Stericated 5-cubes

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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.

More information Orthogonal projections in B5 Coxeter plane ...
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5-cube
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Stericated 5-cube
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Steritruncated 5-cube
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Stericantellated 5-cube
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Steritruncated 5-orthoplex
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Stericantitruncated 5-cube
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Steriruncitruncated 5-cube
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Stericantitruncated 5-orthoplex
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Omnitruncated 5-cube
Orthogonal projections in B5 Coxeter plane
Close

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

Summarize
Perspective
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 Thumb
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.

Dissections

The stericated 5-cube can be dissected into two tesseractic cupolae and a runcinated tesseract between them. This dissection can be seen as analogous to the 4D runcinated tesseract being dissected into two cubic cupolae and a central rhombicuboctahedral prism between them, and also the 3D rhombicuboctahedron being dissected into two square cupolae with a central octagonal prism between them.

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

Steritruncated 5-cube

Summarize
Perspective
More information 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 figureThumb
Coxeter groupsB5, [3,3,3,4]
Propertiesconvex
Close

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

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

Stericantellated 5-cube

Summarize
Perspective
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 Thumb
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

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

Stericantitruncated 5-cube

Summarize
Perspective
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 Thumb
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 a stericantitruncated 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

Steriruncitruncated 5-cube

Summarize
Perspective
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 Thumb
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 a steriruncitruncated penteract 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

Steritruncated 5-orthoplex

Summarize
Perspective
More information 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 figureThumb
Coxeter groupB5, [3,3,3,4]
Propertiesconvex
Close

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

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

Stericantitruncated 5-orthoplex

Summarize
Perspective
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 Thumb
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 a stericantitruncated 5-orthoplex 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

Omnitruncated 5-cube

Summarize
Perspective
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 Thumb
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

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

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.

Notes

References

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