Cantellated 5-cubes

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Cantellated 5-cubes

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

More information Orthogonal projections in B5 Coxeter plane ...
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5-cube
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Cantellated 5-cube
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Bicantellated 5-cube
Thumb
Cantellated 5-orthoplex
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5-orthoplex
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Cantitruncated 5-cube
Thumb
Bicantitruncated 5-cube
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Cantitruncated 5-orthoplex
Orthogonal projections in B5 Coxeter plane
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There are 6 unique cantellation for the 5-cube, including truncations. Half of them are more easily constructed from the dual 5-orthoplex

Cantellated 5-cube

Summarize
Perspective
Cantellated 5-cube
Type Uniform 5-polytope
Schläfli symbol rr{4,3,3,3} =
Coxeter-Dynkin diagram =
4-faces 122 10
80
32
Cells 680 40
320
160
160
Faces 1520 80
480
320
640
Edges 1280 320+960
Vertices 320
Vertex figure
Coxeter group B5 [4,3,3,3]
Properties convex, uniform

Alternate names

  • Small rhombated penteract (Acronym: sirn) (Jonathan Bowers)

Coordinates

The Cartesian coordinates of the vertices of a cantellated 5-cube having edge length 2 are all permutations of:

Images

More information Coxeter plane, B5 ...
orthographic projections
Coxeter plane B5 B4 / D5 B3 / D4 / A2
Graph Thumb Thumb Thumb
Dihedral symmetry [10] [8] [6]
Coxeter plane B2 A3
Graph Thumb Thumb
Dihedral symmetry [4] [4]
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Bicantellated 5-cube

Summarize
Perspective
Bicantellated 5-cube
Type Uniform 5-polytope
Schläfli symbols 2rr{4,3,3,3} =
r{32,1,1} =
Coxeter-Dynkin diagrams =
4-faces 122 10
80
32
Cells 840 40
240
160
320
80
Faces 2160 240
320
960
320
320
Edges 1920 960+960
Vertices 480
Vertex figure
Coxeter groups B5, [3,3,3,4]
D5, [32,1,1]
Properties convex, uniform

In five-dimensional geometry, a bicantellated 5-cube is a uniform 5-polytope.

Alternate names

  • Bicantellated penteract, bicantellated 5-orthoplex, or bicantellated pentacross
  • Small birhombated penteractitriacontiditeron (Acronym: sibrant) (Jonathan Bowers)

Coordinates

The Cartesian coordinates of the vertices of a bicantellated 5-cube having edge length 2 are all permutations of:

(0,1,1,2,2)

Images

More information Coxeter plane, B5 ...
orthographic projections
Coxeter plane B5 B4 / D5 B3 / D4 / A2
Graph Thumb Thumb Thumb
Dihedral symmetry [10] [8] [6]
Coxeter plane B2 A3
Graph Thumb Thumb
Dihedral symmetry [4] [4]
Close




Cantitruncated 5-cube

Summarize
Perspective
Cantitruncated 5-cube
Type Uniform 5-polytope
Schläfli symbol tr{4,3,3,3} =
Coxeter-Dynkin
diagram
=
4-faces 122 10
80
32
Cells 680 40
320
160
160
Faces 1520 80
480
320
640
Edges 1600 320+320+960
Vertices 640
Vertex figure Thumb
Coxeter group B5 [4,3,3,3]
Properties convex, uniform

Alternate names

  • Tricantitruncated 5-orthoplex / tricantitruncated pentacross
  • Great rhombated penteract (girn) (Jonathan Bowers)

Coordinates

The Cartesian coordinates of the vertices of a cantitruncated 5-cube having an edge length of 2 are given by all permutations of coordinates and sign of:

Images

More information Coxeter plane, B5 ...
orthographic projections
Coxeter plane B5 B4 / D5 B3 / D4 / A2
Graph Thumb Thumb Thumb
Dihedral symmetry [10] [8] [6]
Coxeter plane B2 A3
Graph Thumb Thumb
Dihedral symmetry [4] [4]
Close

It is third in a series of cantitruncated hypercubes:

Petrie polygon projections
Truncated cuboctahedron Cantitruncated tesseract Cantitruncated 5-cube Cantitruncated 6-cube Cantitruncated 7-cube Cantitruncated 8-cube

Bicantitruncated 5-cube

Summarize
Perspective
More information 2tr{3,3,3,4} = ...
Bicantitruncated 5-cube
Type uniform 5-polytope
Schläfli symbol 2tr{3,3,3,4} =
t{32,1,1} =
Coxeter-Dynkin diagrams =
4-faces 122 10
80
32
Cells 840 40
240
160
320
80
Faces 2160 240
320
960
320
320
Edges 2400 960+480+960
Vertices 960
Vertex figure
Coxeter groups B5, [3,3,3,4]
D5, [32,1,1]
Properties convex, uniform
Close

Alternate names

  • Bicantitruncated penteract
  • Bicantitruncated pentacross
  • Great birhombated penteractitriacontiditeron (Acronym: gibrant) (Jonathan Bowers)

Coordinates

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

(±3,±3,±2,±1,0)

Images

More information Coxeter plane, B5 ...
orthographic projections
Coxeter plane B5 B4 / D5 B3 / D4 / A2
Graph Thumb Thumb Thumb
Dihedral symmetry [10] [8] [6]
Coxeter plane B2 A3
Graph Thumb Thumb
Dihedral symmetry [4] [4]
Close

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

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, editied 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)". o3o3x3o4x - sirn, o3x3o3x4o - sibrant, o3o3x3x4x - girn, o3x3x3x4o - gibrant
More information Family, Regular polygon ...
Family An Bn I2(p) / Dn E6 / E7 / E8 / F4 / G2 Hn
Regular polygon Triangle Square p-gon Hexagon Pentagon
Uniform polyhedron Tetrahedron OctahedronCube Demicube DodecahedronIcosahedron
Uniform polychoron Pentachoron 16-cellTesseract Demitesseract 24-cell 120-cell600-cell
Uniform 5-polytope 5-simplex 5-orthoplex5-cube 5-demicube
Uniform 6-polytope 6-simplex 6-orthoplex6-cube 6-demicube 122221
Uniform 7-polytope 7-simplex 7-orthoplex7-cube 7-demicube 132231321
Uniform 8-polytope 8-simplex 8-orthoplex8-cube 8-demicube 142241421
Uniform 9-polytope 9-simplex 9-orthoplex9-cube 9-demicube
Uniform 10-polytope 10-simplex 10-orthoplex10-cube 10-demicube
Uniform n-polytope n-simplex n-orthoplexn-cube n-demicube 1k22k1k21 n-pentagonal polytope
Topics: Polytope familiesRegular polytopeList of regular polytopes and compounds
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