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Concept
Uniform 8-polytope
Summary
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:
{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.
The topology of any given 8-polytope is defined by its Betti numbers and torsion coefficients.
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.
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.
Uniform 8-polytopes with reflective symmetry can be generated by these four Coxeter groups, represented by permutations of rings of the Coxeter-Dynkin diagrams:
Selected regular and uniform 8-polytopes from each family include:
Simplex family: A8 [37] -
135 uniform 8-polytopes as permutations of rings in the group diagram, including one regular:
{37} - 8-simplex or ennea-9-tope or enneazetton -
Hypercube/orthoplex family: B8 [4,36] -
255 uniform 8-polytopes as permutations of rings in the group diagram, including two regular ones:
{4,36} - 8-cube or octeract-
{36,4} - 8-orthoplex or octacross -
Demihypercube D8 family: [35,1,1] -
191 uniform 8-polytopes as permutations of rings in the group diagram, including:
{3,35,1} - 8-demicube or demiocteract, 151 - ; also as h{4,36} .
Official source
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