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
Orthogonal projections in B6 Coxeter plane

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

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 ${\displaystyle r\left\{{\begin{array}{l}3,3,3,3\\4\end{array}}\right\}}$
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

Images

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]

Bicantellated 6-cube

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

Images

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]

Cantitruncated 6-cube

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

Images

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:

Petrie polygon projections

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

Bicantitruncated 6-cube

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

Images

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]

Related polytopes

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.

B6 polytopes
β6	t1β6	t2β6	t2γ6	t1γ6	γ6	t0,1β6	t0,2β6
t1,2β6	t0,3β6	t1,3β6	t2,3γ6	t0,4β6	t1,4γ6	t1,3γ6	t1,2γ6
t0,5γ6	t0,4γ6	t0,3γ6	t0,2γ6	t0,1γ6	t0,1,2β6	t0,1,3β6	t0,2,3β6
t1,2,3β6	t0,1,4β6	t0,2,4β6	t1,2,4β6	t0,3,4β6	t1,2,4γ6	t1,2,3γ6	t0,1,5β6
t0,2,5β6	t0,3,4γ6	t0,2,5γ6	t0,2,4γ6	t0,2,3γ6	t0,1,5γ6	t0,1,4γ6	t0,1,3γ6
t0,1,2γ6	t0,1,2,3β6	t0,1,2,4β6	t0,1,3,4β6	t0,2,3,4β6	t1,2,3,4γ6	t0,1,2,5β6	t0,1,3,5β6
t0,2,3,5γ6	t0,2,3,4γ6	t0,1,4,5γ6	t0,1,3,5γ6	t0,1,3,4γ6	t0,1,2,5γ6	t0,1,2,4γ6	t0,1,2,3γ6
t0,1,2,3,4β6	t0,1,2,3,5β6	t0,1,2,4,5β6	t0,1,2,4,5γ6	t0,1,2,3,5γ6	t0,1,2,3,4γ6	t0,1,2,3,4,5γ6

Notes

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
    • (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

External links

• Polytopes of Various Dimensions
• Multi-dimensional Glossary