Stericated 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 t04.svg
Stericated 5-cube
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
5-cube t014.svg
Steritruncated 5-cube
CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
5-cube t024.svg
Stericantellated 5-cube
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
5-cube t034.svg
Steritruncated 5-orthoplex
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
5-cube t0124.svg
Stericantitruncated 5-cube
CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
5-cube t0134.svg
Steriruncitruncated 5-cube
CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
5-cube t0234.svg
Stericantitruncated 5-orthoplex
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
5-cube t01234.svg
Omnitruncated 5-cube
CDel node 1.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 stericated 5-cube is a convex uniform 5-polytope with fourth-order truncations (sterication) of the regular 5-cube.

There are eight degrees of sterication for the 5-cube, including permutations of runcination, cantellation, and truncation. The simple stericated 5-cube is also called an expanded 5-cube, with the first and last nodes ringed, for being constructible by an expansion operation applied to the regular 5-cube. The highest form, the steriruncicantitruncated 5-cube, is more simply called an omnitruncated 5-cube with all of the nodes ringed.

Stericated 5-cube

Stericated 5-cube
Type Uniform 5-polytope
Schläfli symbol 2r2r{4,3,3,3}
Coxeter-Dynkin diagram CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
CDel node.pngCDel split1.pngCDel nodes.pngCDel 3a4b.pngCDel nodes 11.png
4-faces 242
Cells 800
Faces 1040
Edges 640
Vertices 160
Vertex figure Stericated penteract verf.png
Coxeter group B5 [4,3,3,3]
Properties convex

Alternate names

  • Stericated penteract / Stericated 5-orthoplex / Stericated pentacross
  • Expanded penteract / Expanded 5-orthoplex / Expanded pentacross
  • Small cellated penteract (Acronym: scan) (Jonathan Bowers)[1]

Coordinates

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

Images

The stericated 5-cube is constructed by a sterication operation applied to the 5-cube.

orthographic projections
Coxeter plane B5 B4 / D5 B3 / D4 / A2
Graph 5-cube t04.svg 5-cube t04 B4.svg 5-cube t04 B3.svg
Dihedral symmetry [10] [8] [6]
Coxeter plane B2 A3
Graph 5-cube t04 B2.svg 5-cube t04 A3.svg
Dihedral symmetry [4] [4]

Steritruncated 5-cube

Steritruncated 5-cube
Type uniform 5-polytope
Schläfli symbol t0,1,4{4,3,3,3}
Coxeter-Dynkin diagrams CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
4-faces 242
Cells 1600
Faces 2960
Edges 2240
Vertices 640
Vertex figure Steritruncated 5-cube verf.png
Coxeter groups B5, [3,3,3,4]
Properties convex

Alternate names

  • Steritruncated penteract
  • Prismatotruncated penteract (Acronym: capt) (Jonathan Bowers)[2]

Construction and coordinates

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

Images

orthographic projections
Coxeter plane B5 B4 / D5 B3 / D4 / A2
Graph 5-cube t014.svg 5-cube t014 B4.svg 5-cube t014 B3.svg
Dihedral symmetry [10] [8] [6]
Coxeter plane B2 A3
Graph 5-cube t014 B2.svg 5-cube t014 A3.svg
Dihedral symmetry [4] [4]

Stericantellated 5-cube

Stericantellated 5-cube
Type Uniform 5-polytope
Schläfli symbol t0,2,4{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 1.png
CDel node 1.pngCDel split1.pngCDel nodes.pngCDel 3a4b.pngCDel nodes 11.png
4-faces 242
Cells 2080
Faces 4720
Edges 3840
Vertices 960
Vertex figure Stericantellated 5-cube verf.png
Coxeter group B5 [4,3,3,3]
Properties convex

Alternate names

  • Stericantellated penteract
  • Stericantellated 5-orthoplex, stericantellated pentacross
  • Cellirhombated penteractitriacontiditeron (Acronym: carnit) (Jonathan Bowers)[3]

Coordinates

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

Images

orthographic projections
Coxeter plane B5 B4 / D5 B3 / D4 / A2
Graph 5-cube t024.svg 5-cube t024 B4.svg 5-cube t024 B3.svg
Dihedral symmetry [10] [8] [6]
Coxeter plane B2 A3
Graph 5-cube t024 B2.svg 5-cube t024 A3.svg
Dihedral symmetry [4] [4]

Stericantitruncated 5-cube

Stericantitruncated 5-cube
Type Uniform 5-polytope
Schläfli symbol t0,1,2,4{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 1.png
4-faces 242
Cells 2400
Faces 6000
Edges 5760
Vertices 1920
Vertex figure Stericanitruncated 5-cube verf.png
Coxeter group B5 [4,3,3,3]
Properties convex, isogonal

Alternate names

  • Stericantitruncated penteract
  • Steriruncicantellated 16-cell / Biruncicantitruncated pentacross
  • Celligreatorhombated penteract (cogrin) (Jonathan Bowers)[4]

Coordinates

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

Images

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]

Steriruncitruncated 5-cube

Steriruncitruncated 5-cube
Type Uniform 5-polytope
Schläfli symbol 2t2r{4,3,3,3}
Coxeter-Dynkin
diagram
CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
CDel node.pngCDel split1.pngCDel nodes 11.pngCDel 3a4b.pngCDel nodes 11.png
4-faces 242
Cells 2160
Faces 5760
Edges 5760
Vertices 1920
Vertex figure Steriruncitruncated 5-cube verf.png
Coxeter group B5 [4,3,3,3]
Properties convex, isogonal

Alternate names

  • Steriruncitruncated penteract / Steriruncitruncated 5-orthoplex / Steriruncitruncated pentacross
  • Celliprismatotruncated penteractitriacontiditeron (captint) (Jonathan Bowers)[5]

Coordinates

The Cartesian coordinates of the vertices of an steriruncitruncated penteract having an edge length of 2 are given by all permutations of coordinates and sign of:

Images

orthographic projections
Coxeter plane B5 B4 / D5 B3 / D4 / A2
Graph 5-cube t0134.svg 5-cube t0134 B4.svg 5-cube t0134 B3.svg
Dihedral symmetry [10] [8] [6]
Coxeter plane B2 A3
Graph 5-cube t0134 B2.svg 5-cube t0134 A3.svg
Dihedral symmetry [4] [4]

Steritruncated 5-orthoplex

Steritruncated 5-orthoplex
Type uniform 5-polytope
Schläfli symbol t0,1,4{3,3,3,4}
Coxeter-Dynkin diagrams CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
4-faces 242
Cells 1520
Faces 2880
Edges 2240
Vertices 640
Vertex figure Steritruncated 5-orthoplex verf.png
Coxeter group B5, [3,3,3,4]
Properties convex

Alternate names

  • Steritruncated pentacross
  • Celliprismated penteract (Acronym: cappin) (Jonathan Bowers)[6]

Coordinates

Cartesian coordinates for the vertices of a steritruncated 5-orthoplex, centered at the origin, are all permutations of

Images

orthographic projections
Coxeter plane B5 B4 / D5 B3 / D4 / A2
Graph 5-cube t034.svg 5-cube t034 B4.svg 5-cube t034 B3.svg
Dihedral symmetry [10] [8] [6]
Coxeter plane B2 A3
Graph 5-cube t034 B2.svg 5-cube t034 A3.svg
Dihedral symmetry [4] [4]

Stericantitruncated 5-orthoplex

Stericantitruncated 5-orthoplex
Type Uniform 5-polytope
Schläfli symbol t0,2,3,4{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 1.png
4-faces 242
Cells 2320
Faces 5920
Edges 5760
Vertices 1920
Vertex figure Stericanitruncated 5-orthoplex verf.png
Coxeter group B5 [4,3,3,3]
Properties convex, isogonal

Alternate names

  • Stericantitruncated pentacross
  • Celligreatorhombated pentacross (cogart) (Jonathan Bowers)[7]

Coordinates

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

Images

orthographic projections
Coxeter plane B5 B4 / D5 B3 / D4 / A2
Graph 5-cube t0234.svg 5-cube t0234 B4.svg 5-cube t0234 B3.svg
Dihedral symmetry [10] [8] [6]
Coxeter plane B2 A3
Graph 5-cube t0234 B2.svg 5-cube t0234 A3.svg
Dihedral symmetry [4] [4]

Omnitruncated 5-cube

Omnitruncated 5-cube
Type Uniform 5-polytope
Schläfli symbol tr2r{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 1.png
CDel node 1.pngCDel split1.pngCDel nodes 11.pngCDel 3a4b.pngCDel nodes 11.png
4-faces 242
Cells 2640
Faces 8160
Edges 9600
Vertices 3840
Vertex figure Omnitruncated 5-cube verf.png
irr. {3,3,3}
Coxeter group B5 [4,3,3,3]
Properties convex, isogonal

Alternate names

  • Steriruncicantitruncated 5-cube (Full expansion of omnitruncation for 5-polytopes by Johnson)
  • Omnitruncated penteract
  • Omnitruncated 16-cell / omnitruncated pentacross
  • Great cellated penteractitriacontiditeron (Jonathan Bowers)[8]

Coordinates

The Cartesian coordinates of the vertices of an omnitruncated tesseract having an edge length of 2 are given by all permutations of coordinates and sign of:

Images

orthographic projections
Coxeter plane B5 B4 / D5 B3 / D4 / A2
Graph 5-cube t01234.svg 5-cube t01234 B4.svg 5-cube t01234 B3.svg
Dihedral symmetry [10] [8] [6]
Coxeter plane B2 A3
Graph 5-cube t01234 B2.svg 5-cube t01234 A3.svg
Dihedral symmetry [4] [4]

Related polytopes

This polytope is one of 31 uniform 5-polytopes generated from the regular 5-cube or 5-orthoplex.

Notes

  1. ^ Klitzing, (x3o3o3o4x - scan)
  2. ^ Klitzing, (x3o3o3x4x - capt)
  3. ^ Klitzing, (x3o3x3o4x - carnit)
  4. ^ Klitzing, (x3o3x3x4x - cogrin)
  5. ^ Klitzing, (x3x3o3x4x - captint)
  6. ^ Klitzing, (x3x3o3o4x - cappin)
  7. ^ Klitzing, (x3x3x3o4x - cogart)
  8. ^ Klitzing, (x3x3x3x4x - gacnet)

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 [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)".  x3o3o3o4x - scan, x3o3o3x4x - capt, x3o3x3o4x - carnit, x3o3x3x4x - cogrin, x3x3o3x4x - captint, x3x3x3x4x - gacnet, x3x3x3o4x - cogart

External links

  • Glossary for hyperspace, George Olshevsky.
  • Polytopes of Various Dimensions, Jonathan Bowers
  • Multi-dimensional Glossary
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
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