8-simplex |
Rectified 8-simplex | ||
Birectified 8-simplex |
Trirectified 8-simplex | ||
Orthogonal projections in A_{8} Coxeter plane |
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In eight-dimensional geometry, a rectified 8-simplex is a convex uniform 8-polytope, being a rectification of the regular 8-simplex.
There are unique 3 degrees of rectifications in regular 8-polytopes. Vertices of the rectified 8-simplex are located at the edge-centers of the 8-simplex. Vertices of the birectified 8-simplex are located in the triangular face centers of the 8-simplex. Vertices of the trirectified 8-simplex are located in the tetrahedral cell centers of the 8-simplex.
Rectified 8-simplex | |
---|---|
Type | uniform 8-polytope |
Coxeter symbol | 0_{61} |
Schläfli symbol | t_{1}{3^{7}} r{3^{7}} = {3^{6,1}} or |
Coxeter-Dynkin diagrams | or |
7-faces | 18 |
6-faces | 108 |
5-faces | 336 |
4-faces | 630 |
Cells | 756 |
Faces | 588 |
Edges | 252 |
Vertices | 36 |
Vertex figure | 7-simplex prism, {}×{3,3,3,3,3} |
Petrie polygon | enneagon |
Coxeter group | A_{8}, [3^{7}], order 362880 |
Properties | convex |
E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as S^{1}
_{8}. It is also called 0_{6,1} for its branching Coxeter-Dynkin diagram, shown as .
The Cartesian coordinates of the vertices of the rectified 8-simplex can be most simply positioned in 9-space as permutations of (0,0,0,0,0,0,0,1,1). This construction is based on facets of the rectified 9-orthoplex.
A_{k} Coxeter plane | A_{8} | A_{7} | A_{6} | A_{5} |
---|---|---|---|---|
Graph | ||||
Dihedral symmetry | [9] | [8] | [7] | [6] |
A_{k} Coxeter plane | A_{4} | A_{3} | A_{2} | |
Graph | ||||
Dihedral symmetry | [5] | [4] | [3] |
Birectified 8-simplex | |
---|---|
Type | uniform 8-polytope |
Coxeter symbol | 0_{52} |
Schläfli symbol | t_{2}{3^{7}} 2r{3^{7}} = {3^{5,2}} or |
Coxeter-Dynkin diagrams | or |
7-faces | 18 |
6-faces | 144 |
5-faces | 588 |
4-faces | 1386 |
Cells | 2016 |
Faces | 1764 |
Edges | 756 |
Vertices | 84 |
Vertex figure | {3}×{3,3,3,3} |
Coxeter group | A_{8}, [3^{7}], order 362880 |
Properties | convex |
E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as S^{2}
_{8}. It is also called 0_{5,2} for its branching Coxeter-Dynkin diagram, shown as .
The birectified 8-simplex is the vertex figure of the 1_{52} honeycomb.
The Cartesian coordinates of the vertices of the birectified 8-simplex can be most simply positioned in 9-space as permutations of (0,0,0,0,0,0,1,1,1). This construction is based on facets of the birectified 9-orthoplex.
A_{k} Coxeter plane | A_{8} | A_{7} | A_{6} | A_{5} |
---|---|---|---|---|
Graph | ||||
Dihedral symmetry | [9] | [8] | [7] | [6] |
A_{k} Coxeter plane | A_{4} | A_{3} | A_{2} | |
Graph | ||||
Dihedral symmetry | [5] | [4] | [3] |
Trirectified 8-simplex | |
---|---|
Type | uniform 8-polytope |
Coxeter symbol | 0_{43} |
Schläfli symbol | t_{3}{3^{7}} 3r{3^{7}} = {3^{4,3}} or |
Coxeter-Dynkin diagrams | or |
7-faces | 9 + 9 |
6-faces | 36 + 72 + 36 |
5-faces | 84 + 252 + 252 + 84 |
4-faces | 126 + 504 + 756 + 504 |
Cells | 630 + 1260 + 1260 |
Faces | 1260 + 1680 |
Edges | 1260 |
Vertices | 126 |
Vertex figure | {3,3}×{3,3,3} |
Petrie polygon | enneagon |
Coxeter group | A_{7}, [3^{7}], order 362880 |
Properties | convex |
E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as S^{3}
_{8}. It is also called 0_{4,3} for its branching Coxeter-Dynkin diagram, shown as .
The Cartesian coordinates of the vertices of the trirectified 8-simplex can be most simply positioned in 9-space as permutations of (0,0,0,0,0,1,1,1,1). This construction is based on facets of the trirectified 9-orthoplex.
A_{k} Coxeter plane | A_{8} | A_{7} | A_{6} | A_{5} |
---|---|---|---|---|
Graph | ||||
Dihedral symmetry | [9] | [8] | [7] | [6] |
A_{k} Coxeter plane | A_{4} | A_{3} | A_{2} | |
Graph | ||||
Dihedral symmetry | [5] | [4] | [3] |
This polytope is the vertex figure of the 9-demicube, and the edge figure of the uniform 2_{61} honeycomb.
It is also one of 135 uniform 8-polytopes with A_{8} symmetry.