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Runcinated 5-orthoplexes

Orthogonal projections in B5 Coxeter plane

In five-dimensional geometry, a runcinated 5-orthoplex is a convex uniform 5-polytope with 3rd order truncation (runcination) of the regular 5-orthoplex.

There are 8 runcinations of the 5-orthoplex with permutations of truncations, and cantellations. Four are more simply constructed relative to the 5-cube.

Runcinated 5-orthoplex

Propertiesconvex

Alternate names

  • Runcinated pentacross
  • Small prismated triacontiditeron (Acronym: spat) (Jonathan Bowers)

Coordinates

The vertices of the can be made in 5-space, as permutations and sign combinations of: : (0,1,1,1,2)

Images

Runcitruncated 5-orthoplex

Runcitruncated 5-orthoplex
Type
Schläfli symbol
Coxeter-Dynkin diagrams
4-faces
Cells
Faces
Edges
Vertices
Vertex figure
Coxeter groups
Properties

Alternate names

  • Runcitruncated pentacross
  • Prismatotruncated triacontiditeron (Acronym: pattit) (Jonathan Bowers)

Coordinates

Cartesian coordinates for the vertices of a runcitruncated 5-orthoplex, centered at the origin, are all 80 vertices are sign (4) and coordinate (20) permutations of : (±3,±2,±1,±1,0)

Images

Runcicantellated 5-orthoplex

Propertiesconvex

Alternate names

  • Runcicantellated pentacross
  • Prismatorhombated triacontiditeron (Acronym: pirt) (Jonathan Bowers)

Coordinates

The vertices of the runcicantellated 5-orthoplex can be made in 5-space, as permutations and sign combinations of: : (0,1,2,2,3)

Images

Runcicantitruncated 5-orthoplex

Propertiesconvex, isogonal

Alternate names

  • Runcicantitruncated pentacross
  • Great prismated triacontiditeron (gippit) (Jonathan Bowers)

Coordinates

The Cartesian coordinates of the vertices of a runcicantitruncated 5-orthoplex having an edge length of are given by all permutations of coordinates and sign of:

:\left(0, 1, 2, 3, 4\right)

Images

Snub 5-demicube

The snub 5-demicube defined as an alternation of the omnitruncated 5-demicube is not uniform, but it can be given Coxeter diagram or and symmetry [32,1,1]+ or [4,(3,3,3)+], and constructed from 10 snub 24-cells, 32 snub 5-cells, 40 snub tetrahedral antiprisms, 80 2-3 duoantiprisms, and 960 irregular 5-cells filling the gaps at the deleted vertices.

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, wiley.com,
      • (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.
  • x3o3o3x4o - spat, x3x3o3x4o - pattit, x3o3x3x4o - pirt, x3x3x3x4o - gippit

References

  1. Klitzing, (x3o3o3x4o - spat)
  2. Klitzing, (x3x3o3x4o - pattit)
  3. Klitzing, (x3o3x3x4o - pirt)
  4. Klitzing, (x3x3x3x4o - gippit)
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