Cantellated 6-cubes

In six-dimensional geometry, a cantellated 6-cube is a convex uniform 6-polytope, being a cantellation of the regular 6-cube.

Orthogonal projections in B₆ Coxeter plane
6-cube		Cantellated 6-cube		Bicantellated 6-cube
6-orthoplex		Cantellated 6-orthoplex		Bicantellated 6-orthoplex
Cantitruncated 6-cube	Bicantitruncated 6-cube		Bicantitruncated 6-orthoplex		Cantitruncated 6-orthoplex

There are 8 cantellations for the 6-cube, including truncations. Half of them are more easily constructed from the dual 5-orthoplex.

Cantellated 6-cube

Cantellated 6-cube
Type	uniform 6-polytope
Schläfli symbol	rr{4,3,3,3,3} or $r\left\{{\begin{array}{l}3,3,3,3\\4\end{array}}\right\}$
Coxeter-Dynkin diagrams
5-faces
4-faces
Cells
Faces
Edges	4800
Vertices	960
Vertex figure
Coxeter groups	B₆, [3,3,3,3,4]
Properties	convex

orthographic projections
Coxeter plane	B₆	B₅	B₄
Graph
Dihedral symmetry	[12]	[10]	[8]
Coxeter plane	B₃	B₂
Graph
Dihedral symmetry	[6]	[4]
Coxeter plane	A₅	A₃
Graph
Dihedral symmetry	[6]	[4]

Cantellated 6-cube
Type	uniform 6-polytope
Schläfli symbol	2rr{4,3,3,3,3} or $r\left\{{\begin{array}{l}3,3,3\\3,4\end{array}}\right\}$
Coxeter-Dynkin diagrams
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure
Coxeter groups	B₆, [3,3,3,3,4]
Properties	convex

orthographic projections
Coxeter plane	B₆	B₅	B₄
Graph
Dihedral symmetry	[12]	[10]	[8]
Coxeter plane	B₃	B₂
Graph
Dihedral symmetry	[6]	[4]
Coxeter plane	A₅	A₃
Graph
Dihedral symmetry	[6]	[4]

Cantellated 6-cube
Type	uniform 6-polytope
Schläfli symbol	tr{4,3,3,3,3} or $t\left\{{\begin{array}{l}3,3,3,3\\4\end{array}}\right\}$
Coxeter-Dynkin diagrams
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure
Coxeter groups	B₆, [3,3,3,3,4]
Properties	convex

orthographic projections
Coxeter plane	B₆	B₅	B₄
Graph
Dihedral symmetry	[12]	[10]	[8]
Coxeter plane	B₃	B₂
Graph
Dihedral symmetry	[6]	[4]
Coxeter plane	A₅	A₃
Graph
Dihedral symmetry	[6]	[4]

It is fourth in a series of cantitruncated hypercubes:

Cantellated 6-cube
Type	uniform 6-polytope
Schläfli symbol	2tr{4,3,3,3,3} or $t\left\{{\begin{array}{l}3,3,3\\3,4\end{array}}\right\}$
Coxeter-Dynkin diagrams
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure
Coxeter groups	B₆, [3,3,3,3,4]
Properties	convex

orthographic projections
Coxeter plane	B₆	B₅	B₄
Graph
Dihedral symmetry	[12]	[10]	[8]
Coxeter plane	B₃	B₂
Graph
Dihedral symmetry	[6]	[4]
Coxeter plane	A₅	A₃
Graph
Dihedral symmetry	[6]	[4]

These polytopes are part of a set of 63 uniform 6-polytopes generated from the B₆ Coxeter plane, including the regular 6-cube or 6-orthoplex.

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. "6D uniform polytopes (polypeta)". o3o3o3x3o4x - srox, o3o3x3o3x4o - saborx, o3o3o3x3x4x - grox, o3o3x3x3x4o - gaborx

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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