Truncated 6-cubes

6-cube Template:CDD	Truncated 6-cube Template:CDD	Bitruncated 6-cube Template:CDD	Tritruncated 6-cube Template:CDD
6-orthoplex Template:CDD	Truncated 6-orthoplex Template:CDD	Bitruncated 6-orthoplex Template:CDD
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. Template:TOC left Template:-

Truncated 6-cube

Truncated 6-cube
Type	uniform 6-polytope
Class	B6 polytope
Schläfli symbol	t{4,3,3,3,3}
Coxeter-Dynkin diagrams	Template:CDD
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:

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

Related polytopes

The truncated 6-cube, is fifth in a sequence of truncated hypercubes: Template:Truncated hypercube polytopes

Bitruncated 6-cube

Bitruncated 6-cube
Type	uniform 6-polytope
Class	B6 polytope
Schläfli symbol	2t{4,3,3,3,3}
Coxeter-Dynkin diagrams	Template:CDD
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:

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

Related polytopes

The bitruncated 6-cube is fourth in a sequence of bitruncated hypercubes: Template:Bitruncated hypercube polytopes

Tritruncated 6-cube

Tritruncated 6-cube
Type	uniform 6-polytope
Class	B6 polytope
Schläfli symbol	3t{4,3,3,3,3}
Coxeter-Dynkin diagrams	Template:CDD
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:

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

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, Template:ISBN [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.
• Template:KlitzingPolytopes o3o3o3o3x4x - tox, o3o3o3x3x4o - botox, o3o3x3x3o4o - xog

External links

• Template:MathWorld
• Polytopes of Various Dimensions
• Multi-dimensional Glossary