4,6	Bipentiruncicantitruncated 8-simplex	(0,0,1,1,2,3,4,5,5)							226800	45360
98		t_0,1,2,5,6	Hexipenticantitruncated 8-simplex	(0,0,1,2,2,2,3,4,5)							151200	30240
99		t_0,1,3,5,6	Hexipentiruncitruncated 8-simplex	(0,0,1,2,2,3,3,4,5)							249480	45360
100		t_0,2,3,5,6	Hexipentiruncicantellated 8-simplex	(0,0,1,2,2,3,4,4,5)							226800	45360
101		t_1,2,3,5,6	Bipentistericantitruncated 8-simplex	(0,0,1,2,2,3,4,5,5)							204120	45360
102		t_0,1,4,5,6	Hexipentisteritruncated 8-simplex	(0,0,1,2,3,3,3,4,5)							151200	30240
103		t_0,2,4,5,6	Hexipentistericantellated 8-simplex	(0,0,1,2,3,3,4,4,5)							249480	45360
104		t_0,3,4,5,6	Hexipentisteriruncinated 8-simplex	(0,0,1,2,3,4,4,4,5)							151200	30240
105		t_0,1,2,3,7	Heptiruncicantitruncated 8-simplex	(0,1,1,1,1,2,3,4,5)							83160	15120
106		t_0,1,2,4,7	Heptistericantitruncated 8-simplex	(0,1,1,1,2,2,3,4,5)							196560	30240
107		t_0,1,3,4,7	Heptisteriruncitruncated 8-simplex	(0,1,1,1,2,3,3,4,5)							166320	30240
108		t_0,2,3,4,7	Heptisteriruncicantellated 8-simplex	(0,1,1,1,2,3,4,4,5)							166320	30240
109		t_0,1,2,5,7	Heptipenticantitruncated 8-simplex	(0,1,1,2,2,2,3,4,5)							196560	30240
110		t_0,1,3,5,7	Heptipentiruncitruncated 8-simplex	(0,1,1,2,2,3,3,4,5)							294840	45360
111		t_0,2,3,5,7	Heptipentiruncicantellated 8-simplex	(0,1,1,2,2,3,4,4,5)							272160	45360
112		t_0,1,4,5,7	Heptipentisteritruncated 8-simplex	(0,1,1,2,3,3,3,4,5)							166320	30240
113		t_0,1,2,6,7	Heptihexicantitruncated 8-simplex	(0,1,2,2,2,2,3,4,5)							83160	15120
114		t_0,1,3,6,7	Heptihexiruncitruncated 8-simplex	(0,1,2,2,2,3,3,4,5)							196560	30240
115		t_0,1,2,3,4,5	Pentisteriruncicantitruncated 8-simplex	(0,0,0,1,2,3,4,5,6)							241920	60480
116		t_0,1,2,3,4,6	Hexisteriruncicantitruncated 8-simplex	(0,0,1,1,2,3,4,5,6)							453600	90720
117		t_0,1,2,3,5,6	Hexipentiruncicantitruncated 8-simplex	(0,0,1,2,2,3,4,5,6)							408240	90720
118		t_0,1,2,4,5,6	Hexipentistericantitruncated 8-simplex	(0,0,1,2,3,3,4,5,6)							408240	90720
119		t_0,1,3,4,5,6	Hexipentisteriruncitruncated 8-simplex	(0,0,1,2,3,4,4,5,6)							408240	90720
120		t_0,2,3,4,5,6	Hexipentisteriruncicantellated 8-simplex	(0,0,1,2,3,4,5,5,6)							408240	90720
121		t_1,2,3,4,5,6	Bipentisteriruncicantitruncated 8-simplex	(0,0,1,2,3,4,5,6,6)							362880	90720
122		t_0,1,2,3,4,7	Heptisteriruncicantitruncated 8-simplex	(0,1,1,1,2,3,4,5,6)							302400	60480
123		t_0,1,2,3,5,7	Heptipentiruncicantitruncated 8-simplex	(0,1,1,2,2,3,4,5,6)							498960	90720
124		t_0,1,2,4,5,7	Heptipentistericantitruncated 8-simplex	(0,1,1,2,3,3,4,5,6)							453600	90720
125		t_0,1,3,4,5,7	Heptipentisteriruncitruncated 8-simplex	(0,1,1,2,3,4,4,5,6)							453600	90720
126		t_0,2,3,4,5,7	Heptipentisteriruncicantellated 8-simplex	(0,1,1,2,3,4,5,5,6)							453600	90720
127		t_0,1,2,3,6,7	Heptihexiruncicantitruncated 8-simplex	(0,1,2,2,2,3,4,5,6)							302400	60480
128		t_0,1,2,4,6,7	Heptihexistericantitruncated 8-simplex	(0,1,2,2,3,3,4,5,6)							498960	90720
129		t_0,1,3,4,6,7	Heptihexisteriruncitruncated 8-simplex	(0,1,2,2,3,4,4,5,6)							453600	90720
130		t_0,1,2,5,6,7	Heptihexipenticantitruncated 8-simplex	(0,1,2,3,3,3,4,5,6)							302400	60480
131		t_{0,1,2,3,4,5,6}	Hexipentisteriruncicantitruncated 8-simplex	(0,0,1,2,3,4,5,6,7)							725760	181440
132		t_{0,1,2,3,4,5,7}	Heptipentisteriruncicantitruncated 8-simplex	(0,1,1,2,3,4,5,6,7)							816480	181440
133		t_{0,1,2,3,4,6,7}	Heptihexisteriruncicantitruncated 8-simplex	(0,1,2,2,3,4,5,6,7)							816480	181440
134		t_{0,1,2,3,5,6,7}	Heptihexipentiruncicantitruncated 8-simplex	(0,1,2,3,3,4,5,6,7)							816480	181440
135		t_{0,1,2,3,4,5,6,7}	Omnitruncated 8-simplex	(0,1,2,3,4,5,6,7,8)							1451520	362880

The B₈ family

The B₈ family has symmetry of order 10321920 (8 factorial x 2⁸). There are 255 forms based on all permutations of the Coxeter-Dynkin diagrams with one or more rings.

See also a list of B8 polytopes for symmetric Coxeter plane graphs of these polytopes.

B₈ uniform polytopes
#	Coxeter-Dynkin diagram	Schläfli symbol	Name	Element counts
#	Coxeter-Dynkin diagram	Schläfli symbol	Name	7	6	5	4	3	2	1	0
1		t₀{3⁶,4}	8-orthoplex Diacosipentacontahexazetton (ek)	256	1024	1792	1792	1120	448	112	16
2		t₁{3⁶,4}	Rectified 8-orthoplex Rectified diacosipentacontahexazetton (rek)	272	3072	8960	12544	10080	4928	1344	112
3		t₂{3⁶,4}	Birectified 8-orthoplex Birectified diacosipentacontahexazetton (bark)	272	3184	16128	34048	36960	22400	6720	448
4		t₃{3⁶,4}	Trirectified 8-orthoplex Trirectified diacosipentacontahexazetton (tark)	272	3184	16576	48384	71680	53760	17920	1120
5		t₃{4,3⁶}	Trirectified 8-cube Trirectified octeract (tro)	272	3184	16576	47712	80640	71680	26880	1792
6		t₂{4,3⁶}	Birectified 8-cube Birectified octeract (bro)	272	3184	14784	36960	55552	50176	21504	1792
7		t₁{4,3⁶}	Rectified 8-cube Rectified octeract (recto)	272	2160	7616	15456	19712	16128	7168	1024
8		t₀{4,3⁶}	8-cube Octeract (octo)	16	112	448	1120	1792	1792	1024	256
9		t_0,1{3⁶,4}	Truncated 8-orthoplex Truncated diacosipentacontahexazetton (tek)							1456	224
10		t_0,2{3⁶,4}	Cantellated 8-orthoplex Small rhombated diacosipentacontahexazetton (srek)							14784	1344
11		t_1,2{3⁶,4}	Bitruncated 8-orthoplex Bitruncated diacosipentacontahexazetton (batek)							8064	1344
12		t_0,3{3⁶,4}	Runcinated 8-orthoplex Small prismated diacosipentacontahexazetton (spek)							60480	4480
13		t_1,3{3⁶,4}	Bicantellated 8-orthoplex Small birhombated diacosipentacontahexazetton (sabork)							67200	6720
14		t_2,3{3⁶,4}	Tritruncated 8-orthoplex Tritruncated diacosipentacontahexazetton (tatek)							24640	4480
15		t_0,4{3⁶,4}	Stericated 8-orthoplex Small cellated diacosipentacontahexazetton (scak)							125440	8960
16		t_1,4{3⁶,4}	Biruncinated 8-orthoplex Small biprismated diacosipentacontahexazetton (sabpek)							215040	17920
17		t_2,4{3⁶,4}	Tricantellated 8-orthoplex Small trirhombated diacosipentacontahexazetton (satrek)							161280	17920
18		t_3,4{4,3⁶}	Quadritruncated 8-cube Octeractidiacosipentacontahexazetton (oke)							44800	8960
19		t_0,5{3⁶,4}	Pentellated 8-orthoplex Small terated diacosipentacontahexazetton (setek)							134400	10752
20		t_1,5{3⁶,4}	Bistericated 8-orthoplex Small bicellated diacosipentacontahexazetton (sibcak)							322560	26880
21		t_2,5{4,3⁶}	Triruncinated 8-cube Small triprismato-octeractidiacosipentacontahexazetton (sitpoke)							376320	35840
22		t_2,4{4,3⁶}	Tricantellated 8-cube Small trirhombated octeract (satro)							215040	26880
23		t_2,3{4,3⁶}	Tritruncated 8-cube Tritruncated octeract (tato)							48384	10752
24		t_0,6{3⁶,4}	Hexicated 8-orthoplex Small petated diacosipentacontahexazetton (supek)							64512	7168
25		t_1,6{4,3⁶}	Bipentellated 8-cube Small biteri-octeractidiacosipentacontahexazetton (sabtoke)							215040	21504
26		t_1,5{4,3⁶}	Bistericated 8-cube Small bicellated octeract (sobco)							358400	35840
27		t_1,4{4,3⁶}	Biruncinated 8-cube Small biprismated octeract (sabepo)							322560	35840
28		t_1,3{4,3⁶}	Bicantellated 8-cube Small birhombated octeract (subro)							150528	21504
29		t_1,2{4,3⁶}	Bitruncated 8-cube Bitruncated octeract (bato)							28672	7168
30		t_0,7{4,3⁶}	Heptellated 8-cube Small exi-octeractidiacosipentacontahexazetton (saxoke)							14336	2048
31		t_0,6{4,3⁶}	Hexicated 8-cube Small petated octeract (supo)							64512	7168
32		t_0,5{4,3⁶}	Pentellated 8-cube Small terated octeract (soto)							143360	14336
33		t_0,4{4,3⁶}	Stericated 8-cube Small cellated octeract (soco)							179200	17920
34		t_0,3{4,3⁶}	Runcinated 8-cube Small prismated octeract (sopo)							129024	14336
35		t_0,2{4,3⁶}	Cantellated 8-cube Small rhombated octeract (soro)							50176	7168
36		t_0,1{4,3⁶}	Truncated 8-cube Truncated octeract (tocto)							8192	2048
37		t_0,1,2{3⁶,4}	Cantitruncated 8-orthoplex Great rhombated diacosipentacontahexazetton							16128	2688
38		t_0,1,3{3⁶,4}	Runcitruncated 8-orthoplex Prismatotruncated diacosipentacontahexazetton							127680	13440
39		t_0,2,3{3⁶,4}	Runcicantellated 8-orthoplex Prismatorhombated diacosipentacontahexazetton							80640	13440
40		t_1,2,3{3⁶,4}	Bicantitruncated 8-orthoplex Great birhombated diacosipentacontahexazetton							73920	13440
41		t_0,1,4{3⁶,4}	Steritruncated 8-orthoplex Cellitruncated diacosipentacontahexazetton							394240	35840
42		t_0,2,4{3⁶,4}	Stericantellated 8-orthoplex Cellirhombated diacosipentacontahexazetton							483840	53760
43		t_1,2,4{3⁶,4}	Biruncitruncated 8-orthoplex Biprismatotruncated diacosipentacontahexazetton							430080	53760
44		t_0,3,4{3⁶,4}	Steriruncinated 8-orthoplex Celliprismated diacosipentacontahexazetton							215040	35840
45		t_1,3,4{3⁶,4}	Biruncicantellated 8-orthoplex Biprismatorhombated diacosipentacontahexazetton							322560	53760
46		t_2,3,4{3⁶,4}	Tricantitruncated 8-orthoplex Great trirhombated diacosipentacontahexazetton							179200	35840
47		t_0,1,5{3⁶,4}	Pentitruncated 8-orthoplex Teritruncated diacosipentacontahexazetton							564480	53760
48		t_0,2,5{3⁶,4}	Penticantellated 8-orthoplex Terirhombated diacosipentacontahexazetton							1075200	107520
49		t_1,2,5{3⁶,4}	Bisteritruncated 8-orthoplex Bicellitruncated diacosipentacontahexazetton							913920	107520
50		t_0,3,5{3⁶,4}	Pentiruncinated 8-orthoplex Teriprismated diacosipentacontahexazetton							913920	107520
51		t_1,3,5{3⁶,4}	Bistericantellated 8-orthoplex Bicellirhombated diacosipentacontahexazetton							1290240	161280
52		t_2,3,5{3⁶,4}	Triruncitruncated 8-orthoplex Triprismatotruncated diacosipentacontahexazetton							698880	107520
53		t_0,4,5{3⁶,4}	Pentistericated 8-orthoplex Tericellated diacosipentacontahexazetton							322560	53760
54		t_1,4,5{3⁶,4}	Bisteriruncinated 8-orthoplex Bicelliprismated diacosipentacontahexazetton							698880	107520
55		t_2,3,5{4,3⁶}	Triruncitruncated 8-cube Triprismatotruncated octeract							645120	107520
56		t_2,3,4{4,3⁶}	Tricantitruncated 8-cube Great trirhombated octeract							241920	53760
57		t_0,1,6{3⁶,4}	Hexitruncated 8-orthoplex Petitruncated diacosipentacontahexazetton							344064	43008
58		t_0,2,6{3⁶,4}	Hexicantellated 8-orthoplex Petirhombated diacosipentacontahexazetton							967680	107520
59		t_1,2,6{3⁶,4}	Bipentitruncated 8-orthoplex Biteritruncated diacosipentacontahexazetton							752640	107520
60		t_0,3,6{3⁶,4}	Hexiruncinated 8-orthoplex Petiprismated diacosipentacontahexazetton							1290240	143360
61		t_1,3,6{3⁶,4}	Bipenticantellated 8-orthoplex Biterirhombated diacosipentacontahexazetton							1720320	215040
62		t_1,4,5{4,3⁶}	Bisteriruncinated 8-cube Bicelliprismated octeract							860160	143360
63		t_0,4,6{3⁶,4}	Hexistericated 8-orthoplex Peticellated diacosipentacontahexazetton							860160	107520
64		t_1,3,6{4,3⁶}	Bipenticantellated 8-cube Biterirhombated octeract							1720320	215040
65		t_1,3,5{4,3⁶}	Bistericantellated 8-cube Bicellirhombated octeract							1505280	215040
66		t_1,3,4{4,3⁶}	Biruncicantellated 8-cube Biprismatorhombated octeract							537600	107520
67		t_0,5,6{3⁶,4}	Hexipentellated 8-orthoplex Petiterated diacosipentacontahexazetton							258048	43008
68		t_1,2,6{4,3⁶}	Bipentitruncated 8-cube Biteritruncated octeract							752640	107520
69		t_1,2,5{4,3⁶}	Bisteritruncated 8-cube Bicellitruncated octeract							1003520	143360
70		t_1,2,4{4,3⁶}	Biruncitruncated 8-cube Biprismatotruncated octeract							645120	107520
71		t_1,2,3{4,3⁶}	Bicantitruncated 8-cube Great birhombated octeract							172032	43008
72		t_0,1,7{3⁶,4}	Heptitruncated 8-orthoplex Exitruncated diacosipentacontahexazetton							93184	14336
73		t_0,2,7{3⁶,4}	Hepticantellated 8-orthoplex Exirhombated diacosipentacontahexazetton							365568	43008
74		t_0,5,6{4,3⁶}	Hexipentellated 8-cube Petiterated octeract							258048	43008
75		t_0,3,7{3⁶,4}	Heptiruncinated 8-orthoplex Exiprismated diacosipentacontahexazetton							680960	71680
76		t_0,4,6{4,3⁶}	Hexistericated 8-cube Peticellated octeract							860160	107520
77		t_0,4,5{4,3⁶}	Pentistericated 8-cube Tericellated octeract							394240	71680
78		t_0,3,7{4,3⁶}	Heptiruncinated 8-cube Exiprismated octeract							680960	71680
79		t_0,3,6{4,3⁶}	Hexiruncinated 8-cube Petiprismated octeract							1290240	143360
80		t_0,3,5{4,3⁶}	Pentiruncinated 8-cube Teriprismated octeract							1075200	143360
81		t_0,3,4{4,3⁶}	Steriruncinated 8-cube Celliprismated octeract							358400	71680
82		t_0,2,7{4,3⁶}	Hepticantellated 8-cube Exirhombated octeract							365568	43008
83		t_0,2,6{4,3⁶}	Hexicantellated 8-cube Petirhombated octeract							967680	107520
84		t_0,2,5{4,3⁶}	Penticantellated 8-cube Terirhombated octeract							1218560	143360
85		t_0,2,4{4,3⁶}	Stericantellated 8-cube Cellirhombated octeract							752640	107520
86		t_0,2,3{4,3⁶}	Runcicantellated 8-cube Prismatorhombated octeract							193536	43008
87		t_0,1,7{4,3⁶}	Heptitruncated 8-cube Exitruncated octeract							93184	14336
88		t_0,1,6{4,3⁶}	Hexitruncated 8-cube Petitruncated octeract							344064	43008
89		t_0,1,5{4,3⁶}	Pentitruncated 8-cube Teritruncated octeract							609280	71680
90		t_0,1,4{4,3⁶}	Steritruncated 8-cube Cellitruncated octeract							573440	71680
91		t_0,1,3{4,3⁶}	Runcitruncated 8-cube Prismatotruncated octeract							279552	43008
92		t_0,1,2{4,3⁶}	Cantitruncated 8-cube Great rhombated octeract							57344	14336
93		t_0,1,2,3{3⁶,4}	Runcicantitruncated 8-orthoplex Great prismated diacosipentacontahexazetton							147840	26880
94		t_0,1,2,4{3⁶,4}	Stericantitruncated 8-orthoplex Celligreatorhombated diacosipentacontahexazetton							860160	107520
95		t_0,1,3,4{3⁶,4}	Steriruncitruncated 8-orthoplex Celliprismatotruncated diacosipentacontahexazetton							591360	107520
96		t_0,2,3,4{3⁶,4}	Steriruncicantellated 8-orthoplex Celliprismatorhombated diacosipentacontahexazetton							591360	107520
97		t_1,2,3,4{3⁶,4}	Biruncicantitruncated 8-orthoplex Great biprismated diacosipentacontahexazetton							537600	107520
98		t_0,1,2,5{3⁶,4}	Penticantitruncated 8-orthoplex Terigreatorhombated diacosipentacontahexazetton							1827840	215040
99		t_0,1,3,5{3⁶,4}	Pentiruncitruncated 8-orthoplex Teriprismatotruncated diacosipentacontahexazetton							2419200	322560
100		t_0,2,3,5{3⁶,4}	Pentiruncicantellated 8-orthoplex Teriprismatorhombated diacosipentacontahexazetton							2257920	322560
101		t_1,2,3,5{3⁶,4}	Bistericantitruncated 8-orthoplex Bicelligreatorhombated diacosipentacontahexazetton							2096640	322560
102		t_0,1,4,5{3⁶,4}	Pentisteritruncated 8-orthoplex Tericellitruncated diacosipentacontahexazetton							1182720	215040
103		t_0,2,4,5{3⁶,4}	Pentistericantellated 8-orthoplex Tericellirhombated diacosipentacontahexazetton							1935360	322560
104		t_1,2,4,5{3⁶,4}	Bisteriruncitruncated 8-orthoplex Bicelliprismatotruncated diacosipentacontahexazetton							1612800	322560
105		t_0,3,4,5{3⁶,4}	Pentisteriruncinated 8-orthoplex Tericelliprismated diacosipentacontahexazetton							1182720	215040
106		t_1,3,4,5{3⁶,4}	Bisteriruncicantellated 8-orthoplex Bicelliprismatorhombated diacosipentacontahexazetton							1774080	322560
107		t_2,3,4,5{4,3⁶}	Triruncicantitruncated 8-cube Great triprismato-octeractidiacosipentacontahexazetton							967680	215040
108		t_0,1,2,6{3⁶,4}	Hexicantitruncated 8-orthoplex Petigreatorhombated diacosipentacontahexazetton							1505280	215040
109		t_0,1,3,6{3⁶,4}	Hexiruncitruncated 8-orthoplex Petiprismatotruncated diacosipentacontahexazetton							3225600	430080
110		t_0,2,3,6{3⁶,4}	Hexiruncicantellated 8-orthoplex Petiprismatorhombated diacosipentacontahexazetton							2795520	430080
111		t_1,2,3,6{3⁶,4}	Bipenticantitruncated 8-orthoplex Biterigreatorhombated diacosipentacontahexazetton							2580480	430080
112		t_0,1,4,6{3⁶,4}	Hexisteritruncated 8-orthoplex Peticellitruncated diacosipentacontahexazetton							3010560	430080
113		t_0,2,4,6{3⁶,4}	Hexistericantellated 8-orthoplex Peticellirhombated diacosipentacontahexazetton							4515840	645120
114		t_1,2,4,6{3⁶,4}	Bipentiruncitruncated 8-orthoplex Biteriprismatotruncated diacosipentacontahexazetton							3870720	645120
115		t_0,3,4,6{3⁶,4}	Hexisteriruncinated 8-orthoplex Peticelliprismated diacosipentacontahexazetton							2580480	430080
116		t_1,3,4,6{4,3⁶}	Bipentiruncicantellated 8-cube Biteriprismatorhombi-octeractidiacosipentacontahexazetton							3870720	645120
117		t_1,3,4,5{4,3⁶}	Bisteriruncicantellated 8-cube Bicelliprismatorhombated octeract							2150400	430080
118		t_0,1,5,6{3⁶,4}	Hexipentitruncated 8-orthoplex Petiteritruncated diacosipentacontahexazetton							1182720	215040
119		t_0,2,5,6{3⁶,4}	Hexipenticantellated 8-orthoplex Petiterirhombated diacosipentacontahexazetton							2795520	430080
120		t_1,2,5,6{4,3⁶}	Bipentisteritruncated 8-cube Bitericellitrunki-octeractidiacosipentacontahexazetton							2150400	430080
121		t_0,3,5,6{3⁶,4}	Hexipentiruncinated 8-orthoplex Petiteriprismated diacosipentacontahexazetton							2795520	430080
122		t_1,2,4,6{4,3⁶}	Bipentiruncitruncated 8-cube Biteriprismatotruncated octeract							3870720	645120
123		t_1,2,4,5{4,3⁶}	Bisteriruncitruncated 8-cube Bicelliprismatotruncated octeract							1935360	430080
124		t_0,4,5,6{3⁶,4}	Hexipentistericated 8-orthoplex Petitericellated diacosipentacontahexazetton							1182720	215040
125		t_1,2,3,6{4,3⁶}	Bipenticantitruncated 8-cube Biterigreatorhombated octeract							2580480	430080
126		t_1,2,3,5{4,3⁶}	Bistericantitruncated 8-cube Bicelligreatorhombated octeract							2365440	430080
127		t_1,2,3,4{4,3⁶}	Biruncicantitruncated 8-cube Great biprismated octeract							860160	215040
128		t_0,1,2,7{3⁶,4}	Hepticantitruncated 8-orthoplex Exigreatorhombated diacosipentacontahexazetton							516096	86016
129		t_0,1,3,7{3⁶,4}	Heptiruncitruncated 8-orthoplex Exiprismatotruncated diacosipentacontahexazetton							1612800	215040
130		t_0,2,3,7{3⁶,4}	Heptiruncicantellated 8-orthoplex Exiprismatorhombated diacosipentacontahexazetton							1290240	215040
131		t_0,4,5,6{4,3⁶}	Hexipentistericated 8-cube Petitericellated octeract							1182720	215040
132		t_0,1,4,7{3⁶,4}	Heptisteritruncated 8-orthoplex Exicellitruncated diacosipentacontahexazetton							2293760	286720
133		t_0,2,4,7{3⁶,4}	Heptistericantellated 8-orthoplex Exicellirhombated diacosipentacontahexazetton							3225600	430080
134		t_0,3,5,6{4,3⁶}	Hexipentiruncinated 8-cube Petiteriprismated octeract							2795520	430080
135		t_0,3,4,7{4,3⁶}	Heptisteriruncinated 8-cube Exicelliprismato-octeractidiacosipentacontahexazetton							1720320	286720
136		t_0,3,4,6{4,3⁶}	Hexisteriruncinated 8-cube Peticelliprismated octeract							2580480	430080
137		t_0,3,4,5{4,3⁶}	Pentisteriruncinated 8-cube Tericelliprismated octeract							1433600	286720
138		t_0,1,5,7{3⁶,4}	Heptipentitruncated 8-orthoplex Exiteritruncated diacosipentacontahexazetton							1612800	215040
139		t_0,2,5,7{4,3⁶}	Heptipenticantellated 8-cube Exiterirhombi-octeractidiacosipentacontahexazetton							3440640	430080
140		t_0,2,5,6{4,3⁶}	Hexipenticantellated 8-cube Petiterirhombated octeract							2795520	430080
141		t_0,2,4,7{4,3⁶}	Heptistericantellated 8-cube Exicellirhombated octeract							3225600	430080
142		t_0,2,4,6{4,3⁶}	Hexistericantellated 8-cube Peticellirhombated octeract							4515840	645120
143		t_0,2,4,5{4,3⁶}	Pentistericantellated 8-cube Tericellirhombated octeract							2365440	430080
144		t_0,2,3,7{4,3⁶}	Heptiruncicantellated 8-cube Exiprismatorhombated octeract							1290240	215040
145		t_0,2,3,6{4,3⁶}	Hexiruncicantellated 8-cube Petiprismatorhombated octeract							2795520	430080
146		t_0,2,3,5{4,3⁶}	Pentiruncicantellated 8-cube Teriprismatorhombated octeract							2580480	430080
147		t_0,2,3,4{4,3⁶}	Steriruncicantellated 8-cube Celliprismatorhombated octeract							967680	215040
148		t_0,1,6,7{4,3⁶}	Heptihexitruncated 8-cube Exipetitrunki-octeractidiacosipentacontahexazetton							516096	86016
149		t_0,1,5,7{4,3⁶}	Heptipentitruncated 8-cube Exiteritruncated octeract							1612800	215040
150		t_0,1,5,6{4,3⁶}	Hexipentitruncated 8-cube Petiteritruncated octeract							1182720	215040
151		t_0,1,4,7{4,3⁶}	Heptisteritruncated 8-cube Exicellitruncated octeract							2293760	286720
152		t_0,1,4,6{4,3⁶}	Hexisteritruncated 8-cube Peticellitruncated octeract							3010560	430080
153		t_0,1,4,5{4,3⁶}	Pentisteritruncated 8-cube Tericellitruncated octeract							1433600	286720
154		t_0,1,3,7{4,3⁶}	Heptiruncitruncated 8-cube Exiprismatotruncated octeract							1612800	215040
155		t_0,1,3,6{4,3⁶}	Hexiruncitruncated 8-cube Petiprismatotruncated octeract							3225600	430080
156		t_0,1,3,5{4,3⁶}	Pentiruncitruncated 8-cube Teriprismatotruncated octeract							2795520	430080
157		t_0,1,3,4{4,3⁶}	Steriruncitruncated 8-cube Celliprismatotruncated octeract							967680	215040
158		t_0,1,2,7{4,3⁶}	Hepticantitruncated 8-cube Exigreatorhombated octeract							516096	86016
159		t_0,1,2,6{4,3⁶}	Hexicantitruncated 8-cube Petigreatorhombated octeract							1505280	215040
160		t_0,1,2,5{4,3⁶}	Penticantitruncated 8-cube Terigreatorhombated octeract							2007040	286720
161		t_0,1,2,4{4,3⁶}	Stericantitruncated 8-cube Celligreatorhombated octeract							1290240	215040
162		t_0,1,2,3{4,3⁶}	Runcicantitruncated 8-cube Great prismated octeract							344064	86016
163		t_0,1,2,3,4{3⁶,4}	Steriruncicantitruncated 8-orthoplex Great cellated diacosipentacontahexazetton							1075200	215040
164		t_0,1,2,3,5{3⁶,4}	Pentiruncicantitruncated 8-orthoplex Terigreatoprismated diacosipentacontahexazetton							4193280	645120
165		t_0,1,2,4,5{3⁶,4}	Pentistericantitruncated 8-orthoplex Tericelligreatorhombated diacosipentacontahexazetton							3225600	645120
166		t_0,1,3,4,5{3⁶,4}	Pentisteriruncitruncated 8-orthoplex Tericelliprismatotruncated diacosipentacontahexazetton							3225600	645120
167		t_0,2,3,4,5{3⁶,4}	Pentisteriruncicantellated 8-orthoplex Tericelliprismatorhombated diacosipentacontahexazetton							3225600	645120
168		t_1,2,3,4,5{3⁶,4}	Bisteriruncicantitruncated 8-orthoplex Great bicellated diacosipentacontahexazetton							2903040	645120
169		t_0,1,2,3,6{3⁶,4}	Hexiruncicantitruncated 8-orthoplex Petigreatoprismated diacosipentacontahexazetton							5160960	860160
170		t_0,1,2,4,6{3⁶,4}	Hexistericantitruncated 8-orthoplex Peticelligreatorhombated diacosipentacontahexazetton							7741440	1290240
171		t_0,1,3,4,6{3⁶,4}	Hexisteriruncitruncated 8-orthoplex Peticelliprismatotruncated diacosipentacontahexazetton							7096320	1290240
172		t_0,2,3,4,6{3⁶,4}	Hexisteriruncicantellated 8-orthoplex Peticelliprismatorhombated diacosipentacontahexazetton							7096320	1290240
173		t_1,2,3,4,6{3⁶,4}	Bipentiruncicantitruncated 8-orthoplex Biterigreatoprismated diacosipentacontahexazetton							6451200	1290240
174		t_0,1,2,5,6{3⁶,4}	Hexipenticantitruncated 8-orthoplex Petiterigreatorhombated diacosipentacontahexazetton							4300800	860160
175		t_0,1,3,5,6{3⁶,4}	Hexipentiruncitruncated 8-orthoplex Petiteriprismatotruncated diacosipentacontahexazetton							7096320	1290240
176		t_0,2,3,5,6{3⁶,4}	Hexipentiruncicantellated 8-orthoplex Petiteriprismatorhombated diacosipentacontahexazetton							6451200	1290240
177		t_1,2,3,5,6{3⁶,4}	Bipentistericantitruncated 8-orthoplex Bitericelligreatorhombated diacosipentacontahexazetton							5806080	1290240
178		t_0,1,4,5,6{3⁶,4}	Hexipentisteritruncated 8-orthoplex Petitericellitruncated diacosipentacontahexazetton							4300800	860160
179		t_0,2,4,5,6{3⁶,4}	Hexipentistericantellated 8-orthoplex Petitericellirhombated diacosipentacontahexazetton							7096320	1290240
180		t_1,2,3,5,6{4,3⁶}	Bipentistericantitruncated 8-cube Bitericelligreatorhombated octeract							5806080	1290240
181		t_0,3,4,5,6{3⁶,4}	Hexipentisteriruncinated 8-orthoplex Petitericelliprismated diacosipentacontahexazetton							4300800	860160
182		t_1,2,3,4,6{4,3⁶}	Bipentiruncicantitruncated 8-cube Biterigreatoprismated octeract							6451200	1290240
183		t_1,2,3,4,5{4,3⁶}	Bisteriruncicantitruncated 8-cube Great bicellated octeract							3440640	860160
184		t_0,1,2,3,7{3⁶,4}	Heptiruncicantitruncated 8-orthoplex Exigreatoprismated diacosipentacontahexazetton							2365440	430080
185		t_0,1,2,4,7{3⁶,4}	Heptistericantitruncated 8-orthoplex Exicelligreatorhombated diacosipentacontahexazetton							5591040	860160
186		t_0,1,3,4,7{3⁶,4}	Heptisteriruncitruncated 8-orthoplex Exicelliprismatotruncated diacosipentacontahexazetton							4730880	860160
187		t_0,2,3,4,7{3⁶,4}	Heptisteriruncicantellated 8-orthoplex Exicelliprismatorhombated diacosipentacontahexazetton							4730880	860160
188		t_0,3,4,5,6{4,3⁶}	Hexipentisteriruncinated 8-cube Petitericelliprismated octeract							4300800	860160
189		t_0,1,2,5,7{3⁶,4}	Heptipenticantitruncated 8-orthoplex Exiterigreatorhombated diacosipentacontahexazetton							5591040	860160
190		t_0,1,3,5,7{3⁶,4}	Heptipentiruncitruncated 8-orthoplex Exiteriprismatotruncated diacosipentacontahexazetton							8386560	1290240
191		t_0,2,3,5,7{3⁶,4}	Heptipentiruncicantellated 8-orthoplex Exiteriprismatorhombated diacosipentacontahexazetton							7741440	1290240
192		t_0,2,4,5,6{4,3⁶}	Hexipentistericantellated 8-cube Petitericellirhombated octeract							7096320	1290240
193		t_0,1,4,5,7{3⁶,4}	Heptipentisteritruncated 8-orthoplex Exitericellitruncated diacosipentacontahexazetton							4730880	860160
194		t_0,2,3,5,7{4,3⁶}	Heptipentiruncicantellated 8-cube Exiteriprismatorhombated octeract							7741440	1290240
195		t_0,2,3,5,6{4,3⁶}	Hexipentiruncicantellated 8-cube Petiteriprismatorhombated octeract							6451200	1290240
196		t_0,2,3,4,7{4,3⁶}	Heptisteriruncicantellated 8-cube Exicelliprismatorhombated octeract							4730880	860160
197		t_0,2,3,4,6{4,3⁶}	Hexisteriruncicantellated 8-cube Peticelliprismatorhombated octeract							7096320	1290240
198		t_0,2,3,4,5{4,3⁶}	Pentisteriruncicantellated 8-cube Tericelliprismatorhombated octeract							3870720	860160
199		t_0,1,2,6,7{3⁶,4}	Heptihexicantitruncated 8-orthoplex Exipetigreatorhombated diacosipentacontahexazetton							2365440	430080
200		t_0,1,3,6,7{3⁶,4}	Heptihexiruncitruncated 8-orthoplex Exipetiprismatotruncated diacosipentacontahexazetton							5591040	860160
201		t_0,1,4,5,7{4,3⁶}	Heptipentisteritruncated 8-cube Exitericellitruncated octeract							4730880	860160
202		t_0,1,4,5,6{4,3⁶}	Hexipentisteritruncated 8-cube Petitericellitruncated octeract							4300800	860160
203		t_0,1,3,6,7{4,3⁶}	Heptihexiruncitruncated 8-cube Exipetiprismatotruncated octeract							5591040	860160
204		t_0,1,3,5,7{4,3⁶}	Heptipentiruncitruncated 8-cube Exiteriprismatotruncated octeract							8386560	1290240
205		t_0,1,3,5,6{4,3⁶}	Hexipentiruncitruncated 8-cube Petiteriprismatotruncated octeract							7096320	1290240
206		t_0,1,3,4,7{4,3⁶}	Heptisteriruncitruncated 8-cube Exicelliprismatotruncated octeract							4730880	860160
207		t_0,1,3,4,6{4,3⁶}	Hexisteriruncitruncated 8-cube Peticelliprismatotruncated octeract							7096320	1290240
208		t_0,1,3,4,5{4,3⁶}	Pentisteriruncitruncated 8-cube Tericelliprismatotruncated octeract							3870720	860160
209		t_0,1,2,6,7{4,3⁶}	Heptihexicantitruncated 8-cube Exipetigreatorhombated octeract							2365440	430080
210		t_0,1,2,5,7{4,3⁶}	Heptipenticantitruncated 8-cube Exiterigreatorhombated octeract							5591040	860160
211		t_0,1,2,5,6{4,3⁶}	Hexipenticantitruncated 8-cube Petiterigreatorhombated octeract							4300800	860160
212		t_0,1,2,4,7{4,3⁶}	Heptistericantitruncated 8-cube Exicelligreatorhombated octeract							5591040	860160
213		t_0,1,2,4,6{4,3⁶}	Hexistericantitruncated 8-cube Peticelligreatorhombated octeract							7741440	1290240
214		t_0,1,2,4,5{4,3⁶}	Pentistericantitruncated 8-cube Tericelligreatorhombated octeract							3870720	860160
215		t_0,1,2,3,7{4,3⁶}	Heptiruncicantitruncated 8-cube Exigreatoprismated octeract							2365440	430080
216		t_0,1,2,3,6{4,3⁶}	Hexiruncicantitruncated 8-cube Petigreatoprismated octeract							5160960	860160
217		t_0,1,2,3,5{4,3⁶}	Pentiruncicantitruncated 8-cube Terigreatoprismated octeract							4730880	860160
218		t_0,1,2,3,4{4,3⁶}	Steriruncicantitruncated 8-cube Great cellated octeract							1720320	430080
219		t_0,1,2,3,4,5{3⁶,4}	Pentisteriruncicantitruncated 8-orthoplex Great terated diacosipentacontahexazetton							5806080	1290240
220		t_0,1,2,3,4,6{3⁶,4}	Hexisteriruncicantitruncated 8-orthoplex Petigreatocellated diacosipentacontahexazetton							12902400	2580480
221		t_0,1,2,3,5,6{3⁶,4}	Hexipentiruncicantitruncated 8-orthoplex Petiterigreatoprismated diacosipentacontahexazetton							11612160	2580480
222		t_0,1,2,4,5,6{3⁶,4}	Hexipentistericantitruncated 8-orthoplex Petitericelligreatorhombated diacosipentacontahexazetton							11612160	2580480
223		t_0,1,3,4,5,6{3⁶,4}	Hexipentisteriruncitruncated 8-orthoplex Petitericelliprismatotruncated diacosipentacontahexazetton							11612160	2580480
224		t_0,2,3,4,5,6{3⁶,4}	Hexipentisteriruncicantellated 8-orthoplex Petitericelliprismatorhombated diacosipentacontahexazetton							11612160	2580480
225		t_1,2,3,4,5,6{4,3⁶}	Bipentisteriruncicantitruncated 8-cube Great biteri-octeractidiacosipentacontahexazetton							10321920	2580480
226		t_0,1,2,3,4,7{3⁶,4}	Heptisteriruncicantitruncated 8-orthoplex Exigreatocellated diacosipentacontahexazetton							8601600	1720320
227		t_0,1,2,3,5,7{3⁶,4}	Heptipentiruncicantitruncated 8-orthoplex Exiterigreatoprismated diacosipentacontahexazetton							14192640	2580480
228		t_0,1,2,4,5,7{3⁶,4}	Heptipentistericantitruncated 8-orthoplex Exitericelligreatorhombated diacosipentacontahexazetton							12902400	2580480
229		t_0,1,3,4,5,7{3⁶,4}	Heptipentisteriruncitruncated 8-orthoplex Exitericelliprismatotruncated diacosipentacontahexazetton							12902400	2580480
230		t_0,2,3,4,5,7{4,3⁶}	Heptipentisteriruncicantellated 8-cube Exitericelliprismatorhombi-octeractidiacosipentacontahexazetton							12902400	2580480
231		t_0,2,3,4,5,6{4,3⁶}	Hexipentisteriruncicantellated 8-cube Petitericelliprismatorhombated octeract							11612160	2580480
232		t_0,1,2,3,6,7{3⁶,4}	Heptihexiruncicantitruncated 8-orthoplex Exipetigreatoprismated diacosipentacontahexazetton							8601600	1720320
233		t_0,1,2,4,6,7{3⁶,4}	Heptihexistericantitruncated 8-orthoplex Exipeticelligreatorhombated diacosipentacontahexazetton							14192640	2580480
234		t_0,1,3,4,6,7{4,3⁶}	Heptihexisteriruncitruncated 8-cube Exipeticelliprismatotrunki-octeractidiacosipentacontahexazetton							12902400	2580480
235		t_0,1,3,4,5,7{4,3⁶}	Heptipentisteriruncitruncated 8-cube Exitericelliprismatotruncated octeract							12902400	2580480
236		t_0,1,3,4,5,6{4,3⁶}	Hexipentisteriruncitruncated 8-cube Petitericelliprismatotruncated octeract							11612160	2580480
237		t_0,1,2,5,6,7{4,3⁶}	Heptihexipenticantitruncated 8-cube Exipetiterigreatorhombi-octeractidiacosipentacontahexazetton							8601600	1720320
238		t_0,1,2,4,6,7{4,3⁶}	Heptihexistericantitruncated 8-cube Exipeticelligreatorhombated octeract							14192640	2580480
239		t_0,1,2,4,5,7{4,3⁶}	Heptipentistericantitruncated 8-cube Exitericelligreatorhombated octeract							12902400	2580480
240		t_0,1,2,4,5,6{4,3⁶}	Hexipentistericantitruncated 8-cube Petitericelligreatorhombated octeract							11612160	2580480
241		t_0,1,2,3,6,7{4,3⁶}	Heptihexiruncicantitruncated 8-cube Exipetigreatoprismated octeract							8601600	1720320
242		t_0,1,2,3,5,7{4,3⁶}	Heptipentiruncicantitruncated 8-cube Exiterigreatoprismated octeract							14192640	2580480
243		t_0,1,2,3,5,6{4,3⁶}	Hexipentiruncicantitruncated 8-cube Petiterigreatoprismated octeract							11612160	2580480
244		t_0,1,2,3,4,7{4,3⁶}	Heptisteriruncicantitruncated 8-cube Exigreatocellated octeract							8601600	1720320
245		t_0,1,2,3,4,6{4,3⁶}	Hexisteriruncicantitruncated 8-cube Petigreatocellated octeract							12902400	2580480
246		t_0,1,2,3,4,5{4,3⁶}	Pentisteriruncicantitruncated 8-cube Great terated octeract							6881280	1720320
247		t_{0,1,2,3,4,5,6}{3⁶,4}	Hexipentisteriruncicantitruncated 8-orthoplex Great petated diacosipentacontahexazetton							20643840	5160960
248		t_{0,1,2,3,4,5,7}{3⁶,4}	Heptipentisteriruncicantitruncated 8-orthoplex Exigreatoterated diacosipentacontahexazetton							23224320	5160960
249		t_{0,1,2,3,4,6,7}{3⁶,4}	Heptihexisteriruncicantitruncated 8-orthoplex Exipetigreatocellated diacosipentacontahexazetton							23224320	5160960
250		t_{0,1,2,3,5,6,7}{3⁶,4}	Heptihexipentiruncicantitruncated 8-orthoplex Exipetiterigreatoprismated diacosipentacontahexazetton							23224320	5160960
251		t_{0,1,2,3,5,6,7}{4,3⁶}	Heptihexipentiruncicantitruncated 8-cube Exipetiterigreatoprismated octeract							23224320	5160960
252		t_{0,1,2,3,4,6,7}{4,3⁶}	Heptihexisteriruncicantitruncated 8-cube Exipetigreatocellated octeract							23224320	5160960
253		t_{0,1,2,3,4,5,7}{4,3⁶}	Heptipentisteriruncicantitruncated 8-cube Exigreatoterated octeract							23224320	5160960
254		t_{0,1,2,3,4,5,6}{4,3⁶}	Hexipentisteriruncicantitruncated 8-cube Great petated octeract							20643840	5160960
255		t_{0,1,2,3,4,5,6,7}{4,3⁶}	Omnitruncated 8-cube Great exi-octeractidiacosipentacontahexazetton							41287680	10321920

The D₈ family

The D₈ family has symmetry of order 5,160,960 (8 factorial x 2⁷).

This family has 191 Wythoffian uniform polytopes, from 3x64-1 permutations of the D₈ Coxeter-Dynkin diagram with one or more rings. 127 (2x64-1) are repeated from the B₈ family and 64 are unique to this family, all listed below.

See list of D8 polytopes for Coxeter plane graphs of these polytopes.

D₈ uniform polytopes
#	Coxeter-Dynkin diagram	Name	Base point (Alternately signed)	Element counts								Circumrad
#	Coxeter-Dynkin diagram	Name	Base point (Alternately signed)	7	6	5	4	3	2	1	0	Circumrad
1	=	8-demicube h{4,3,3,3,3,3,3}	(1,1,1,1,1,1,1,1)	144	1136	4032	8288	10752	7168	1792	128	1.0000000
2	=	cantic 8-cube h₂{4,3,3,3,3,3,3}	(1,1,3,3,3,3,3,3)							23296	3584	2.6457512
3	=	runcic 8-cube h₃{4,3,3,3,3,3,3}	(1,1,1,3,3,3,3,3)							64512	7168	2.4494896
4	=	steric 8-cube h₄{4,3,3,3,3,3,3}	(1,1,1,1,3,3,3,3)							98560	8960	2.2360678
5	=	pentic 8-cube h₅{4,3,3,3,3,3,3}	(1,1,1,1,1,3,3,3)							89600	7168	1.9999999
6	=	hexic 8-cube h₆{4,3,3,3,3,3,3}	(1,1,1,1,1,1,3,3)							48384	3584	1.7320508
7	=	heptic 8-cube h₇{4,3,3,3,3,3,3}	(1,1,1,1,1,1,1,3)							14336	1024	1.4142135
8	=	runcicantic 8-cube h_2,3{4,3,3,3,3,3,3}	(1,1,3,5,5,5,5,5)							86016	21504	4.1231055
9	=	stericantic 8-cube h_2,4{4,3,3,3,3,3,3}	(1,1,3,3,5,5,5,5)							349440	53760	3.8729835
10	=	steriruncic 8-cube h_3,4{4,3,3,3,3,3,3}	(1,1,1,3,5,5,5,5)							179200	35840	3.7416575
11	=	penticantic 8-cube h_2,5{4,3,3,3,3,3,3}	(1,1,3,3,3,5,5,5)							573440	71680	3.6055512
12	=	pentiruncic 8-cube h_3,5{4,3,3,3,3,3,3}	(1,1,1,3,3,5,5,5)							537600	71680	3.4641016
13	=	pentisteric 8-cube h_4,5{4,3,3,3,3,3,3}	(1,1,1,1,3,5,5,5)							232960	35840	3.3166249
14	=	hexicantic 8-cube h_2,6{4,3,3,3,3,3,3}	(1,1,3,3,3,3,5,5)							456960	53760	3.3166249
15	=	hexicruncic 8-cube h_3,6{4,3,3,3,3,3,3}	(1,1,1,3,3,3,5,5)							645120	71680	3.1622777
16	=	hexisteric 8-cube h_4,6{4,3,3,3,3,3,3}	(1,1,1,1,3,3,5,5)							483840	53760	3
17	=	hexipentic 8-cube h_5,6{4,3,3,3,3,3,3}	(1,1,1,1,1,3,5,5)							182784	21504	2.8284271
18	=	hepticantic 8-cube h_2,7{4,3,3,3,3,3,3}	(1,1,3,3,3,3,3,5)							172032	21504	3
19	=	heptiruncic 8-cube h_3,7{4,3,3,3,3,3,3}	(1,1,1,3,3,3,3,5)							340480	35840	2.8284271
20	=	heptsteric 8-cube h_4,7{4,3,3,3,3,3,3}	(1,1,1,1,3,3,3,5)							376320	35840	2.6457512
21	=	heptipentic 8-cube h_5,7{4,3,3,3,3,3,3}	(1,1,1,1,1,3,3,5)							236544	21504	2.4494898
22	=	heptihexic 8-cube h_6,7{4,3,3,3,3,3,3}	(1,1,1,1,1,1,3,5)							78848	7168	2.236068
23	=	steriruncicantic 8-cube h_2,3,4{4,3⁶}	(1,1,3,5,7,7,7,7)							430080	107520	5.3851647
24	=	pentiruncicantic 8-cube h_2,3,5{4,3⁶}	(1,1,3,5,5,7,7,7)							1182720	215040	5.0990195
25	=	pentistericantic 8-cube h_2,4,5{4,3⁶}	(1,1,3,3,5,7,7,7)							1075200	215040	4.8989797
26	=	pentisterirunic 8-cube h_3,4,5{4,3⁶}	(1,1,1,3,5,7,7,7)							716800	143360	4.7958317
27	=	hexiruncicantic 8-cube h_2,3,6{4,3⁶}	(1,1,3,5,5,5,7,7)							1290240	215040	4.7958317
28	=	hexistericantic 8-cube h_2,4,6{4,3⁶}	(1,1,3,3,5,5,7,7)							2096640	322560	4.5825758
29	=	hexisterirunic 8-cube h_3,4,6{4,3⁶}	(1,1,1,3,5,5,7,7)							1290240	215040	4.472136
30	=	hexipenticantic 8-cube h_2,5,6{4,3⁶}	(1,1,3,3,3,5,7,7)							1290240	215040	4.3588991
31	=	hexipentirunic 8-cube h_3,5,6{4,3⁶}	(1,1,1,3,3,5,7,7)							1397760	215040	4.2426405
32	=	hexipentisteric 8-cube h_4,5,6{4,3⁶}	(1,1,1,1,3,5,7,7)							698880	107520	4.1231055
33	=	heptiruncicantic 8-cube h_2,3,7{4,3⁶}	(1,1,3,5,5,5,5,7)							591360	107520	4.472136
34	=	heptistericantic 8-cube h_2,4,7{4,3⁶}	(1,1,3,3,5,5,5,7)							1505280	215040	4.2426405
35	=	heptisterruncic 8-cube h_3,4,7{4,3⁶}	(1,1,1,3,5,5,5,7)							860160	143360	4.1231055
36	=	heptipenticantic 8-cube h_2,5,7{4,3⁶}	(1,1,3,3,3,5,5,7)							1612800	215040	4
37	=	heptipentiruncic 8-cube h_3,5,7{4,3⁶}	(1,1,1,3,3,5,5,7)							1612800	215040	3.8729835
38	=	heptipentisteric 8-cube h_4,5,7{4,3⁶}	(1,1,1,1,3,5,5,7)							752640	107520	3.7416575
39	=	heptihexicantic 8-cube h_2,6,7{4,3⁶}	(1,1,3,3,3,3,5,7)							752640	107520	3.7416575
40	=	heptihexiruncic 8-cube h_3,6,7{4,3⁶}	(1,1,1,3,3,3,5,7)							1146880	143360	3.6055512
41	=	heptihexisteric 8-cube h_4,6,7{4,3⁶}	(1,1,1,1,3,3,5,7)							913920	107520	3.4641016
42	=	heptihexipentic 8-cube h_5,6,7{4,3⁶}	(1,1,1,1,1,3,5,7)							365568	43008	3.3166249
43	=	pentisteriruncicantic 8-cube h_2,3,4,5{4,3⁶}	(1,1,3,5,7,9,9,9)							1720320	430080	6.4031243
44	=	hexisteriruncicantic 8-cube h_2,3,4,6{4,3⁶}	(1,1,3,5,7,7,9,9)							3225600	645120	6.0827627
45	=	hexipentiruncicantic 8-cube h_2,3,5,6{4,3⁶}	(1,1,3,5,5,7,9,9)							2903040	645120	5.8309517
46	=	hexipentistericantic 8-cube h_2,4,5,6{4,3⁶}	(1,1,3,3,5,7,9,9)							3225600	645120	5.6568542
47	=	hexipentisteriruncic 8-cube h_3,4,5,6{4,3⁶}	(1,1,1,3,5,7,9,9)							2150400	430080	5.5677648
48	=	heptsteriruncicantic 8-cube h_2,3,4,7{4,3⁶}	(1,1,3,5,7,7,7,9)							2150400	430080	5.7445626
49	=	heptipentiruncicantic 8-cube h_2,3,5,7{4,3⁶}	(1,1,3,5,5,7,7,9)							3548160	645120	5.4772258
50	=	heptipentistericantic 8-cube h_2,4,5,7{4,3⁶}	(1,1,3,3,5,7,7,9)							3548160	645120	5.291503
51	=	heptipentisteriruncic 8-cube h_3,4,5,7{4,3⁶}	(1,1,1,3,5,7,7,9)							2365440	430080	5.1961527
52	=	heptihexiruncicantic 8-cube h_2,3,6,7{4,3⁶}	(1,1,3,5,5,5,7,9)							2150400	430080	5.1961527
53	=	heptihexistericantic 8-cube h_2,4,6,7{4,3⁶}	(1,1,3,3,5,5,7,9)							3870720	645120	5
54	=	heptihexisteriruncic 8-cube h_3,4,6,7{4,3⁶}	(1,1,1,3,5,5,7,9)							2365440	430080	4.8989797
55	=	heptihexipenticantic 8-cube h_2,5,6,7{4,3⁶}	(1,1,3,3,3,5,7,9)							2580480	430080	4.7958317
56	=	heptihexipentiruncic 8-cube h_3,5,6,7{4,3⁶}	(1,1,1,3,3,5,7,9)							2795520	430080	4.6904159
57	=	heptihexipentisteric 8-cube h_4,5,6,7{4,3⁶}	(1,1,1,1,3,5,7,9)							1397760	215040	4.5825758
58	=	hexipentisteriruncicantic 8-cube h_2,3,4,5,6{4,3⁶}	(1,1,3,5,7,9,11,11)							5160960	1290240	7.1414285
59	=	heptipentisteriruncicantic 8-cube h_2,3,4,5,7{4,3⁶}	(1,1,3,5,7,9,9,11)							5806080	1290240	6.78233
60	=	heptihexisteriruncicantic 8-cube h_2,3,4,6,7{4,3⁶}	(1,1,3,5,7,7,9,11)							5806080	1290240	6.480741
61	=	heptihexipentiruncicantic 8-cube h_2,3,5,6,7{4,3⁶}	(1,1,3,5,5,7,9,11)							5806080	1290240	6.244998
62	=	heptihexipentistericantic 8-cube h_2,4,5,6,7{4,3⁶}	(1,1,3,3,5,7,9,11)							6451200	1290240	6.0827627
63	=	heptihexipentisteriruncic 8-cube h_3,4,5,6,7{4,3⁶}	(1,1,1,3,5,7,9,11)							4300800	860160	6.0000000
64	=	heptihexipentisteriruncicantic 8-cube h_2,3,4,5,6,7{4,3⁶}	(1,1,3,5,7,9,11,13)							2580480	10321920	7.5498347

The E₈ family

The E₈ family has symmetry order 696,729,600.

There are 255 forms based on all permutations of the Coxeter-Dynkin diagrams with one or more rings. Eight forms are shown below, 4 single-ringed, 3 truncations (2 rings), and the final omnitruncation are given below. Bowers-style acronym names are given for cross-referencing.

See also list of E8 polytopes for Coxeter plane graphs of this family.

E₈ uniform polytopes
#	Coxeter-Dynkin diagram	Names	Element counts
#	Coxeter-Dynkin diagram	Names	7-faces	6-faces	5-faces	4-faces	Cells	Faces	Edges	Vertices
1		4₂₁ (fy)	19440	207360	483840	483840	241920	60480	6720	240
2		Truncated 4₂₁ (tiffy)							188160	13440
3		Rectified 4₂₁ (riffy)	19680	375840	1935360	3386880	2661120	1028160	181440	6720
4		Birectified 4₂₁ (borfy)	19680	382560	2600640	7741440	9918720	5806080	1451520	60480
5		Trirectified 4₂₁ (torfy)	19680	382560	2661120	9313920	16934400	14515200	4838400	241920
6		Rectified 1₄₂ (buffy)	19680	382560	2661120	9072000	16934400	16934400	7257600	483840
7		Rectified 2₄₁ (robay)	19680	313440	1693440	4717440	7257600	5322240	1451520	69120
8		2₄₁ (bay)	17520	144960	544320	1209600	1209600	483840	69120	2160
9		Truncated 2₄₁								138240
10		1₄₂ (bif)	2400	106080	725760	2298240	3628800	2419200	483840	17280
11		Truncated 1₄₂								967680
12		Omnitruncated 4₂₁								696729600

Regular and uniform honeycombs

There are five fundamental affine Coxeter groups that generate regular and uniform tessellations in 7-space:

#	Coxeter group		Forms
1	${\tilde {A}}_{7}$	[3^[8]]	29
2	${\tilde {C}}_{7}$	[4,3⁵,4]	135
3	${\tilde {B}}_{7}$	[4,3⁴,3^1,1]	191 (64 new)
4	${\tilde {D}}_{7}$	[3^1,1,3³,3^1,1]	77 (10 new)
5	${\tilde {E}}_{7}$	[3^3,3,1]	143

Regular and uniform tessellations include:

${\tilde {A}}_{7}$ ${\tilde {A}}_{7}$ 29 uniquely ringed forms, including:
- 7-simplex honeycomb: {3^[8]}
${\tilde {C}}_{7}$ ${\tilde {C}}_{7}$ 135 uniquely ringed forms, including:
- Regular 7-cube honeycomb: {4,3⁴,4} = {4,3⁴,3^1,1}, =
${\tilde {B}}_{7}$ ${\tilde {B}}_{7}$ 191 uniquely ringed forms, 127 shared with ${\tilde {C}}_{7}$ ${\tilde {C}}_{7}$ , and 64 new, including:
- 7-demicube honeycomb: h{4,3⁴,4} = {3^1,1,3⁴,4}, =
${\tilde {D}}_{7}$ ${\tilde {D}}_{7}$ , [3^1,1,3³,3^1,1]: 77 unique ring permutations, and 10 are new, the first Coxeter called a quarter 7-cubic honeycomb.
- , , , , , , , , ,
${\tilde {E}}_{7}$ ${\tilde {E}}_{7}$ 143 uniquely ringed forms, including:
- 1₃₃ honeycomb: {3,3^3,3},
- 3₃₁ honeycomb: {3,3,3,3^3,1},

Regular and uniform hyperbolic honeycombs

There are no compact hyperbolic Coxeter groups of rank 8, groups that can generate honeycombs with all finite facets, and a finite vertex figure. However, there are 4 paracompact hyperbolic Coxeter groups of rank 8, each generating uniform honeycombs in 7-space as permutations of rings of the Coxeter diagrams.

${\bar {P}}_{7}$ = [3,3^[7]]:	${\bar {Q}}_{7}$ = [3^1,1,3²,3^2,1]:	${\bar {S}}_{7}$ = [4,3³,3^2,1]:	${\bar {T}}_{7}$ = [3^3,2,2]:

References

^ ^a ^b ^c 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 Wiley::Kaleidoscopes: Selected Writings of H.S.M. Coxeter
- (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. "8D uniform polytopes (polyzetta)".

External links

v t e Fundamental convex regular and uniform polytopes in dimensions 2–10
Family	A_n	B_n	I₂(p) / D_n	E₆ / E₇ / E₈ / F₄ / G₂	H_n
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	1_k2 • 2_k1 • k₂₁	n-pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds

[richeson-1] Richeson, D.; Euler's Gem: The Polyhedron Formula and the Birth of Topoplogy, Princeton, 2008.

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