Stericated 5-cubes

In five-dimensional geometry, a stericated 5-cube is a convex uniform 5-polytope with fourth-order truncations (sterication) of the regular 5-cube.


5-cube

Stericated 5-cube

Steritruncated 5-cube

Stericantellated 5-cube

Steritruncated 5-orthoplex

Stericantitruncated 5-cube

Steriruncitruncated 5-cube

Stericantitruncated 5-orthoplex

Omnitruncated 5-cube
Orthogonal projections in B5 Coxeter plane

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
4-faces 242
Cells 800
Faces 1040
Edges 640
Vertices 160
Vertex figure
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 penteractitriacontaditeron (Acronym: scant) (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
Dihedral symmetry [10] [8] [6]
Coxeter plane B2 A3
Graph
Dihedral symmetry [4] [4]

Steritruncated 5-cube

Steritruncated 5-cube
Typeuniform 5-polytope
Schläfli symbolt0,1,4{4,3,3,3}
Coxeter-Dynkin diagrams
4-faces242
Cells1600
Faces2960
Edges2240
Vertices640
Vertex figure
Coxeter groupsB5, [3,3,3,4]
Propertiesconvex

Alternate names

  • Steritruncated penteract
  • Celliprismated triacontaditeron (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
Dihedral symmetry [10] [8] [6]
Coxeter plane B2 A3
Graph
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
4-faces242
Cells2080
Faces4720
Edges3840
Vertices960
Vertex figure
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
Dihedral symmetry [10] [8] [6]
Coxeter plane B2 A3
Graph
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
4-faces242
Cells2400
Faces6000
Edges5760
Vertices1920
Vertex figure
Coxeter group B5 [4,3,3,3]
Properties convex, isogonal

Alternate names

  • Stericantitruncated penteract
  • Steriruncicantellated triacontiditeron / 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
Dihedral symmetry [10] [8] [6]
Coxeter plane B2 A3
Graph
Dihedral symmetry [4] [4]

Steriruncitruncated 5-cube

Steriruncitruncated 5-cube
Type Uniform 5-polytope
Schläfli symbol 2t2r{4,3,3,3}
Coxeter-Dynkin
diagram

4-faces242
Cells2160
Faces5760
Edges5760
Vertices1920
Vertex figure
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
Dihedral symmetry [10] [8] [6]
Coxeter plane B2 A3
Graph
Dihedral symmetry [4] [4]

Steritruncated 5-orthoplex

Steritruncated 5-orthoplex
Typeuniform 5-polytope
Schläfli symbolt0,1,4{3,3,3,4}
Coxeter-Dynkin diagrams
4-faces242
Cells1520
Faces2880
Edges2240
Vertices640
Vertex figure
Coxeter groupB5, [3,3,3,4]
Propertiesconvex

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
Dihedral symmetry [10] [8] [6]
Coxeter plane B2 A3
Graph
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
4-faces242
Cells2320
Faces5920
Edges5760
Vertices1920
Vertex figure
Coxeter group B5 [4,3,3,3]
Properties convex, isogonal

Alternate names

  • Stericantitruncated pentacross
  • Celligreatorhombated triacontaditeron (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
Dihedral symmetry [10] [8] [6]
Coxeter plane B2 A3
Graph
Dihedral symmetry [4] [4]

Omnitruncated 5-cube

Omnitruncated 5-cube
Type Uniform 5-polytope
Schläfli symbol tr2r{4,3,3,3}
Coxeter-Dynkin
diagram

4-faces242
Cells2640
Faces8160
Edges9600
Vertices3840
Vertex figure
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 triacontiditeron / omnitruncated pentacross
  • Great cellated penteractitriacontiditeron (Jonathan Bowers)[8]

Coordinates

The Cartesian coordinates of the vertices of an omnitruncated 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
Dihedral symmetry [10] [8] [6]
Coxeter plane B2 A3
Graph
Dihedral symmetry [4] [4]

Full snub 5-cube

The full snub 5-cube or omnisnub 5-cube, defined as an alternation of the omnitruncated 5-cube is not uniform, but it can be given Coxeter diagram and symmetry [4,3,3,3]+, and constructed from 10 snub tesseracts, 32 snub 5-cells, 40 snub cubic antiprisms, 80 snub tetrahedral antiprisms, 80 3-4 duoantiprisms, and 1920 irregular 5-cells filling the gaps at the deleted vertices.

This polytope is one 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. Klitzing, (x3o3o3o4x - scant)
  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
      • (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
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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