Runcinated 5-cubes

Orthogonal projections in B5 Coxeter plane
5-cube	Runcinated 5-cube	Runcinated 5-orthoplex
Runcitruncated 5-cube	Runcicantellated 5-cube	Runcicantitruncated 5-cube
Runcitruncated 5-orthoplex	Runcicantellated 5-orthoplex	Runcicantitruncated 5-orthoplex

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
4-faces	202	10 Runcinated tesseract 80 3-4 duoprism 80 Octahedral prism 32 Rectified 5-cell
Cells	1240	40 Cube 240 Cube 320 Triangular prism 160 Tetrahedron 320 Triangular prism 160 Octahedron
Faces	2160	240 Square 960 Square 640 Triangle 320 Triangle
Edges	1440	480+960
Vertices	320
Vertex figure
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:

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

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]

Runcitruncated 5-cube

Runcitruncated 5-cube
Type	Uniform 5-polytope
Schläfli symbol	t0,1,3{4,3,3,3}
Coxeter-Dynkin diagrams
4-faces	202	10 Runcitruncated tesseract 80 3-8 duoprism 80 Octahedral prism 32 Cantellated 5-cell
Cells	1560	40 Truncated cube 240 Octagonal prism 320 Triangular prism 320 Triangular prism 160 Cuboctahedron 320 Triangular prism 160 Octahedron
Faces	3760	240 Octagon 960 Square 320 Equilateral triangle 960 Square 640 Equilateral triangle 640 Equilateral triangle
Edges	3360	480+960+1920
Vertices	960
Vertex figure
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:

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

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]

Runcicantellated 5-cube

Runcicantellated 5-cube
Type	Uniform 5-polytope
Schläfli symbol	t0,2,3{4,3,3,3}
Coxeter-Dynkin diagram
4-faces	202	10 Runcitruncated 16-cell 80 3-4 duoprism 80 Truncated tetrahedral prism 32 Bitruncated 5-cell
Cells	1240	40 Rhombicuboctahedron 240 Cube 320 Hexagonal prism 320 Triangular prism 160 Truncated tetrahedron 160 Truncated tetrahedron
Faces	2960	240 Square 480 Square 960 Square 320 Equilateral triangle 640 Hexagon 320 Equilateral triangle
Edges	2880	960+960+960
Vertices	960
Vertex figure
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:

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

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]

Runcicantitruncated 5-cube

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

Alternate names

Runcicantitruncated penteract
Biruncicantitruncated pentacross
Great prismated penteract (Acronym: 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 + 2 , 1 + 2 2 , 1 + 3 2 , 1 + 3 2 ) {\displaystyle \left(1,\ 1+{\sqrt {2}},\ 1+2{\sqrt {2}},\ 1+3{\sqrt {2}},\ 1+3{\sqrt {2}}\right)}

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]

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
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

Notes

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 (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. . o3x3o3o4x - span, o3x3o3x4x - pattin, o3x3x3o4x - prin, o3x3x3x4x - gippin

External links

, George Olshevsky.
, Jonathan Bowers (spid), Jonathan Bowers

vteFundamental convex regular and uniform polytopes in dimensions 2–10
Family	An	Bn	I2(p) / Dn	E6 / E7 / E8 / F4 / G2	Hn
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	1k2 • 2k1 • k21	n-pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds • Polytope operations