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 | |
---|---|
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 |
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 are all permutations of:
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 |
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 are all permutations of:
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 |
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 are all permutations of:
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 |
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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} |