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
8-cube				Rectified 8-cube				Truncated 8-cube
8-demicube				Truncated 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.^[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	[35,1,1]		191 (64 unique)
4	E8	[34,2,1]		255

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

1. 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 -
2. 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 -
3. 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 -
4. 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	[34,1,1]×[ ]
4	E7A1	[33,2,1]×[ ]
6+2
1	A6I2(p)	[3,3,3,3,3]×[p]
2	B6I2(p)	[4,3,3,3,3]×[p]
3	D6I2(p)	[33,1,1]×[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	[33,1,1]×[ ]x[ ]
4	E6A1A1	[3,3,3,3,3]×[ ]x[ ]
5+3
1	A5A3	[34]×[3,3]
2	B5A3	[4,33]×[3,3]
3	D5A3	[32,1,1]×[3,3]
4	A5B3	[34]×[4,3]
5	B5B3	[4,33]×[4,3]
6	D5B3	[32,1,1]×[4,3]
7	A5H3	[34]×[5,3]
8	B5H3	[4,33]×[5,3]
9	D5H3	[32,1,1]×[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	[32,1,1]×[p]×[ ]
5+1+1+1
1	A5A1A1A1	[3,3,3]×[ ]×[ ]×[ ]
2	B5A1A1A1	[4,3,3]×[ ]×[ ]×[ ]
3	D5A1A1A1	[32,1,1]×[ ]×[ ]×[ ]
4+4
1	A4A4	[3,3,3]×[3,3,3]
2	B4A4	[4,3,3]×[3,3,3]
3	D4A4	[31,1,1]×[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	[31,1,1]×[4,3,3]
8	F4B4	[3,4,3]×[4,3,3]
9	H4B4	[5,3,3]×[4,3,3]
10	D4D4	[31,1,1]×[31,1,1]
11	F4D4	[3,4,3]×[31,1,1]
12	H4D4	[5,3,3]×[31,1,1]
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	[31,1,1]×[3,3]×[ ]
14	D4B3A1	[31,1,1]×[4,3]×[ ]
15	D4H3A1	[31,1,1]×[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
					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		(0,0,0,1,1,1,1,2,3)							21420	2520
32		t0,2,5		(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		(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		(0,0,0,1,1,2,3,3,3)							27720	5040
37		t0,4,5		(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		(0,0,1,1,1,1,1,2,3)							12096	1512
40		t0,2,6		(0,0,1,1,1,1,2,2,3)							34020	3780
41		t1,2,6		(0,0,1,1,1,1,2,3,3)							26460	3780
42		t0,3,6		(0,0,1,1,1,2,2,2,3)							45360	5040
43		t1,3,6		(0,0,1,1,1,2,2,3,3)							60480	7560
44		t0,4,6		(0,0,1,1,2,2,2,2,3)							30240	3780
45		t0,5,6		(0,0,1,2,2,2,2,2,3)							9072	1512
46		t0,1,7		(0,1,1,1,1,1,1,2,3)							3276	504
47		t0,2,7		(0,1,1,1,1,1,2,2,3)							12852	1512
48		t0,3,7		(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		(0,0,0,1,1,1,2,3,4)							70560	10080
55		t0,1,3,5		(0,0,0,1,1,2,2,3,4)							98280	15120
56		t0,2,3,5		(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		(0,0,0,1,2,2,2,3,4)							50400	10080
59		t0,2,4,5		(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		(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		(0,0,1,1,1,1,2,3,4)							52920	7560
65		t0,1,3,6		(0,0,1,1,1,2,2,3,4)							113400	15120
66		t0,2,3,6		(0,0,1,1,1,2,3,3,4)							98280	15120
67		t1,2,3,6		(0,0,1,1,1,2,3,4,4)							90720	15120
68		t0,1,4,6		(0,0,1,1,2,2,2,3,4)							105840	15120
69		t0,2,4,6		(0,0,1,1,2,2,3,3,4)							158760	22680
70		t1,2,4,6		(0,0,1,1,2,2,3,4,4)							136080	22680
71		t0,3,4,6		(0,0,1,1,2,3,3,3,4)							90720	15120
72		t1,3,4,6		(0,0,1,1,2,3,3,4,4)							136080	22680
73		t0,1,5,6		(0,0,1,2,2,2,2,3,4)							41580	7560
74		t0,2,5,6		(0,0,1,2,2,2,3,3,4)							98280	15120
75		t1,2,5,6		(0,0,1,2,2,2,3,4,4)							75600	15120
76		t0,3,5,6		(0,0,1,2,2,3,3,3,4)							98280	15120
77		t0,4,5,6		(0,0,1,2,3,3,3,3,4)							41580	7560
78		t0,1,2,7		(0,1,1,1,1,1,2,3,4)							18144	3024
79		t0,1,3,7		(0,1,1,1,1,2,2,3,4)							56700	7560
80		t0,2,3,7		(0,1,1,1,1,2,3,3,4)							45360	7560
81		t0,1,4,7		(0,1,1,1,2,2,2,3,4)							80640	10080
82		t0,2,4,7		(0,1,1,1,2,2,3,3,4)							113400	15120
83		t0,3,4,7		(0,1,1,1,2,3,3,3,4)							60480	10080
84		t0,1,5,7		(0,1,1,2,2,2,2,3,4)							56700	7560
85		t0,2,5,7		(0,1,1,2,2,2,3,3,4)							120960	15120
86		t0,1,6,7		(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		(0,0,0,1,1,2,3,4,5)							166320	30240
89		t0,1,2,4,5		(0,0,0,1,2,2,3,4,5)							136080	30240
90		t0,1,3,4,5		(0,0,0,1,2,3,3,4,5)							136080	30240
91		t0,2,3,4,5		(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		(0,0,1,1,1,2,3,4,5)							181440	30240
94		t0,1,2,4,6		(0,0,1,1,2,2,3,4,5)							272160	45360
95		t0,1,3,4,6		(0,0,1,1,2,3,3,4,5)							249480	45360
96		t0,2,3,4,6		(0,0,1,1,2,3,4,4,5)							249480	45360
97		t1,2,3,4,6		(0,0,1,1,2,3,4,5,5)							226800	45360
98		t0,1,2,5,6		(0,0,1,2,2,2,3,4,5)							151200	30240
99		t0,1,3,5,6		(0,0,1,2,2,3,3,4,5)							249480	45360
100		t0,2,3,5,6		(0,0,1,2,2,3,4,4,5)							226800	45360
101		t1,2,3,5,6		(0,0,1,2,2,3,4,5,5)							204120	45360
102		t0,1,4,5,6		(0,0,1,2,3,3,3,4,5)							151200	30240
103		t0,2,4,5,6		(0,0,1,2,3,3,4,4,5)							249480	45360
104		t0,3,4,5,6		(0,0,1,2,3,4,4,4,5)							151200	30240
105		t0,1,2,3,7		(0,1,1,1,1,2,3,4,5)							83160	15120
106		t0,1,2,4,7		(0,1,1,1,2,2,3,4,5)							196560	30240
107		t0,1,3,4,7		(0,1,1,1,2,3,3,4,5)							166320	30240
108		t0,2,3,4,7		(0,1,1,1,2,3,4,4,5)							166320	30240
109		t0,1,2,5,7		(0,1,1,2,2,2,3,4,5)							196560	30240
110		t0,1,3,5,7		(0,1,1,2,2,3,3,4,5)							294840	45360
111		t0,2,3,5,7		(0,1,1,2,2,3,4,4,5)							272160	45360
112		t0,1,4,5,7		(0,1,1,2,3,3,3,4,5)							166320	30240
113		t0,1,2,6,7		(0,1,2,2,2,2,3,4,5)							83160	15120
114		t0,1,3,6,7		(0,1,2,2,2,3,3,4,5)							196560	30240
115		t0,1,2,3,4,5		(0,0,0,1,2,3,4,5,6)							241920	60480
116		t0,1,2,3,4,6		(0,0,1,1,2,3,4,5,6)							453600	90720
117		t0,1,2,3,5,6		(0,0,1,2,2,3,4,5,6)							408240	90720
118		t0,1,2,4,5,6		(0,0,1,2,3,3,4,5,6)							408240	90720
119		t0,1,3,4,5,6		(0,0,1,2,3,4,4,5,6)							408240	90720
120		t0,2,3,4,5,6		(0,0,1,2,3,4,5,5,6)							408240	90720
121		t1,2,3,4,5,6		(0,0,1,2,3,4,5,6,6)							362880	90720
122		t0,1,2,3,4,7		(0,1,1,1,2,3,4,5,6)							302400	60480
123		t0,1,2,3,5,7		(0,1,1,2,2,3,4,5,6)							498960	90720
124		t0,1,2,4,5,7		(0,1,1,2,3,3,4,5,6)							453600	90720
125		t0,1,3,4,5,7		(0,1,1,2,3,4,4,5,6)							453600	90720
126		t0,2,3,4,5,7		(0,1,1,2,3,4,5,5,6)							453600	90720
127		t0,1,2,3,6,7		(0,1,2,2,2,3,4,5,6)							302400	60480
128		t0,1,2,4,6,7		(0,1,2,2,3,3,4,5,6)							498960	90720
129		t0,1,3,4,6,7		(0,1,2,2,3,4,4,5,6)							453600	90720
130		t0,1,2,5,6,7		(0,1,2,3,3,3,4,5,6)							302400	60480
131		t0,1,2,3,4,5,6		(0,0,1,2,3,4,5,6,7)							725760	181440
132		t0,1,2,3,4,5,7		(0,1,1,2,3,4,5,6,7)							816480	181440
133		t0,1,2,3,4,6,7		(0,1,2,2,3,4,5,6,7)							816480	181440
134		t0,1,2,3,5,6,7		(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		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		t0,1{36,4}	Truncated 8-orthoplex Truncated diacosipentacontahexazetton (tek)							1456	224
10		t0,2{36,4}	Cantellated 8-orthoplex Small rhombated diacosipentacontahexazetton (srek)							14784	1344
11		t1,2{36,4}	Bitruncated 8-orthoplex Bitruncated diacosipentacontahexazetton (batek)							8064	1344
12		t0,3{36,4}	Small prismated diacosipentacontahexazetton (spek)							60480	4480
13		t1,3{36,4}	Small birhombated diacosipentacontahexazetton (sabork)							67200	6720
14		t2,3{36,4}	Tritruncated 8-orthoplex Tritruncated diacosipentacontahexazetton (tatek)							24640	4480
15		t0,4{36,4}	Small cellated diacosipentacontahexazetton (scak)							125440	8960
16		t1,4{36,4}	Small biprismated diacosipentacontahexazetton (sabpek)							215040	17920
17		t2,4{36,4}	Small trirhombated diacosipentacontahexazetton (satrek)							161280	17920
18		t3,4{4,36}	Quadritruncated 8-cube Octeractidiacosipentacontahexazetton (oke)							44800	8960
19		t0,5{36,4}	Small terated diacosipentacontahexazetton (setek)							134400	10752
20		t1,5{36,4}	Small bicellated diacosipentacontahexazetton (sibcak)							322560	26880
21		t2,5{4,36}	Small triprismato-octeractidiacosipentacontahexazetton (sitpoke)							376320	35840
22		t2,4{4,36}	Small trirhombated octeract (satro)							215040	26880
23		t2,3{4,36}	Tritruncated 8-cube Tritruncated octeract (tato)							48384	10752
24		t0,6{36,4}	Small petated diacosipentacontahexazetton (supek)							64512	7168
25		t1,6{4,36}	Small biteri-octeractidiacosipentacontahexazetton (sabtoke)							215040	21504
26		t1,5{4,36}	Small bicellated octeract (sobco)							358400	35840
27		t1,4{4,36}	Small biprismated octeract (sabepo)							322560	35840
28		t1,3{4,36}	Small birhombated octeract (subro)							150528	21504
29		t1,2{4,36}	Bitruncated 8-cube Bitruncated octeract (bato)							28672	7168
30		t0,7{4,36}	Small exi-octeractidiacosipentacontahexazetton (saxoke)							14336	2048
31		t0,6{4,36}	Small petated octeract (supo)							64512	7168
32		t0,5{4,36}	Small terated octeract (soto)							143360	14336
33		t0,4{4,36}	Small cellated octeract (soco)							179200	17920
34		t0,3{4,36}	Small prismated octeract (sopo)							129024	14336
35		t0,2{4,36}	Small rhombated octeract (soro)							50176	7168
36		t0,1{4,36}	Truncated 8-cube Truncated octeract (tocto)							8192	2048
37		t0,1,2{36,4}	Great rhombated diacosipentacontahexazetton							16128	2688
38		t0,1,3{36,4}	Prismatotruncated diacosipentacontahexazetton							127680	13440
39		t0,2,3{36,4}	Prismatorhombated diacosipentacontahexazetton							80640	13440
40		t1,2,3{36,4}	Great birhombated diacosipentacontahexazetton							73920	13440
41		t0,1,4{36,4}	Cellitruncated diacosipentacontahexazetton							394240	35840
42		t0,2,4{36,4}	Cellirhombated diacosipentacontahexazetton							483840	53760
43		t1,2,4{36,4}	Biprismatotruncated diacosipentacontahexazetton							430080	53760
44		t0,3,4{36,4}	Celliprismated diacosipentacontahexazetton							215040	35840
45		t1,3,4{36,4}	Biprismatorhombated diacosipentacontahexazetton							322560	53760
46		t2,3,4{36,4}	Great trirhombated diacosipentacontahexazetton							179200	35840
47		t0,1,5{36,4}	Teritruncated diacosipentacontahexazetton							564480	53760
48		t0,2,5{36,4}	Terirhombated diacosipentacontahexazetton							1075200	107520
49		t1,2,5{36,4}	Bicellitruncated diacosipentacontahexazetton							913920	107520
50		t0,3,5{36,4}	Teriprismated diacosipentacontahexazetton							913920	107520
51		t1,3,5{36,4}	Bicellirhombated diacosipentacontahexazetton							1290240	161280
52		t2,3,5{36,4}	Triprismatotruncated diacosipentacontahexazetton							698880	107520
53		t0,4,5{36,4}	Tericellated diacosipentacontahexazetton							322560	53760
54		t1,4,5{36,4}	Bicelliprismated diacosipentacontahexazetton							698880	107520
55		t2,3,5{4,36}	Triprismatotruncated octeract							645120	107520
56		t2,3,4{4,36}	Great trirhombated octeract							241920	53760
57		t0,1,6{36,4}	Petitruncated diacosipentacontahexazetton							344064	43008
58		t0,2,6{36,4}	Petirhombated diacosipentacontahexazetton							967680	107520
59		t1,2,6{36,4}	Biteritruncated diacosipentacontahexazetton							752640	107520
60		t0,3,6{36,4}	Petiprismated diacosipentacontahexazetton							1290240	143360
61		t1,3,6{36,4}	Biterirhombated diacosipentacontahexazetton							1720320	215040
62		t1,4,5{4,36}	Bicelliprismated octeract							860160	143360
63		t0,4,6{36,4}	Peticellated diacosipentacontahexazetton							860160	107520
64		t1,3,6{4,36}	Biterirhombated octeract							1720320	215040
65		t1,3,5{4,36}	Bicellirhombated octeract							1505280	215040
66		t1,3,4{4,36}	Biprismatorhombated octeract							537600	107520
67		t0,5,6{36,4}	Petiterated diacosipentacontahexazetton							258048	43008
68		t1,2,6{4,36}	Biteritruncated octeract							752640	107520
69		t1,2,5{4,36}	Bicellitruncated octeract							1003520	143360
70		t1,2,4{4,36}	Biprismatotruncated octeract							645120	107520
71		t1,2,3{4,36}	Great birhombated octeract							172032	43008
72		t0,1,7{36,4}	Exitruncated diacosipentacontahexazetton							93184	14336
73		t0,2,7{36,4}	Exirhombated diacosipentacontahexazetton							365568	43008
74		t0,5,6{4,36}	Petiterated octeract							258048	43008
75		t0,3,7{36,4}	Exiprismated diacosipentacontahexazetton							680960	71680
76		t0,4,6{4,36}	Peticellated octeract							860160	107520
77		t0,4,5{4,36}	Tericellated octeract							394240	71680
78		t0,3,7{4,36}	Exiprismated octeract							680960	71680
79		t0,3,6{4,36}	Petiprismated octeract							1290240	143360
80		t0,3,5{4,36}	Teriprismated octeract							1075200	143360
81		t0,3,4{4,36}	Celliprismated octeract							358400	71680
82		t0,2,7{4,36}	Exirhombated octeract							365568	43008
83		t0,2,6{4,36}	Petirhombated octeract							967680	107520
84		t0,2,5{4,36}	Terirhombated octeract							1218560	143360
85		t0,2,4{4,36}	Cellirhombated octeract							752640	107520
86		t0,2,3{4,36}	Prismatorhombated octeract							193536	43008
87		t0,1,7{4,36}	Exitruncated octeract							93184	14336
88		t0,1,6{4,36}	Petitruncated octeract							344064	43008
89		t0,1,5{4,36}	Teritruncated octeract							609280	71680
90		t0,1,4{4,36}	Cellitruncated octeract							573440	71680
91		t0,1,3{4,36}	Prismatotruncated octeract							279552	43008
92		t0,1,2{4,36}	Great rhombated octeract							57344	14336
93		t0,1,2,3{36,4}	Great prismated diacosipentacontahexazetton							147840	26880
94		t0,1,2,4{36,4}	Celligreatorhombated diacosipentacontahexazetton							860160	107520
95		t0,1,3,4{36,4}	Celliprismatotruncated diacosipentacontahexazetton							591360	107520
96		t0,2,3,4{36,4}	Celliprismatorhombated diacosipentacontahexazetton							591360	107520
97		t1,2,3,4{36,4}	Great biprismated diacosipentacontahexazetton							537600	107520
98		t0,1,2,5{36,4}	Terigreatorhombated diacosipentacontahexazetton							1827840	215040
99		t0,1,3,5{36,4}	Teriprismatotruncated diacosipentacontahexazetton							2419200	322560
100		t0,2,3,5{36,4}	Teriprismatorhombated diacosipentacontahexazetton							2257920	322560
101		t1,2,3,5{36,4}	Bicelligreatorhombated diacosipentacontahexazetton							2096640	322560
102		t0,1,4,5{36,4}	Tericellitruncated diacosipentacontahexazetton							1182720	215040
103		t0,2,4,5{36,4}	Tericellirhombated diacosipentacontahexazetton							1935360	322560
104		t1,2,4,5{36,4}	Bicelliprismatotruncated diacosipentacontahexazetton							1612800	322560
105		t0,3,4,5{36,4}	Tericelliprismated diacosipentacontahexazetton							1182720	215040
106		t1,3,4,5{36,4}	Bicelliprismatorhombated diacosipentacontahexazetton							1774080	322560
107		t2,3,4,5{4,36}	Great triprismato-octeractidiacosipentacontahexazetton							967680	215040
108		t0,1,2,6{36,4}	Petigreatorhombated diacosipentacontahexazetton							1505280	215040
109		t0,1,3,6{36,4}	Petiprismatotruncated diacosipentacontahexazetton							3225600	430080
110		t0,2,3,6{36,4}	Petiprismatorhombated diacosipentacontahexazetton							2795520	430080
111		t1,2,3,6{36,4}	Biterigreatorhombated diacosipentacontahexazetton							2580480	430080
112		t0,1,4,6{36,4}	Peticellitruncated diacosipentacontahexazetton							3010560	430080
113		t0,2,4,6{36,4}	Peticellirhombated diacosipentacontahexazetton							4515840	645120
114		t1,2,4,6{36,4}	Biteriprismatotruncated diacosipentacontahexazetton							3870720	645120
115		t0,3,4,6{36,4}	Peticelliprismated diacosipentacontahexazetton							2580480	430080
116		t1,3,4,6{4,36}	Biteriprismatorhombi-octeractidiacosipentacontahexazetton							3870720	645120
117		t1,3,4,5{4,36}	Bicelliprismatorhombated octeract							2150400	430080
118		t0,1,5,6{36,4}	Petiteritruncated diacosipentacontahexazetton							1182720	215040
119		t0,2,5,6{36,4}	Petiterirhombated diacosipentacontahexazetton							2795520	430080
120		t1,2,5,6{4,36}	Bitericellitrunki-octeractidiacosipentacontahexazetton							2150400	430080
121		t0,3,5,6{36,4}	Petiteriprismated diacosipentacontahexazetton							2795520	430080
122		t1,2,4,6{4,36}	Biteriprismatotruncated octeract							3870720	645120
123		t1,2,4,5{4,36}	Bicelliprismatotruncated octeract							1935360	430080
124		t0,4,5,6{36,4}	Petitericellated diacosipentacontahexazetton							1182720	215040
125		t1,2,3,6{4,36}	Biterigreatorhombated octeract							2580480	430080
126		t1,2,3,5{4,36}	Bicelligreatorhombated octeract							2365440	430080
127		t1,2,3,4{4,36}	Great biprismated octeract							860160	215040
128		t0,1,2,7{36,4}	Exigreatorhombated diacosipentacontahexazetton							516096	86016
129		t0,1,3,7{36,4}	Exiprismatotruncated diacosipentacontahexazetton							1612800	215040
130		t0,2,3,7{36,4}	Exiprismatorhombated diacosipentacontahexazetton							1290240	215040
131		t0,4,5,6{4,36}	Petitericellated octeract							1182720	215040
132		t0,1,4,7{36,4}	Exicellitruncated diacosipentacontahexazetton							2293760	286720
133		t0,2,4,7{36,4}	Exicellirhombated diacosipentacontahexazetton							3225600	430080
134		t0,3,5,6{4,36}	Petiteriprismated octeract							2795520	430080
135		t0,3,4,7{4,36}	Exicelliprismato-octeractidiacosipentacontahexazetton							1720320	286720
136		t0,3,4,6{4,36}	Peticelliprismated octeract							2580480	430080
137		t0,3,4,5{4,36}	Tericelliprismated octeract							1433600	286720
138		t0,1,5,7{36,4}	Exiteritruncated diacosipentacontahexazetton							1612800	215040
139		t0,2,5,7{4,36}	Exiterirhombi-octeractidiacosipentacontahexazetton							3440640	430080
140		t0,2,5,6{4,36}	Petiterirhombated octeract							2795520	430080
141		t0,2,4,7{4,36}	Exicellirhombated octeract							3225600	430080
142		t0,2,4,6{4,36}	Peticellirhombated octeract							4515840	645120
143		t0,2,4,5{4,36}	Tericellirhombated octeract							2365440	430080
144		t0,2,3,7{4,36}	Exiprismatorhombated octeract							1290240	215040
145		t0,2,3,6{4,36}	Petiprismatorhombated octeract							2795520	430080
146		t0,2,3,5{4,36}	Teriprismatorhombated octeract							2580480	430080
147		t0,2,3,4{4,36}	Celliprismatorhombated octeract							967680	215040
148		t0,1,6,7{4,36}	Exipetitrunki-octeractidiacosipentacontahexazetton							516096	86016
149		t0,1,5,7{4,36}	Exiteritruncated octeract							1612800	215040
150		t0,1,5,6{4,36}	Petiteritruncated octeract							1182720	215040
151		t0,1,4,7{4,36}	Exicellitruncated octeract							2293760	286720
152		t0,1,4,6{4,36}	Peticellitruncated octeract							3010560	430080
153		t0,1,4,5{4,36}	Tericellitruncated octeract							1433600	286720
154		t0,1,3,7{4,36}	Exiprismatotruncated octeract							1612800	215040
155		t0,1,3,6{4,36}	Petiprismatotruncated octeract							3225600	430080
156		t0,1,3,5{4,36}	Teriprismatotruncated octeract							2795520	430080
157		t0,1,3,4{4,36}	Celliprismatotruncated octeract							967680	215040
158		t0,1,2,7{4,36}	Exigreatorhombated octeract							516096	86016
159		t0,1,2,6{4,36}	Petigreatorhombated octeract							1505280	215040
160		t0,1,2,5{4,36}	Terigreatorhombated octeract							2007040	286720
161		t0,1,2,4{4,36}	Celligreatorhombated octeract							1290240	215040
162		t0,1,2,3{4,36}	Great prismated octeract							344064	86016
163		t0,1,2,3,4{36,4}	Great cellated diacosipentacontahexazetton							1075200	215040
164		t0,1,2,3,5{36,4}	Terigreatoprismated diacosipentacontahexazetton							4193280	645120
165		t0,1,2,4,5{36,4}	Tericelligreatorhombated diacosipentacontahexazetton							3225600	645120
166		t0,1,3,4,5{36,4}	Tericelliprismatotruncated diacosipentacontahexazetton							3225600	645120
167		t0,2,3,4,5{36,4}	Tericelliprismatorhombated diacosipentacontahexazetton							3225600	645120
168		t1,2,3,4,5{36,4}	Great bicellated diacosipentacontahexazetton							2903040	645120
169		t0,1,2,3,6{36,4}	Petigreatoprismated diacosipentacontahexazetton							5160960	860160
170		t0,1,2,4,6{36,4}	Peticelligreatorhombated diacosipentacontahexazetton							7741440	1290240
171		t0,1,3,4,6{36,4}	Peticelliprismatotruncated diacosipentacontahexazetton							7096320	1290240
172		t0,2,3,4,6{36,4}	Peticelliprismatorhombated diacosipentacontahexazetton							7096320	1290240
173		t1,2,3,4,6{36,4}	Biterigreatoprismated diacosipentacontahexazetton							6451200	1290240
174		t0,1,2,5,6{36,4}	Petiterigreatorhombated diacosipentacontahexazetton							4300800	860160
175		t0,1,3,5,6{36,4}	Petiteriprismatotruncated diacosipentacontahexazetton							7096320	1290240
176		t0,2,3,5,6{36,4}	Petiteriprismatorhombated diacosipentacontahexazetton							6451200	1290240
177		t1,2,3,5,6{36,4}	Bitericelligreatorhombated diacosipentacontahexazetton							5806080	1290240
178		t0,1,4,5,6{36,4}	Petitericellitruncated diacosipentacontahexazetton							4300800	860160
179		t0,2,4,5,6{36,4}	Petitericellirhombated diacosipentacontahexazetton							7096320	1290240
180		t1,2,3,5,6{4,36}	Bitericelligreatorhombated octeract							5806080	1290240
181		t0,3,4,5,6{36,4}	Petitericelliprismated diacosipentacontahexazetton							4300800	860160
182		t1,2,3,4,6{4,36}	Biterigreatoprismated octeract							6451200	1290240
183		t1,2,3,4,5{4,36}	Great bicellated octeract							3440640	860160
184		t0,1,2,3,7{36,4}	Exigreatoprismated diacosipentacontahexazetton							2365440	430080
185		t0,1,2,4,7{36,4}	Exicelligreatorhombated diacosipentacontahexazetton							5591040	860160
186		t0,1,3,4,7{36,4}	Exicelliprismatotruncated diacosipentacontahexazetton							4730880	860160
187		t0,2,3,4,7{36,4}	Exicelliprismatorhombated diacosipentacontahexazetton							4730880	860160
188		t0,3,4,5,6{4,36}	Petitericelliprismated octeract							4300800	860160
189		t0,1,2,5,7{36,4}	Exiterigreatorhombated diacosipentacontahexazetton							5591040	860160
190		t0,1,3,5,7{36,4}	Exiteriprismatotruncated diacosipentacontahexazetton							8386560	1290240
191		t0,2,3,5,7{36,4}	Exiteriprismatorhombated diacosipentacontahexazetton							7741440	1290240
192		t0,2,4,5,6{4,36}	Petitericellirhombated octeract							7096320	1290240
193		t0,1,4,5,7{36,4}	Exitericellitruncated diacosipentacontahexazetton							4730880	860160
194		t0,2,3,5,7{4,36}	Exiteriprismatorhombated octeract							7741440	1290240
195		t0,2,3,5,6{4,36}	Petiteriprismatorhombated octeract							6451200	1290240
196		t0,2,3,4,7{4,36}	Exicelliprismatorhombated octeract							4730880	860160
197		t0,2,3,4,6{4,36}	Peticelliprismatorhombated octeract							7096320	1290240
198		t0,2,3,4,5{4,36}	Tericelliprismatorhombated octeract							3870720	860160
199		t0,1,2,6,7{36,4}	Exipetigreatorhombated diacosipentacontahexazetton							2365440	430080
200		t0,1,3,6,7{36,4}	Exipetiprismatotruncated diacosipentacontahexazetton							5591040	860160
201		t0,1,4,5,7{4,36}	Exitericellitruncated octeract							4730880	860160
202		t0,1,4,5,6{4,36}	Petitericellitruncated octeract							4300800	860160
203		t0,1,3,6,7{4,36}	Exipetiprismatotruncated octeract							5591040	860160
204		t0,1,3,5,7{4,36}	Exiteriprismatotruncated octeract							8386560	1290240
205		t0,1,3,5,6{4,36}	Petiteriprismatotruncated octeract							7096320	1290240
206		t0,1,3,4,7{4,36}	Exicelliprismatotruncated octeract							4730880	860160
207		t0,1,3,4,6{4,36}	Peticelliprismatotruncated octeract							7096320	1290240
208		t0,1,3,4,5{4,36}	Tericelliprismatotruncated octeract							3870720	860160
209		t0,1,2,6,7{4,36}	Exipetigreatorhombated octeract							2365440	430080
210		t0,1,2,5,7{4,36}	Exiterigreatorhombated octeract							5591040	860160
211		t0,1,2,5,6{4,36}	Petiterigreatorhombated octeract							4300800	860160
212		t0,1,2,4,7{4,36}	Exicelligreatorhombated octeract							5591040	860160
213		t0,1,2,4,6{4,36}	Peticelligreatorhombated octeract							7741440	1290240
214		t0,1,2,4,5{4,36}	Tericelligreatorhombated octeract							3870720	860160
215		t0,1,2,3,7{4,36}	Exigreatoprismated octeract							2365440	430080
216		t0,1,2,3,6{4,36}	Petigreatoprismated octeract							5160960	860160
217		t0,1,2,3,5{4,36}	Terigreatoprismated octeract							4730880	860160
218		t0,1,2,3,4{4,36}	Great cellated octeract							1720320	430080
219		t0,1,2,3,4,5{36,4}	Great terated diacosipentacontahexazetton							5806080	1290240
220		t0,1,2,3,4,6{36,4}	Petigreatocellated diacosipentacontahexazetton							12902400	2580480
221		t0,1,2,3,5,6{36,4}	Petiterigreatoprismated diacosipentacontahexazetton							11612160	2580480
222		t0,1,2,4,5,6{36,4}	Petitericelligreatorhombated diacosipentacontahexazetton							11612160	2580480
223		t0,1,3,4,5,6{36,4}	Petitericelliprismatotruncated diacosipentacontahexazetton							11612160	2580480
224		t0,2,3,4,5,6{36,4}	Petitericelliprismatorhombated diacosipentacontahexazetton							11612160	2580480
225		t1,2,3,4,5,6{4,36}	Great biteri-octeractidiacosipentacontahexazetton							10321920	2580480
226		t0,1,2,3,4,7{36,4}	Exigreatocellated diacosipentacontahexazetton							8601600	1720320
227		t0,1,2,3,5,7{36,4}	Exiterigreatoprismated diacosipentacontahexazetton							14192640	2580480
228		t0,1,2,4,5,7{36,4}	Exitericelligreatorhombated diacosipentacontahexazetton							12902400	2580480
229		t0,1,3,4,5,7{36,4}	Exitericelliprismatotruncated diacosipentacontahexazetton							12902400	2580480
230		t0,2,3,4,5,7{4,36}	Exitericelliprismatorhombi-octeractidiacosipentacontahexazetton							12902400	2580480
231		t0,2,3,4,5,6{4,36}	Petitericelliprismatorhombated octeract							11612160	2580480
232		t0,1,2,3,6,7{36,4}	Exipetigreatoprismated diacosipentacontahexazetton							8601600	1720320
233		t0,1,2,4,6,7{36,4}	Exipeticelligreatorhombated diacosipentacontahexazetton							14192640	2580480
234		t0,1,3,4,6,7{4,36}	Exipeticelliprismatotrunki-octeractidiacosipentacontahexazetton							12902400	2580480
235		t0,1,3,4,5,7{4,36}	Exitericelliprismatotruncated octeract							12902400	2580480
236		t0,1,3,4,5,6{4,36}	Petitericelliprismatotruncated octeract							11612160	2580480
237		t0,1,2,5,6,7{4,36}	Exipetiterigreatorhombi-octeractidiacosipentacontahexazetton							8601600	1720320
238		t0,1,2,4,6,7{4,36}	Exipeticelligreatorhombated octeract							14192640	2580480
239		t0,1,2,4,5,7{4,36}	Exitericelligreatorhombated octeract							12902400	2580480
240		t0,1,2,4,5,6{4,36}	Petitericelligreatorhombated octeract							11612160	2580480
241		t0,1,2,3,6,7{4,36}	Exipetigreatoprismated octeract							8601600	1720320
242		t0,1,2,3,5,7{4,36}	Exiterigreatoprismated octeract							14192640	2580480
243		t0,1,2,3,5,6{4,36}	Petiterigreatoprismated octeract							11612160	2580480
244		t0,1,2,3,4,7{4,36}	Exigreatocellated octeract							8601600	1720320
245		t0,1,2,3,4,6{4,36}	Petigreatocellated octeract							12902400	2580480
246		t0,1,2,3,4,5{4,36}	Great terated octeract							6881280	1720320
247		t0,1,2,3,4,5,6{36,4}	Great petated diacosipentacontahexazetton							20643840	5160960
248		t0,1,2,3,4,5,7{36,4}	Exigreatoterated diacosipentacontahexazetton							23224320	5160960
249		t0,1,2,3,4,6,7{36,4}	Exipetigreatocellated diacosipentacontahexazetton							23224320	5160960
250		t0,1,2,3,5,6,7{36,4}	Exipetiterigreatoprismated diacosipentacontahexazetton							23224320	5160960
251		t0,1,2,3,5,6,7{4,36}	Exipetiterigreatoprismated octeract							23224320	5160960
252		t0,1,2,3,4,6,7{4,36}	Exipetigreatocellated octeract							23224320	5160960
253		t0,1,2,3,4,5,7{4,36}	Exigreatoterated octeract							23224320	5160960
254		t0,1,2,3,4,5,6{4,36}	Great petated octeract							20643840	5160960
255		t0,1,2,3,4,5,6,7{4,36}	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	=	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 h2,3{4,3,3,3,3,3,3}	(1,1,3,5,5,5,5,5)							86016	21504	4.1231055
9	=	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	=	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	=	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	=	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	=	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	=	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	=	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	=	hexisteric 8-cube h4,6{4,3,3,3,3,3,3}	(1,1,1,1,3,3,5,5)							483840	53760	3
17	=	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	=	hepticantic 8-cube h2,7{4,3,3,3,3,3,3}	(1,1,3,3,3,3,3,5)							172032	21504	3
19	=	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	=	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	=	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	=	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	=	steriruncicantic 8-cube h2,3,4{4,36}	(1,1,3,5,7,7,7,7)							430080	107520	5.3851647
24	=	pentiruncicantic 8-cube h2,3,5{4,36}	(1,1,3,5,5,7,7,7)							1182720	215040	5.0990195
25	=	pentistericantic 8-cube h2,4,5{4,36}	(1,1,3,3,5,7,7,7)							1075200	215040	4.8989797
26	=	pentisterirunic 8-cube h3,4,5{4,36}	(1,1,1,3,5,7,7,7)							716800	143360	4.7958317
27	=	hexiruncicantic 8-cube h2,3,6{4,36}	(1,1,3,5,5,5,7,7)							1290240	215040	4.7958317
28	=	hexistericantic 8-cube h2,4,6{4,36}	(1,1,3,3,5,5,7,7)							2096640	322560	4.5825758
29	=	hexisterirunic 8-cube h3,4,6{4,36}	(1,1,1,3,5,5,7,7)							1290240	215040	4.472136
30	=	hexipenticantic 8-cube h2,5,6{4,36}	(1,1,3,3,3,5,7,7)							1290240	215040	4.3588991
31	=	hexipentirunic 8-cube h3,5,6{4,36}	(1,1,1,3,3,5,7,7)							1397760	215040	4.2426405
32	=	hexipentisteric 8-cube h4,5,6{4,36}	(1,1,1,1,3,5,7,7)							698880	107520	4.1231055
33	=	heptiruncicantic 8-cube h2,3,7{4,36}	(1,1,3,5,5,5,5,7)							591360	107520	4.472136
34	=	heptistericantic 8-cube h2,4,7{4,36}	(1,1,3,3,5,5,5,7)							1505280	215040	4.2426405
35	=	heptisterruncic 8-cube h3,4,7{4,36}	(1,1,1,3,5,5,5,7)							860160	143360	4.1231055
36	=	heptipenticantic 8-cube h2,5,7{4,36}	(1,1,3,3,3,5,5,7)							1612800	215040	4
37	=	heptipentiruncic 8-cube h3,5,7{4,36}	(1,1,1,3,3,5,5,7)							1612800	215040	3.8729835
38	=	heptipentisteric 8-cube h4,5,7{4,36}	(1,1,1,1,3,5,5,7)							752640	107520	3.7416575
39	=	heptihexicantic 8-cube h2,6,7{4,36}	(1,1,3,3,3,3,5,7)							752640	107520	3.7416575
40	=	heptihexiruncic 8-cube h3,6,7{4,36}	(1,1,1,3,3,3,5,7)							1146880	143360	3.6055512
41	=	heptihexisteric 8-cube h4,6,7{4,36}	(1,1,1,1,3,3,5,7)							913920	107520	3.4641016
42	=	heptihexipentic 8-cube h5,6,7{4,36}	(1,1,1,1,1,3,5,7)							365568	43008	3.3166249
43	=	pentisteriruncicantic 8-cube h2,3,4,5{4,36}	(1,1,3,5,7,9,9,9)							1720320	430080	6.4031243
44	=	hexisteriruncicantic 8-cube h2,3,4,6{4,36}	(1,1,3,5,7,7,9,9)							3225600	645120	6.0827627
45	=	hexipentiruncicantic 8-cube h2,3,5,6{4,36}	(1,1,3,5,5,7,9,9)							2903040	645120	5.8309517
46	=	hexipentistericantic 8-cube h2,4,5,6{4,36}	(1,1,3,3,5,7,9,9)							3225600	645120	5.6568542
47	=	hexipentisteriruncic 8-cube h3,4,5,6{4,36}	(1,1,1,3,5,7,9,9)							2150400	430080	5.5677648
48	=	heptsteriruncicantic 8-cube h2,3,4,7{4,36}	(1,1,3,5,7,7,7,9)							2150400	430080	5.7445626
49	=	heptipentiruncicantic 8-cube h2,3,5,7{4,36}	(1,1,3,5,5,7,7,9)							3548160	645120	5.4772258
50	=	heptipentistericantic 8-cube h2,4,5,7{4,36}	(1,1,3,3,5,7,7,9)							3548160	645120	5.291503
51	=	heptipentisteriruncic 8-cube h3,4,5,7{4,36}	(1,1,1,3,5,7,7,9)							2365440	430080	5.1961527
52	=	heptihexiruncicantic 8-cube h2,3,6,7{4,36}	(1,1,3,5,5,5,7,9)							2150400	430080	5.1961527
53	=	heptihexistericantic 8-cube h2,4,6,7{4,36}	(1,1,3,3,5,5,7,9)							3870720	645120	5
54	=	heptihexisteriruncic 8-cube h3,4,6,7{4,36}	(1,1,1,3,5,5,7,9)							2365440	430080	4.8989797
55	=	heptihexipenticantic 8-cube h2,5,6,7{4,36}	(1,1,3,3,3,5,7,9)							2580480	430080	4.7958317
56	=	heptihexipentiruncic 8-cube h3,5,6,7{4,36}	(1,1,1,3,3,5,7,9)							2795520	430080	4.6904159
57	=	heptihexipentisteric 8-cube h4,5,6,7{4,36}	(1,1,1,1,3,5,7,9)							1397760	215040	4.5825758
58	=	hexipentisteriruncicantic 8-cube h2,3,4,5,6{4,36}	(1,1,3,5,7,9,11,11)							5160960	1290240	7.1414285
59	=	heptipentisteriruncicantic 8-cube h2,3,4,5,7{4,36}	(1,1,3,5,7,9,9,11)							5806080	1290240	6.78233
60	=	heptihexisteriruncicantic 8-cube h2,3,4,6,7{4,36}	(1,1,3,5,7,7,9,11)							5806080	1290240	6.480741
61	=	heptihexipentiruncicantic 8-cube h2,3,5,6,7{4,36}	(1,1,3,5,5,7,9,11)							5806080	1290240	6.244998
62	=	heptihexipentistericantic 8-cube h2,4,5,6,7{4,36}	(1,1,3,3,5,7,9,11)							6451200	1290240	6.0827627
63	=	heptihexipentisteriruncic 8-cube h3,4,5,6,7{4,36}	(1,1,1,3,5,7,9,11)							4300800	860160	6.0000000
64	=	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		421 (fy)	19440	207360	483840	483840	241920	60480	6720	240
2		(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										138240
10		142 (bif)	2400	106080	725760	2298240	3628800	2419200	483840	17280
11										967680
12										696729600

Regular and uniform honeycombs[]

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

#	Coxeter group		Coxeter diagram	Forms
1	${\displaystyle {\tilde {A}}_{7}}$	[3[8]]		29
2	${\displaystyle {\tilde {C}}_{7}}$	[4,35,4]		135
3	${\displaystyle {\tilde {B}}_{7}}$	[4,34,31,1]		191 (64 new)
4	${\displaystyle {\tilde {D}}_{7}}$	[31,1,33,31,1]		77 (10 new)
5	${\displaystyle {\tilde {E}}_{7}}$	[33,3,1]		143

Regular and uniform tessellations include:

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

${\displaystyle {\bar {P}}_{7}}$ = [3,3[7]]:

${\displaystyle {\bar {Q}}_{7}}$ = [31,1,32,32,1]:

${\displaystyle {\bar {S}}_{7}}$ = [4,33,32,1]:

${\displaystyle {\bar {T}}_{7}}$ = [33,2,2]:

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, 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[]

• Polytope names
• Polytopes of Various Dimensions
• Multi-dimensional Glossary
• Glossary for hyperspace, George Olshevsky.

v t Fundamental convex regular and uniform polytopes in dimensions 2–10
Family	An	Bn	I2(p) / Dn	E6 / E7 / E8 / F4 / G2	Hn
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	1k2 • 2k1 • k21	n-pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds