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 sterirunci​cantitruncated​ 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 penteractitriacontaditeron (Acronym: scant) (Jonathan Bowers)

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

[math]\displaystyle{ \left(\pm1,\ \pm1,\ \pm1,\ \pm1,\ \pm(1+\sqrt{2})\right) }[/math]

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
• Celliprismated triacontaditeron (Acronym: capt) (Jonathan Bowers)

Construction and coordinates

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

[math]\displaystyle{ \left(\pm1,\ \pm(1+\sqrt{2}),\ \pm(1+\sqrt{2}),\ \pm(1+\sqrt{2}),\ \pm(1+2\sqrt{2})\right) }[/math]

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)

Coordinates

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

[math]\displaystyle{ \left(\pm1,\ \pm1,\ \pm1,\ \pm(1+\sqrt{2}),\ \pm(1+2\sqrt{2})\right) }[/math]

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-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 triacontiditeron / Biruncicantitruncated pentacross
• Celligreatorhombated penteract (cogrin) (Jonathan Bowers)

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:

[math]\displaystyle{ \left(1,\ 1+\sqrt{2},\ 1+2\sqrt{2},\ 1+2\sqrt{2},\ 1+3\sqrt{2}\right) }[/math]

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)

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:

[math]\displaystyle{ \left(1,\ 1+\sqrt{2},\ 1+1\sqrt{2},\ 1+2\sqrt{2},\ 1+3\sqrt{2}\right) }[/math]

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)

Coordinates

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

[math]\displaystyle{ \left(\pm1,\ \pm1,\ \pm1,\ \pm1,\ \pm(1+\sqrt{2})\right) }[/math]

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-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 triacontaditeron (cogart) (Jonathan Bowers)

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:

[math]\displaystyle{ \left(1,\ 1,\ 1+\sqrt{2},\ 1+2\sqrt{2},\ 1+3\sqrt{2}\right) }[/math]

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 triacontiditeron / omnitruncated pentacross
• Great cellated penteractitriacontiditeron (Jonathan Bowers)

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:

[math]\displaystyle{ \left(1,\ 1+\sqrt{2},\ 1+2\sqrt{2},\ 1+3\sqrt{2},\ 1+4\sqrt{2}\right) }[/math]

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.

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)". https://bendwavy.org/klitzing/dimensions/polytera.htm.  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