Uniform 8-polytope

Graphs of three regular and related uniform polytopes.

8-simplex				Rectified 8-simplex				Truncated 8-simplex
Cantellated 8-simplex				Runcinated 8-simplex				Stericated 8-simplex
Pentellated 8-simplex				Hexicated 8-simplex				Heptellated 8-simplex
8-orthoplex				Rectified 8-orthoplex				Truncated 8-orthoplex
Cantellated 8-orthoplex						Runcinated 8-orthoplex
Hexicated 8-orthoplex						Cantellated 8-cube
Runcinated 8-cube				Stericated 8-cube				Pentellated 8-cube
Hexicated 8-cube						Heptellated 8-cube
8-cube				Rectified 8-cube				Truncated 8-cube
8-demicube				Truncated 8-demicube				Cantellated 8-demicube
Runcinated 8-demicube						Stericated 8-demicube
Pentellated 8-demicube						Hexicated 8-demicube
421				142				241

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

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

Regular 8-polytopes 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.

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 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,

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 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.