Rectified 7-cubes

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

Orthogonal projections in B₇ Coxeter plane
7-cube	Rectified 7-cube	Birectified 7-cube	Trirectified 7-cube
Birectified 7-orthoplex	Rectified 7-orthoplex	7-orthoplex

There are unique 7 degrees of rectifications, the zeroth being the 7-cube, and the 6th and last being the 7-cube. Vertices of the rectified 7-cube are located at the edge-centers of the 7-ocube. Vertices of the birectified 7-cube are located in the square face centers of the 7-cube. Vertices of the trirectified 7-cube are located in the cube cell centers of the 7-cube.

Rectified 7-cube

Rectified 7-cube
Type	uniform 7-polytope
Schläfli symbol	r{4,3,3,3,3,3}
Coxeter-Dynkin diagrams
6-faces	128 + 14
5-faces	896 + 84
4-faces	2688 + 280
Cells	4480 + 560
Faces	4480 + 672
Edges	2688
Vertices	448
Vertex figure	5-simplex prism
Coxeter groups	B₇, [3,3,3,3,3,4]
Properties	convex

Alternate names

rectified hepteract (Acronym rasa) (Jonathan Bowers)[1]

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]

Cartesian coordinates

Cartesian coordinates for the vertices of a rectified 7-cube, centered at the origin, edge length ${\sqrt {2}}\$ are all permutations of:

(±1,±1,±1,±1,±1,±1,0)

Birectified 7-cube

Birectified 7-cube
Type	uniform 7-polytope
Coxeter symbol	0₄₁₁
Schläfli symbol	2r{4,3,3,3,3,3}
Coxeter-Dynkin diagrams
6-faces	128 + 14
5-faces	448 + 896 + 84
4-faces	2688 + 2688 + 280
Cells	6720 + 4480 + 560
Faces	8960 + 4480
Edges	6720
Vertices	672
Vertex figure	{3}x{3,3,3}
Coxeter groups	B₇, [3,3,3,3,3,4]
Properties	convex

Alternate names

Birectified hepteract (Acronym bersa) (Jonathan Bowers)[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]

Cartesian coordinates

Cartesian coordinates for the vertices of a birectified 7-cube, centered at the origin, edge length ${\sqrt {2}}\$ are all permutations of:

(±1,±1,±1,±1,±1,0,0)

Trirectified 7-cube

Trirectified 7-cube
Type	uniform 7-polytope
Schläfli symbol	3r{4,3,3,3,3,3}
Coxeter-Dynkin diagrams
6-faces	128 + 14
5-faces	448 + 896 + 84
4-faces	672 + 2688 + 2688 + 280
Cells	3360 + 6720 + 4480
Faces	6720 + 8960
Edges	6720
Vertices	560
Vertex figure	{3,3}x{3,3}
Coxeter groups	B₇, [3,3,3,3,3,4]
Properties	convex

Alternate names

Trirectified hepteract
Trirectified 7-orthoplex
Trirectified heptacross (Acronym sez) (Jonathan Bowers)[3]

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]

Cartesian coordinates

Cartesian coordinates for the vertices of a trirectified 7-cube, centered at the origin, edge length ${\sqrt {2}}\$ are all permutations of:

(±1,±1,±1,±1,0,0,0)

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}

Notes

Klitzing, (o3o3o3o3o3x4o - rasa)
Klitzing, (o3o3o3o3x3o4o - bersa)
Klitzing, (o3o3o3x3o3o4o - sez)

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)". o3o3o3x3o3o4o - sez, o3o3o3o3x3o4o - bersa, o3o3o3o3o3x4o - rasa

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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[1] Klitzing, (o3o3o3o3o3x4o - rasa)

[2] Klitzing, (o3o3o3o3x3o4o - bersa)

[3] Klitzing, (o3o3o3x3o3o4o - sez)