Cantellated 5-cubes

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5-cube t0.svg
5-cube
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
5-cube t02.svg
Cantellated 5-cube
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
5-cube t13.svg
Bicantellated 5-cube
CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
5-cube t24.svg
Cantellated 5-orthoplex
CDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
5-cube t4.svg
5-orthoplex
CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
5-cube t012.svg
Cantitruncated 5-cube
CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
5-cube t123.svg
Bicantitruncated 5-cube
CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
5-cube t234.svg
Cantitruncated 5-orthoplex
CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
Orthogonal projections in B5 Coxeter plane

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

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[edit]

Cantellated 5-cube
Type Uniform 5-polytope
Schläfli symbol rr{4,3,3,3} =
Coxeter-Dynkin diagram CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png = CDel node.pngCDel split1-43.pngCDel nodes 11.pngCDel 3b.pngCDel nodeb.pngCDel 3b.pngCDel nodeb.png
4-faces 122
Cells 680
Faces 1520
Edges 1280
Vertices 320
Vertex figure Cantellated 5-cube vertf.png
Coxeter group B5 [4,3,3,3]
Properties convex

Alternate names[edit]

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

Coordinates[edit]

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

Images[edit]

orthographic projections
Coxeter plane B5 B4 / D5 B3 / D4 / A2
Graph 5-cube t02.svg 5-cube t02 B4.svg 5-cube t02 B3.svg
Dihedral symmetry [10] [8] [6]
Coxeter plane B2 A3
Graph 5-cube t02 B2.svg 5-cube t02 A3.svg
Dihedral symmetry [4] [4]

Bicantellated 5-cube[edit]

Bicantellated 5-cube
Type Uniform 5-polytope
Schläfli symbols 2rr{4,3,3,3} =
r{32,1,1} =
Coxeter-Dynkin diagrams CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png = CDel node.pngCDel split1.pngCDel nodes 11.pngCDel 4a3b.pngCDel nodes.png
CDel nodes 11.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
4-faces 122
Cells 840
Faces 2160
Edges 1920
Vertices 480
Vertex figure Bicantellated penteract verf.png
Coxeter group B5 [4,3,3,3]
Properties convex

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

Alternate names[edit]

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

Coordinates[edit]

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[edit]

orthographic projections
Coxeter plane B5 B4 / D5 B3 / D4 / A2
Graph 5-cube t13.svg 5-cube t13 B4.svg 5-cube t13 B3.svg
Dihedral symmetry [10] [8] [6]
Coxeter plane B2 A3
Graph 5-cube t13 B2.svg 5-cube t13 A3.svg
Dihedral symmetry [4] [4]

Cantitruncated 5-cube[edit]

Cantitruncated 5-cube
Type Uniform 5-polytope
Schläfli symbol tr{4,3,3,3} =
Coxeter-Dynkin
diagram
CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png = CDel node 1.pngCDel split1-43.pngCDel nodes 11.pngCDel 3b.pngCDel nodeb.pngCDel 3b.pngCDel nodeb.png
4-faces 122
Cells 680
Faces 1520
Edges 1600
Vertices 640
Vertex figure Canitruncated 5-cube verf.png
Irr. 5-cell
Coxeter group B5 [4,3,3,3]
Properties convex, isogonal

Alternate names[edit]

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

Coordinates[edit]

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

Images[edit]

orthographic projections
Coxeter plane B5 B4 / D5 B3 / D4 / A2
Graph 5-cube t012.svg 5-cube t012 B4.svg 5-cube t012 B3.svg
Dihedral symmetry [10] [8] [6]
Coxeter plane B2 A3
Graph 5-cube t012 B2.svg 5-cube t012 A3.svg
Dihedral symmetry [4] [4]

Bicantitruncated 5-cube[edit]

Bicantitruncated 5-cube
Type uniform 5-polytope
Schläfli symbol 2tr{3,3,3,4} =
t{32,1,1} =
Coxeter-Dynkin diagrams CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png = CDel node 1.pngCDel split1.pngCDel nodes 11.pngCDel 4a3b.pngCDel nodes.png
CDel nodes 11.pngCDel split2.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
4-faces 122
Cells 840
Faces 2160
Edges 2400
Vertices 960
Vertex figure Bicanitruncated 5-cube verf.png
Coxeter groups B5, [3,3,3,4]
D5, [32,1,1]
Properties convex

Alternate names[edit]

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

Coordinates[edit]

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[edit]

orthographic projections
Coxeter plane B5 B4 / D5 B3 / D4 / A2
Graph 5-cube t123.svg 5-cube t123 B4.svg 5-cube t123 B3.svg
Dihedral symmetry [10] [8] [6]
Coxeter plane B2 A3
Graph 5-cube t123 B2.svg 5-cube t123 A3.svg
Dihedral symmetry [4] [4]

Related polytopes[edit]

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

References[edit]

  • 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 [1]
      • (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

External links[edit]

Fundamental convex regular and uniform polytopes in dimensions 2–10
Family An Bn I2(p) / Dn E6 / E7 / E8 / E9 / E10 / F4 / G2 Hn
Regular polygon Triangle Square p-gon Hexagon Pentagon
Uniform polyhedron Tetrahedron OctahedronCube Demicube DodecahedronIcosahedron
Uniform 4-polytope 5-cell 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