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Rectified 8-simplexes
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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.
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Rectified 8-simplex
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Rectified 8-simplex | |
---|---|
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.
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Birectified 8-simplex
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Perspective
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.
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Trirectified 8-simplex
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Perspective
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
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.
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Notes
References
External links
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