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Rectified 7-cubes

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Source updated April 13, 2026

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

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.

01Rectified 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)

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)

02Birectified 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)

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)

03Trirectified 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)

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}

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Sources and credits

This article is adapted from the Wikipedia article “Rectified 7-cubes”, written by its contributors and licensed under CC BY-SA 4.0. Fathomly has changed the layout, removed citation markers, navigation and maintenance notices, and adjusted punctuation. This adapted version is shared under the same license. For references, see the original article.

Images, from Wikimedia Commons:

7-cube t0.svg by self, Public domain

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