Truncated 6-cubes

6-cube	Truncated 6-cube	Bitruncated 6-cube	Tritruncated 6-cube
6-orthoplex	Truncated 6-orthoplex	Bitruncated 6-orthoplex
Orthogonal projections in B6 Coxeter plane

In six-dimensional geometry, a truncated 6-cube (or truncated hexeract) is a convex uniform 6-polytope, being a truncation of the regular 6-cube.

There are 5 truncations for the 6-cube. Vertices of the truncated 6-cube are located as pairs on the edge of the 6-cube. Vertices of the bitruncated 6-cube are located on the square faces of the 6-cube. Vertices of the tritruncated 6-cube are located inside the cubic cells of the 6-cube.

Truncated 6-cube

Truncated 6-cube
Type	uniform 6-polytope
Class	B6 polytope
Schläfli symbol	t{4,3,3,3,3}
Coxeter-Dynkin diagrams
5-faces	76
4-faces	464
Cells	1120
Faces	1520
Edges	1152
Vertices	384
Vertex figure	( )v{3,3,3}
Coxeter groups	B6, [3,3,3,3,4]
Properties	convex

Alternate names

• Truncated hexeract (Acronym: tox) (Jonathan Bowers)

Construction and coordinates

The truncated 6-cube may be constructed by truncating the vertices of the 6-cube at $1/({\sqrt {2}}+2)$ of the edge length. A regular 5-simplex replaces each original vertex.

The Cartesian coordinates of the vertices of a truncated 6-cube having edge length 2 are the permutations of:

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

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

The truncated 6-cube, is fifth in a sequence of truncated hypercubes:

Truncated hypercubes

Image								...
Name	Octagon	Truncated cube	Truncated tesseract	Truncated 5-cube	Truncated 6-cube	Truncated 7-cube	Truncated 8-cube
Coxeter diagram
Vertex figure	( )v( )	( )v{ }	( )v{3}	( )v{3,3}	( )v{3,3,3}	( )v{3,3,3,3}	( )v{3,3,3,3,3}

Bitruncated 6-cube

Bitruncated 6-cube
Type	uniform 6-polytope
Class	B6 polytope
Schläfli symbol	2t{4,3,3,3,3}
Coxeter-Dynkin diagrams
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure	{ }v{3,3}
Coxeter groups	B6, [3,3,3,3,4]
Properties	convex

Alternate names

• Bitruncated hexeract (Acronym: botox) (Jonathan Bowers)

Construction and coordinates

The Cartesian coordinates of the vertices of a bitruncated 6-cube having edge length 2 are the permutations of:

$\left(0,\ \pm 1,\ \pm 2,\ \pm 2,\ \pm 2,\ \pm 2\right)$

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

The bitruncated 6-cube is fourth in a sequence of bitruncated hypercubes:

Bitruncated hypercubes

Image							...
Name	Bitruncated cube	Bitruncated tesseract	Bitruncated 5-cube	Bitruncated 6-cube	Bitruncated 7-cube	Bitruncated 8-cube
Coxeter
Vertex figure	( )v{ }	{ }v{ }	{ }v{3}	{ }v{3,3}	{ }v{3,3,3}	{ }v{3,3,3,3}

Tritruncated 6-cube

Tritruncated 6-cube
Type	uniform 6-polytope
Class	B6 polytope
Schläfli symbol	3t{4,3,3,3,3}
Coxeter-Dynkin diagrams
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure	{3}v{4}
Coxeter groups	B6, [3,3,3,3,4]
Properties	convex

Alternate names

• Tritruncated hexeract (Acronym: xog) (Jonathan Bowers)

Construction and coordinates

The Cartesian coordinates of the vertices of a tritruncated 6-cube having edge length 2 are the permutations of:

$\left(0,\ 0,\ \pm 1,\ \pm 2,\ \pm 2,\ \pm 2\right)$

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

2-isotopic hypercubes

Dim.	2	3	4	5	6	7	8	n
Name	t{4}	r{4,3}	2t{4,3,3}	2r{4,3,3,3}	3t{4,3,3,3,3}	3r{4,3,3,3,3,3}	4t{4,3,3,3,3,3,3}	...
Coxeter diagram
Images
Facets		{3} {4}	t{3,3} t{3,4}	r{3,3,3} r{3,3,4}	2t{3,3,3,3} 2t{3,3,3,4}	2r{3,3,3,3,3} 2r{3,3,3,3,4}	3t{3,3,3,3,3,3} 3t{3,3,3,3,3,4}
Vertex figure	( )v( )	{ }×{ }	{ }v{ }	{3}×{4}	{3}v{4}	{3,3}×{3,4}	{3,3}v{3,4}

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)". o3o3o3o3x4x - tox, o3o3o3x3x4o - botox, o3o3x3x3o4o - xog

External links

• Weisstein, Eric W. "Hypercube". MathWorld.
• Polytopes of Various Dimensions
• Multi-dimensional Glossary