Cantellated 5-orthoplexes


5-orthoplex
Cantellated 5-orthoplex
Bicantellated 5-cube
Cantellated 5-cube

5-cube
Cantitruncated 5-orthoplex
Bicantitruncated 5-cube
Cantitruncated 5-cube
Orthogonal projections in B5 Coxeter plane

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

There are 6 cantellation for the 5-orthoplex, including truncations. Some of them are more easily constructed from the dual 5-cube.

Cantellated 5-orthoplex

Cantellated 5-orthoplex
Type Uniform 5-polytope
Schläfli symbol rr{3,3,3,4}
rr{3,3,31,1}
Coxeter-Dynkin diagram
4-faces 82
Cells 640
Faces 1520
Edges 1200
Vertices 240
Vertex figure
Coxeter group B5 [4,3,3,3]
D5 [32,1,1]
Properties convex

Alternate names

  • Cantellated 5-orthoplex
  • Bicantellated 5-demicube
  • Small rhombated triacontiditeron (Acronym: sart) (Jonathan Bowers)[1]

Coordinates

The vertices of the can be made in 5-space, as permutations and sign combinations of:

(0,0,1,1,2)

Images

The cantellated 5-orthoplex is constructed by a cantellation operation applied to the 5-orthoplex.

orthographic projections
Coxeter plane B5 B4 / D5 B3 / D4 / A2
Graph
Dihedral symmetry [10] [8] [6]
Coxeter plane B2 A3
Graph
Dihedral symmetry [4] [4]

Cantitruncated 5-orthoplex

Cantitruncated 5-orthoplex
Typeuniform 5-polytope
Schläfli symboltr{3,3,3,4}
tr{3,31,1}
Coxeter-Dynkin diagrams
4-faces82
Cells640
Faces1520
Edges1440
Vertices480
Vertex figure
Coxeter groupsB5, [3,3,3,4]
D5, [32,1,1]
Propertiesconvex

Alternate names

  • Cantitruncated pentacross
  • Cantitruncated triacontiditeron (Acronym: gart) (Jonathan Bowers)[2]

Coordinates

Cartesian coordinates for the vertices of a cantitruncated 5-orthoplex, centered at the origin, are all sign and coordinate permutations of

(±3,±2,±1,0,0)

Images

orthographic projections
Coxeter plane B5 B4 / D5 B3 / D4 / A2
Graph
Dihedral symmetry [10] [8] [6]
Coxeter plane B2 A3
Graph
Dihedral symmetry [4] [4]

These polytopes are from a set of 31 uniform 5-polytopes generated from the regular 5-cube or 5-orthoplex.

Notes

  1. ↑ Klitizing, (x3o3x3o4o - sart)
  2. ↑ Klitizing, (x3x3x3o4o - gart)

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, ISBN 978-0-471-01003-6
      • (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.
  • Klitzing, Richard. "5D uniform polytopes (polytera)". x3o3x3o4o - sart, x3x3x3o4o - gart
Fundamental convex regular and uniform polytopes in dimensions 2–10
Family An Bn I2(p) / Dn E6 / E7 / E8 / F4 / G2 Hn
Regular polygon Triangle Square p-gon Hexagon Pentagon
Uniform polyhedron Tetrahedron Octahedron • Cube Demicube Dodecahedron • Icosahedron
Uniform 4-polytope 5-cell 16-cell • Tesseract Demitesseract 24-cell 120-cell • 600-cell
Uniform 5-polytope 5-simplex 5-orthoplex • 5-cube 5-demicube
Uniform 6-polytope 6-simplex 6-orthoplex • 6-cube 6-demicube 122 • 221
Uniform 7-polytope 7-simplex 7-orthoplex • 7-cube 7-demicube 132 • 231 • 321
Uniform 8-polytope 8-simplex 8-orthoplex • 8-cube 8-demicube 142 • 241 • 421
Uniform 9-polytope 9-simplex 9-orthoplex • 9-cube 9-demicube
Uniform 10-polytope 10-simplex 10-orthoplex • 10-cube 10-demicube
Uniform n-polytope n-simplex n-orthoplex • n-cube n-demicube 1k2 • 2k1 • k21 n-pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds
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