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Concept
Uniform 10-polytope
Summary
In ten-dimensional geometry, a 10-polytope is a 10-dimensional polytope whose boundary consists of 9-polytope facets, exactly two such facets meeting at each 8-polytope ridge.
A uniform 10-polytope is one which is vertex-transitive, and constructed from uniform facets.
Regular 10-polytopes can be represented by the Schläfli symbol {p,q,r,s,t,u,v,w,x}, with x {p,q,r,s,t,u,v,w} 9-polytope facets around each peak.
There are exactly three such convex regular 10-polytopes:
{3,3,3,3,3,3,3,3,3} - 10-simplex
{4,3,3,3,3,3,3,3,3} - 10-cube
{3,3,3,3,3,3,3,3,4} - 10-orthoplex
There are no nonconvex regular 10-polytopes.
The topology of any given 10-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 10-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 10-polytopes with reflective symmetry can be generated by these three Coxeter groups, represented by permutations of rings of the Coxeter-Dynkin diagrams:
Selected regular and uniform 10-polytopes from each family include:
Simplex family: A10 [39] -
527 uniform 10-polytopes as permutations of rings in the group diagram, including one regular:
{39} - 10-simplex -
Hypercube/orthoplex family: B10 [4,38] -
1023 uniform 10-polytopes as permutations of rings in the group diagram, including two regular ones:
{4,38} - 10-cube or dekeract -
{38,4} - 10-orthoplex or decacross -
h{4,38} - 10-demicube .
Demihypercube D10 family: [37,1,1] -
767 uniform 10-polytopes as permutations of rings in the group diagram, including:
17,1 - 10-demicube or demidekeract -
71,1 - 10-orthoplex -
The A10 family has symmetry of order 39,916,800 (11 factorial).
Official source
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