Truncated 7-cubes

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

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

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

Alternate names

Truncated hepteract (Jonathan Bowers)^[1]

Coordinates

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)

Images

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]

Related polytopes

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

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, [3^4,1,1]
Properties	convex

Alternate names

Bitruncated hepteract (Jonathan Bowers)^[2]

Coordinates

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)

Images

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]

Related polytopes

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

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, [3^4,1,1]
Properties	convex

Alternate names

Tritruncated hepteract (Jonathan Bowers)^[3]

Coordinates

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)

Images

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]

Notes

↑ Klitizing (x3x3o3o3o3o4o - taz)
↑ Klitizing (o3x3x3o3o3o4o - botaz)
↑ Klitizing (o3o3x3x3o3o4o - totaz)

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
  - (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

External links

Fundamental convex regular and uniform polytopes in dimensions 2–10
Family	A_n	B_n	I₂(p) / D_n	E₆ / E₇ / E₈ / F₄ / G₂	H_n
Regular polygon	Triangle	Square	p-gon	Hexagon	Pentagon
Uniform polyhedron	Tetrahedron	Octahedron • Cube	Demicube		Dodecahedron • Icosahedron
Uniform polychoron	Pentachoron	16-cell • Tesseract	Demitesseract	24-cell	120-cell • 600-cell
Uniform 5-polytope	5-simplex	5-orthoplex • 5-cube	5-demicube
Uniform 6-polytope	6-simplex	6-orthoplex • 6-cube	6-demicube	1₂₂ • 2₂₁
Uniform 7-polytope	7-simplex	7-orthoplex • 7-cube	7-demicube	1₃₂ • 2₃₁ • 3₂₁
Uniform 8-polytope	8-simplex	8-orthoplex • 8-cube	8-demicube	1₄₂ • 2₄₁ • 4₂₁
Uniform 9-polytope	9-simplex	9-orthoplex • 9-cube	9-demicube
Uniform 10-polytope	10-simplex	10-orthoplex • 10-cube	10-demicube
Uniform n-polytope	n-simplex	n-orthoplex • n-cube	n-demicube	1_k2 • 2_k1 • k₂₁	n-pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds

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