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:

1. {3,3,3,3,3,3,3} - 8-simplex
2. {4,3,3,3,3,3,3} - 8-cube
3. {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.

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.

Similarly, the notion of orientability of a polyhedron is insufficient to characterise the surface twistings of toroidal polytopes, and this led to the use of torsion coefficients.

Uniform 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]	Template:CDD	135
2	BC8	[4,36]	Template:CDD	255
3	D8	[35,1,1]	Template:CDD	191 (64 unique)
4	E8	[34,2,1]	Template:CDD	255

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

1. Simplex family: A₈ [3⁷] - Template:CDD
   • 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 - Template:CDD
2. Hypercube/orthoplex family: B₈ [4,3⁶] - Template:CDD
   • 255 uniform 8-polytopes as permutations of rings in the group diagram, including two regular ones:
     1. {4,3⁶} - 8-cube or octeract- Template:CDD
     2. {3⁶,4} - 8-orthoplex or octacross - Template:CDD
3. Demihypercube D₈ family: [3^5,1,1] - Template:CDD
   • 191 uniform 8-polytopes as permutations of rings in the group diagram, including:
     1. {3,3^5,1} - 8-demicube or demiocteract, 1₅₁ - Template:CDD; also as h{4,3⁶} Template:CDD.
     2. {3,3,3,3,3,3^1,1} - 8-orthoplex, 5₁₁ - Template:CDD
4. E-polytope family E₈ family: [3^4,1,1] - Template:CDD
   • 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, Template:CDD
     2. {3,3^4,2} - the uniform 1₄₂, Template:CDD,
     3. {3,3,3^4,1} - the uniform 2₄₁, Template:CDD

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]×[ ]	Template:CDD
2	B7A1	[4,3,3,3,3,3]×[ ]	Template:CDD
3	D7A1	[34,1,1]×[ ]	Template:CDD
4	E7A1	[33,2,1]×[ ]	Template:CDD
6+2
1	A6I2(p)	[3,3,3,3,3]×[p]	Template:CDD
2	B6I2(p)	[4,3,3,3,3]×[p]	Template:CDD
3	D6I2(p)	[33,1,1]×[p]	Template:CDD
4	E6I2(p)	[3,3,3,3,3]×[p]	Template:CDD
6+1+1
1	A6A1A1	[3,3,3,3,3]×[ ]x[ ]	Template:CDD
2	B6A1A1	[4,3,3,3,3]×[ ]x[ ]	Template:CDD
3	D6A1A1	[33,1,1]×[ ]x[ ]	Template:CDD
4	E6A1A1	[3,3,3,3,3]×[ ]x[ ]	Template:CDD
5+3
1	A5A3	[34]×[3,3]	Template:CDD
2	B5A3	[4,33]×[3,3]	Template:CDD
3	D5A3	[32,1,1]×[3,3]	Template:CDD
4	A5B3	[34]×[4,3]	Template:CDD
5	B5B3	[4,33]×[4,3]	Template:CDD
6	D5B3	[32,1,1]×[4,3]	Template:CDD
7	A5H3	[34]×[5,3]	Template:CDD
8	B5H3	[4,33]×[5,3]	Template:CDD
9	D5H3	[32,1,1]×[5,3]	Template:CDD
5+2+1
1	A5I2(p)A1	[3,3,3]×[p]×[ ]	Template:CDD
2	B5I2(p)A1	[4,3,3]×[p]×[ ]	Template:CDD
3	D5I2(p)A1	[32,1,1]×[p]×[ ]	Template:CDD
5+1+1+1
1	A5A1A1A1	[3,3,3]×[ ]×[ ]×[ ]	Template:CDD
2	B5A1A1A1	[4,3,3]×[ ]×[ ]×[ ]	Template:CDD
3	D5A1A1A1	[32,1,1]×[ ]×[ ]×[ ]	Template:CDD
4+4
1	A4A4	[3,3,3]×[3,3,3]	Template:CDD
2	B4A4	[4,3,3]×[3,3,3]	Template:CDD
3	D4A4	[31,1,1]×[3,3,3]	Template:CDD
4	F4A4	[3,4,3]×[3,3,3]	Template:CDD
5	H4A4	[5,3,3]×[3,3,3]	Template:CDD
6	B4B4	[4,3,3]×[4,3,3]	Template:CDD
7	D4B4	[31,1,1]×[4,3,3]	Template:CDD
8	F4B4	[3,4,3]×[4,3,3]	Template:CDD
9	H4B4	[5,3,3]×[4,3,3]	Template:CDD
10	D4D4	[31,1,1]×[31,1,1]	Template:CDD
11	F4D4	[3,4,3]×[31,1,1]	Template:CDD
12	H4D4	[5,3,3]×[31,1,1]	Template:CDD
13	F4×F4	[3,4,3]×[3,4,3]	Template:CDD
14	H4×F4	[5,3,3]×[3,4,3]	Template:CDD
15	H4H4	[5,3,3]×[5,3,3]	Template:CDD
4+3+1
1	A4A3A1	[3,3,3]×[3,3]×[ ]	Template:CDD
2	A4B3A1	[3,3,3]×[4,3]×[ ]	Template:CDD
3	A4H3A1	[3,3,3]×[5,3]×[ ]	Template:CDD
4	B4A3A1	[4,3,3]×[3,3]×[ ]	Template:CDD
5	B4B3A1	[4,3,3]×[4,3]×[ ]	Template:CDD
6	B4H3A1	[4,3,3]×[5,3]×[ ]	Template:CDD
7	H4A3A1	[5,3,3]×[3,3]×[ ]	Template:CDD
8	H4B3A1	[5,3,3]×[4,3]×[ ]	Template:CDD
9	H4H3A1	[5,3,3]×[5,3]×[ ]	Template:CDD
10	F4A3A1	[3,4,3]×[3,3]×[ ]	Template:CDD
11	F4B3A1	[3,4,3]×[4,3]×[ ]	Template:CDD
12	F4H3A1	[3,4,3]×[5,3]×[ ]	Template:CDD
13	D4A3A1	[31,1,1]×[3,3]×[ ]	Template:CDD
14	D4B3A1	[31,1,1]×[4,3]×[ ]	Template:CDD
15	D4H3A1	[31,1,1]×[5,3]×[ ]	Template:CDD
4+2+2
...
4+2+1+1
...
4+1+1+1+1
...
3+3+2
1	A3A3I2(p)	[3,3]×[3,3]×[p]	Template:CDD
2	B3A3I2(p)	[4,3]×[3,3]×[p]	Template:CDD
3	H3A3I2(p)	[5,3]×[3,3]×[p]	Template:CDD
4	B3B3I2(p)	[4,3]×[4,3]×[p]	Template:CDD
5	H3B3I2(p)	[5,3]×[4,3]×[p]	Template:CDD
6	H3H3I2(p)	[5,3]×[5,3]×[p]	Template:CDD
3+3+1+1
1	A32A12	[3,3]×[3,3]×[ ]×[ ]	Template:CDD
2	B3A3A12	[4,3]×[3,3]×[ ]×[ ]	Template:CDD
3	H3A3A12	[5,3]×[3,3]×[ ]×[ ]	Template:CDD
4	B3B3A12	[4,3]×[4,3]×[ ]×[ ]	Template:CDD
5	H3B3A12	[5,3]×[4,3]×[ ]×[ ]	Template:CDD
6	H3H3A12	[5,3]×[5,3]×[ ]×[ ]	Template:CDD
3+2+2+1
1	A3I2(p)I2(q)A1	[3,3]×[p]×[q]×[ ]	Template:CDD
2	B3I2(p)I2(q)A1	[4,3]×[p]×[q]×[ ]	Template:CDD
3	H3I2(p)I2(q)A1	[5,3]×[p]×[q]×[ ]	Template:CDD
3+2+1+1+1
1	A3I2(p)A13	[3,3]×[p]×[ ]x[ ]×[ ]	Template:CDD
2	B3I2(p)A13	[4,3]×[p]×[ ]x[ ]×[ ]	Template:CDD
3	H3I2(p)A13	[5,3]×[p]×[ ]x[ ]×[ ]	Template:CDD
3+1+1+1+1+1
1	A3A15	[3,3]×[ ]x[ ]×[ ]x[ ]×[ ]	Template:CDD
2	B3A15	[4,3]×[ ]x[ ]×[ ]x[ ]×[ ]	Template:CDD
3	H3A15	[5,3]×[ ]x[ ]×[ ]x[ ]×[ ]	Template:CDD
2+2+2+2
1	I2(p)I2(q)I2(r)I2(s)	[p]×[q]×[r]×[s]	Template:CDD
2+2+2+1+1
1	I2(p)I2(q)I2(r)A12	[p]×[q]×[r]×[ ]×[ ]	Template:CDD
2+2+1+1+1+1
2	I2(p)I2(q)A14	[p]×[q]×[ ]×[ ]×[ ]×[ ]	Template:CDD
2+1+1+1+1+1+1
1	I2(p)A16	[p]×[ ]×[ ]×[ ]×[ ]×[ ]×[ ]	Template:CDD
1+1+1+1+1+1+1+1
1	A18	[ ]×[ ]×[ ]×[ ]×[ ]×[ ]×[ ]×[ ]	Template:CDD

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
					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		t0,1	Truncated 8-simplex (tene)	(0,0,0,0,0,0,0,1,2)							288	72
6		t0,2	Cantellated 8-simplex	(0,0,0,0,0,0,1,1,2)							1764	252
7		t1,2	Bitruncated 8-simplex	(0,0,0,0,0,0,1,2,2)							1008	252
8		t0,3	Runcinated 8-simplex	(0,0,0,0,0,1,1,1,2)							4536	504
9		t1,3	Bicantellated 8-simplex	(0,0,0,0,0,1,1,2,2)							5292	756
10		t2,3	Tritruncated 8-simplex	(0,0,0,0,0,1,2,2,2)							2016	504
11		t0,4	Stericated 8-simplex	(0,0,0,0,1,1,1,1,2)							6300	630
12		t1,4	Biruncinated 8-simplex	(0,0,0,0,1,1,1,2,2)							11340	1260
13		t2,4	Tricantellated 8-simplex	(0,0,0,0,1,1,2,2,2)							8820	1260
14		t3,4	Quadritruncated 8-simplex	(0,0,0,0,1,2,2,2,2)							2520	630
15		t0,5	Pentellated 8-simplex	(0,0,0,1,1,1,1,1,2)							5040	504
16		t1,5	Bistericated 8-simplex	(0,0,0,1,1,1,1,2,2)							12600	1260
17		t2,5	Triruncinated 8-simplex	(0,0,0,1,1,1,2,2,2)							15120	1680
18		t0,6	Hexicated 8-simplex	(0,0,1,1,1,1,1,1,2)							2268	252
19		t1,6	Bipentellated 8-simplex	(0,0,1,1,1,1,1,2,2)							7560	756
20		t0,7	Heptellated 8-simplex	(0,1,1,1,1,1,1,1,2)							504	72
21		t0,1,2	Cantitruncated 8-simplex	(0,0,0,0,0,0,1,2,3)							2016	504
22		t0,1,3	Runcitruncated 8-simplex	(0,0,0,0,0,1,1,2,3)							9828	1512
23		t0,2,3	Runcicantellated 8-simplex	(0,0,0,0,0,1,2,2,3)							6804	1512
24		t1,2,3	Bicantitruncated 8-simplex	(0,0,0,0,0,1,2,3,3)							6048	1512
25		t0,1,4	Steritruncated 8-simplex	(0,0,0,0,1,1,1,2,3)							20160	2520
26		t0,2,4	Stericantellated 8-simplex	(0,0,0,0,1,1,2,2,3)							26460	3780
27		t1,2,4	Biruncitruncated 8-simplex	(0,0,0,0,1,1,2,3,3)							22680	3780
28		t0,3,4	Steriruncinated 8-simplex	(0,0,0,0,1,2,2,2,3)							12600	2520
29		t1,3,4	Biruncicantellated 8-simplex	(0,0,0,0,1,2,2,3,3)							18900	3780
30		t2,3,4	Tricantitruncated 8-simplex	(0,0,0,0,1,2,3,3,3)							10080	2520
31		t0,1,5	Pentitruncated 8-simplex	(0,0,0,1,1,1,1,2,3)							21420	2520
32		t0,2,5	Penticantellated 8-simplex	(0,0,0,1,1,1,2,2,3)							42840	5040
33		t1,2,5	Bisteritruncated 8-simplex	(0,0,0,1,1,1,2,3,3)							35280	5040
34		t0,3,5	Pentiruncinated 8-simplex	(0,0,0,1,1,2,2,2,3)							37800	5040
35		t1,3,5	Bistericantellated 8-simplex	(0,0,0,1,1,2,2,3,3)							52920	7560
36		t2,3,5	Triruncitruncated 8-simplex	(0,0,0,1,1,2,3,3,3)							27720	5040
37		t0,4,5	Pentistericated 8-simplex	(0,0,0,1,2,2,2,2,3)							13860	2520
38		t1,4,5	Bisteriruncinated 8-simplex	(0,0,0,1,2,2,2,3,3)							30240	5040
39		t0,1,6	Hexitruncated 8-simplex	(0,0,1,1,1,1,1,2,3)							12096	1512
40		t0,2,6	Hexicantellated 8-simplex	(0,0,1,1,1,1,2,2,3)							34020	3780
41		t1,2,6	Bipentitruncated 8-simplex	(0,0,1,1,1,1,2,3,3)							26460	3780
42		t0,3,6	Hexiruncinated 8-simplex	(0,0,1,1,1,2,2,2,3)							45360	5040
43		t1,3,6	Bipenticantellated 8-simplex	(0,0,1,1,1,2,2,3,3)							60480	7560
44		t0,4,6	Hexistericated 8-simplex	(0,0,1,1,2,2,2,2,3)							30240	3780
45		t0,5,6	Hexipentellated 8-simplex	(0,0,1,2,2,2,2,2,3)							9072	1512
46		t0,1,7	Heptitruncated 8-simplex	(0,1,1,1,1,1,1,2,3)							3276	504
47		t0,2,7	Hepticantellated 8-simplex	(0,1,1,1,1,1,2,2,3)							12852	1512
48		t0,3,7	Heptiruncinated 8-simplex	(0,1,1,1,1,2,2,2,3)							23940	2520
49		t0,1,2,3	Runcicantitruncated 8-simplex	(0,0,0,0,0,1,2,3,4)							12096	3024
50		t0,1,2,4	Stericantitruncated 8-simplex	(0,0,0,0,1,1,2,3,4)							45360	7560
51		t0,1,3,4	Steriruncitruncated 8-simplex	(0,0,0,0,1,2,2,3,4)							34020	7560
52		t0,2,3,4	Steriruncicantellated 8-simplex	(0,0,0,0,1,2,3,3,4)							34020	7560
53		t1,2,3,4	Biruncicantitruncated 8-simplex	(0,0,0,0,1,2,3,4,4)							30240	7560
54		t0,1,2,5	Penticantitruncated 8-simplex	(0,0,0,1,1,1,2,3,4)							70560	10080
55		t0,1,3,5	Pentiruncitruncated 8-simplex	(0,0,0,1,1,2,2,3,4)							98280	15120
56		t0,2,3,5	Pentiruncicantellated 8-simplex	(0,0,0,1,1,2,3,3,4)							90720	15120
57		t1,2,3,5	Bistericantitruncated 8-simplex	(0,0,0,1,1,2,3,4,4)							83160	15120
58		t0,1,4,5	Pentisteritruncated 8-simplex	(0,0,0,1,2,2,2,3,4)							50400	10080
59		t0,2,4,5	Pentistericantellated 8-simplex	(0,0,0,1,2,2,3,3,4)							83160	15120
60		t1,2,4,5	Bisteriruncitruncated 8-simplex	(0,0,0,1,2,2,3,4,4)							68040	15120
61		t0,3,4,5	Pentisteriruncinated 8-simplex	(0,0,0,1,2,3,3,3,4)							50400	10080
62		t1,3,4,5	Bisteriruncicantellated 8-simplex	(0,0,0,1,2,3,3,4,4)							75600	15120
63		t2,3,4,5	Triruncicantitruncated 8-simplex	(0,0,0,1,2,3,4,4,4)							40320	10080
64		t0,1,2,6	Hexicantitruncated 8-simplex	(0,0,1,1,1,1,2,3,4)							52920	7560
65		t0,1,3,6	Hexiruncitruncated 8-simplex	(0,0,1,1,1,2,2,3,4)							113400	15120
66		t0,2,3,6	Hexiruncicantellated 8-simplex	(0,0,1,1,1,2,3,3,4)							98280	15120
67		t1,2,3,6	Bipenticantitruncated 8-simplex	(0,0,1,1,1,2,3,4,4)							90720	15120
68		t0,1,4,6	Hexisteritruncated 8-simplex	(0,0,1,1,2,2,2,3,4)							105840	15120
69		t0,2,4,6	Hexistericantellated 8-simplex	(0,0,1,1,2,2,3,3,4)							158760	22680
70		t1,2,4,6	Bipentiruncitruncated 8-simplex	(0,0,1,1,2,2,3,4,4)							136080	22680
71		t0,3,4,6	Hexisteriruncinated 8-simplex	(0,0,1,1,2,3,3,3,4)							90720	15120
72		t1,3,4,6	Bipentiruncicantellated 8-simplex	(0,0,1,1,2,3,3,4,4)							136080	22680
73		t0,1,5,6	Hexipentitruncated 8-simplex	(0,0,1,2,2,2,2,3,4)							41580	7560
74		t0,2,5,6	Hexipenticantellated 8-simplex	(0,0,1,2,2,2,3,3,4)							98280	15120
75		t1,2,5,6	Bipentisteritruncated 8-simplex	(0,0,1,2,2,2,3,4,4)							75600	15120
76		t0,3,5,6	Hexipentiruncinated 8-simplex	(0,0,1,2,2,3,3,3,4)							98280	15120
77		t0,4,5,6	Hexipentistericated 8-simplex	(0,0,1,2,3,3,3,3,4)							41580	7560
78		t0,1,2,7	Hepticantitruncated 8-simplex	(0,1,1,1,1,1,2,3,4)							18144	3024
79		t0,1,3,7	Heptiruncitruncated 8-simplex	(0,1,1,1,1,2,2,3,4)							56700	7560
80		t0,2,3,7	Heptiruncicantellated 8-simplex	(0,1,1,1,1,2,3,3,4)							45360	7560
81		t0,1,4,7	Heptisteritruncated 8-simplex	(0,1,1,1,2,2,2,3,4)							80640	10080
82		t0,2,4,7	Heptistericantellated 8-simplex	(0,1,1,1,2,2,3,3,4)							113400	15120
83		t0,3,4,7	Heptisteriruncinated 8-simplex	(0,1,1,1,2,3,3,3,4)							60480	10080
84		t0,1,5,7	Heptipentitruncated 8-simplex	(0,1,1,2,2,2,2,3,4)							56700	7560
85		t0,2,5,7	Heptipenticantellated 8-simplex	(0,1,1,2,2,2,3,3,4)							120960	15120
86		t0,1,6,7	Heptihexitruncated 8-simplex	(0,1,2,2,2,2,2,3,4)							18144	3024
87		t0,1,2,3,4	Steriruncicantitruncated 8-simplex	(0,0,0,0,1,2,3,4,5)							60480	15120
88		t0,1,2,3,5	Pentiruncicantitruncated 8-simplex	(0,0,0,1,1,2,3,4,5)							166320	30240
89		t0,1,2,4,5	Pentistericantitruncated 8-simplex	(0,0,0,1,2,2,3,4,5)							136080	30240
90		t0,1,3,4,5	Pentisteriruncitruncated 8-simplex	(0,0,0,1,2,3,3,4,5)							136080	30240
91		t0,2,3,4,5	Pentisteriruncicantellated 8-simplex	(0,0,0,1,2,3,4,4,5)							136080	30240
92		t1,2,3,4,5	Bisteriruncicantitruncated 8-simplex	(0,0,0,1,2,3,4,5,5)							120960	30240
93		t0,1,2,3,6	Hexiruncicantitruncated 8-simplex	(0,0,1,1,1,2,3,4,5)							181440	30240
94		t0,1,2,4,6	Hexistericantitruncated 8-simplex	(0,0,1,1,2,2,3,4,5)							272160	45360
95		t0,1,3,4,6	Hexisteriruncitruncated 8-simplex	(0,0,1,1,2,3,3,4,5)							249480	45360
96		t0,2,3,4,6	Hexisteriruncicantellated 8-simplex	(0,0,1,1,2,3,4,4,5)							249480	45360
97		t1,2,3,4,6	Bipentiruncicantitruncated 8-simplex	(0,0,1,1,2,3,4,5,5)							226800	45360
98		t0,1,2,5,6	Hexipenticantitruncated 8-simplex	(0,0,1,2,2,2,3,4,5)							151200	30240
99		t0,1,3,5,6	Hexipentiruncitruncated 8-simplex	(0,0,1,2,2,3,3,4,5)							249480	45360
100		t0,2,3,5,6	Hexipentiruncicantellated 8-simplex	(0,0,1,2,2,3,4,4,5)							226800	45360
101		t1,2,3,5,6	Bipentistericantitruncated 8-simplex	(0,0,1,2,2,3,4,5,5)							204120	45360
102		t0,1,4,5,6	Hexipentisteritruncated 8-simplex	(0,0,1,2,3,3,3,4,5)							151200	30240
103		t0,2,4,5,6	Hexipentistericantellated 8-simplex	(0,0,1,2,3,3,4,4,5)							249480	45360
104		t0,3,4,5,6	Hexipentisteriruncinated 8-simplex	(0,0,1,2,3,4,4,4,5)							151200	30240
105		t0,1,2,3,7	Heptiruncicantitruncated 8-simplex	(0,1,1,1,1,2,3,4,5)							83160	15120
106		t0,1,2,4,7	Heptistericantitruncated 8-simplex	(0,1,1,1,2,2,3,4,5)							196560	30240
107		t0,1,3,4,7	Heptisteriruncitruncated 8-simplex	(0,1,1,1,2,3,3,4,5)							166320	30240
108		t0,2,3,4,7	Heptisteriruncicantellated 8-simplex	(0,1,1,1,2,3,4,4,5)							166320	30240
109		t0,1,2,5,7	Heptipenticantitruncated 8-simplex	(0,1,1,2,2,2,3,4,5)							196560	30240
110		t0,1,3,5,7	Heptipentiruncitruncated 8-simplex	(0,1,1,2,2,3,3,4,5)							294840	45360
111		t0,2,3,5,7	Heptipentiruncicantellated 8-simplex	(0,1,1,2,2,3,4,4,5)							272160	45360
112		t0,1,4,5,7	Heptipentisteritruncated 8-simplex	(0,1,1,2,3,3,3,4,5)							166320	30240
113		t0,1,2,6,7	Heptihexicantitruncated 8-simplex	(0,1,2,2,2,2,3,4,5)							83160	15120
114		t0,1,3,6,7	Heptihexiruncitruncated 8-simplex	(0,1,2,2,2,3,3,4,5)							196560	30240
115		t0,1,2,3,4,5	Pentisteriruncicantitruncated 8-simplex	(0,0,0,1,2,3,4,5,6)							241920	60480
116		t0,1,2,3,4,6	Hexisteriruncicantitruncated 8-simplex	(0,0,1,1,2,3,4,5,6)							453600	90720
117		t0,1,2,3,5,6	Hexipentiruncicantitruncated 8-simplex	(0,0,1,2,2,3,4,5,6)							408240	90720
118		t0,1,2,4,5,6	Hexipentistericantitruncated 8-simplex	(0,0,1,2,3,3,4,5,6)							408240	90720
119		t0,1,3,4,5,6	Hexipentisteriruncitruncated 8-simplex	(0,0,1,2,3,4,4,5,6)							408240	90720
120		t0,2,3,4,5,6	Hexipentisteriruncicantellated 8-simplex	(0,0,1,2,3,4,5,5,6)							408240	90720
121		t1,2,3,4,5,6	Bipentisteriruncicantitruncated 8-simplex	(0,0,1,2,3,4,5,6,6)							362880	90720
122		t0,1,2,3,4,7	Heptisteriruncicantitruncated 8-simplex	(0,1,1,1,2,3,4,5,6)							302400	60480
123		t0,1,2,3,5,7	Heptipentiruncicantitruncated 8-simplex	(0,1,1,2,2,3,4,5,6)							498960	90720
124		t0,1,2,4,5,7	Heptipentistericantitruncated 8-simplex	(0,1,1,2,3,3,4,5,6)							453600	90720
125		t0,1,3,4,5,7	Heptipentisteriruncitruncated 8-simplex	(0,1,1,2,3,4,4,5,6)							453600	90720
126		t0,2,3,4,5,7	Heptipentisteriruncicantellated 8-simplex	(0,1,1,2,3,4,5,5,6)							453600	90720
127		t0,1,2,3,6,7	Heptihexiruncicantitruncated 8-simplex	(0,1,2,2,2,3,4,5,6)							302400	60480
128		t0,1,2,4,6,7	Heptihexistericantitruncated 8-simplex	(0,1,2,2,3,3,4,5,6)							498960	90720
129		t0,1,3,4,6,7	Heptihexisteriruncitruncated 8-simplex	(0,1,2,2,3,4,4,5,6)							453600	90720
130		t0,1,2,5,6,7	Heptihexipenticantitruncated 8-simplex	(0,1,2,3,3,3,4,5,6)							302400	60480
131		t0,1,2,3,4,5,6	Hexipentisteriruncicantitruncated 8-simplex	(0,0,1,2,3,4,5,6,7)							725760	181440
132		t0,1,2,3,4,5,7	Heptipentisteriruncicantitruncated 8-simplex	(0,1,1,2,3,4,5,6,7)							816480	181440
133		t0,1,2,3,4,6,7	Heptihexisteriruncicantitruncated 8-simplex	(0,1,2,2,3,4,5,6,7)							816480	181440
134		t0,1,2,3,5,6,7	Heptihexipentiruncicantitruncated 8-simplex	(0,1,2,3,3,4,5,6,7)							816480	181440
135		t0,1,2,3,4,5,6,7	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
				7	6	5	4	3	2	1	0
1	Template:CDD	t0{36,4}	8-orthoplex Diacosipentacontahexazetton (ek)	256	1024	1792	1792	1120	448	112	16
2	Template:CDD	t1{36,4}	Rectified 8-orthoplex Rectified diacosipentacontahexazetton (rek)	272	3072	8960	12544	10080	4928	1344	112
3	Template:CDD	t2{36,4}	Birectified 8-orthoplex Birectified diacosipentacontahexazetton (bark)	272	3184	16128	34048	36960	22400	6720	448
4	Template:CDD	t3{36,4}	Trirectified 8-orthoplex Trirectified diacosipentacontahexazetton (tark)	272	3184	16576	48384	71680	53760	17920	1120
5	Template:CDD	t3{4,36}	Trirectified 8-cube Trirectified octeract (tro)	272	3184	16576	47712	80640	71680	26880	1792
6	Template:CDD	t2{4,36}	Birectified 8-cube Birectified octeract (bro)	272	3184	14784	36960	55552	50176	21504	1792
7	Template:CDD	t1{4,36}	Rectified 8-cube Rectified octeract (recto)	272	2160	7616	15456	19712	16128	7168	1024
8	Template:CDD	t0{4,36}	8-cube Octeract (octo)	16	112	448	1120	1792	1792	1024	256
9	Template:CDD	t0,1{36,4}	Truncated 8-orthoplex Truncated diacosipentacontahexazetton (tek)							1456	224
10	Template:CDD	t0,2{36,4}	Cantellated 8-orthoplex Small rhombated diacosipentacontahexazetton (srek)							14784	1344
11	Template:CDD	t1,2{36,4}	Bitruncated 8-orthoplex Bitruncated diacosipentacontahexazetton (batek)							8064	1344
12	Template:CDD	t0,3{36,4}	Runcinated 8-orthoplex Small prismated diacosipentacontahexazetton (spek)							60480	4480
13	Template:CDD	t1,3{36,4}	Bicantellated 8-orthoplex Small birhombated diacosipentacontahexazetton (sabork)							67200	6720
14	Template:CDD	t2,3{36,4}	Tritruncated 8-orthoplex Tritruncated diacosipentacontahexazetton (tatek)							24640	4480
15	Template:CDD	t0,4{36,4}	Stericated 8-orthoplex Small cellated diacosipentacontahexazetton (scak)							125440	8960
16	Template:CDD	t1,4{36,4}	Biruncinated 8-orthoplex Small biprismated diacosipentacontahexazetton (sabpek)							215040	17920
17	Template:CDD	t2,4{36,4}	Tricantellated 8-orthoplex Small trirhombated diacosipentacontahexazetton (satrek)							161280	17920
18	Template:CDD	t3,4{4,36}	Quadritruncated 8-cube Octeractidiacosipentacontahexazetton (oke)							44800	8960
19	Template:CDD	t0,5{36,4}	Pentellated 8-orthoplex Small terated diacosipentacontahexazetton (setek)							134400	10752
20	Template:CDD	t1,5{36,4}	Bistericated 8-orthoplex Small bicellated diacosipentacontahexazetton (sibcak)							322560	26880
21	Template:CDD	t2,5{4,36}	Triruncinated 8-cube Small triprismato-octeractidiacosipentacontahexazetton (sitpoke)							376320	35840
22	Template:CDD	t2,4{4,36}	Tricantellated 8-cube Small trirhombated octeract (satro)							215040	26880
23	Template:CDD	t2,3{4,36}	Tritruncated 8-cube Tritruncated octeract (tato)							48384	10752
24	Template:CDD	t0,6{36,4}	Hexicated 8-orthoplex Small petated diacosipentacontahexazetton (supek)							64512	7168
25	Template:CDD	t1,6{4,36}	Bipentellated 8-cube Small biteri-octeractidiacosipentacontahexazetton (sabtoke)							215040	21504
26	Template:CDD	t1,5{4,36}	Bistericated 8-cube Small bicellated octeract (sobco)							358400	35840
27	Template:CDD	t1,4{4,36}	Biruncinated 8-cube Small biprismated octeract (sabepo)							322560	35840
28	Template:CDD	t1,3{4,36}	Bicantellated 8-cube Small birhombated octeract (subro)							150528	21504
29	Template:CDD	t1,2{4,36}	Bitruncated 8-cube Bitruncated octeract (bato)							28672	7168
30	Template:CDD	t0,7{4,36}	Heptellated 8-cube Small exi-octeractidiacosipentacontahexazetton (saxoke)							14336	2048
31	Template:CDD	t0,6{4,36}	Hexicated 8-cube Small petated octeract (supo)							64512	7168
32	Template:CDD	t0,5{4,36}	Pentellated 8-cube Small terated octeract (soto)							143360	14336
33	Template:CDD	t0,4{4,36}	Stericated 8-cube Small cellated octeract (soco)							179200	17920
34	Template:CDD	t0,3{4,36}	Runcinated 8-cube Small prismated octeract (sopo)							129024	14336
35	Template:CDD	t0,2{4,36}	Cantellated 8-cube Small rhombated octeract (soro)							50176	7168
36	Template:CDD	t0,1{4,36}	Truncated 8-cube Truncated octeract (tocto)							8192	2048
37	Template:CDD	t0,1,2{36,4}	Cantitruncated 8-orthoplex Great rhombated diacosipentacontahexazetton							16128	2688
38	Template:CDD	t0,1,3{36,4}	Runcitruncated 8-orthoplex Prismatotruncated diacosipentacontahexazetton							127680	13440
39	Template:CDD	t0,2,3{36,4}	Runcicantellated 8-orthoplex Prismatorhombated diacosipentacontahexazetton							80640	13440
40	Template:CDD	t1,2,3{36,4}	Bicantitruncated 8-orthoplex Great birhombated diacosipentacontahexazetton							73920	13440
41	Template:CDD	t0,1,4{36,4}	Steritruncated 8-orthoplex Cellitruncated diacosipentacontahexazetton							394240	35840
42	Template:CDD	t0,2,4{36,4}	Stericantellated 8-orthoplex Cellirhombated diacosipentacontahexazetton							483840	53760
43	Template:CDD	t1,2,4{36,4}	Biruncitruncated 8-orthoplex Biprismatotruncated diacosipentacontahexazetton							430080	53760
44	Template:CDD	t0,3,4{36,4}	Steriruncinated 8-orthoplex Celliprismated diacosipentacontahexazetton							215040	35840
45	Template:CDD	t1,3,4{36,4}	Biruncicantellated 8-orthoplex Biprismatorhombated diacosipentacontahexazetton							322560	53760
46	Template:CDD	t2,3,4{36,4}	Tricantitruncated 8-orthoplex Great trirhombated diacosipentacontahexazetton							179200	35840
47	Template:CDD	t0,1,5{36,4}	Pentitruncated 8-orthoplex Teritruncated diacosipentacontahexazetton							564480	53760
48	Template:CDD	t0,2,5{36,4}	Penticantellated 8-orthoplex Terirhombated diacosipentacontahexazetton							1075200	107520
49	Template:CDD	t1,2,5{36,4}	Bisteritruncated 8-orthoplex Bicellitruncated diacosipentacontahexazetton							913920	107520
50	Template:CDD	t0,3,5{36,4}	Pentiruncinated 8-orthoplex Teriprismated diacosipentacontahexazetton							913920	107520
51	Template:CDD	t1,3,5{36,4}	Bistericantellated 8-orthoplex Bicellirhombated diacosipentacontahexazetton							1290240	161280
52	Template:CDD	t2,3,5{36,4}	Triruncitruncated 8-orthoplex Triprismatotruncated diacosipentacontahexazetton							698880	107520
53	Template:CDD	t0,4,5{36,4}	Pentistericated 8-orthoplex Tericellated diacosipentacontahexazetton							322560	53760
54	Template:CDD	t1,4,5{36,4}	Bisteriruncinated 8-orthoplex Bicelliprismated diacosipentacontahexazetton							698880	107520
55	Template:CDD	t2,3,5{4,36}	Triruncitruncated 8-cube Triprismatotruncated octeract							645120	107520
56	Template:CDD	t2,3,4{4,36}	Tricantitruncated 8-cube Great trirhombated octeract							241920	53760
57	Template:CDD	t0,1,6{36,4}	Hexitruncated 8-orthoplex Petitruncated diacosipentacontahexazetton							344064	43008
58	Template:CDD	t0,2,6{36,4}	Hexicantellated 8-orthoplex Petirhombated diacosipentacontahexazetton							967680	107520
59	Template:CDD	t1,2,6{36,4}	Bipentitruncated 8-orthoplex Biteritruncated diacosipentacontahexazetton							752640	107520
60	Template:CDD	t0,3,6{36,4}	Hexiruncinated 8-orthoplex Petiprismated diacosipentacontahexazetton							1290240	143360
61	Template:CDD	t1,3,6{36,4}	Bipenticantellated 8-orthoplex Biterirhombated diacosipentacontahexazetton							1720320	215040
62	Template:CDD	t1,4,5{4,36}	Bisteriruncinated 8-cube Bicelliprismated octeract							860160	143360
63	Template:CDD	t0,4,6{36,4}	Hexistericated 8-orthoplex Peticellated diacosipentacontahexazetton							860160	107520
64	Template:CDD	t1,3,6{4,36}	Bipenticantellated 8-cube Biterirhombated octeract							1720320	215040
65	Template:CDD	t1,3,5{4,36}	Bistericantellated 8-cube Bicellirhombated octeract							1505280	215040
66	Template:CDD	t1,3,4{4,36}	Biruncicantellated 8-cube Biprismatorhombated octeract							537600	107520
67	Template:CDD	t0,5,6{36,4}	Hexipentellated 8-orthoplex Petiterated diacosipentacontahexazetton							258048	43008
68	Template:CDD	t1,2,6{4,36}	Bipentitruncated 8-cube Biteritruncated octeract							752640	107520
69	Template:CDD	t1,2,5{4,36}	Bisteritruncated 8-cube Bicellitruncated octeract							1003520	143360
70	Template:CDD	t1,2,4{4,36}	Biruncitruncated 8-cube Biprismatotruncated octeract							645120	107520
71	Template:CDD	t1,2,3{4,36}	Bicantitruncated 8-cube Great birhombated octeract							172032	43008
72	Template:CDD	t0,1,7{36,4}	Heptitruncated 8-orthoplex Exitruncated diacosipentacontahexazetton							93184	14336
73	Template:CDD	t0,2,7{36,4}	Hepticantellated 8-orthoplex Exirhombated diacosipentacontahexazetton							365568	43008
74	Template:CDD	t0,5,6{4,36}	Hexipentellated 8-cube Petiterated octeract							258048	43008
75	Template:CDD	t0,3,7{36,4}	Heptiruncinated 8-orthoplex Exiprismated diacosipentacontahexazetton							680960	71680
76	Template:CDD	t0,4,6{4,36}	Hexistericated 8-cube Peticellated octeract							860160	107520
77	Template:CDD	t0,4,5{4,36}	Pentistericated 8-cube Tericellated octeract							394240	71680
78	Template:CDD	t0,3,7{4,36}	Heptiruncinated 8-cube Exiprismated octeract							680960	71680
79	Template:CDD	t0,3,6{4,36}	Hexiruncinated 8-cube Petiprismated octeract							1290240	143360
80	Template:CDD	t0,3,5{4,36}	Pentiruncinated 8-cube Teriprismated octeract							1075200	143360
81	Template:CDD	t0,3,4{4,36}	Steriruncinated 8-cube Celliprismated octeract							358400	71680
82	Template:CDD	t0,2,7{4,36}	Hepticantellated 8-cube Exirhombated octeract							365568	43008
83	Template:CDD	t0,2,6{4,36}	Hexicantellated 8-cube Petirhombated octeract							967680	107520
84	Template:CDD	t0,2,5{4,36}	Penticantellated 8-cube Terirhombated octeract							1218560	143360
85	Template:CDD	t0,2,4{4,36}	Stericantellated 8-cube Cellirhombated octeract							752640	107520
86	Template:CDD	t0,2,3{4,36}	Runcicantellated 8-cube Prismatorhombated octeract							193536	43008
87	Template:CDD	t0,1,7{4,36}	Heptitruncated 8-cube Exitruncated octeract							93184	14336
88	Template:CDD	t0,1,6{4,36}	Hexitruncated 8-cube Petitruncated octeract							344064	43008
89	Template:CDD	t0,1,5{4,36}	Pentitruncated 8-cube Teritruncated octeract							609280	71680
90	Template:CDD	t0,1,4{4,36}	Steritruncated 8-cube Cellitruncated octeract							573440	71680
91	Template:CDD	t0,1,3{4,36}	Runcitruncated 8-cube Prismatotruncated octeract							279552	43008
92	Template:CDD	t0,1,2{4,36}	Cantitruncated 8-cube Great rhombated octeract							57344	14336
93	Template:CDD	t0,1,2,3{36,4}	Runcicantitruncated 8-orthoplex Great prismated diacosipentacontahexazetton							147840	26880
94	Template:CDD	t0,1,2,4{36,4}	Stericantitruncated 8-orthoplex Celligreatorhombated diacosipentacontahexazetton							860160	107520
95	Template:CDD	t0,1,3,4{36,4}	Steriruncitruncated 8-orthoplex Celliprismatotruncated diacosipentacontahexazetton							591360	107520
96	Template:CDD	t0,2,3,4{36,4}	Steriruncicantellated 8-orthoplex Celliprismatorhombated diacosipentacontahexazetton							591360	107520
97	Template:CDD	t1,2,3,4{36,4}	Biruncicantitruncated 8-orthoplex Great biprismated diacosipentacontahexazetton							537600	107520
98	Template:CDD	t0,1,2,5{36,4}	Penticantitruncated 8-orthoplex Terigreatorhombated diacosipentacontahexazetton							1827840	215040
99	Template:CDD	t0,1,3,5{36,4}	Pentiruncitruncated 8-orthoplex Teriprismatotruncated diacosipentacontahexazetton							2419200	322560
100	Template:CDD	t0,2,3,5{36,4}	Pentiruncicantellated 8-orthoplex Teriprismatorhombated diacosipentacontahexazetton							2257920	322560
101	Template:CDD	t1,2,3,5{36,4}	Bistericantitruncated 8-orthoplex Bicelligreatorhombated diacosipentacontahexazetton							2096640	322560
102	Template:CDD	t0,1,4,5{36,4}	Pentisteritruncated 8-orthoplex Tericellitruncated diacosipentacontahexazetton							1182720	215040
103	Template:CDD	t0,2,4,5{36,4}	Pentistericantellated 8-orthoplex Tericellirhombated diacosipentacontahexazetton							1935360	322560
104	Template:CDD	t1,2,4,5{36,4}	Bisteriruncitruncated 8-orthoplex Bicelliprismatotruncated diacosipentacontahexazetton							1612800	322560
105	Template:CDD	t0,3,4,5{36,4}	Pentisteriruncinated 8-orthoplex Tericelliprismated diacosipentacontahexazetton							1182720	215040
106	Template:CDD	t1,3,4,5{36,4}	Bisteriruncicantellated 8-orthoplex Bicelliprismatorhombated diacosipentacontahexazetton							1774080	322560
107	Template:CDD	t2,3,4,5{4,36}	Triruncicantitruncated 8-cube Great triprismato-octeractidiacosipentacontahexazetton							967680	215040
108	Template:CDD	t0,1,2,6{36,4}	Hexicantitruncated 8-orthoplex Petigreatorhombated diacosipentacontahexazetton							1505280	215040
109	Template:CDD	t0,1,3,6{36,4}	Hexiruncitruncated 8-orthoplex Petiprismatotruncated diacosipentacontahexazetton							3225600	430080
110	Template:CDD	t0,2,3,6{36,4}	Hexiruncicantellated 8-orthoplex Petiprismatorhombated diacosipentacontahexazetton							2795520	430080
111	Template:CDD	t1,2,3,6{36,4}	Bipenticantitruncated 8-orthoplex Biterigreatorhombated diacosipentacontahexazetton							2580480	430080
112	Template:CDD	t0,1,4,6{36,4}	Hexisteritruncated 8-orthoplex Peticellitruncated diacosipentacontahexazetton							3010560	430080
113	Template:CDD	t0,2,4,6{36,4}	Hexistericantellated 8-orthoplex Peticellirhombated diacosipentacontahexazetton							4515840	645120
114	Template:CDD	t1,2,4,6{36,4}	Bipentiruncitruncated 8-orthoplex Biteriprismatotruncated diacosipentacontahexazetton							3870720	645120
115	Template:CDD	t0,3,4,6{36,4}	Hexisteriruncinated 8-orthoplex Peticelliprismated diacosipentacontahexazetton							2580480	430080
116	Template:CDD	t1,3,4,6{4,36}	Bipentiruncicantellated 8-cube Biteriprismatorhombi-octeractidiacosipentacontahexazetton							3870720	645120
117	Template:CDD	t1,3,4,5{4,36}	Bisteriruncicantellated 8-cube Bicelliprismatorhombated octeract							2150400	430080
118	Template:CDD	t0,1,5,6{36,4}	Hexipentitruncated 8-orthoplex Petiteritruncated diacosipentacontahexazetton							1182720	215040
119	Template:CDD	t0,2,5,6{36,4}	Hexipenticantellated 8-orthoplex Petiterirhombated diacosipentacontahexazetton							2795520	430080
120	Template:CDD	t1,2,5,6{4,36}	Bipentisteritruncated 8-cube Bitericellitrunki-octeractidiacosipentacontahexazetton							2150400	430080
121	Template:CDD	t0,3,5,6{36,4}	Hexipentiruncinated 8-orthoplex Petiteriprismated diacosipentacontahexazetton							2795520	430080
122	Template:CDD	t1,2,4,6{4,36}	Bipentiruncitruncated 8-cube Biteriprismatotruncated octeract							3870720	645120
123	Template:CDD	t1,2,4,5{4,36}	Bisteriruncitruncated 8-cube Bicelliprismatotruncated octeract							1935360	430080
124	Template:CDD	t0,4,5,6{36,4}	Hexipentistericated 8-orthoplex Petitericellated diacosipentacontahexazetton							1182720	215040
125	Template:CDD	t1,2,3,6{4,36}	Bipenticantitruncated 8-cube Biterigreatorhombated octeract							2580480	430080
126	Template:CDD	t1,2,3,5{4,36}	Bistericantitruncated 8-cube Bicelligreatorhombated octeract							2365440	430080
127	Template:CDD	t1,2,3,4{4,36}	Biruncicantitruncated 8-cube Great biprismated octeract							860160	215040
128	Template:CDD	t0,1,2,7{36,4}	Hepticantitruncated 8-orthoplex Exigreatorhombated diacosipentacontahexazetton							516096	86016
129	Template:CDD	t0,1,3,7{36,4}	Heptiruncitruncated 8-orthoplex Exiprismatotruncated diacosipentacontahexazetton							1612800	215040
130	Template:CDD	t0,2,3,7{36,4}	Heptiruncicantellated 8-orthoplex Exiprismatorhombated diacosipentacontahexazetton							1290240	215040
131	Template:CDD	t0,4,5,6{4,36}	Hexipentistericated 8-cube Petitericellated octeract							1182720	215040
132	Template:CDD	t0,1,4,7{36,4}	Heptisteritruncated 8-orthoplex Exicellitruncated diacosipentacontahexazetton							2293760	286720
133	Template:CDD	t0,2,4,7{36,4}	Heptistericantellated 8-orthoplex Exicellirhombated diacosipentacontahexazetton							3225600	430080
134	Template:CDD	t0,3,5,6{4,36}	Hexipentiruncinated 8-cube Petiteriprismated octeract							2795520	430080
135	Template:CDD	t0,3,4,7{4,36}	Heptisteriruncinated 8-cube Exicelliprismato-octeractidiacosipentacontahexazetton							1720320	286720
136	Template:CDD	t0,3,4,6{4,36}	Hexisteriruncinated 8-cube Peticelliprismated octeract							2580480	430080
137	Template:CDD	t0,3,4,5{4,36}	Pentisteriruncinated 8-cube Tericelliprismated octeract							1433600	286720
138	Template:CDD	t0,1,5,7{36,4}	Heptipentitruncated 8-orthoplex Exiteritruncated diacosipentacontahexazetton							1612800	215040
139	Template:CDD	t0,2,5,7{4,36}	Heptipenticantellated 8-cube Exiterirhombi-octeractidiacosipentacontahexazetton							3440640	430080
140	Template:CDD	t0,2,5,6{4,36}	Hexipenticantellated 8-cube Petiterirhombated octeract							2795520	430080
141	Template:CDD	t0,2,4,7{4,36}	Heptistericantellated 8-cube Exicellirhombated octeract							3225600	430080
142	Template:CDD	t0,2,4,6{4,36}	Hexistericantellated 8-cube Peticellirhombated octeract							4515840	645120
143	Template:CDD	t0,2,4,5{4,36}	Pentistericantellated 8-cube Tericellirhombated octeract							2365440	430080
144	Template:CDD	t0,2,3,7{4,36}	Heptiruncicantellated 8-cube Exiprismatorhombated octeract							1290240	215040
145	Template:CDD	t0,2,3,6{4,36}	Hexiruncicantellated 8-cube Petiprismatorhombated octeract							2795520	430080
146	Template:CDD	t0,2,3,5{4,36}	Pentiruncicantellated 8-cube Teriprismatorhombated octeract							2580480	430080
147	Template:CDD	t0,2,3,4{4,36}	Steriruncicantellated 8-cube Celliprismatorhombated octeract							967680	215040
148	Template:CDD	t0,1,6,7{4,36}	Heptihexitruncated 8-cube Exipetitrunki-octeractidiacosipentacontahexazetton							516096	86016
149	Template:CDD	t0,1,5,7{4,36}	Heptipentitruncated 8-cube Exiteritruncated octeract							1612800	215040
150	Template:CDD	t0,1,5,6{4,36}	Hexipentitruncated 8-cube Petiteritruncated octeract							1182720	215040
151	Template:CDD	t0,1,4,7{4,36}	Heptisteritruncated 8-cube Exicellitruncated octeract							2293760	286720
152	Template:CDD	t0,1,4,6{4,36}	Hexisteritruncated 8-cube Peticellitruncated octeract							3010560	430080
153	Template:CDD	t0,1,4,5{4,36}	Pentisteritruncated 8-cube Tericellitruncated octeract							1433600	286720
154	Template:CDD	t0,1,3,7{4,36}	Heptiruncitruncated 8-cube Exiprismatotruncated octeract							1612800	215040
155	Template:CDD	t0,1,3,6{4,36}	Hexiruncitruncated 8-cube Petiprismatotruncated octeract							3225600	430080
156	Template:CDD	t0,1,3,5{4,36}	Pentiruncitruncated 8-cube Teriprismatotruncated octeract							2795520	430080
157	Template:CDD	t0,1,3,4{4,36}	Steriruncitruncated 8-cube Celliprismatotruncated octeract							967680	215040
158	Template:CDD	t0,1,2,7{4,36}	Hepticantitruncated 8-cube Exigreatorhombated octeract							516096	86016
159	Template:CDD	t0,1,2,6{4,36}	Hexicantitruncated 8-cube Petigreatorhombated octeract							1505280	215040
160	Template:CDD	t0,1,2,5{4,36}	Penticantitruncated 8-cube Terigreatorhombated octeract							2007040	286720
161	Template:CDD	t0,1,2,4{4,36}	Stericantitruncated 8-cube Celligreatorhombated octeract							1290240	215040
162	Template:CDD	t0,1,2,3{4,36}	Runcicantitruncated 8-cube Great prismated octeract							344064	86016
163	Template:CDD	t0,1,2,3,4{36,4}	Steriruncicantitruncated 8-orthoplex Great cellated diacosipentacontahexazetton							1075200	215040
164	Template:CDD	t0,1,2,3,5{36,4}	Pentiruncicantitruncated 8-orthoplex Terigreatoprismated diacosipentacontahexazetton							4193280	645120
165	Template:CDD	t0,1,2,4,5{36,4}	Pentistericantitruncated 8-orthoplex Tericelligreatorhombated diacosipentacontahexazetton							3225600	645120
166	Template:CDD	t0,1,3,4,5{36,4}	Pentisteriruncitruncated 8-orthoplex Tericelliprismatotruncated diacosipentacontahexazetton							3225600	645120
167	Template:CDD	t0,2,3,4,5{36,4}	Pentisteriruncicantellated 8-orthoplex Tericelliprismatorhombated diacosipentacontahexazetton							3225600	645120
168	Template:CDD	t1,2,3,4,5{36,4}	Bisteriruncicantitruncated 8-orthoplex Great bicellated diacosipentacontahexazetton							2903040	645120
169	Template:CDD	t0,1,2,3,6{36,4}	Hexiruncicantitruncated 8-orthoplex Petigreatoprismated diacosipentacontahexazetton							5160960	860160
170	Template:CDD	t0,1,2,4,6{36,4}	Hexistericantitruncated 8-orthoplex Peticelligreatorhombated diacosipentacontahexazetton							7741440	1290240
171	Template:CDD	t0,1,3,4,6{36,4}	Hexisteriruncitruncated 8-orthoplex Peticelliprismatotruncated diacosipentacontahexazetton							7096320	1290240
172	Template:CDD	t0,2,3,4,6{36,4}	Hexisteriruncicantellated 8-orthoplex Peticelliprismatorhombated diacosipentacontahexazetton							7096320	1290240
173	Template:CDD	t1,2,3,4,6{36,4}	Bipentiruncicantitruncated 8-orthoplex Biterigreatoprismated diacosipentacontahexazetton							6451200	1290240
174	Template:CDD	t0,1,2,5,6{36,4}	Hexipenticantitruncated 8-orthoplex Petiterigreatorhombated diacosipentacontahexazetton							4300800	860160
175	Template:CDD	t0,1,3,5,6{36,4}	Hexipentiruncitruncated 8-orthoplex Petiteriprismatotruncated diacosipentacontahexazetton							7096320	1290240
176	Template:CDD	t0,2,3,5,6{36,4}	Hexipentiruncicantellated 8-orthoplex Petiteriprismatorhombated diacosipentacontahexazetton							6451200	1290240
177	Template:CDD	t1,2,3,5,6{36,4}	Bipentistericantitruncated 8-orthoplex Bitericelligreatorhombated diacosipentacontahexazetton							5806080	1290240
178	Template:CDD	t0,1,4,5,6{36,4}	Hexipentisteritruncated 8-orthoplex Petitericellitruncated diacosipentacontahexazetton							4300800	860160
179	Template:CDD	t0,2,4,5,6{36,4}	Hexipentistericantellated 8-orthoplex Petitericellirhombated diacosipentacontahexazetton							7096320	1290240
180	Template:CDD	t1,2,3,5,6{4,36}	Bipentistericantitruncated 8-cube Bitericelligreatorhombated octeract							5806080	1290240
181	Template:CDD	t0,3,4,5,6{36,4}	Hexipentisteriruncinated 8-orthoplex Petitericelliprismated diacosipentacontahexazetton							4300800	860160
182	Template:CDD	t1,2,3,4,6{4,36}	Bipentiruncicantitruncated 8-cube Biterigreatoprismated octeract							6451200	1290240
183	Template:CDD	t1,2,3,4,5{4,36}	Bisteriruncicantitruncated 8-cube Great bicellated octeract							3440640	860160
184	Template:CDD	t0,1,2,3,7{36,4}	Heptiruncicantitruncated 8-orthoplex Exigreatoprismated diacosipentacontahexazetton							2365440	430080
185	Template:CDD	t0,1,2,4,7{36,4}	Heptistericantitruncated 8-orthoplex Exicelligreatorhombated diacosipentacontahexazetton							5591040	860160
186	Template:CDD	t0,1,3,4,7{36,4}	Heptisteriruncitruncated 8-orthoplex Exicelliprismatotruncated diacosipentacontahexazetton							4730880	860160
187	Template:CDD	t0,2,3,4,7{36,4}	Heptisteriruncicantellated 8-orthoplex Exicelliprismatorhombated diacosipentacontahexazetton							4730880	860160
188	Template:CDD	t0,3,4,5,6{4,36}	Hexipentisteriruncinated 8-cube Petitericelliprismated octeract							4300800	860160
189	Template:CDD	t0,1,2,5,7{36,4}	Heptipenticantitruncated 8-orthoplex Exiterigreatorhombated diacosipentacontahexazetton							5591040	860160
190	Template:CDD	t0,1,3,5,7{36,4}	Heptipentiruncitruncated 8-orthoplex Exiteriprismatotruncated diacosipentacontahexazetton							8386560	1290240
191	Template:CDD	t0,2,3,5,7{36,4}	Heptipentiruncicantellated 8-orthoplex Exiteriprismatorhombated diacosipentacontahexazetton							7741440	1290240
192	Template:CDD	t0,2,4,5,6{4,36}	Hexipentistericantellated 8-cube Petitericellirhombated octeract							7096320	1290240
193	Template:CDD	t0,1,4,5,7{36,4}	Heptipentisteritruncated 8-orthoplex Exitericellitruncated diacosipentacontahexazetton							4730880	860160
194	Template:CDD	t0,2,3,5,7{4,36}	Heptipentiruncicantellated 8-cube Exiteriprismatorhombated octeract							7741440	1290240
195	Template:CDD	t0,2,3,5,6{4,36}	Hexipentiruncicantellated 8-cube Petiteriprismatorhombated octeract							6451200	1290240
196	Template:CDD	t0,2,3,4,7{4,36}	Heptisteriruncicantellated 8-cube Exicelliprismatorhombated octeract							4730880	860160
197	Template:CDD	t0,2,3,4,6{4,36}	Hexisteriruncicantellated 8-cube Peticelliprismatorhombated octeract							7096320	1290240
198	Template:CDD	t0,2,3,4,5{4,36}	Pentisteriruncicantellated 8-cube Tericelliprismatorhombated octeract							3870720	860160
199	Template:CDD	t0,1,2,6,7{36,4}	Heptihexicantitruncated 8-orthoplex Exipetigreatorhombated diacosipentacontahexazetton							2365440	430080
200	Template:CDD	t0,1,3,6,7{36,4}	Heptihexiruncitruncated 8-orthoplex Exipetiprismatotruncated diacosipentacontahexazetton							5591040	860160
201	Template:CDD	t0,1,4,5,7{4,36}	Heptipentisteritruncated 8-cube Exitericellitruncated octeract							4730880	860160
202	Template:CDD	t0,1,4,5,6{4,36}	Hexipentisteritruncated 8-cube Petitericellitruncated octeract							4300800	860160
203	Template:CDD	t0,1,3,6,7{4,36}	Heptihexiruncitruncated 8-cube Exipetiprismatotruncated octeract							5591040	860160
204	Template:CDD	t0,1,3,5,7{4,36}	Heptipentiruncitruncated 8-cube Exiteriprismatotruncated octeract							8386560	1290240
205	Template:CDD	t0,1,3,5,6{4,36}	Hexipentiruncitruncated 8-cube Petiteriprismatotruncated octeract							7096320	1290240
206	Template:CDD	t0,1,3,4,7{4,36}	Heptisteriruncitruncated 8-cube Exicelliprismatotruncated octeract							4730880	860160
207	Template:CDD	t0,1,3,4,6{4,36}	Hexisteriruncitruncated 8-cube Peticelliprismatotruncated octeract							7096320	1290240
208	Template:CDD	t0,1,3,4,5{4,36}	Pentisteriruncitruncated 8-cube Tericelliprismatotruncated octeract							3870720	860160
209	Template:CDD	t0,1,2,6,7{4,36}	Heptihexicantitruncated 8-cube Exipetigreatorhombated octeract							2365440	430080
210	Template:CDD	t0,1,2,5,7{4,36}	Heptipenticantitruncated 8-cube Exiterigreatorhombated octeract							5591040	860160
211	Template:CDD	t0,1,2,5,6{4,36}	Hexipenticantitruncated 8-cube Petiterigreatorhombated octeract							4300800	860160
212	Template:CDD	t0,1,2,4,7{4,36}	Heptistericantitruncated 8-cube Exicelligreatorhombated octeract							5591040	860160
213	Template:CDD	t0,1,2,4,6{4,36}	Hexistericantitruncated 8-cube Peticelligreatorhombated octeract							7741440	1290240
214	Template:CDD	t0,1,2,4,5{4,36}	Pentistericantitruncated 8-cube Tericelligreatorhombated octeract							3870720	860160
215	Template:CDD	t0,1,2,3,7{4,36}	Heptiruncicantitruncated 8-cube Exigreatoprismated octeract							2365440	430080
216	Template:CDD	t0,1,2,3,6{4,36}	Hexiruncicantitruncated 8-cube Petigreatoprismated octeract							5160960	860160
217	Template:CDD	t0,1,2,3,5{4,36}	Pentiruncicantitruncated 8-cube Terigreatoprismated octeract							4730880	860160
218	Template:CDD	t0,1,2,3,4{4,36}	Steriruncicantitruncated 8-cube Great cellated octeract							1720320	430080
219	Template:CDD	t0,1,2,3,4,5{36,4}	Pentisteriruncicantitruncated 8-orthoplex Great terated diacosipentacontahexazetton							5806080	1290240
220	Template:CDD	t0,1,2,3,4,6{36,4}	Hexisteriruncicantitruncated 8-orthoplex Petigreatocellated diacosipentacontahexazetton							12902400	2580480
221	Template:CDD	t0,1,2,3,5,6{36,4}	Hexipentiruncicantitruncated 8-orthoplex Petiterigreatoprismated diacosipentacontahexazetton							11612160	2580480
222	Template:CDD	t0,1,2,4,5,6{36,4}	Hexipentistericantitruncated 8-orthoplex Petitericelligreatorhombated diacosipentacontahexazetton							11612160	2580480
223	Template:CDD	t0,1,3,4,5,6{36,4}	Hexipentisteriruncitruncated 8-orthoplex Petitericelliprismatotruncated diacosipentacontahexazetton							11612160	2580480
224	Template:CDD	t0,2,3,4,5,6{36,4}	Hexipentisteriruncicantellated 8-orthoplex Petitericelliprismatorhombated diacosipentacontahexazetton							11612160	2580480
225	Template:CDD	t1,2,3,4,5,6{4,36}	Bipentisteriruncicantitruncated 8-cube Great biteri-octeractidiacosipentacontahexazetton							10321920	2580480
226	Template:CDD	t0,1,2,3,4,7{36,4}	Heptisteriruncicantitruncated 8-orthoplex Exigreatocellated diacosipentacontahexazetton							8601600	1720320
227	Template:CDD	t0,1,2,3,5,7{36,4}	Heptipentiruncicantitruncated 8-orthoplex Exiterigreatoprismated diacosipentacontahexazetton							14192640	2580480
228	Template:CDD	t0,1,2,4,5,7{36,4}	Heptipentistericantitruncated 8-orthoplex Exitericelligreatorhombated diacosipentacontahexazetton							12902400	2580480
229	Template:CDD	t0,1,3,4,5,7{36,4}	Heptipentisteriruncitruncated 8-orthoplex Exitericelliprismatotruncated diacosipentacontahexazetton							12902400	2580480
230	Template:CDD	t0,2,3,4,5,7{4,36}	Heptipentisteriruncicantellated 8-cube Exitericelliprismatorhombi-octeractidiacosipentacontahexazetton							12902400	2580480
231	Template:CDD	t0,2,3,4,5,6{4,36}	Hexipentisteriruncicantellated 8-cube Petitericelliprismatorhombated octeract							11612160	2580480
232	Template:CDD	t0,1,2,3,6,7{36,4}	Heptihexiruncicantitruncated 8-orthoplex Exipetigreatoprismated diacosipentacontahexazetton							8601600	1720320
233	Template:CDD	t0,1,2,4,6,7{36,4}	Heptihexistericantitruncated 8-orthoplex Exipeticelligreatorhombated diacosipentacontahexazetton							14192640	2580480
234	Template:CDD	t0,1,3,4,6,7{4,36}	Heptihexisteriruncitruncated 8-cube Exipeticelliprismatotrunki-octeractidiacosipentacontahexazetton							12902400	2580480
235	Template:CDD	t0,1,3,4,5,7{4,36}	Heptipentisteriruncitruncated 8-cube Exitericelliprismatotruncated octeract							12902400	2580480
236	Template:CDD	t0,1,3,4,5,6{4,36}	Hexipentisteriruncitruncated 8-cube Petitericelliprismatotruncated octeract							11612160	2580480
237	Template:CDD	t0,1,2,5,6,7{4,36}	Heptihexipenticantitruncated 8-cube Exipetiterigreatorhombi-octeractidiacosipentacontahexazetton							8601600	1720320
238	Template:CDD	t0,1,2,4,6,7{4,36}	Heptihexistericantitruncated 8-cube Exipeticelligreatorhombated octeract							14192640	2580480
239	Template:CDD	t0,1,2,4,5,7{4,36}	Heptipentistericantitruncated 8-cube Exitericelligreatorhombated octeract							12902400	2580480
240	Template:CDD	t0,1,2,4,5,6{4,36}	Hexipentistericantitruncated 8-cube Petitericelligreatorhombated octeract							11612160	2580480
241	Template:CDD	t0,1,2,3,6,7{4,36}	Heptihexiruncicantitruncated 8-cube Exipetigreatoprismated octeract							8601600	1720320
242	Template:CDD	t0,1,2,3,5,7{4,36}	Heptipentiruncicantitruncated 8-cube Exiterigreatoprismated octeract							14192640	2580480
243	Template:CDD	t0,1,2,3,5,6{4,36}	Hexipentiruncicantitruncated 8-cube Petiterigreatoprismated octeract							11612160	2580480
244	Template:CDD	t0,1,2,3,4,7{4,36}	Heptisteriruncicantitruncated 8-cube Exigreatocellated octeract							8601600	1720320
245	Template:CDD	t0,1,2,3,4,6{4,36}	Hexisteriruncicantitruncated 8-cube Petigreatocellated octeract							12902400	2580480
246	Template:CDD	t0,1,2,3,4,5{4,36}	Pentisteriruncicantitruncated 8-cube Great terated octeract							6881280	1720320
247	Template:CDD	t0,1,2,3,4,5,6{36,4}	Hexipentisteriruncicantitruncated 8-orthoplex Great petated diacosipentacontahexazetton							20643840	5160960
248	Template:CDD	t0,1,2,3,4,5,7{36,4}	Heptipentisteriruncicantitruncated 8-orthoplex Exigreatoterated diacosipentacontahexazetton							23224320	5160960
249	Template:CDD	t0,1,2,3,4,6,7{36,4}	Heptihexisteriruncicantitruncated 8-orthoplex Exipetigreatocellated diacosipentacontahexazetton							23224320	5160960
250	Template:CDD	t0,1,2,3,5,6,7{36,4}	Heptihexipentiruncicantitruncated 8-orthoplex Exipetiterigreatoprismated diacosipentacontahexazetton							23224320	5160960
251	Template:CDD	t0,1,2,3,5,6,7{4,36}	Heptihexipentiruncicantitruncated 8-cube Exipetiterigreatoprismated octeract							23224320	5160960
252	Template:CDD	t0,1,2,3,4,6,7{4,36}	Heptihexisteriruncicantitruncated 8-cube Exipetigreatocellated octeract							23224320	5160960
253	Template:CDD	t0,1,2,3,4,5,7{4,36}	Heptipentisteriruncicantitruncated 8-cube Exigreatoterated octeract							23224320	5160960
254	Template:CDD	t0,1,2,3,4,5,6{4,36}	Hexipentisteriruncicantitruncated 8-cube Great petated octeract							20643840	5160960
255	Template:CDD	t0,1,2,3,4,5,6,7{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
				7	6	5	4	3	2	1	0
1	Template:CDD = Template:CDD	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	Template:CDD = Template:CDD	cantic 8-cube h2{4,3,3,3,3,3,3}	(1,1,3,3,3,3,3,3)							23296	3584	2.6457512
3	Template:CDD = Template:CDD	runcic 8-cube h3{4,3,3,3,3,3,3}	(1,1,1,3,3,3,3,3)							64512	7168	2.4494896
4	Template:CDD = Template:CDD	steric 8-cube h4{4,3,3,3,3,3,3}	(1,1,1,1,3,3,3,3)							98560	8960	2.2360678
5	Template:CDD = Template:CDD	pentic 8-cube h5{4,3,3,3,3,3,3}	(1,1,1,1,1,3,3,3)							89600	7168	1.9999999
6	Template:CDD = Template:CDD	hexic 8-cube h6{4,3,3,3,3,3,3}	(1,1,1,1,1,1,3,3)							48384	3584	1.7320508
7	Template:CDD = Template:CDD	heptic 8-cube h7{4,3,3,3,3,3,3}	(1,1,1,1,1,1,1,3)							14336	1024	1.4142135
8	Template:CDD = Template:CDD	runcicantic 8-cube h2,3{4,3,3,3,3,3,3}	(1,1,3,5,5,5,5,5)							86016	21504	4.1231055
9	Template:CDD = Template:CDD	stericantic 8-cube h2,4{4,3,3,3,3,3,3}	(1,1,3,3,5,5,5,5)							349440	53760	3.8729835
10	Template:CDD = Template:CDD	steriruncic 8-cube h3,4{4,3,3,3,3,3,3}	(1,1,1,3,5,5,5,5)							179200	35840	3.7416575
11	Template:CDD = Template:CDD	penticantic 8-cube h2,5{4,3,3,3,3,3,3}	(1,1,3,3,3,5,5,5)							573440	71680	3.6055512
12	Template:CDD = Template:CDD	pentiruncic 8-cube h3,5{4,3,3,3,3,3,3}	(1,1,1,3,3,5,5,5)							537600	71680	3.4641016
13	Template:CDD = Template:CDD	pentisteric 8-cube h4,5{4,3,3,3,3,3,3}	(1,1,1,1,3,5,5,5)							232960	35840	3.3166249
14	Template:CDD = Template:CDD	hexicantic 8-cube h2,6{4,3,3,3,3,3,3}	(1,1,3,3,3,3,5,5)							456960	53760	3.3166249
15	Template:CDD = Template:CDD	hexicruncic 8-cube h3,6{4,3,3,3,3,3,3}	(1,1,1,3,3,3,5,5)							645120	71680	3.1622777
16	Template:CDD = Template:CDD	hexisteric 8-cube h4,6{4,3,3,3,3,3,3}	(1,1,1,1,3,3,5,5)							483840	53760	3
17	Template:CDD = Template:CDD	hexipentic 8-cube h5,6{4,3,3,3,3,3,3}	(1,1,1,1,1,3,5,5)							182784	21504	2.8284271
18	Template:CDD = Template:CDD	hepticantic 8-cube h2,7{4,3,3,3,3,3,3}	(1,1,3,3,3,3,3,5)							172032	21504	3
19	Template:CDD = Template:CDD	heptiruncic 8-cube h3,7{4,3,3,3,3,3,3}	(1,1,1,3,3,3,3,5)							340480	35840	2.8284271
20	Template:CDD = Template:CDD	heptsteric 8-cube h4,7{4,3,3,3,3,3,3}	(1,1,1,1,3,3,3,5)							376320	35840	2.6457512
21	Template:CDD = Template:CDD	heptipentic 8-cube h5,7{4,3,3,3,3,3,3}	(1,1,1,1,1,3,3,5)							236544	21504	2.4494898
22	Template:CDD = Template:CDD	heptihexic 8-cube h6,7{4,3,3,3,3,3,3}	(1,1,1,1,1,1,3,5)							78848	7168	2.236068
23	Template:CDD = Template:CDD	steriruncicantic 8-cube h2,3,4{4,36}	(1,1,3,5,7,7,7,7)							430080	107520	5.3851647
24	Template:CDD = Template:CDD	pentiruncicantic 8-cube h2,3,5{4,36}	(1,1,3,5,5,7,7,7)							1182720	215040	5.0990195
25	Template:CDD = Template:CDD	pentistericantic 8-cube h2,4,5{4,36}	(1,1,3,3,5,7,7,7)							1075200	215040	4.8989797
26	Template:CDD = Template:CDD	pentisterirunic 8-cube h3,4,5{4,36}	(1,1,1,3,5,7,7,7)							716800	143360	4.7958317
27	Template:CDD = Template:CDD	hexiruncicantic 8-cube h2,3,6{4,36}	(1,1,3,5,5,5,7,7)							1290240	215040	4.7958317
28	Template:CDD = Template:CDD	hexistericantic 8-cube h2,4,6{4,36}	(1,1,3,3,5,5,7,7)							2096640	322560	4.5825758
29	Template:CDD = Template:CDD	hexisterirunic 8-cube h3,4,6{4,36}	(1,1,1,3,5,5,7,7)							1290240	215040	4.472136
30	Template:CDD = Template:CDD	hexipenticantic 8-cube h2,5,6{4,36}	(1,1,3,3,3,5,7,7)							1290240	215040	4.3588991
31	Template:CDD = Template:CDD	hexipentirunic 8-cube h3,5,6{4,36}	(1,1,1,3,3,5,7,7)							1397760	215040	4.2426405
32	Template:CDD = Template:CDD	hexipentisteric 8-cube h4,5,6{4,36}	(1,1,1,1,3,5,7,7)							698880	107520	4.1231055
33	Template:CDD = Template:CDD	heptiruncicantic 8-cube h2,3,7{4,36}	(1,1,3,5,5,5,5,7)							591360	107520	4.472136
34	Template:CDD = Template:CDD	heptistericantic 8-cube h2,4,7{4,36}	(1,1,3,3,5,5,5,7)							1505280	215040	4.2426405
35	Template:CDD = Template:CDD	heptisterruncic 8-cube h3,4,7{4,36}	(1,1,1,3,5,5,5,7)							860160	143360	4.1231055
36	Template:CDD = Template:CDD	heptipenticantic 8-cube h2,5,7{4,36}	(1,1,3,3,3,5,5,7)							1612800	215040	4
37	Template:CDD = Template:CDD	heptipentiruncic 8-cube h3,5,7{4,36}	(1,1,1,3,3,5,5,7)							1612800	215040	3.8729835
38	Template:CDD = Template:CDD	heptipentisteric 8-cube h4,5,7{4,36}	(1,1,1,1,3,5,5,7)							752640	107520	3.7416575
39	Template:CDD = Template:CDD	heptihexicantic 8-cube h2,6,7{4,36}	(1,1,3,3,3,3,5,7)							752640	107520	3.7416575
40	Template:CDD = Template:CDD	heptihexiruncic 8-cube h3,6,7{4,36}	(1,1,1,3,3,3,5,7)							1146880	143360	3.6055512
41	Template:CDD = Template:CDD	heptihexisteric 8-cube h4,6,7{4,36}	(1,1,1,1,3,3,5,7)							913920	107520	3.4641016
42	Template:CDD = Template:CDD	heptihexipentic 8-cube h5,6,7{4,36}	(1,1,1,1,1,3,5,7)							365568	43008	3.3166249
43	Template:CDD = Template:CDD	pentisteriruncicantic 8-cube h2,3,4,5{4,36}	(1,1,3,5,7,9,9,9)							1720320	430080	6.4031243
44	Template:CDD = Template:CDD	hexisteriruncicantic 8-cube h2,3,4,6{4,36}	(1,1,3,5,7,7,9,9)							3225600	645120	6.0827627
45	Template:CDD = Template:CDD	hexipentiruncicantic 8-cube h2,3,5,6{4,36}	(1,1,3,5,5,7,9,9)							2903040	645120	5.8309517
46	Template:CDD = Template:CDD	hexipentistericantic 8-cube h2,4,5,6{4,36}	(1,1,3,3,5,7,9,9)							3225600	645120	5.6568542
47	Template:CDD = Template:CDD	hexipentisteriruncic 8-cube h3,4,5,6{4,36}	(1,1,1,3,5,7,9,9)							2150400	430080	5.5677648
48	Template:CDD = Template:CDD	heptsteriruncicantic 8-cube h2,3,4,7{4,36}	(1,1,3,5,7,7,7,9)							2150400	430080	5.7445626
49	Template:CDD = Template:CDD	heptipentiruncicantic 8-cube h2,3,5,7{4,36}	(1,1,3,5,5,7,7,9)							3548160	645120	5.4772258
50	Template:CDD = Template:CDD	heptipentistericantic 8-cube h2,4,5,7{4,36}	(1,1,3,3,5,7,7,9)							3548160	645120	5.291503
51	Template:CDD = Template:CDD	heptipentisteriruncic 8-cube h3,4,5,7{4,36}	(1,1,1,3,5,7,7,9)							2365440	430080	5.1961527
52	Template:CDD = Template:CDD	heptihexiruncicantic 8-cube h2,3,6,7{4,36}	(1,1,3,5,5,5,7,9)							2150400	430080	5.1961527
53	Template:CDD = Template:CDD	heptihexistericantic 8-cube h2,4,6,7{4,36}	(1,1,3,3,5,5,7,9)							3870720	645120	5
54	Template:CDD = Template:CDD	heptihexisteriruncic 8-cube h3,4,6,7{4,36}	(1,1,1,3,5,5,7,9)							2365440	430080	4.8989797
55	Template:CDD = Template:CDD	heptihexipenticantic 8-cube h2,5,6,7{4,36}	(1,1,3,3,3,5,7,9)							2580480	430080	4.7958317
56	Template:CDD = Template:CDD	heptihexipentiruncic 8-cube h3,5,6,7{4,36}	(1,1,1,3,3,5,7,9)							2795520	430080	4.6904159
57	Template:CDD = Template:CDD	heptihexipentisteric 8-cube h4,5,6,7{4,36}	(1,1,1,1,3,5,7,9)							1397760	215040	4.5825758
58	Template:CDD = Template:CDD	hexipentisteriruncicantic 8-cube h2,3,4,5,6{4,36}	(1,1,3,5,7,9,11,11)							5160960	1290240	7.1414285
59	Template:CDD = Template:CDD	heptipentisteriruncicantic 8-cube h2,3,4,5,7{4,36}	(1,1,3,5,7,9,9,11)							5806080	1290240	6.78233
60	Template:CDD = Template:CDD	heptihexisteriruncicantic 8-cube h2,3,4,6,7{4,36}	(1,1,3,5,7,7,9,11)							5806080	1290240	6.480741
61	Template:CDD = Template:CDD	heptihexipentiruncicantic 8-cube h2,3,5,6,7{4,36}	(1,1,3,5,5,7,9,11)							5806080	1290240	6.244998
62	Template:CDD = Template:CDD	heptihexipentistericantic 8-cube h2,4,5,6,7{4,36}	(1,1,3,3,5,7,9,11)							6451200	1290240	6.0827627
63	Template:CDD = Template:CDD	heptihexipentisteriruncic 8-cube h3,4,5,6,7{4,36}	(1,1,1,3,5,7,9,11)							4300800	860160	6.0000000
64	Template:CDD = Template:CDD	heptihexipentisteriruncicantic 8-cube h2,3,4,5,6,7{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
			7-faces	6-faces	5-faces	4-faces	Cells	Faces	Edges	Vertices
1	Template:CDD	421 (fy)	19440	207360	483840	483840	241920	60480	6720	240
2	Template:CDD	Truncated 421 (tiffy)							188160	13440
3	Template:CDD	Rectified 421 (riffy)	19680	375840	1935360	3386880	2661120	1028160	181440	6720
4	Template:CDD	Birectified 421 (borfy)	19680	382560	2600640	7741440	9918720	5806080	1451520	60480
5	Template:CDD	Trirectified 421 (torfy)	19680	382560	2661120	9313920	16934400	14515200	4838400	241920
6	Template:CDD	Rectified 142 (buffy)	19680	382560	2661120	9072000	16934400	16934400	7257600	483840
7	Template:CDD	Rectified 241 (robay)	19680	313440	1693440	4717440	7257600	5322240	1451520	69120
8	Template:CDD	241 (bay)	17520	144960	544320	1209600	1209600	483840	69120	2160
9	Template:CDD	Truncated 241								138240
10	Template:CDD	142 (bif)	2400	106080	725760	2298240	3628800	2419200	483840	17280
11	Template:CDD	Truncated 142								967680
12	Template:CDD	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		Coxeter diagram	Forms
1	${\tilde{A}}_7$	[3[8]]	Template:CDD	29
2	${\tilde{C}}_7$	[4,35,4]	Template:CDD	135
3	${\tilde{B}}_7$	[4,34,31,1]	Template:CDD	191 (64 new)
4	${\tilde{D}}_7$	[31,1,33,31,1]	Template:CDD	77 (10 new)
5	${\tilde{E}}_7$	[33,3,1]	Template:CDD	143

Regular and uniform tessellations include:

• ${\tilde{A}}_7$ 29 uniquely ringed forms, including:
  • 7-simplex honeycomb: {3^[8]} Template:CDD
• ${\tilde{C}}_7$ 135 uniquely ringed forms, including:
  • Regular 7-cube honeycomb: {4,3⁴,4} = {4,3⁴,3^1,1}, Template:CDD = Template:CDD
• ${\tilde{B}}_7$ 191 uniquely ringed forms, 127 shared with ${\tilde{C}}_7$, and 64 new, including:
  • 7-demicube honeycomb: h{4,3⁴,4} = {3^1,1,3⁴,4}, Template:CDD = Template:CDD
• ${\tilde{D}}_7$, [3^1,1,3³,3^1,1]: 77 unique ring permutations, and 10 are new, the first Coxeter called a quarter 7-cubic honeycomb.
  • Template:CDD, Template:CDD, Template:CDD, Template:CDD, Template:CDD, Template:CDD, Template:CDD, Template:CDD, Template:CDD, Template:CDD
• ${\tilde{E}}_7$ 143 uniquely ringed forms, including:
  • 1₃₃ honeycomb: {3,3^3,3}, Template:CDD
  • 3₃₁ honeycomb: {3,3,3,3^3,1}, Template:CDD

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.

${\overline{P}}_7$ = [3,3[7]]: Template:CDD

${\overline{Q}}_7$ = [31,1,32,32,1]: Template:CDD

${\overline{S}}_7$ = [4,33,32,1]: Template:CDD

${\overline{T}}_7$ = [33,2,2]: Template:CDD

References

• T. Gosset: On the Regular and Semi-Regular Figures in Space of n Dimensions, Messenger of Mathematics, Macmillan, 1900
• A. Boole Stott: Geometrical deduction of semiregular from regular polytopes and space fillings, Verhandelingen of the Koninklijke academy van Wetenschappen width unit Amsterdam, Eerste Sectie 11,1, Amsterdam, 1910
• H.S.M. Coxeter:
  • H.S.M. Coxeter, M.S. Longuet-Higgins und J.C.P. Miller: Uniform Polyhedra, Philosophical Transactions of the Royal Society of London, Londne, 1954
  • H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973
• Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, Template:ISBN 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
• Template:KlitzingPolytopes

External links

• Polytope names
• Polytopes of Various Dimensions
• Multi-dimensional Glossary