Stericated 5-cubes

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: scan) (Jonathan Bowers)^[1]

Coordinates

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

\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	t_0,1,4{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:

\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	t_0,2,4{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:

\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	t_0,1,2,4{4,3,3,3}
Coxeter-Dynkin diagram
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 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:

\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-Dynkin diagram
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:

\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	t_0,1,4{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

\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	t_0,2,3,4{4,3,3,3}
Coxeter-Dynkin diagram
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:

\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-Dynkin diagram
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 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:

\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]

Related polytopes

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	t_0,1β₅	t_0,2β₅	t_1,2β₅
t_0,3β₅	t_1,3γ₅	t_1,2γ₅	t_0,4γ₅	t_0,3γ₅	t_0,2γ₅	t_0,1γ₅	t_0,1,2β₅
t_0,1,3β₅	t_0,2,3β₅	t_1,2,3γ₅	t_0,1,4β₅	t_0,2,4γ₅	t_0,2,3γ₅	t_0,1,4γ₅	t_0,1,3γ₅
t_0,1,2γ₅	t_0,1,2,3β₅	t_0,1,2,4β₅	t_0,1,3,4γ₅	t_0,1,2,4γ₅	t_0,1,2,3γ₅	t_0,1,2,3,4γ₅

Notes

↑ Klitzing, (x3o3o3o4x - scan)
↑ Klitzing, (x3o3o3x4x - capt)
↑ Klitzing, (x3o3x3o4x - carnit)
↑ Klitzing, (x3o3x3x4x - cogrin)
↑ Klitzing, (x3x3o3x4x - captint)
↑ Klitzing, (x3x3o3o4x - cappin)
↑ Klitzing, (x3x3x3o4x - cogart)
↑ 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

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	A_n	B_n	I₂(p) / D_n	E₆ / E₇ / E₈ / E₉ / E₁₀ / F₄ / G₂	H_n
Regular polygon	Triangle	Square	p-gon	Hexagon	Pentagon
Uniform polyhedron	Tetrahedron	Octahedron • Cube	Demicube		Dodecahedron • Icosahedron
Uniform 4-polytope	5-cell	16-cell • Tesseract	Demitesseract	24-cell	120-cell • 600-cell
Uniform 5-polytope	5-simplex	5-orthoplex • 5-cube	5-demicube
Uniform 6-polytope	6-simplex	6-orthoplex • 6-cube	6-demicube	1₂₂ • 2₂₁
Uniform 7-polytope	7-simplex	7-orthoplex • 7-cube	7-demicube	1₃₂ • 2₃₁ • 3₂₁
Uniform 8-polytope	8-simplex	8-orthoplex • 8-cube	8-demicube	1₄₂ • 2₄₁ • 4₂₁
Uniform 9-polytope	9-simplex	9-orthoplex • 9-cube	9-demicube
Uniform 10-polytope	10-simplex	10-orthoplex • 10-cube	10-demicube
Uniform n-polytope	n-simplex	n-orthoplex • n-cube	n-demicube	1_k2 • 2_k1 • k₂₁	n-pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds

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