Cantellated 5-cubes

In six-dimensional geometry, a cantellated 5-cube is a convex uniform 5-polytope, being a cantellation of the regular 5-cube.

Orthogonal projections in B₅ Coxeter plane
5-cube	Cantellated 5-cube	Bicantellated 5-cube	Cantellated 5-orthoplex
5-orthoplex	Cantitruncated 5-cube	Bicantitruncated 5-cube	Cantitruncated 5-orthoplex

There are 6 unique cantellation for the 5-cube, including truncations. Half of them are more easily constructed from the dual 5-orthoplex

Cantellated 5-cube

Cantellated 5-cube
Type	Uniform 5-polytope
Schläfli symbol	rr{4,3,3,3} = $r\left\{{\begin{array}{l}4\\3,3,3\end{array}}\right\}$
Coxeter-Dynkin diagram	=
4-faces	122	10 80 32
Cells	680	40 320 160 160
Faces	1520	80 480 320 640
Edges	1280	320+960
Vertices	320
Vertex figure
Coxeter group	B₅ [4,3,3,3]
Properties	convex, uniform

Alternate names

Small rhombated penteract (Acronym: sirn) (Jonathan Bowers)

Coordinates

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

\left(\pm 1,\ \pm 1,\ \pm (1+{\sqrt {2}}),\ \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]

Bicantellated 5-cube

Bicantellated 5-cube
Type	Uniform 5-polytope
Schläfli symbols	2rr{4,3,3,3} = $r\left\{{\begin{array}{l}3,4\\3,3\end{array}}\right\}$ r{3^2,1,1} = $r\left\{{\begin{array}{l}3,3\\3\\3\end{array}}\right\}$
Coxeter-Dynkin diagrams	=
4-faces	122	10 80 32
Cells	840	40 240 160 320 80
Faces	2160	240 320 960 320 320
Edges	1920	960+960
Vertices	480
Vertex figure
Coxeter groups	B₅, [3,3,3,4] D₅, [3^2,1,1]
Properties	convex, uniform

In five-dimensional geometry, a bicantellated 5-cube is a uniform 5-polytope.

Alternate names

Bicantellated penteract, bicantellated 5-orthoplex, or bicantellated pentacross
Small birhombated penteractitriacontiditeron (Acronym: sibrant) (Jonathan Bowers)

Coordinates

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

(0,1,1,2,2)

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]

Cantitruncated 5-cube

Cantitruncated 5-cube
Type	Uniform 5-polytope
Schläfli symbol	tr{4,3,3,3} = $t\left\{{\begin{array}{l}4\\3,3,3\end{array}}\right\}$
Coxeter-Dynkin diagram	=
4-faces	122	10 80 32
Cells	680	40 320 160 160
Faces	1520	80 480 320 640
Edges	1600	320+320+960
Vertices	640
Vertex figure
Coxeter group	B₅ [4,3,3,3]
Properties	convex, uniform

Alternate names

Tricantitruncated 5-orthoplex / tricantitruncated pentacross
Great rhombated penteract (girn) (Jonathan Bowers)

Coordinates

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

\left(1,\ 1+{\sqrt {2}},\ 1+2{\sqrt {2}},\ 1+2{\sqrt {2}},\ 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]

Related polytopes

It is third in a series of cantitruncated hypercubes:

Petrie polygon projections

Truncated cuboctahedron	Cantitruncated tesseract	Cantitruncated 5-cube	Cantitruncated 6-cube	Cantitruncated 7-cube	Cantitruncated 8-cube

Bicantitruncated 5-cube

Bicantitruncated 5-cube
Type	uniform 5-polytope
Schläfli symbol	2tr{3,3,3,4} = $t\left\{{\begin{array}{l}3,4\\3,3\end{array}}\right\}$ t{3^2,1,1} = $t\left\{{\begin{array}{l}3,3\\3\\3\end{array}}\right\}$
Coxeter-Dynkin diagrams	=
4-faces	122	10 80 32
Cells	840	40 240 160 320 80
Faces	2160	240 320 960 320 320
Edges	2400	960+480+960
Vertices	960
Vertex figure
Coxeter groups	B₅, [3,3,3,4] D₅, [3^2,1,1]
Properties	convex, uniform

Alternate names

Bicantitruncated penteract
Bicantitruncated pentacross
Great birhombated penteractitriacontiditeron (Acronym: gibrant) (Jonathan Bowers)

Coordinates

Cartesian coordinates for the vertices of a bicantitruncated 5-cube, centered at the origin, are all sign and coordinate permutations of

(±3,±3,±2,±1,0)

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 from a set 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
  - (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)". o3o3x3o4x - sirn, o3x3o3x4o - sibrant, o3o3x3x4x - girn, o3x3x3x4o - gibrant

External links

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

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

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