Stericated 5-cubes

Orthogonal projections in B₅ Coxeter plane
File:5-cube t0.svg 5-cube File:CDel node 1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png	File:5-cube t04.svg Stericated 5-cube File:CDel node 1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.png	File:5-cube t014.svg Steritruncated 5-cube File:CDel node 1.pngFile:CDel 4.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.png
File:5-cube t024.svg Stericantellated 5-cube File:CDel node 1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.png	File:5-cube t034.svg Steritruncated 5-orthoplex File:CDel node 1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.png	File:5-cube t0124.svg Stericantitruncated 5-cube File:CDel node 1.pngFile:CDel 4.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.png
File:5-cube t0134.svg Steriruncitruncated 5-cube File:CDel node 1.pngFile:CDel 4.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.png	File:5-cube t0234.svg Stericantitruncated 5-orthoplex File:CDel node 1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.png	File:5-cube t01234.svg Omnitruncated 5-cube File:CDel node 1.pngFile:CDel 4.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.png

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	File:CDel node 1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.png File:CDel node.pngFile:CDel split1.pngFile:CDel nodes.pngFile:CDel 3a4b.pngFile:CDel nodes 11.png
4-faces	242
Cells	800
Faces	1040
Edges	640
Vertices	160
Vertex figure	File:Stericated penteract verf.png
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:

(\pm 1, \pm 1, \pm 1, \pm 1, \pm (1 + \sqrt{2}))

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.

orthographic projections
Coxeter plane	B₅	B₄ / D₅	B₃ / D₄ / A₂
Graph	File:5-cube t04.svg	File:5-cube t04 B4.svg	File:5-cube t04 B3.svg
Dihedral symmetry	[10]	[8]	[6]
Coxeter plane	B₂	A₃
Graph	File:5-cube t04 B2.svg	File:5-cube t04 A3.svg
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	File:CDel node 1.pngFile:CDel 4.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.png
4-faces	242
Cells	1600
Faces	2960
Edges	2240
Vertices	640
Vertex figure	File:Steritruncated 5-cube verf.png
Coxeter groups	B₅, [3,3,3,4]
Properties	convex

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:

(\pm 1, \pm (1 + \sqrt{2}), \pm (1 + \sqrt{2}), \pm (1 + \sqrt{2}), \pm (1 + 2 \sqrt{2}))

Images

orthographic projections
Coxeter plane	B₅	B₄ / D₅	B₃ / D₄ / A₂
Graph	File:5-cube t014.svg	File:5-cube t014 B4.svg	File:5-cube t014 B3.svg
Dihedral symmetry	[10]	[8]	[6]
Coxeter plane	B₂	A₃
Graph	File:5-cube t014 B2.svg	File:5-cube t014 A3.svg
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	File:CDel node 1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.png File:CDel node 1.pngFile:CDel split1.pngFile:CDel nodes.pngFile:CDel 3a4b.pngFile:CDel nodes 11.png
4-faces	242
Cells	2080
Faces	4720
Edges	3840
Vertices	960
Vertex figure	File:Stericantellated 5-cube verf.png
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:

(\pm 1, \pm 1, \pm 1, \pm (1 + \sqrt{2}), \pm (1 + 2 \sqrt{2}))

Images

orthographic projections
Coxeter plane	B₅	B₄ / D₅	B₃ / D₄ / A₂
Graph	File:5-cube t024.svg	File:5-cube t024 B4.svg	File:5-cube t024 B3.svg
Dihedral symmetry	[10]	[8]	[6]
Coxeter plane	B₂	A₃
Graph	File:5-cube t024 B2.svg	File:5-cube t024 A3.svg
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	File:CDel node 1.pngFile:CDel 4.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.png
4-faces	242
Cells	2400
Faces	6000
Edges	5760
Vertices	1920
Vertex figure	File:Stericanitruncated 5-cube verf.png
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:

(1, 1 + \sqrt{2}, 1 + 2 \sqrt{2}, 1 + 2 \sqrt{2}, 1 + 3 \sqrt{2})

Images

orthographic projections
Coxeter plane	B₅	B₄ / D₅	B₃ / D₄ / A₂
Graph	File:5-cube t013.svg	File:5-cube t013 B4.svg	File:5-cube t013 B3.svg
Dihedral symmetry	[10]	[8]	[6]
Coxeter plane	B₂	A₃
Graph	File:5-cube t013 B2.svg	File:5-cube t013 A3.svg
Dihedral symmetry	[4]	[4]

Steriruncitruncated 5-cube

Steriruncitruncated 5-cube
Type	Uniform 5-polytope
Schläfli symbol	2t2r{4,3,3,3}
Coxeter-Dynkin diagram	File:CDel node 1.pngFile:CDel 4.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.png File:CDel node.pngFile:CDel split1.pngFile:CDel nodes 11.pngFile:CDel 3a4b.pngFile:CDel nodes 11.png
4-faces	242
Cells	2160
Faces	5760
Edges	5760
Vertices	1920
Vertex figure	File:Steriruncitruncated 5-cube verf.png
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:

(1, 1 + \sqrt{2}, 1 + 1 \sqrt{2}, 1 + 2 \sqrt{2}, 1 + 3 \sqrt{2})

Images

orthographic projections
Coxeter plane	B₅	B₄ / D₅	B₃ / D₄ / A₂
Graph	File:5-cube t0134.svg	File:5-cube t0134 B4.svg	File:5-cube t0134 B3.svg
Dihedral symmetry	[10]	[8]	[6]
Coxeter plane	B₂	A₃
Graph	File:5-cube t0134 B2.svg	File:5-cube t0134 A3.svg
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	File:CDel node 1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.png
4-faces	242
Cells	1520
Faces	2880
Edges	2240
Vertices	640
Vertex figure	File:Steritruncated 5-orthoplex verf.png
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

(\pm 1, \pm 1, \pm 1, \pm 1, \pm (1 + \sqrt{2}))

Images

orthographic projections
Coxeter plane	B₅	B₄ / D₅	B₃ / D₄ / A₂
Graph	File:5-cube t034.svg	File:5-cube t034 B4.svg	File:5-cube t034 B3.svg
Dihedral symmetry	[10]	[8]	[6]
Coxeter plane	B₂	A₃
Graph	File:5-cube t034 B2.svg	File:5-cube t034 A3.svg
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	File:CDel node 1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.png
4-faces	242
Cells	2320
Faces	5920
Edges	5760
Vertices	1920
Vertex figure	File:Stericanitruncated 5-orthoplex verf.png
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:

(1, 1, 1 + \sqrt{2}, 1 + 2 \sqrt{2}, 1 + 3 \sqrt{2})

Images

orthographic projections
Coxeter plane	B₅	B₄ / D₅	B₃ / D₄ / A₂
Graph	File:5-cube t0234.svg	File:5-cube t0234 B4.svg	File:5-cube t0234 B3.svg
Dihedral symmetry	[10]	[8]	[6]
Coxeter plane	B₂	A₃
Graph	File:5-cube t0234 B2.svg	File:5-cube t0234 A3.svg
Dihedral symmetry	[4]	[4]

Omnitruncated 5-cube

Omnitruncated 5-cube
Type	Uniform 5-polytope
Schläfli symbol	tr2r{4,3,3,3}
Coxeter-Dynkin diagram	File:CDel node 1.pngFile:CDel 4.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.png File:CDel node 1.pngFile:CDel split1.pngFile:CDel nodes 11.pngFile:CDel 3a4b.pngFile:CDel nodes 11.png
4-faces	242
Cells	2640
Faces	8160
Edges	9600
Vertices	3840
Vertex figure	File:Omnitruncated 5-cube verf.png 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:

(1, 1 + \sqrt{2}, 1 + 2 \sqrt{2}, 1 + 3 \sqrt{2}, 1 + 4 \sqrt{2})

Images

orthographic projections
Coxeter plane	B₅	B₄ / D₅	B₃ / D₄ / A₂
Graph	File:5-cube t01234.svg	File:5-cube t01234 B4.svg	File:5-cube t01234 B3.svg
Dihedral symmetry	[10]	[8]	[6]
Coxeter plane	B₂	A₃
Graph	File:5-cube t01234 B2.svg	File:5-cube t01234 A3.svg
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 File:CDel node h.pngFile:CDel 4.pngFile:CDel node h.pngFile:CDel 3.pngFile:CDel node h.pngFile:CDel 3.pngFile:CDel node h.pngFile:CDel 3.pngFile:CDel node h.png 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.

B5 polytopes
File:5-cube t4.svg β₅	File:5-cube t3.svg t₁β₅	File:5-cube t2.svg t₂γ₅	File:5-cube t1.svg t₁γ₅	File:5-cube t0.svg γ₅	File:5-cube t34.svg t_0,1β₅	File:5-cube t24.svg t_0,2β₅	File:5-cube t23.svg t_1,2β₅
File:5-cube t14.svg t_0,3β₅	File:5-cube t13.svg t_1,3γ₅	File:5-cube t12.svg t_1,2γ₅	File:5-cube t04.svg t_0,4γ₅	File:5-cube t03.svg t_0,3γ₅	File:5-cube t02.svg t_0,2γ₅	File:5-cube t01.svg t_0,1γ₅	File:5-cube t234.svg t_0,1,2β₅
File:5-cube t134.svg t_0,1,3β₅	File:5-cube t124.svg t_0,2,3β₅	File:5-cube t123.svg t_1,2,3γ₅	File:5-cube t034.svg t_0,1,4β₅	File:5-cube t024.svg t_0,2,4γ₅	File:5-cube t023.svg t_0,2,3γ₅	File:5-cube t014.svg t_0,1,4γ₅	File:5-cube t013.svg t_0,1,3γ₅
File:5-cube t012.svg t_0,1,2γ₅	File:5-cube t1234.svg t_0,1,2,3β₅	File:5-cube t0234.svg t_0,1,2,4β₅	File:5-cube t0134.svg t_0,1,3,4γ₅	File:5-cube t0124.svg t_0,1,2,4γ₅	File:5-cube t0123.svg t_0,1,2,3γ₅	File:5-cube t01234.svg t_0,1,2,3,4γ₅

Notes

^ Klitzing, (x3o3o3o4x - scant)
^ 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, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, wiley.com, Lua error in Module:Citation/CS1/Configuration at line 2172: attempt to index field '?' (a nil value).
  - (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.
Lua error in Module:Citation/CS1/Configuration at line 2172: attempt to index field '?' (a nil value). 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

v t e Fundamental convex regular and uniform polytopes in dimensions 2–10
Family	$A n$	$B n$	$I 2 (p)$ / $D n$	$E 6$ / $E 7$ / $E 8$ / $F 4$ / $G 2$	$H n$
Regular polygon	Triangle	Square	p-gon	Hexagon	Pentagon
Uniform polyhedron	Tetrahedron	Octahedron • Cube	Demicube		Dodecahedron • Icosahedron
Uniform polychoron	Pentachoron	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 • Polytope operations

[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)

[1]

[2]

[3]

[4]

[5]

[6]

[7]

[8]

Stericated 5-cubes