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Cantellated 5-simplexes

Orthogonal projections in A5 Coxeter plane

In five-dimensional geometry, a cantellated 5-simplex is a convex uniform 5-polytope, being a cantellation of the regular 5-simplex.

There are unique 4 degrees of cantellation for the 5-simplex, including truncations.

Cantellated 5-simplex

Propertiesconvex

The cantellated 5-simplex has 60 vertices, 240 edges, 290 faces (200 triangles and 90 squares), 135 cells (30 tetrahedra, 30 octahedra, 15 cuboctahedra and 60 triangular prisms), and 27 4-faces (6 cantellated 5-cell, 6 rectified 5-cells, and 15 tetrahedral prisms).

Alternate names

  • Cantellated hexateron
  • Small rhombated hexateron (Acronym: sarx) (Jonathan Bowers)

Coordinates

The vertices of the cantellated 5-simplex can be most simply constructed on a hyperplane in 6-space as permutations of (0,0,0,1,1,2) or of (0,1,1,2,2,2). These represent positive orthant facets of the cantellated hexacross and bicantellated hexeract respectively.

Images

Bicantellated 5-simplex

Propertiesconvex, isogonal

Alternate names

  • Bicantellated hexateron
  • Small birhombated dodecateron (Acronym: sibrid) (Jonathan Bowers)

Coordinates

The coordinates can be made in 6-space, as 90 permutations of: : (0,0,1,1,2,2)

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

Images

Cantitruncated 5-simplex

Propertiesconvex

Alternate names

  • Cantitruncated hexateron
  • Great rhombated hexateron (Acronym: garx) (Jonathan Bowers)

Coordinates

The vertices of the cantitruncated 5-simplex can be most simply constructed on a hyperplane in 6-space as permutations of (0,0,0,1,2,3) or of (0,1,2,3,3,3). These construction can be seen as facets of the cantitruncated 6-orthoplex or bicantitruncated 6-cube respectively.

Images

Bicantitruncated 5-simplex

Propertiesconvex, isogonal

Alternate names

  • Bicantitruncated hexateron
  • Great birhombated dodecateron(Acronym: gibrid) (Jonathan Bowers)

Coordinates

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

This construction exists as one of 64 orthant facets of the bicantitruncated 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.
  • x3o3x3o3o - sarx, o3x3o3x3o - sibrid, x3x3x3o3o - garx, o3x3x3x3o - gibrid

References

  1. Klitizing, (x3o3x3o3o - sarx)
  2. Klitizing, (o3x3o3x3o - {{not a typo. sibrid)
  3. Klitizing, (x3x3x3o3o - {{not a typo. garx)
  4. Klitizing, (o3x3x3x3o - {{not a typo. gibrid)
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