Cantellated 6-cubes

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 [math]\displaystyle{ r\left\{\begin{array}{l}3, 3, 3, 3\\4\end{array}\right\} }[/math]
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

Alternate names

• Cantellated hexeract
• Small rhombated hexeract (acronym: srox) (Jonathan Bowers)

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 [math]\displaystyle{ r\left\{\begin{array}{l}3, 3, 3\\3, 4\end{array}\right\} }[/math]
Coxeter-Dynkin diagrams
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure
Coxeter groups	B6, [3,3,3,3,4]
Properties	convex

Alternate names

• Bicantellated hexeract
• Small birhombated hexeract (acronym: saborx) (Jonathan Bowers)

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 [math]\displaystyle{ t\left\{\begin{array}{l}3, 3, 3, 3\\4\end{array}\right\} }[/math]
Coxeter-Dynkin diagrams
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure
Coxeter groups	B6, [3,3,3,3,4]
Properties	convex

Alternate names

• Cantitruncated hexeract
• Great rhombihexeract (acronym: grox) (Jonathan Bowers)

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:

Bicantitruncated 6-cube

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

Alternate names

• Bicantitruncated hexeract
• Great birhombihexeract (acronym: gaborx) (Jonathan Bowers)

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.

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. "6D uniform polytopes (polypeta)". https://bendwavy.org/klitzing/dimensions/polypeta.htm.  o3o3o3x3o4x - srox, o3o3x3o3x4o - saborx, o3o3o3x3x4x - grox, o3o3x3x3x4o - gaborx

External links

• Polytopes of Various Dimensions
• Multi-dimensional Glossary