Runcinated 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 t03.svg Runcinated 5-cube 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.png	File:5-cube t14.svg Runcinated 5-orthoplex File:CDel node.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 t013.svg Runcitruncated 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.png	File:5-cube t023.svg Runcicantellated 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 1.pngFile:CDel 3.pngFile:CDel node.png	File:5-cube t0123.svg Runcicantitruncated 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.png
File:5-cube t134.svg Runcitruncated 5-orthoplex File:CDel node.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 t124.svg Runcicantellated 5-orthoplex File:CDel node.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 t1234.svg Runcicantitruncated 5-orthoplex File:CDel node.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 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	t_0,3{4,3,3,3}
Coxeter diagram	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.png
4-faces	202	10 File:CDel node 1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.png File:Schlegel half-solid runcinated 8-cell.png 80 File:CDel node 1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 2.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.png File:4-3 duoprism.png 80 File:CDel node 1.pngFile:CDel 2.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.png File:Triangular antiprismatic prism.png 32 File:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.png File:Schlegel half-solid rectified 5-cell.png
Cells	1240	40 File:CDel node 1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png File:Uniform polyhedron-43-t0.png 240 File:CDel node 1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 2.pngFile:CDel node 1.png File:Tetragonal prism.png 320 File:CDel node 1.pngFile:CDel 2.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.png File:Triangular prism wedge.png 160 File:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.png File:Uniform polyhedron-33-t0.png 320 File:CDel node 1.pngFile:CDel 2.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.png File:Triangular prism wedge.png 160 File:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.png File:Uniform polyhedron-33-t1.svg
Faces	2160	240 File:CDel node 1.pngFile:CDel 4.pngFile:CDel node.png File:Regular quadrilateral.svg 960 File:CDel node 1.pngFile:CDel 2.pngFile:CDel node 1.png File:Regular quadrilateral.svg 640 File:CDel node.pngFile:CDel 3.pngFile:CDel node 1.png File:Regular triangle.svg 320 File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.png File:Regular triangle.svg
Edges	1440	480+960
Vertices	320
Vertex figure	File:Runcinated penteract verf.png
Coxeter group	B₅ [4,3,3,3]
Properties	convex

Alternate names

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

Coordinates

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}))

Images

orthographic projections
Coxeter plane	B₅	B₄ / D₅	B₃ / D₄ / A₂
Graph	File:5-cube t03.svg	File:5-cube t03 B4.svg	File:5-cube t03 B3.svg
Dihedral symmetry	[10]	[8]	[6]
Coxeter plane	B₂	A₃
Graph	File:5-cube t03 B2.svg	File:5-cube t03 A3.svg
Dihedral symmetry	[4]	[4]

Runcitruncated 5-cube

Runcitruncated 5-cube
Type	Uniform 5-polytope
Schläfli symbol	t_0,1,3{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 1.pngFile:CDel 3.pngFile:CDel node.png
4-faces	202	10 File:CDel node 1.pngFile:CDel 4.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.png File:Schlegel half-solid runcitruncated 8-cell.png 80 File:CDel node 1.pngFile:CDel 4.pngFile:CDel node 1.pngFile:CDel 2.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.png File:3-8 duoprism.png 80 File:CDel node 1.pngFile:CDel 2.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.png File:Triangular antiprismatic prism.png 32 File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.png File:Schlegel half-solid cantellated 5-cell.png
Cells	1560	40 File:CDel node 1.pngFile:CDel 4.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.png File:Uniform polyhedron-43-t01.png 240 File:CDel node 1.pngFile:CDel 4.pngFile:CDel node 1.pngFile:CDel 2.pngFile:CDel node 1.png File:Octagonal prism.png 320 File:CDel node 1.pngFile:CDel 2.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.png File:Triangular prism wedge.png 320 File:CDel node 1.pngFile:CDel 2.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.png File:Triangular prism wedge.png 160 File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.png File:Uniform polyhedron-33-t02.png 320 File:CDel node 1.pngFile:CDel 2.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.png File:Triangular prism wedge.png 160 File:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.png File:Uniform polyhedron-33-t1.svg
Faces	3760	240 File:CDel node 1.pngFile:CDel 4.pngFile:CDel node 1.png File:Regular octagon.svg 960 File:CDel node 1.pngFile:CDel 2.pngFile:CDel node 1.png File:Regular quadrilateral.svg 320 File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.png File:Regular triangle.svg 960 File:CDel node 1.pngFile:CDel 2.pngFile:CDel node 1.png File:Regular quadrilateral.svg 640 File:CDel node.pngFile:CDel 3.pngFile:CDel node 1.png File:Regular triangle.svg 640 File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.png 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

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]

Runcicantellated 5-cube

Runcicantellated 5-cube
Type	Uniform 5-polytope
Schläfli symbol	t_0,2,3{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.png
4-faces	202	10 File:CDel node 1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.png File:Runcitruncated 16-cell.png 80 File:CDel node 1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 2.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.png File:4-3 duoprism.png 80 File:CDel node 1.pngFile:CDel 2.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.png File:Truncated tetrahedral prism.png 32 File:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.png File:Schlegel half-solid bitruncated 5-cell.png
Cells	1240	40 File:CDel node 1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.png File:Uniform polyhedron-43-t02.png 240 File:CDel node 1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 2.pngFile:CDel node 1.png File:Tetragonal prism.png 320 File:CDel node 1.pngFile:CDel 2.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.png File:Hexagonal prism.png 320 File:CDel node 1.pngFile:CDel 2.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.png File:Triangular prism wedge.png 160 File:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.png File:Uniform polyhedron-33-t01.png 160 File:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.png File:Uniform polyhedron-33-t01.png
Faces	2960	240 File:CDel node 1.pngFile:CDel 4.pngFile:CDel node.png File:Regular quadrilateral.svg 480 File:CDel node 1.pngFile:CDel 2.pngFile:CDel node 1.png File:Regular quadrilateral.svg 960 File:CDel node 1.pngFile:CDel 2.pngFile:CDel node 1.png File:Regular quadrilateral.svg 320 File:CDel node.pngFile:CDel 3.pngFile:CDel node 1.png File:Regular triangle.svg 640 File:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.png File:Regular hexagon.svg 320 File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.png 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

orthographic projections
Coxeter plane	B₅	B₄ / D₅	B₃ / D₄ / A₂
Graph	File:5-cube t023.svg	File:5-cube t023 B4.svg	File:5-cube t023 B3.svg
Dihedral symmetry	[10]	[8]	[6]
Coxeter plane	B₂	A₃
Graph	File:5-cube t023 B2.svg	File:5-cube t023 A3.svg
Dihedral symmetry	[4]	[4]

Runcicantitruncated 5-cube

Runcicantitruncated 5-cube
Type	Uniform 5-polytope
Schläfli symbol	t_0,1,2,3{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.png
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 (gippin) (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})

Images

orthographic projections
Coxeter plane	B₅	B₄ / D₅	B₃ / D₄ / A₂
Graph	File:5-cube t0123.svg	File:5-cube t0123 B4.svg	File:5-cube t0123 B3.svg
Dihedral symmetry	[10]	[8]	[6]
Coxeter plane	B₂	A₃
Graph	File:5-cube t0123 B2.svg	File:5-cube t0123 A3.svg
Dihedral symmetry	[4]	[4]

Related polytopes

These polytopes are a part of a set of 31 uniform polytera 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

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, 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)". o3x3o3o4x - span, o3x3o3x4x - pattin, o3x3x3o4x - prin, o3x3x3x4x - gippin

External links

Glossary for hyperspace, George Olshevsky.
Polytopes of Various Dimensions, Jonathan Bowers
- Runcinated uniform polytera (spid), Jonathan Bowers
Multi-dimensional Glossary

v t e Fundamental convex regular and uniform polytopes in dimensions 2–10
Family	A_n	B_n	I₂(p) / D_n	E₆ / E₇ / E₈ / F₄ / G₂	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

Runcinated 5-cubes

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