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

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 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

1 2 3 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

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

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[richeson-1] 1 2 3 Richeson, D.; Euler's Gem: The Polyhedron Formula and the Birth of Topoplogy, Princeton, 2008.

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