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Runcinated 5-simplexes

Orthogonal projections in A5 Coxeter plane

In six-dimensional geometry, a runcinated 5-simplex is a convex uniform 5-polytope with 3rd order truncations (Runcination) of the regular 5-simplex.

There are 4 unique runcinations of the 5-simplex with permutations of truncations, and cantellations.

Runcinated 5-simplex

Propertiesconvex

Alternate names

  • Runcinated hexateron
  • Small prismated hexateron (Acronym: spix) (Jonathan Bowers)

Coordinates

The vertices of the runcinated 5-simplex can be most simply constructed on a hyperplane in 6-space as permutations of (0,0,1,1,1,2) or of (0,1,1,1,2,2), seen as facets of a runcinated 6-orthoplex, or a biruncinated 6-cube respectively.

Images

Runcitruncated 5-simplex

Propertiesconvex, isogonal

Alternate names

  • Runcitruncated hexateron
  • Prismatotruncated hexateron (Acronym: pattix) (Jonathan Bowers)

Coordinates

The coordinates can be made in 6-space, as 180 permutations of: : (0,0,1,1,2,3)

This construction exists as one of 64 orthant facets of the runcitruncated 6-orthoplex.

Images

Runcicantellated 5-simplex

Propertiesconvex, isogonal

Alternate names

  • Runcicantellated hexateron
  • Biruncitruncated 5-simplex/hexateron
  • Prismatorhombated hexateron (Acronym: pirx) (Jonathan Bowers)

Coordinates

The coordinates can be made in 6-space, as 180 permutations of: : (0,0,1,2,2,3)

This construction exists as one of 64 orthant facets of the runcicantellated 6-orthoplex.

Images

Runcicantitruncated 5-simplex

Propertiesconvex, isogonal

Alternate names

  • Runcicantitruncated hexateron
  • Great prismated hexateron (Acronym: gippix) (Jonathan Bowers)

Coordinates

The coordinates can be made in 6-space, as 360 permutations of: : (0,0,1,2,3,4)

This construction exists as one of 64 orthant facets of the runcicantitruncated 6-orthoplex.

Images

Notes

References

  • H.S.M. Coxeter:
    • H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973
    • Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471010030.html
      • (Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380–407, MR 2,10]
      • (Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559-591]
      • (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
  • Norman Johnson Uniform Polytopes, Manuscript (1991)
    • N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D.
  • x3o3o3x3o - spidtix, x3x3o3x3o - pattix, x3o3x3x3o - pirx, x3x3x3x3o - gippix

References

  1. Klitizing, (x3o3o3x3o - spidtix)
  2. Klitizing, (x3x3o3x3o - pattix)
  3. Klitizing, (x3o3x3x3o - pirx)
  4. Klitizing, (x3x3x3x3o - gippix)
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