Cantellated 5-orthoplexes

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


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

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 diagrams
4-faces 82 10
40
32
Cells 640 80
160
320
80
Faces 1520 640
320
480
80
Edges 1200 960
240
Vertices 240
Vertex figure Square pyramidal prism
Coxeter group B5, [4,3,3,3], order 3840
D5, [32,1,1], order 1920
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-faces 82 10
40
32
Cells 640 80
160
320
80
Faces 1520 640
320
480
80
Edges 1440 960
240
240
Vertices 480
Vertex figureSquare pyramidal pyramid
Coxeter groupsB5, [3,3,3,4], order 3840
D5, [32,1,1], order 1920
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.

B5 polytopes

β5
t1β5
t2γ5
t1γ5
γ5
t0,1β5
t0,2β5
t1,2β5

t0,3β5
t1,3γ5
t1,2γ5
t0,4γ5
t0,3γ5
t0,2γ5
t0,1γ5
t0,1,2β5

t0,1,3β5
t0,2,3β5
t1,2,3γ5
t0,1,4β5
t0,2,4γ5
t0,2,3γ5
t0,1,4γ5
t0,1,3γ5

t0,1,2γ5
t0,1,2,3β5
t0,1,2,4β5
t0,1,3,4γ5
t0,1,2,4γ5
t0,1,2,3γ5
t0,1,2,3,4γ5

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 polychoron Pentachoron 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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