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Uniform 8-polytope

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Regular 8-polytopesEdit

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.

CharacteristicsEdit

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 groupsEdit

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: [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 formsEdit

There are many uniform prismatic families, including:

The A8 familyEdit

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.