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 BC7 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

Rectified 7-cube
Type	uniform 7-polytope
Schläfli symbol	r{4,3,3,3,3,3}
Coxeter-Dynkin diagrams
6-faces
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure	5-simplex prism
Coxeter groups	BC7, [3,3,3,3,3,4]
Properties	convex

Alternate names

• rectified hepteract (Acronym resa) (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
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure	{3}x{3,3,3}
Coxeter groups	BC7, [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
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure	{3,3}x{3,3}
Coxeter groups	BC7, [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
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		Rectangle	Disphenoid	{3}×{4} duoprism		{3,3}×{3,4} duoprism

Notes

1. Klitzing, (o3o3o3o3o3x4o - rasa)
2. Klitzing, (o3o3o3o3x3o4o - bersa)
3. 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.
• Richard Klitzing, 7D, uniform polytopes (polyexa) o3o3o3x3o3o4o - sez, o3o3o3o3x3o4o - bersa, o3o3o3o3o3x4o - rasa

External links

• Olshevsky, George, Cross polytope at Glossary for Hyperspace.
• Polytopes of Various Dimensions
• Multi-dimensional Glossary