8-polytope - AbsoluteAstronomy.com

Graphs of three regular and related uniform polytope
Uniform polytope
A uniform polytope is a vertex-transitive polytope made from uniform polytope facets of a lower dimension. Uniform polytopes of 2 dimensions are the regular polygons....

s.
8-simplex	Rectified 8-simplex Rectified 8-simplex In eight-dimensional geometry, a rectified 8-simplex is a convex uniform 8-polytope, being a rectification of the regular 8-simplex.There are unique 3 degrees of rectifications. Vertices of the rectified 8-simplex are located at the edge-centers of the 8-simplex. Vertices of the birectified...		Truncated 8-simplex Truncated 8-simplex In eight-dimensional geometry, a truncated 8-simplex is a convex uniform 8-polytope, being a truncation of the regular 8-simplex.There are a four unique degrees of truncation. Vertices of the truncation 8-simplex are located as pairs on the edge of the 8-simplex. Vertices of the bitruncated...
Cantellated 8-simplex Cantellated 8-simplex In eight-dimensional geometry, a cantellated 8-simplex is a convex uniform 8-polytope, being a cantellation of the regular 8-simplex.There are six unique cantellations for the 8-simplex, including permutations of truncation.- Cantellated 8-simplex:...	Runcinated 8-simplex Runcinated 8-simplex In eight-dimensional geometry, a runcinated 8-simplex is a convex uniform 8-polytope with 3rd order truncations of the regular 8-simplex....		Stericated 8-simplex Stericated 8-simplex In eight-dimensional geometry, a stericated 8-simplex is a convex uniform 8-polytope with 4th order truncations of the regular 8-simplex...
Pentellated 8-simplex Pentellated 8-simplex In eight-dimensional geometry, a pentellated 8-simplex is a convex uniform 8-polytope with 5th order truncations of the regular 8-simplex.There are two unique pentellations of the 8-simplex. Including truncations, canetellations, runcinations, and sterications, there are 32 more pentallations....	Hexicated 8-simplex Hexicated 8-simplex In eight-dimensional geometry, a hexicated 8-simplex is a uniform 8-polytope, being a hexication of the regular 8-simplex.- Coordinates :...		Heptellated 8-simplex Heptellated 8-simplex In eight-dimensional geometry, a heptellated 8-simplex is a convex uniform 8-polytope, including 7th-order truncations from the regular 8-simplex....
8-orthoplex	Rectified 8-orthoplex		Truncated 8-orthoplex Truncated 8-orthoplex In eight-dimensional geometry, a truncated 8-orthoplex is a convex uniform 8-polytope, being a truncation of the regular 8-orthoplex.There are 7 truncation for the 8-orthoplex. Vertices of the truncation 8-orthoplex are located as pairs on the edge of the 8-orthoplex. Vertices of the bitruncated...
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 Rectified 8-cube In eight-dimensional geometry, a rectified 8-cube is a convex uniform 8-polytope, being a rectification of the regular 8-cube.There are unique 8 degrees of rectifications, the zeroth being the 8-cube, and the 7th and last being the 8-orthoplex. Vertices of the rectified 8-cube are located at the...		Truncated 8-cube Truncated 8-cube In eight-dimensional geometry, a truncated 8-cube is a convex uniform 8-polytope, being a truncation of the regular 8-cube.There are unique 7 degrees of truncation for the 8-cube. Vertices of the truncation 8-cube are located as pairs on the edge of the 8-cube. Vertices of the bitruncated 8-cube...
8-demicube	Truncated 8-demicube Truncated 8-demicube In eight-dimensional geometry, a truncated 8-demicube is a uniform 8-polytope, being a truncation of the 8-demicube.- Cartesian coordinates :The Cartesian coordinates for the vertices of a truncated 8-demicube centered at the origin and edge length 6√2 are coordinate permutations:with an odd number...		Cantellated 8-demicube
Runcinated 8-demicube		Stericated 8-demicube
Pentellated 8-demicube		Hexicated 8-demicube
4₂₁	1₄₂		2₄₁

In eight-dimensional

Eight-dimensional space

In physics and mathematics, a sequence of n numbers can also be understood as a location in n-dimensional space. When n = 8, the set of all such locations is called 8-dimensional Euclidean space...

geometry

Geometry

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....

, a polyzetton (or 8-polytope) is a polytope

Polytope

In elementary geometry, a polytope is a geometric object with flat sides, which exists in any general number of dimensions. A polygon is a polytope in two dimensions, a polyhedron in three dimensions, and so on in higher dimensions...

contained by 7-polytope facets. Each 6-polytope

6-polytope

In six-dimensional geometry, a uniform polypeton is a six-dimensional uniform polytope. A uniform polypeton is vertex-transitive, and all facets are uniform polytera....

ridge

Ridge (geometry)

In geometry, a ridge is an -dimensional element of an n-dimensional polytope. It is also sometimes called a subfacet for having one lower dimension than a facet.By dimension, this corresponds to:*a vertex of a polygon;...

being shared by exactly two 7-polytope

7-polytope

In seven-dimensional geometry, a 7-polytope is a polytope contained by 6-polytope facets. Each 5-polytope ridge being shared by exactly two 6-polytope facets....

facets

Facet (mathematics)

A facet of a simplicial complex is a maximal simplex.In the general theory of polyhedra and polytopes, two conflicting meanings are currently jostling for acceptability:...

.

A uniform polyzetton is one which is vertex-transitive

Vertex-transitive

In geometry, a polytope is isogonal or vertex-transitive if, loosely speaking, all its vertices are the same...

, and constructed from uniform (7-polytope) facets.

Regular 8-polytopes

Regular polyzetta 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

Facet (mathematics)

A facet of a simplicial complex is a maximal simplex.In the general theory of polyhedra and polytopes, two conflicting meanings are currently jostling for acceptability:...

around each peak

Peak (geometry)

In geometry, a peak is an -face of an n-dimensional polytope. A peak attaches at least three facets ....

.

There are exactly three such convex regular 8-polytopes:

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

There are no nonconvex regular 8-polytopes.

Euler characteristic

The Euler characteristic

Euler characteristic

In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic is a topological invariant, a number that describes a topological space's shape or structure regardless of the way it is bent...

for 8-polytopes that are topological 7-spheres (including all convex 8-polytopes) is zero. χ=V-E+F-C+f₄-f₅+f₆-f₇=0.

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 diagram

Coxeter-Dynkin diagram

In geometry, a Coxeter–Dynkin diagram is a graph with numerically labeled edges representing the spatial relations between a collection of mirrors...

#	Coxeter group Coxeter group In mathematics, a Coxeter group, named after H.S.M. Coxeter, is an abstract group that admits a formal description in terms of mirror symmetries. Indeed, the finite Coxeter groups are precisely the finite Euclidean reflection groups; the symmetry groups of regular polyhedra are an example...		Forms
1	A₈	[3⁷]	135
2	BC₈	[4,3⁶]	255
3	D₈	[3^5,1,1]	191 (64 unique)
4	E₈ E8 (mathematics) In mathematics, E8 is any of several closely related exceptional simple Lie groups, linear algebraic groups or Lie algebras of dimension 248; the same notation is used for the corresponding root lattice, which has rank 8...	[3^4,2,1]	255

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

Simplex
Simplex
In geometry, a simplex is a generalization of the notion of a triangle or tetrahedron to arbitrary dimension. Specifically, an n-simplex is an n-dimensional polytope which is the convex hull of its n + 1 vertices. For example, a 2-simplex is a triangle, a 3-simplex is a tetrahedron,...

family: A₈ [3⁷] -
- 135 uniform 8-polytopes as permutations of rings in the group diagram, including one regular:
  1. {3⁷} - 8-simplex or ennea-9-tope or enneazetton -
Hypercube
Hypercube
In geometry, a hypercube is an n-dimensional analogue of a square and a cube . It is a closed, compact, convex figure whose 1-skeleton consists of groups of opposite parallel line segments aligned in each of the space's dimensions, perpendicular to each other and of the same length.An...

/orthoplex family: B₈ [4,3⁶] -
- 255 uniform 8-polytopes as permutations of rings in the group diagram, including two regular ones:
  1. {4,3⁶} - 8-cube or octeract-
  2. {3⁶,4} - 8-orthoplex or octacross -
Demihypercube D₈ family: [3^5,1,1] -
- 191 uniform 8-polytopes as permutations of rings in the group diagram, including:
  1. {3^1,5,1} - 8-demicube or demiocteract, 1₅₁ - ; also as h{4,3⁶} .
  2. {3^5,1,1} - 8-orthoplex, 5₁₁ -
E-polytope family
Semiregular E-polytope
In geometry, a uniform k21 polytope is a polytope in k + 4 dimensions constructed from the En Coxeter group, and having only regular polytope facets...

E₈ family: [3^4,1,1] -
- 255 uniform 8-polytopes as permutations of rings in the group diagram, including:
  1. {3^4,2,1} - Thorold Gosset
    Thorold Gosset
    Thorold Gosset was an English lawyer and an amateur mathematician. In mathematics, he is noted for discovering and classifying the semiregular polytopes in dimensions four and higher.According to H. S. M...
    
    's semiregular 4₂₁
    Gosset 4 21 polytope
    In 8-dimensional geometry, the 421 is a semiregular uniform 8-polytope, constructed within the symmetry of the E8 group. It was discovered by Thorold Gosset, published in his 1900 paper...
    
    ,
  2. {3^1,4,2} - the uniform 1₄₂
    Gosset 1 42 polytope
    In 8-dimensional geometry, the 142 is a uniform 8-polytope, constructed within the symmetry of the E8 group.Coxeter named it 142 by its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of the 1-node sequences....
    
    , ,
  3. {3^2,4,1} - the uniform 2₄₁
    Gosset 2 41 polytope
    In 8-dimensional geometry, the 241 is a uniform 8-polytope, constructed within the symmetry of the E8 group.Coxeter named it 241 by its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of the 2-node sequences....
    
    ,

Uniform prismatic forms

There are many uniform

Uniform polytope

A uniform polytope is a vertex-transitive polytope made from uniform polytope facets of a lower dimension. Uniform polytopes of 2 dimensions are the regular polygons....

prismatic families, including:

#	Coxeter group Coxeter group In mathematics, a Coxeter group, named after H.S.M. Coxeter, is an abstract group that admits a formal description in terms of mirror symmetries. Indeed, the finite Coxeter groups are precisely the finite Euclidean reflection groups; the symmetry groups of regular polyhedra are an example...
7+1
1	A₇×A₁	[3,3,3,3,3,3]×[ ]
2	BC₇×A₁	[4,3,3,3,3,3]×[ ]
3	D₇×A₁	[3^4,1,1]×[ ]
4	E₇×A₁	[3^3,2,1]×[ ]
6+2
1	A₆×I₂(p)	[3,3,3,3,3]×[p]
2	BC₆×I₂(p)	[4,3,3,3,3]×[p]
3	D₆×I₂(p)	[3^3,1,1]×[p]
4	E₆×I₂(p)	[3,3,3,3,3]×[p]
6+1+1
1	A₆×A₁×A₁	[3,3,3,3,3]×[ ]x[ ]
2	BC₆×A₁×A₁	[4,3,3,3,3]×[ ]x[ ]
3	D₆×A₁×A₁	[3^3,1,1]×[ ]x[ ]
4	E₆×A₁×A₁	[3,3,3,3,3]×[ ]x[ ]
5+3
1	A₅×A₃	[3⁴]×[3,3]
2	BC₅×A₃	[4,3³]×[3,3]
3	D₅×A₃	[3^2,1,1]×[3,3]
4	A₅×BC₃	[3⁴]×[4,3]
5	BC₅×BC₃	[4,3³]×[4,3]
6	D₅×BC₃	[3^2,1,1]×[4,3]
7	A₅×H₃	[3⁴]×[5,3]
8	BC₅×H₃	[4,3³]×[5,3]
9	D₅×H₃	[3^2,1,1]×[5,3]
5+2+1
1	A₅×I₂(p)×A₁	[3,3,3]×[p]×[ ]
2	BC₅×I₂(p)×A₁	[4,3,3]×[p]×[ ]
3	D₅×I₂(p)×A₁	[3^2,1,1]×[p]×[ ]
5+1+1+1
1	A₅×A₁×A₁×A₁	[3,3,3]×[ ]×[ ]×[ ]
2	BC₅×A₁×A₁×A₁	[4,3,3]×[ ]×[ ]×[ ]
3	D₅×A₁×A₁×A₁	[3^2,1,1]×[ ]×[ ]×[ ]
4+4
1	A₄×A₄	[3,3,3]×[3,3,3]
2	BC₄×A₄	[4,3,3]×[3,3,3]
3	D₄×A₄	[3^1,1,1]×[3,3,3]
4	F₄×A₄	[3,4,3]×[3,3,3]
5	H₄×A₄	[5,3,3]×[3,3,3]
6	BC₄×BC₄	[4,3,3]×[4,3,3]
7	D₄×BC₄	[3^1,1,1]×[4,3,3]
8	F₄×BC₄	[3,4,3]×[4,3,3]
9	H₄×BC₄	[5,3,3]×[4,3,3]
10	D₄×D₄	[3^1,1,1]×[3^1,1,1]
11	F₄×D₄	[3,4,3]×[3^1,1,1]
12	H₄×D₄	[5,3,3]×[3^1,1,1]
13	F₄×F₄	[3,4,3]×[3,4,3]
14	H₄×F₄	[5,3,3]×[3,4,3]
15	H₄×H₄	[5,3,3]×[5,3,3]
4+3+1
1	A₄×A₃×A₁	[3,3,3]×[3,3]×[ ]
2	A₄×BC₃×A₁	[3,3,3]×[4,3]×[ ]
3	A₄×H₃×A₁	[3,3,3]×[5,3]×[ ]
4	BC₄×A₃×A₁	[4,3,3]×[3,3]×[ ]
5	BC₄×BC₃×A₁	[4,3,3]×[4,3]×[ ]
6	BC₄×H₃×A₁	[4,3,3]×[5,3]×[ ]
7	H₄×A₃×A₁	[5,3,3]×[3,3]×[ ]
8	H₄×BC₃×A₁	[5,3,3]×[4,3]×[ ]
9	H₄×H₃×A₁	[5,3,3]×[5,3]×[ ]
10	F₄×A₃×A₁	[3,4,3]×[3,3]×[ ]
11	F₄×BC₃×A₁	[3,4,3]×[4,3]×[ ]
12	F₄×H₃×A₁	[3,4,3]×[5,3]×[ ]
13	D₄×A₃×A₁	[3^1,1,1]×[3,3]×[ ]
14	D₄×BC₃×A₁	[3^1,1,1]×[4,3]×[ ]
15	D₄×H₃×A₁	[3^1,1,1]×[5,3]×[ ]
4+2+2
...
4+2+1+1
...
4+1+1+1+1
...
3+3+2
1	A₃×A₃×I₂(p)	[3,3]×[3,3]×[p]
2	BC₃×A₃×I₂(p)	[4,3]×[3,3]×[p]
3	H₃×A₃×I₂(p)	[5,3]×[3,3]×[p]
4	BC₃×BC₃×I₂(p)	[4,3]×[4,3]×[p]
5	H₃×BC₃×I₂(p)	[5,3]×[4,3]×[p]
6	H₃×H₃×I₂(p)	[5,3]×[5,3]×[p]
3+3+1+1
1	A₃×A₃×A₁×A₁	[3,3]×[3,3]×[ ]×[ ]
2	BC₃×A₃×A₁×A₁	[4,3]×[3,3]×[ ]×[ ]
3	H₃×A₃×A₁×A₁	[5,3]×[3,3]×[ ]×[ ]
4	BC₃×BC₃×A₁×A₁	[4,3]×[4,3]×[ ]×[ ]
5	H₃×BC₃×A₁×A₁	[5,3]×[4,3]×[ ]×[ ]
6	H₃×H₃×A₁×A₁	[5,3]×[5,3]×[ ]×[ ]
3+2+2+1
1	A₃×I₂(p)×I₂(q)×A₁	[3,3]×[p]×[q]×[ ]
2	BC₃×I₂(p)×I₂(q)×A₁	[4,3]×[p]×[q]×[ ]
3	H₃×I₂(p)×I₂(q)×A₁	[5,3]×[p]×[q]×[ ]
3+2+1+1+1
1	A₃×I₂(p)×A₁×A₁×A₁	[3,3]×[p]×[ ]x[ ]×[ ]
2	BC₃×I₂(p)×A₁×A₁×A₁	[4,3]×[p]×[ ]x[ ]×[ ]
3	H₃×I₂(p)×A₁×A₁×A₁	[5,3]×[p]×[ ]x[ ]×[ ]
3+1+1+1+1+1
1	A₃×A₁×A₁×A₁×A₁×A₁	[3,3]×[ ]x[ ]×[ ]x[ ]×[ ]
2	BC₃×A₁×A₁×A₁×A₁×A₁	[4,3]×[ ]x[ ]×[ ]x[ ]×[ ]
3	H₃×A₁×A₁×A₁×A₁×A₁	[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)×A₁×A₁	[p]×[q]×[r]×[ ]×[ ]
2+2+1+1+1+1
2	I₂(p)×I₂(q)×A₁×A₁×A₁×A₁	[p]×[q]×[ ]×[ ]×[ ]×[ ]
2+1+1+1+1+1+1
1	I₂(p)×A₁×A₁×A₁×A₁×A₁×A₁	[p]×[ ]×[ ]×[ ]×[ ]×[ ]×[ ]
1+1+1+1+1+1+1+1
1	A₁×A₁×A₁×A₁×A₁×A₁×A₁×A₁	[ ]×[ ]×[ ]×[ ]×[ ]×[ ]×[ ]×[ ]

The A₈ family

The A₈ family has symmetry of order 362880 (9 factorial

Factorial

In mathematics, the factorial of a non-negative integer n, denoted by n!, is the product of all positive integers less than or equal to n...

).

There are 135 forms based on all permutations of the Coxeter-Dynkin diagram

Coxeter-Dynkin diagram

In geometry, a Coxeter–Dynkin diagram is a graph with numerically labeled edges representing the spatial relations between a collection of mirrors...

s 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.

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

The B₈ family

The B₈ family has symmetry of order 10321920 (8 factorial

Factorial

In mathematics, the factorial of a non-negative integer n, denoted by n!, is the product of all positive integers less than or equal to n...

x 2⁸). There are 255 forms based on all permutations of the Coxeter-Dynkin diagram

Coxeter-Dynkin diagram

In geometry, a Coxeter–Dynkin diagram is a graph with numerically labeled edges representing the spatial relations between a collection of mirrors...

s with one or more rings.

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

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

The D₈ family

The D₈ family has symmetry of order 5,160,960 (8 factorial

Factorial

In mathematics, the factorial of a non-negative integer n, denoted by n!, is the product of all positive integers less than or equal to n...

x 2⁷).

This family has 191 Wythoffian uniform polytopes, from 3x64-1 permutations of the D₈ Coxeter-Dynkin diagram

Coxeter-Dynkin diagram

In geometry, a Coxeter–Dynkin diagram is a graph with numerically labeled edges representing the spatial relations between a collection of mirrors...

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.

#	Coxeter-Dynkin diagram Coxeter-Dynkin diagram In geometry, a Coxeter–Dynkin diagram is a graph with numerically labeled edges representing the spatial relations between a collection of mirrors... Name	Base point (Alternately signed)	Element counts								Circumrad
#		Base point (Alternately signed)	7	6	5	4	3	2	1	0	Circumrad
1	8-demicube	(1,1,1,1,1,1,1,1)	144	1136	4032	8288	10752	7168	1792	128	1.0000000
2	Truncated 8-demicube Truncated 8-demicube In eight-dimensional geometry, a truncated 8-demicube is a uniform 8-polytope, being a truncation of the 8-demicube.- Cartesian coordinates :The Cartesian coordinates for the vertices of a truncated 8-demicube centered at the origin and edge length 6√2 are coordinate permutations:with an odd number...	(1,1,3,3,3,3,3,3)							23296	3584	2.6457512
3		(1,1,1,3,3,3,3,3)							64512	7168	2.4494896
4		(1,1,1,1,3,3,3,3)							98560	8960	2.2360678
5		(1,1,1,1,1,3,3,3)							89600	7168	1.9999999
6		(1,1,1,1,1,1,3,3)							48384	3584	1.7320508
7		(1,1,1,1,1,1,1,3)							14336	1024	1.4142135
8		(1,1,3,5,5,5,5,5)							86016	21504	4.1231055
9		(1,1,3,3,5,5,5,5)							349440	53760	3.8729835
10		(1,1,1,3,5,5,5,5)							179200	35840	3.7416575
11		(1,1,3,3,3,5,5,5)							573440	71680	3.6055512
12		(1,1,1,3,3,5,5,5)							537600	71680	3.4641016
13		(1,1,1,1,3,5,5,5)							232960	35840	3.3166249
14		(1,1,3,3,3,3,5,5)							456960	53760	3.3166249
15		(1,1,1,3,3,3,5,5)							645120	71680	3.1622777
16		(1,1,1,1,3,3,5,5)							483840	53760	3
17		(1,1,1,1,1,3,5,5)							182784	21504	2.8284271
18		(1,1,3,3,3,3,3,5)							172032	21504	3
19		(1,1,1,3,3,3,3,5)							340480	35840	2.8284271
20		(1,1,1,1,3,3,3,5)							376320	35840	2.6457512
21		(1,1,1,1,1,3,3,5)							236544	21504	2.4494898
22		(1,1,1,1,1,1,3,5)							78848	7168	2.236068
23		(1,1,3,5,7,7,7,7)							430080	107520	5.3851647
24		(1,1,3,5,5,7,7,7)							1182720	215040	5.0990195
25		(1,1,3,3,5,7,7,7)							1075200	215040	4.8989797
26		(1,1,1,3,5,7,7,7)							716800	143360	4.7958317
27		(1,1,3,5,5,5,7,7)							1290240	215040	4.7958317
28		(1,1,3,3,5,5,7,7)							2096640	322560	4.5825758
29		(1,1,1,3,5,5,7,7)							1290240	215040	4.472136
30		(1,1,3,3,3,5,7,7)							1290240	215040	4.3588991
31		(1,1,1,3,3,5,7,7)							1397760	215040	4.2426405
32		(1,1,1,1,3,5,7,7)							698880	107520	4.1231055
33		(1,1,3,5,5,5,5,7)							591360	107520	4.472136
34		(1,1,3,3,5,5,5,7)							1505280	215040	4.2426405
35		(1,1,1,3,5,5,5,7)							860160	143360	4.1231055
36		(1,1,3,3,3,5,5,7)							1612800	215040	4
37		(1,1,1,3,3,5,5,7)							1612800	215040	3.8729835
38		(1,1,1,1,3,5,5,7)							752640	107520	3.7416575
39		(1,1,3,3,3,3,5,7)							752640	107520	3.7416575
40		(1,1,1,3,3,3,5,7)							1146880	143360	3.6055512
41		(1,1,1,1,3,3,5,7)							913920	107520	3.4641016
42		(1,1,1,1,1,3,5,7)							365568	43008	3.3166249
43		(1,1,3,5,7,9,9,9)							1720320	430080	6.4031243
44		(1,1,3,5,7,7,9,9)							3225600	645120	6.0827627
45		(1,1,3,5,5,7,9,9)							2903040	645120	5.8309517
46		(1,1,3,3,5,7,9,9)							3225600	645120	5.6568542
47		(1,1,1,3,5,7,9,9)							2150400	430080	5.5677648
48		(1,1,3,5,7,7,7,9)							2150400	430080	5.7445626
49		(1,1,3,5,5,7,7,9)							3548160	645120	5.4772258
50		(1,1,3,3,5,7,7,9)							3548160	645120	5.291503
51		(1,1,1,3,5,7,7,9)							2365440	430080	5.1961527
52		(1,1,3,5,5,5,7,9)							2150400	430080	5.1961527
53		(1,1,3,3,5,5,7,9)							3870720	645120	5
54		(1,1,1,3,5,5,7,9)							2365440	430080	4.8989797
55		(1,1,3,3,3,5,7,9)							2580480	430080	4.7958317
56		(1,1,1,3,3,5,7,9)							2795520	430080	4.6904159
57		(1,1,1,1,3,5,7,9)							1397760	215040	4.5825758
58		(1,1,3,5,7,9,11,11)							5160960	1290240	7.1414285
59		(1,1,3,5,7,9,9,11)							5806080	1290240	6.78233
60		(1,1,3,5,7,7,9,11)							5806080	1290240	6.480741
61		(1,1,3,5,5,7,9,11)							5806080	1290240	6.244998
62		(1,1,3,3,5,7,9,11)							6451200	1290240	6.0827627
63		(1,1,1,3,5,7,9,11)							4300800	860160	6.0000000
64		(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 diagram

Coxeter-Dynkin diagram

In geometry, a Coxeter–Dynkin diagram is a graph with numerically labeled edges representing the spatial relations between a collection of mirrors...

s 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.

#	Names	Element counts
#	Names	7-faces	6-faces	5-faces	4-faces	Cells	Faces	Edges	Vertices
1	4₂₁ Gosset 4 21 polytope In 8-dimensional geometry, the 421 is a semiregular uniform 8-polytope, constructed within the symmetry of the E8 group. It was discovered by Thorold Gosset, published in his 1900 paper... (fy)	19440	207360	483840	483840	241920	60480	6720	240
2	Truncated 4₂₁ (tiffy)							188160	13440
3	Rectified 4₂₁ (riffy)	19680	375840	1935360	3386880	2661120	1028160	181440	6720
4	Birectified 4₂₁ (borfy)	19680	382560	2600640	7741440	9918720	5806080	1451520	60480
5	Trirectified 4₂₁ (torfy)	19680	382560	2661120	9313920	16934400	14515200	4838400	241920
6	Rectified 1₄₂ (buffy)	19680	382560	2661120	9072000	16934400	16934400	7257600	483840
7	Rectified 2₄₁ (robay)	19680	313440	1693440	4717440	7257600	5322240	1451520	69120
8	2₄₁ (bay)	17520	144960	544320	1209600	1209600	483840	69120	2160
9	Truncated 2₄₁								138240
10	1₄₂ (bif)	2400	106080	725760	2298240	3628800	2419200	483840	17280
11	Truncated 1₄₂								967680
12	Omnitruncated 4₂₁								696729600

Regular and uniform honeycombs

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

#	Coxeter group Coxeter group In mathematics, a Coxeter group, named after H.S.M. Coxeter, is an abstract group that admits a formal description in terms of mirror symmetries. Indeed, the finite Coxeter groups are precisely the finite Euclidean reflection groups; the symmetry groups of regular polyhedra are an example...		Coxeter-Dynkin diagram Coxeter-Dynkin diagram In geometry, a Coxeter–Dynkin diagram is a graph with numerically labeled edges representing the spatial relations between a collection of mirrors...
1		[3^[8]]
2		[4,3⁵,4]
3		h[4,3⁵,4] [4,3⁴,3^1,1]
4		q[4,3⁵,4] [3^1,1,3³,3^1,1]
5		[3^3,3,1]

Regular and uniform tessellations include:

- Regular hepteractic honeycomb: {4,3⁴,4} = {4,3⁴,3^1,1}, =
- Uniform demihepteractic honeycomb: h{4,3⁴,4} = {3^1,1,3⁴,4}, =
- Uniform 3₃₁ honeycomb: {3^3,3,1},
- Uniform 1₃₃ honeycomb: {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

Vertex figure

In geometry a vertex figure is, broadly speaking, the figure exposed when a corner of a polyhedron or polytope is sliced off.-Definitions - theme and variations:...

. However there are 4 noncompact hyperbolic Coxeter groups of rank 8, each generating uniform honeycombs in 7-space as permutations of rings of the Coxeter diagrams.