Rectified 7-cubes

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

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

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
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

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

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

(±1,±1,±1,±1,±1,±1,0)

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

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

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

(±1,±1,±1,±1,±1,0,0)

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

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

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

(±1,±1,±1,±1,0,0,0)

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}

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

• Polytopes of Various Dimensions
• Multi-dimensional Glossary