# Runcinated 5-cube

 Orthogonal projections in BC5Coxeter plane 5-cube Runcinated 5-cube Runcinated 5-orthoplex Runcitruncated 5-cube Runcicantellated 5-cube Runcicantitruncated 5-cube Runcitruncated 5-orthoplex Runcicantellated 5-orthoplex Runcicantitruncated 5-orthoplex

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

 Runcinated 5-cube Type Uniform 5-polytope Schläfli symbol t0,3{4,3,3,3} Coxeter-Dynkin diagram 4-faces 162 Cells 1200 Faces 2160 Edges 1440 Vertices 320 Vertex figure Coxeter group BC5 [4,3,3,3] Properties convex

### Alternate names

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

### Coordinates

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

$\left(\pm1,\ \pm1,\ \pm1,\ \pm(1+\sqrt{2}),\ \pm(1+\sqrt{2})\right)$

### 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]
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## Runcitruncated 5-cube

Runcitruncated 5-cube
Type uniform polyteron
Schläfli symbol t0,1,3{4,3,3,3}
Coxeter-Dynkin diagrams
4-faces 162
Cells 1440
Faces 3680
Edges 3360
Vertices 960
Vertex figure
Coxeter groups BC5, [3,3,3,4]
Properties convex

### Alternate names

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

### Construction and coordinates

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

$\left(\pm1,\ \pm(1+\sqrt{2}),\ \pm(1+\sqrt{2}),\ \pm(1+2\sqrt{2}),\ \pm(1+2\sqrt{2})\right)$

### 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]
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## Runcicantellated 5-cube

 Runcicantellated 5-cube Type Uniform 5-polytope Schläfli symbol t0,2,3{4,3,3,3} Coxeter-Dynkin diagram 4-faces 162 Cells 1200 Faces 2960 Edges 2880 Vertices 960 Vertex figure Coxeter group BC5 [4,3,3,3] Properties convex

### Alternate names

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

### Coordinates

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

$\left(\pm1,\ \pm1,\ \pm(1+\sqrt{2}),\ \pm(1+2\sqrt{2}),\ \pm(1+2\sqrt{2})\right)$

### 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]
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## Runcicantitruncated 5-cube

 Runcicantitruncated 5-cube Type Uniform 5-polytope Schläfli symbol t0,1,2,3{4,3,3,3} Coxeter-Dynkin diagram 4-faces 162 Cells 1440 Faces 4160 Edges 4800 Vertices 1920 Vertex figure Coxeter group BC5 [4,3,3,3] Properties convex, isogonal

### Alternate names

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

### Coordinates

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

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

### 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]
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## Related polytopes

These polytopes are a part of a set of 31 uniform polytera generated from the regular 5-cube or 5-orthoplex.

 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
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## 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.
• Richard Klitzing, 5D, uniform polytopes (polytera) o3x3o3o4x - span, o3x3o3x4x - pattin, o3x3x3o4x - prin, o3x3x3x4x - gippin
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