7-simplex |
Rectified 7-simplex | |
Birectified 7-simplex |
Trirectified 7-simplex | |
Orthogonal projections in A_{7} Coxeter plane |
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In seven-dimensional geometry, a rectified 7-simplex is a convex uniform 7-polytope, being a rectification of the regular 7-simplex.
There are four unique degrees of rectifications, including the zeroth, the 7-simplex itself. Vertices of the rectified 7-simplex are located at the edge-centers of the 7-simplex. Vertices of the birectified 7-simplex are located in the triangular face centers of the 7-simplex. Vertices of the trirectified 7-simplex are located in the tetrahedral cell centers of the 7-simplex.
Rectified 7-simplex | |
---|---|
Type | uniform 7-polytope |
Coxeter symbol | 0_{51} |
Schläfli symbol | r{3^{6}} = {3^{5,1}} or |
Coxeter diagrams | Or |
6-faces | 16 |
5-faces | 84 |
4-faces | 224 |
Cells | 350 |
Faces | 336 |
Edges | 168 |
Vertices | 28 |
Vertex figure | 6-simplex prism |
Petrie polygon | Octagon |
Coxeter group | A_{7}, [3^{6}], order 40320 |
Properties | convex |
The rectified 7-simplex is the edge figure of the 2_{51} honeycomb. It is called 0_{5,1} for its branching Coxeter-Dynkin diagram, shown as .
E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as S^{1}
_{7}.
The vertices of the rectified 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,0,0,0,1,1). This construction is based on facets of the rectified 8-orthoplex.
A_{k} Coxeter plane | A_{7} | A_{6} | A_{5} |
---|---|---|---|
Graph | |||
Dihedral symmetry | [8] | [7] | [6] |
A_{k} Coxeter plane | A_{4} | A_{3} | A_{2} |
Graph | |||
Dihedral symmetry | [5] | [4] | [3] |
Birectified 7-simplex | |
---|---|
Type | uniform 7-polytope |
Coxeter symbol | 0_{42} |
Schläfli symbol | 2r{3,3,3,3,3,3} = {3^{4,2}} or |
Coxeter diagrams | Or |
6-faces | 16: 8 r{3^{5}} 8 2r{3^{5}} |
5-faces | 112: 28 {3^{4}} 56 r{3^{4}} 28 2r{3^{4}} |
4-faces | 392: 168 {3^{3}} (56+168) r{3^{3}} |
Cells | 770: (420+70) {3,3} 280 {3,4} |
Faces | 840: (280+560) {3} |
Edges | 420 |
Vertices | 56 |
Vertex figure | {3}x{3,3,3} |
Coxeter group | A_{7}, [3^{6}], order 40320 |
Properties | convex |
E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as S^{2}
_{7}. It is also called 0_{4,2} for its branching Coxeter-Dynkin diagram, shown as .
The vertices of the birectified 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,0,0,1,1,1). This construction is based on facets of the birectified 8-orthoplex.
A_{k} Coxeter plane | A_{7} | A_{6} | A_{5} |
---|---|---|---|
Graph | |||
Dihedral symmetry | [8] | [7] | [6] |
A_{k} Coxeter plane | A_{4} | A_{3} | A_{2} |
Graph | |||
Dihedral symmetry | [5] | [4] | [3] |
Trirectified 7-simplex | |
---|---|
Type | uniform 7-polytope |
Coxeter symbol | 0_{33} |
Schläfli symbol | 3r{3^{6}} = {3^{3,3}} or |
Coxeter diagrams | Or |
6-faces | 16 2r{3^{5}} |
5-faces | 112 |
4-faces | 448 |
Cells | 980 |
Faces | 1120 |
Edges | 560 |
Vertices | 70 |
Vertex figure | {3,3}x{3,3} |
Coxeter group | A_{7}×2, [[3^{6}]], order 80640 |
Properties | convex, isotopic |
The trirectified 7-simplex is the intersection of two regular 7-simplexes in dual configuration.
E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as S^{3}
_{7}.
This polytope is the vertex figure of the 1_{33} honeycomb. It is called 0_{3,3} for its branching Coxeter-Dynkin diagram, shown as .
The vertices of the trirectified 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,0,1,1,1,1). This construction is based on facets of the trirectified 8-orthoplex.
The trirectified 7-simplex is the intersection of two regular 7-simplices in dual configuration. This characterization yields simple coordinates for the vertices of a trirectified 7-simplex in 8-space: the 70 distinct permutations of (1,1,1,1,−1,−1,−1,-1).
A_{k} Coxeter plane | A_{7} | A_{6} | A_{5} |
---|---|---|---|
Graph | |||
Dihedral symmetry | [8] | [[7]] | [6] |
A_{k} Coxeter plane | A_{4} | A_{3} | A_{2} |
Graph | |||
Dihedral symmetry | [[5]] | [4] | [[3]] |
Dim. | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
---|---|---|---|---|---|---|---|
Name Coxeter |
Hexagon = t{3} = {6} |
Octahedron = r{3,3} = {3^{1,1}} = {3,4} |
Decachoron 2t{3^{3}} |
Dodecateron 2r{3^{4}} = {3^{2,2}} |
Tetradecapeton 3t{3^{5}} |
Hexadecaexon 3r{3^{6}} = {3^{3,3}} |
Octadecazetton 4t{3^{7}} |
Images | |||||||
Vertex figure | ( )v( ) | { }×{ } |
{ }v{ } |
{3}×{3} |
{3}v{3} |
{3,3}x{3,3} | {3,3}v{3,3} |
Facets | {3} | t{3,3} | r{3,3,3} | 2t{3,3,3,3} | 2r{3,3,3,3,3} | 3t{3,3,3,3,3,3} | |
As intersecting dual simplexes |
∩ |
∩ |
∩ |
∩ |
∩ | ∩ | ∩ |
These polytopes are three of 71 uniform 7-polytopes with A_{7} symmetry.