Stericated 5-cubes

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
Orthogonal projections in B5 Coxeter plane

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
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

• Stericated penteract / Stericated 5-orthoplex / Stericated pentacross
• Expanded penteract / Expanded 5-orthoplex / Expanded pentacross
• Small cellated penteractitriacontaditeron (Acronym: scant) (Jonathan Bowers)^[1]

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

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

The stericated 5-cube is constructed by a sterication operation applied to the 5-cube.

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
Dihedral symmetry	[10]	[8]	[6]
Coxeter plane	B2	A3
Graph
Dihedral symmetry	[4]	[4]

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

• Steritruncated penteract
• Celliprismated triacontaditeron (Acronym: capt) (Jonathan Bowers)^[2]

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

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

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
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

• Stericantellated penteract
• Stericantellated 5-orthoplex, stericantellated pentacross
• Cellirhombated penteractitriacontiditeron (Acronym: carnit) (Jonathan Bowers)^[3]

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

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

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
Type	Uniform 5-polytope
Schläfli symbol	t0,1,2,4{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

• Stericantitruncated penteract
• Steriruncicantellated triacontiditeron / Biruncicantitruncated pentacross
• Celligreatorhombated penteract (cogrin) (Jonathan Bowers)^[4]

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:

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

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
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

• Steriruncitruncated penteract / Steriruncitruncated 5-orthoplex / Steriruncitruncated pentacross
• Celliprismatotruncated penteractitriacontiditeron (captint) (Jonathan Bowers)^[5]

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:

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

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
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

• Steritruncated pentacross
• Celliprismated penteract (Acronym: cappin) (Jonathan Bowers)^[6]

Cartesian coordinates for the vertices of a steritruncated 5-orthoplex, centered at the origin, are all permutations of

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

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
Type	Uniform 5-polytope
Schläfli symbol	t0,2,3,4{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

• Stericantitruncated pentacross
• Celligreatorhombated triacontaditeron (cogart) (Jonathan Bowers)^[7]

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:

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

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
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

• Steriruncicantitruncated 5-cube (Full expansion of omnitruncation for 5-polytopes by Johnson)
• Omnitruncated penteract
• Omnitruncated triacontiditeron / omnitruncated pentacross
• Great cellated penteractitriacontiditeron (Jonathan Bowers)^[8]

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:

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

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]

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.

B5 polytopes
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

• 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

• Glossary for hyperspace, George Olshevsky.
• Polytopes of Various Dimensions, Jonathan Bowers
• Multi-dimensional Glossary