Runcinated 6-cubes

6-cube	Runcinated 6-cube	Biruncinated 6-cube	Runcinated 6-orthoplex	6-orthoplex
Runcitruncated 6-cube	Biruncitruncated 6-cube	Runcicantellated 6-orthoplex	Runcicantellated 6-cube	Biruncitruncated 6-orthoplex
Runcitruncated 6-orthoplex	Runcicanti-truncated 6-cube	Biruncicanti-truncated 6-cube	Runcicanti-truncated 6-orthoplex
Orthogonal projections in B6 Coxeter plane

In six-dimensional geometry, a runcinated 6-cube is a convex uniform 6-polytope with 3rd order truncations (runcination) of the regular 6-cube.

There are 12 unique runcinations of the 6-cube with permutations of truncations, and cantellations. Half are expressed relative to the dual 6-orthoplex.

Runcinated 6-cube

Runcinated 6-cube
Type	Uniform 6-polytope
Schläfli symbol	t0,3{4,3,3,3,3}
Coxeter-Dynkin diagram
4-faces
Cells
Faces
Edges	7680
Vertices	1280
Vertex figure
Coxeter group	B6 [4,3,3,3,3]
Properties	convex

Alternate names

• Small prismated hexeract (spox) (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]

Biruncinated 6-cube

Biruncinated 6-cube
Type	Uniform 6-polytope
Schläfli symbol	t1,4{4,3,3,3,3}
Coxeter-Dynkin diagram
4-faces
Cells
Faces
Edges	11520
Vertices	1920
Vertex figure
Coxeter group	B6 [4,3,3,3,3]
Properties	convex

Alternate names

• Small biprismated hexeractihexacontatetrapeton (sobpoxog) (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]

Runcitruncated 6-cube

Runcitruncated 6-cube
Type	Uniform 6-polytope
Schläfli symbol	t0,1,3{4,3,3,3,3}
Coxeter-Dynkin diagram
4-faces
Cells
Faces
Edges	17280
Vertices	3840
Vertex figure
Coxeter group	B6 [4,3,3,3,3]
Properties	convex

Alternate names

• Prismatotruncated hexeract (potax) (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]

Biruncitruncated 6-cube

Biruncitruncated 6-cube
Type	Uniform 6-polytope
Schläfli symbol	t1,2,4{4,3,3,3,3}
Coxeter-Dynkin diagram
4-faces
Cells
Faces
Edges	23040
Vertices	5760
Vertex figure
Coxeter group	B6 [4,3,3,3,3]
Properties	convex

Alternate names

• Biprismatotruncated hexeract (boprag) (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]

Runcicantellated 6-cube

Runcicantellated 6-cube
Type	Uniform 6-polytope
Schläfli symbol	t0,2,3{4,3,3,3,3}
Coxeter-Dynkin diagram
4-faces
Cells
Faces
Edges	13440
Vertices	3840
Vertex figure
Coxeter group	B6 [4,3,3,3,3]
Properties	convex

Alternate names

• Prismatorhombated hexeract (prox) (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]

Runcicantitruncated 6-cube

Runcicantitruncated 6-cube
Type	Uniform 6-polytope
Schläfli symbol	t0,1,2,3{4,3,3,3,3}
Coxeter-Dynkin diagram
4-faces
Cells
Faces
Edges	23040
Vertices	7680
Vertex figure
Coxeter group	B6 [4,3,3,3,3]
Properties	convex

Alternate names

• Great prismated hexeract (gippox) (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]

Biruncitruncated 6-cube

Biruncitruncated 6-cube
Type	Uniform 6-polytope
Schläfli symbol	t1,2,3,4{4,3,3,3,3}
Coxeter-Dynkin diagram
4-faces
Cells
Faces
Edges	23040
Vertices	5760
Vertex figure
Coxeter group	B6 [4,3,3,3,3]
Properties	convex

Alternate names

• Biprismatotruncated hexeract (boprag) (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 from 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.  o3o3x3o3o4x - spox, o3x3o3o3x4o - sobpoxog, o3o3x3o3x4x - potax, o3x3o3x3x4o - boprag, o3o3x3x3o4x - prox, o3o3x3x3x4x - gippox, o3x3x3x3x4o - boprag

External links

• Weisstein, Eric W.. "Hypercube". http://mathworld.wolfram.com/Hypercube.html. 
• Polytopes of Various Dimensions
• Multi-dimensional Glossary