Runcinated 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 t03.svg
Runcinated 5-cube
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
5-cube t14.svg
Runcinated 5-orthoplex
CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
5-cube t013.svg
Runcitruncated 5-cube
CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
5-cube t023.svg
Runcicantellated 5-cube
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
5-cube t0123.svg
Runcicantitruncated 5-cube
CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
5-cube t134.svg
Runcitruncated 5-orthoplex
CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
5-cube t124.svg
Runcicantellated 5-orthoplex
CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
5-cube t1234.svg
Runcicantitruncated 5-orthoplex
CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
Orthogonal projections in B5 Coxeter plane

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

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

Runcinated 5-cube[edit]

Runcinated 5-cube
Type Uniform 5-polytope
Schläfli symbol t0,3{4,3,3,3}
Coxeter diagram CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
4-faces 202
Cells 1240
Faces 2160
Edges 1440
Vertices 320
Vertex figure Runcinated penteract verf.png
3-3 duoprism
Coxeter group B5 [4,3,3,3]
Properties convex

Alternate names[edit]

  • Small prismated penteract (Acronym: span) (Jonathan Bowers)

Coordinates[edit]

The Cartesian coordinates of the vertices of a runcinated 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 t03.svg 5-cube t03 B4.svg 5-cube t03 B3.svg
Dihedral symmetry [10] [8] [6]
Coxeter plane B2 A3
Graph 5-cube t03 B2.svg 5-cube t03 A3.svg
Dihedral symmetry [4] [4]

Runcitruncated 5-cube[edit]

Runcitruncated 5-cube
Type uniform polyteron
Schläfli symbol t0,1,3{4,3,3,3}
Coxeter-Dynkin diagrams CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
4-faces 202
Cells 1560
Faces 3760
Edges 3360
Vertices 960
Vertex figure Runcitruncated 5-cube verf.png
Coxeter groups B5, [3,3,3,4]
Properties convex

Alternate names[edit]

  • Runcitruncated penteract
  • Prismatotruncated penteract (Acronym: pattin) (Jonathan Bowers)

Construction and coordinates[edit]

The Cartesian coordinates of the vertices of a runcitruncated 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 t013.svg 5-cube t013 B4.svg 5-cube t013 B3.svg
Dihedral symmetry [10] [8] [6]
Coxeter plane B2 A3
Graph 5-cube t013 B2.svg 5-cube t013 A3.svg
Dihedral symmetry [4] [4]

Runcicantellated 5-cube[edit]

Runcicantellated 5-cube
Type Uniform 5-polytope
Schläfli symbol t0,2,3{4,3,3,3}
Coxeter-Dynkin diagram CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
4-faces 202
Cells 1240
Faces 2960
Edges 2880
Vertices 960
Vertex figure Runcicantellated 5-cube verf.png
Coxeter group B5 [4,3,3,3]
Properties convex

Alternate names[edit]

  • Runcicantellated penteract
  • Prismatorhombated penteract (Acronym: prin) (Jonathan Bowers)

Coordinates[edit]

The Cartesian coordinates of the vertices of a runcicantellated 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 t023.svg 5-cube t023 B4.svg 5-cube t023 B3.svg
Dihedral symmetry [10] [8] [6]
Coxeter plane B2 A3
Graph 5-cube t023 B2.svg 5-cube t023 A3.svg
Dihedral symmetry [4] [4]

Runcicantitruncated 5-cube[edit]

Runcicantitruncated 5-cube
Type Uniform 5-polytope
Schläfli symbol t0,1,2,3{4,3,3,3}
Coxeter-Dynkin
diagram
CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
4-faces 202
Cells 1560
Faces 4240
Edges 4800
Vertices 1920
Vertex figure Runcicantitruncated 5-cube verf.png
Irregular 5-cell
Coxeter group B5 [4,3,3,3]
Properties convex, isogonal

Alternate names[edit]

  • Runcicantitruncated penteract
  • Biruncicantitruncated 16-cell / Biruncicantitruncated pentacross
  • great prismated penteract (gippin) (Jonathan Bowers)

Coordinates[edit]

The Cartesian coordinates of the vertices of a runcicantitruncated 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 t0123.svg 5-cube t0123 B4.svg 5-cube t0123 B3.svg
Dihedral symmetry [10] [8] [6]
Coxeter plane B2 A3
Graph 5-cube t0123 B2.svg 5-cube t0123 A3.svg
Dihedral symmetry [4] [4]

Related polytopes[edit]

These polytopes are a part of a set of 31 uniform polytera 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, edited 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)".  o3x3o3o4x - span, o3x3o3x4x - pattin, o3x3x3o4x - prin, o3x3x3x4x - gippin

External links[edit]

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 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