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Uniform 10-polytope

Type of geometrical object From Wikipedia, the free encyclopedia

Uniform 10-polytope
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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.

Graphs of three regular and related uniform polytopes.
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10-simplex
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Truncated 10-simplex
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Rectified 10-simplex
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Cantellated 10-simplex
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Runcinated 10-simplex
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Stericated 10-simplex
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Pentellated 10-simplex
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Hexicated 10-simplex
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Heptellated 10-simplex
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Octellated 10-simplex
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Ennecated 10-simplex
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10-orthoplex
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Truncated 10-orthoplex
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Rectified 10-orthoplex
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10-cube
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Truncated 10-cube
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Rectified 10-cube
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10-demicube
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Truncated 10-demicube

A uniform 10-polytope is one which is vertex-transitive, and constructed from uniform facets.

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Regular 10-polytopes

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:

  1. {3,3,3,3,3,3,3,3,3} - 10-simplex
  2. {4,3,3,3,3,3,3,3,3} - 10-cube
  3. {3,3,3,3,3,3,3,3,4} - 10-orthoplex

There are no nonconvex regular 10-polytopes.

Euler characteristic

The topology of any given 10-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 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.[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]

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Uniform 10-polytopes by fundamental Coxeter groups

Uniform 10-polytopes with reflective symmetry can be generated by these three Coxeter groups, represented by permutations of rings of the Coxeter-Dynkin diagrams:

More information #, Coxeter group ...

Selected regular and uniform 10-polytopes from each family include:

  1. Simplex family: A10 [39] -
    • 527 uniform 10-polytopes as permutations of rings in the group diagram, including one regular:
      1. {39} - 10-simplex -
  2. Hypercube/orthoplex family: B10 [4,38] -
    • 1023 uniform 10-polytopes as permutations of rings in the group diagram, including two regular ones:
      1. {4,38} - 10-cube or dekeract -
      2. {38,4} - 10-orthoplex or decacross -
      3. h{4,38} - 10-demicube .
  3. Demihypercube D10 family: [37,1,1] -
    • 767 uniform 10-polytopes as permutations of rings in the group diagram, including:
      1. 17,1 - 10-demicube or demidekeract -
      2. 71,1 - 10-orthoplex -

The A10 family

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The A10 family has symmetry of order 39,916,800 (11 factorial).

There are 512+16-1=527 forms based on all permutations of the Coxeter-Dynkin diagrams with one or more rings. 31 are shown below: all one and two ringed forms, and the final omnitruncated form. Bowers-style acronym names are given in parentheses for cross-referencing.

More information #, Graph ...

The B10 family

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Perspective

There are 1023 forms based on all permutations of the Coxeter-Dynkin diagrams with one or more rings.

Twelve cases are shown below: ten single-ring (rectified) forms, and two truncations. Bowers-style acronym names are given in parentheses for cross-referencing.

More information #, Graph ...

The D10 family

The D10 family has symmetry of order 1,857,945,600 (10 factorial × 29).

This family has 3×256−1=767 Wythoffian uniform polytopes, generated by marking one or more nodes of the D10 Coxeter-Dynkin diagram. Of these, 511 (2×256−1) are repeated from the B10 family and 256 are unique to this family, with 2 listed below. Bowers-style acronym names are given in parentheses for cross-referencing.

More information #, Graph ...
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Regular and uniform honeycombs

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There are four fundamental affine Coxeter groups that generate regular and uniform tessellations in 9-space:

More information , ...

Regular and uniform tessellations include:

Regular and uniform hyperbolic honeycombs

There are no compact hyperbolic Coxeter groups of rank 10, groups that can generate honeycombs with all finite facets, and a finite vertex figure. However, there are 3 paracompact hyperbolic Coxeter groups of rank 9, each generating uniform honeycombs in 9-space as permutations of rings of the Coxeter diagrams.

= [31,1,34,32,1]:
= [4,35,32,1]:
or = [36,2,1]:

Three honeycombs from the family, generated by end-ringed Coxeter diagrams are:

References

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