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Rectified 8-orthoplexes
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In eight-dimensional geometry, a rectified 8-orthoplex is a convex uniform 8-polytope, being a rectification of the regular 8-orthoplex.
There are unique 8 degrees of rectifications, the zeroth being the 8-orthoplex, and the 7th and last being the 8-cube. Vertices of the rectified 8-orthoplex are located at the edge-centers of the 8-orthoplex. Vertices of the birectified 8-orthoplex are located in the triangular face centers of the 8-orthoplex. Vertices of the trirectified 8-orthoplex are located in the tetrahedral cell centers of the 8-orthoplex.
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Rectified 8-orthoplex
Summarize
Perspective
Rectified 8-orthoplex | |
---|---|
Type | uniform 8-polytope |
Schläfli symbol | t1{3,3,3,3,3,3,4} |
Coxeter-Dynkin diagrams | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
7-faces | 272 |
6-faces | 3072 |
5-faces | 8960 |
4-faces | 12544 |
Cells | 10080 |
Faces | 4928 |
Edges | 1344 |
Vertices | 112 |
Vertex figure | 6-orthoplex prism |
Petrie polygon | hexakaidecagon |
Coxeter groups | C8, [4,36] D8, [35,1,1] |
Properties | convex |
The rectified 8-orthoplex has 112 vertices. These represent the root vectors of the simple Lie group D8. The vertices can be seen in 3 hyperplanes, with the 28 vertices rectified 7-simplexs cells on opposite sides, and 56 vertices of an expanded 7-simplex passing through the center. When combined with the 16 vertices of the 8-orthoplex, these vertices represent the 128 root vectors of the B8 and C8 simple Lie groups.
Related polytopes
The rectified 8-orthoplex is the vertex figure for the demiocteractic honeycomb.
or
Alternate names
- rectified octacross
- rectified diacosipentacontahexazetton (Acronym: rek) (Jonathan Bowers)[1]
Construction
There are two Coxeter groups associated with the rectified 8-orthoplex, one with the C8 or [4,36] Coxeter group, and a lower symmetry with two copies of heptcross facets, alternating, with the D8 or [35,1,1] Coxeter group.
Cartesian coordinates
Cartesian coordinates for the vertices of a rectified 8-orthoplex, centered at the origin, edge length are all permutations of:
- (±1,±1,0,0,0,0,0,0)
Images
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Birectified 8-orthoplex
Alternate names
- birectified octacross
- birectified diacosipentacontahexazetton (Acronym: bark) (Jonathan Bowers)[2]
Cartesian coordinates
Cartesian coordinates for the vertices of a birectified 8-orthoplex, centered at the origin, edge length are all permutations of:
- (±1,±1,±1,0,0,0,0,0)
Images
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Trirectified 8-orthoplex
The trirectified 8-orthoplex can tessellate space in the quadrirectified 8-cubic honeycomb.
Alternate names
- trirectified octacross
- trirectified diacosipentacontahexazetton (acronym: tark) (Jonathan Bowers)[3]
Cartesian coordinates
Cartesian coordinates for the vertices of a trirectified 8-orthoplex, centered at the origin, edge length are all permutations of:
- (±1,±1,±1,±1,0,0,0,0)
Images
Notes
References
External links
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