![]() 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 |
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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
editRectified 7-cube | |
---|---|
Typ | 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
edit- rectified hepteract (Acronym rasa) (Jonathan Bowers)[1]
Images
editCoxeter 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
editCartesian coordinates for the vertices of a rectified 7-cube, centered at the origin, edge length are all permutations of:
- (±1,±1,±1,±1,±1,±1,0)
Birectified 7-cube
editBirectified 7-cube | |
---|---|
Typ | 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
edit- Birectified hepteract (Acronym bersa) (Jonathan Bowers)[2]
Images
editCoxeter 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
editCartesian coordinates for the vertices of a birectified 7-cube, centered at the origin, edge length are all permutations of:
- (±1,±1,±1,±1,±1,0,0)
Trirectified 7-cube
editTrirectified 7-cube | |
---|---|
Typ | 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
edit- Trirectified hepteract
- Trirectified 7-orthoplex
- Trirectified heptacross (Acronym sez) (Jonathan Bowers)[3]
Images
editCoxeter 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
editCartesian coordinates for the vertices of a trirectified 7-cube, centered at the origin, edge length are all permutations of:
- (±1,±1,±1,±1,0,0,0)
Related polytopes
editDim. | 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
editReferences
edit- 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