Runcinated 5-cubes

File:5-cube t0.svg 5-cube Template:CDD	File:5-cube t03.svg Runcinated 5-cube Template:CDD	File:5-cube t14.svg Runcinated 5-orthoplex Template:CDD
File:5-cube t013.svg Runcitruncated 5-cube Template:CDD	File:5-cube t023.svg Runcicantellated 5-cube Template:CDD	File:5-cube t0123.svg Runcicantitruncated 5-cube Template:CDD
Error creating thumbnail: Runcitruncated 5-orthoplex Template:CDD	File:5-cube t124.svg Runcicantellated 5-orthoplex Template:CDD	Runcicantitruncated 5-orthoplex Template:CDD
Orthogonal projections in B5 Coxeter plane

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 8 unique 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 diagram	Template:CDD
4-faces	202	10 Template:CDD Error creating thumbnail: 80 Template:CDD File:4-3 duoprism.png 80 Template:CDD File:Triangular antiprismatic prism.png 32 Template:CDD
Cells	1240	40 Template:CDD File:Uniform polyhedron-43-t0.png 240 Template:CDD File:Tetragonal prism.png 320 Template:CDD Error creating thumbnail: 160 Template:CDD File:Uniform polyhedron-33-t0.png 320 Template:CDD Error creating thumbnail: 160 Template:CDD File:Uniform polyhedron-33-t1.svg
Faces	2160	240 Template:CDD File:Regular quadrilateral.svg 960 Template:CDD File:Regular quadrilateral.svg 640 Template:CDD File:Regular triangle.svg 320 Template:CDD File:Regular triangle.svg
Edges	1440	480+960
Vertices	320
Vertex figure	Error creating thumbnail:
Coxeter group	B5 [4,3,3,3]
Properties	convex

Alternate names

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

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

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

Runcitruncated 5-cube

Runcitruncated 5-cube
Type	Uniform 5-polytope
Schläfli symbol	t0,1,3{4,3,3,3}
Coxeter-Dynkin diagrams	Template:CDD
4-faces	202	10 Template:CDD File:Schlegel half-solid runcitruncated 8-cell.png 80 Template:CDD File:3-8 duoprism.png 80 Template:CDD File:Triangular antiprismatic prism.png 32 Template:CDD File:Schlegel half-solid cantellated 5-cell.png
Cells	1560	40 Template:CDD File:Uniform polyhedron-43-t01.png 240 Template:CDD File:Octagonal prism.png 320 Template:CDD Error creating thumbnail: 320 Template:CDD Error creating thumbnail: 160 Template:CDD File:Uniform polyhedron-33-t02.svg 320 Template:CDD Error creating thumbnail: 160 Template:CDD File:Uniform polyhedron-33-t1.svg
Faces	3760	240 Template:CDD 960 Template:CDD File:Regular quadrilateral.svg 320 Template:CDD File:Regular triangle.svg 960 Template:CDD File:Regular quadrilateral.svg 640 Template:CDD File:Regular triangle.svg 640 Template:CDD File:Regular triangle.svg
Edges	3360	480+960+1920
Vertices	960
Vertex figure	File:Runcitruncated 5-cube verf.png
Coxeter group	B5, [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 runcitruncated 5-cube having edge length 2 are all permutations of:

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

Images

Template:5-cube Coxeter plane graphs

Runcicantellated 5-cube

Runcicantellated 5-cube
Type	Uniform 5-polytope
Schläfli symbol	t0,2,3{4,3,3,3}
Coxeter-Dynkin diagram	Template:CDD
4-faces	202	10 Template:CDD File:Runcitruncated 16-cell.png 80 Template:CDD File:4-3 duoprism.png 80 Template:CDD File:Truncated tetrahedral prism.png 32 Template:CDD Error creating thumbnail:
Cells	1240	40 Template:CDD File:Uniform polyhedron-43-t02.png 240 Template:CDD File:Tetragonal prism.png 320 Template:CDD File:Hexagonal prism.png 320 Template:CDD 160 Template:CDD File:Uniform polyhedron-33-t01.png 160 Template:CDD File:Uniform polyhedron-33-t01.png
Faces	2960	240 Template:CDD File:Regular quadrilateral.svg 480 Template:CDD File:Regular quadrilateral.svg 960 Template:CDD File:Regular quadrilateral.svg 320 Template:CDD File:Regular triangle.svg 640 Template:CDD Error creating thumbnail: 320 Template:CDD File:Regular triangle.svg
Edges	2880	960+960+960
Vertices	960
Vertex figure	File:Runcicantellated 5-cube verf.png
Coxeter group	B5 [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:

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

Images

Template:5-cube Coxeter plane graphs

Template:Anchor Runcicantitruncated 5-cube

Runcicantitruncated 5-cube
Type	Uniform 5-polytope
Schläfli symbol	t0,1,2,3{4,3,3,3}
Coxeter-Dynkin diagram	Template:CDD
4-faces	202
Cells	1560
Faces	4240
Edges	4800
Vertices	1920
Vertex figure	File:Runcicantitruncated 5-cube verf.png Irregular 5-cell
Coxeter group	B5 [4,3,3,3]
Properties	convex, isogonal

Alternate names

• Runcicantitruncated penteract
• Biruncicantitruncated pentacross
• great prismated penteract (Template:Not a typo) (Jonathan Bowers)

Coordinates

The Cartesian coordinates of the vertices of a runcicantitruncated 5-cube having an edge length of 2 are given by all permutations of coordinates and sign of:

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

Related polytopes

These polytopes are a part of a set of 31 uniform polytera 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, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, Template:ISBN [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.
• Template:KlitzingPolytopes o3x3o3o4x - span, o3x3o3x4x - pattin, o3x3x3o4x - prin, o3x3x3x4x - gippin

External links

• Template:PolyCell
• Polytopes of Various Dimensions, Jonathan Bowers
  • Runcinated uniform polytera (spid), Jonathan Bowers
• Multi-dimensional Glossary