# Stericated 5-cubes

(Redirected from Steritruncated 5-orthoplex)
 Orthogonal projections in B5 Coxeter plane 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

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 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 penteract (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:

${\displaystyle \left(\pm 1,\ \pm 1,\ \pm 1,\ \pm 1,\ \pm (1+{\sqrt {2}})\right)}$

### 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
Type uniform 5-polytope
Schläfli symbol t0,1,4{4,3,3,3}
Coxeter-Dynkin diagrams
4-faces 242
Cells 1600
Faces 2960
Edges 2240
Vertices 640
Vertex figure
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:

${\displaystyle \left(\pm 1,\ \pm (1+{\sqrt {2}}),\ \pm (1+{\sqrt {2}}),\ \pm (1+{\sqrt {2}}),\ \pm (1+2{\sqrt {2}})\right)}$

### 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-faces 242 Cells 2080 Faces 4720 Edges 3840 Vertices 960 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:

${\displaystyle \left(\pm 1,\ \pm 1,\ \pm 1,\ \pm (1+{\sqrt {2}}),\ \pm (1+2{\sqrt {2}})\right)}$

### 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-Dynkindiagram 4-faces 242 Cells 2400 Faces 6000 Edges 5760 Vertices 1920 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:

${\displaystyle \left(1,\ 1+{\sqrt {2}},\ 1+2{\sqrt {2}},\ 1+2{\sqrt {2}},\ 1+3{\sqrt {2}}\right)}$

### 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-Dynkindiagram 4-faces 242 Cells 2160 Faces 5760 Edges 5760 Vertices 1920 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:

${\displaystyle \left(1,\ 1+{\sqrt {2}},\ 1+1{\sqrt {2}},\ 1+2{\sqrt {2}},\ 1+3{\sqrt {2}}\right)}$

### 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
Type uniform 5-polytope
Schläfli symbol t0,1,4{3,3,3,4}
Coxeter-Dynkin diagrams
4-faces 242
Cells 1520
Faces 2880
Edges 2240
Vertices 640
Vertex figure
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

${\displaystyle \left(\pm 1,\ \pm 1,\ \pm 1,\ \pm 1,\ \pm (1+{\sqrt {2}})\right)}$

### 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-Dynkindiagram 4-faces 242 Cells 2320 Faces 5920 Edges 5760 Vertices 1920 Vertex figure 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:

${\displaystyle \left(1,\ 1,\ 1+{\sqrt {2}},\ 1+2{\sqrt {2}},\ 1+3{\sqrt {2}}\right)}$

### 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-Dynkindiagram 4-faces 242 Cells 2640 Faces 8160 Edges 9600 Vertices 3840 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:

${\displaystyle \left(1,\ 1+{\sqrt {2}},\ 1+2{\sqrt {2}},\ 1+3{\sqrt {2}},\ 1+4{\sqrt {2}}\right)}$

### 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.

## Related polytopes

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

## 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 [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