6-simplex |
Rectified 6-simplex |
Birectified 6-simplex |
Orthogonal projections in A_{6} Coxeter plane |
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In six-dimensional geometry, a rectified 6-simplex is a convex uniform 6-polytope, being a rectification of the regular 6-simplex.
There are three unique degrees of rectifications, including the zeroth, the 6-simplex itself. Vertices of the rectified 6-simplex are located at the edge-centers of the 6-simplex. Vertices of the birectified 6-simplex are located in the triangular face centers of the 6-simplex.
Rectified 6-simplex | |
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Type | uniform polypeton |
Schläfli symbol | t_{1}{3^{5}} r{3^{5}} = {3^{4,1}} or |
Coxeter diagrams | |
Elements |
f_{5} = 14, f_{4} = 63, C = 140, F = 175, E = 105, V = 21 |
Coxeter group | A_{6}, [3^{5}], order 5040 |
Bowers name and (acronym) |
Rectified heptapeton (ril) |
Vertex figure | 5-cell prism |
Circumradius | 0.845154 |
Properties | convex, isogonal |
E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as S^{1}
_{6}. It is also called 0_{4,1} for its branching Coxeter-Dynkin diagram, shown as .
The vertices of the rectified 6-simplex can be most simply positioned in 7-space as permutations of (0,0,0,0,0,1,1). This construction is based on facets of the rectified 7-orthoplex.
A_{k} Coxeter plane | A_{6} | A_{5} | A_{4} |
---|---|---|---|
Graph | |||
Dihedral symmetry | [7] | [6] | [5] |
A_{k} Coxeter plane | A_{3} | A_{2} | |
Graph | |||
Dihedral symmetry | [4] | [3] |
Birectified 6-simplex | |
---|---|
Type | uniform 6-polytope |
Class | A6 polytope |
Schläfli symbol | t_{2}{3,3,3,3,3} 2r{3^{5}} = {3^{3,2}} or |
Coxeter symbol | 0_{32} |
Coxeter diagrams | |
5-faces | 14 total: 7 t_{1}{3,3,3,3} 7 t_{2}{3,3,3,3} |
4-faces | 84 |
Cells | 245 |
Faces | 350 |
Edges | 210 |
Vertices | 35 |
Vertex figure | {3}x{3,3} |
Petrie polygon | Heptagon |
Coxeter groups | A_{6}, [3,3,3,3,3] |
Properties | convex |
E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as S^{2}
_{6}. It is also called 0_{3,2} for its branching Coxeter-Dynkin diagram, shown as .
The vertices of the birectified 6-simplex can be most simply positioned in 7-space as permutations of (0,0,0,0,1,1,1). This construction is based on facets of the birectified 7-orthoplex.
A_{k} Coxeter plane | A_{6} | A_{5} | A_{4} |
---|---|---|---|
Graph | |||
Dihedral symmetry | [7] | [6] | [5] |
A_{k} Coxeter plane | A_{3} | A_{2} | |
Graph | |||
Dihedral symmetry | [4] | [3] |
The rectified 6-simplex polytope is the vertex figure of the 7-demicube, and the edge figure of the uniform 2_{41} polytope.
These polytopes are a part of 35 uniform 6-polytopes based on the [3,3,3,3,3] Coxeter group, all shown here in A_{6} Coxeter plane orthographic projections.