Truncated 7-cubes

7-cube	Truncated 7-cube	Bitruncated 7-cube	Tritruncated 7-cube
7-orthoplex	Truncated 7-orthoplex	Bitruncated 7-orthoplex	Tritruncated 7-orthoplex
Orthogonal projections in B7 Coxeter plane

In seven-dimensional geometry, a truncated 7-cube is a convex uniform 7-polytope, being a truncation of the regular 7-cube.

There are 6 truncations for the 7-cube. Vertices of the truncated 7-cube are located as pairs on the edge of the 7-cube. Vertices of the bitruncated 7-cube are located on the square faces of the 7-cube. Vertices of the tritruncated 7-cube are located inside the cubic cells of the 7-cube. The final three truncations are best expressed relative to the 7-orthoplex.

Truncated 7-cube
Type	uniform 7-polytope
Schläfli symbol	t{4,35}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges	3136
Vertices	896
Vertex figure	Elongated 5-simplex pyramid
Coxeter groups	B7, [35,4]
Properties	convex

• Truncated hepteract (Jonathan Bowers)^[1]

Cartesian coordinates for the vertices of a truncated 7-cube, centered at the origin, are all sign and coordinate permutations of

(1,1+√2,1+√2,1+√2,1+√2,1+√2,1+√2)

orthographic projections

Coxeter plane	B7 / A6	B6 / D7	B5 / D6 / A4
Graph
Dihedral symmetry	[14]	[12]	[10]
Coxeter plane	B4 / D5	B3 / D4 / A2	B2 / D3
Graph
Dihedral symmetry	[8]	[6]	[4]
Coxeter plane	A5	A3
Graph
Dihedral symmetry	[6]	[4]

The truncated 7-cube, is sixth 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 7-cube
Type	uniform 7-polytope
Schläfli symbol	2t{4,35}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges	9408
Vertices	2688
Vertex figure	{ }v{3,3,3}
Coxeter groups	B7, [35,4] D7, [34,1,1]
Properties	convex

• Bitruncated hepteract (Jonathan Bowers)^[2]

Cartesian coordinates for the vertices of a bitruncated 7-cube, centered at the origin, are all sign and coordinate permutations of

(±2,±2,±2,±2,±2,±1,0)

orthographic projections

Coxeter plane	B7 / A6	B6 / D7	B5 / D6 / A4
Graph
Dihedral symmetry	[14]	[12]	[10]
Coxeter plane	B4 / D5	B3 / D4 / A2	B2 / D3
Graph
Dihedral symmetry	[8]	[6]	[4]
Coxeter plane	A5	A3
Graph
Dihedral symmetry	[6]	[4]

The bitruncated 7-cube is fifth 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 7-cube
Type	uniform 7-polytope
Schläfli symbol	3t{4,35}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges	13440
Vertices	3360
Vertex figure	{4}v{3,3}
Coxeter groups	B7, [35,4] D7, [34,1,1]
Properties	convex

• Tritruncated hepteract (Jonathan Bowers)^[3]

Cartesian coordinates for the vertices of a tritruncated 7-cube, centered at the origin, are all sign and coordinate permutations of

(±2,±2,±2,±2,±1,0,0)

orthographic projections

Coxeter plane	B7 / A6	B6 / D7	B5 / D6 / A4
Graph
Dihedral symmetry	[14]	[12]	[10]
Coxeter plane	B4 / D5	B3 / D4 / A2	B2 / D3
Graph
Dihedral symmetry	[8]	[6]	[4]
Coxeter plane	A5	A3
Graph
Dihedral symmetry	[6]	[4]

• 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. "7D uniform polytopes (polyexa)". o3o3o3o3o3x4x - taz, o3o3o3o3x3x4o - botaz, o3o3o3x3x3o4o - totaz

• Polytopes of Various Dimensions
• Multi-dimensional Glossary