# Uniform 8-polytope

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

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

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

## Regular 8-polytopes

Regular 8-polytopes can be represented by the Schläfli symbol {p,q,r,s,t,u,v}, with v {p,q,r,s,t,u} 7-polytope facets around each peak.

There are exactly three such convex regular 8-polytopes:

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

There are no nonconvex regular 8-polytopes.

## Characteristics

The topology of any given 8-polytope is defined by its Betti numbers and torsion coefficients.[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
1A8 [37]135
2BC8[4,36]255
3D8[35,1,1]191 (64 unique)
4E8[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: [35,1,1] -
• 191 uniform 8-polytopes as permutations of rings in the group diagram, including:
1. {3,35,1} - 8-demicube or demiocteract, 151 - ; also as h{4,36} .
2. {3,3,3,3,3,31,1} - 8-orthoplex, 511 -
4. E-polytope family E8 family: [34,1,1] -
• 255 uniform 8-polytopes as permutations of rings in the group diagram, including:
1. {3,3,3,3,32,1} - Thorold Gosset's semiregular 421,
2. {3,34,2} - the uniform 142, ,
3. {3,3,34,1} - the uniform 241,

### Uniform prismatic forms

There are many uniform prismatic families, including:

### 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.

### 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.