Rectified 8-simplexes
8-simplex |
Rectified 8-simplex | ||
Birectified 8-simplex |
Trirectified 8-simplex | ||
Orthogonal projections in A8 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
Rectified 8-simplex | |
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Type | uniform 8-polytope |
Coxeter symbol | 061 |
Schläfli symbol | t1{37} r{37} = {36,1} 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 | A8, [37], order 362880 |
Properties | convex |
E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as S1
8. It is also called 06,1 for its branching Coxeter-Dynkin diagram, shown as .
Coordinates
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.
Images
Ak Coxeter plane | A8 | A7 | A6 | A5 |
---|---|---|---|---|
Graph | ||||
Dihedral symmetry | [9] | [8] | [7] | [6] |
Ak Coxeter plane | A4 | A3 | A2 | |
Graph | ||||
Dihedral symmetry | [5] | [4] | [3] |
Birectified 8-simplex
Birectified 8-simplex | |
---|---|
Type | uniform 8-polytope |
Coxeter symbol | 052 |
Schläfli symbol | t2{37} 2r{37} = {35,2} 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 | A8, [37], order 362880 |
Properties | convex |
E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as S2
8. It is also called 05,2 for its branching Coxeter-Dynkin diagram, shown as .
The birectified 8-simplex is the vertex figure of the 152 honeycomb.
Coordinates
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.
Images
Ak Coxeter plane | A8 | A7 | A6 | A5 |
---|---|---|---|---|
Graph | ||||
Dihedral symmetry | [9] | [8] | [7] | [6] |
Ak Coxeter plane | A4 | A3 | A2 | |
Graph | ||||
Dihedral symmetry | [5] | [4] | [3] |
Trirectified 8-simplex
Trirectified 8-simplex | |
---|---|
Type | uniform 8-polytope |
Coxeter symbol | 043 |
Schläfli symbol | t3{37} 3r{37} = {34,3} 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 | A7, [37], order 362880 |
Properties | convex |
E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as S3
8. It is also called 04,3 for its branching Coxeter-Dynkin diagram, shown as .
Coordinates
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.
Images
Ak Coxeter plane | A8 | A7 | A6 | A5 |
---|---|---|---|---|
Graph | ||||
Dihedral symmetry | [9] | [8] | [7] | [6] |
Ak Coxeter plane | A4 | A3 | A2 | |
Graph | ||||
Dihedral symmetry | [5] | [4] | [3] |
Related polytopes
This polytope is the vertex figure of the 9-demicube, and the edge figure of the uniform 261 honeycomb.
It is also one of 135 uniform 8-polytopes with A8 symmetry.
Notes
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 [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. "8D Uniform polytopes (polyzetta)". o3x3o3o3o3o3o3o - rene, o3o3x3o3o3o3o3o - brene, o3o3o3x3o3o3o3o - trene