Rectified 8-orthoplexes

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 Orthogonal projections in A8 Coxeter plane 8-orthoplex Rectified 8-orthoplex Birectified 8-orthoplex Trirectified 8-orthoplex Trirectified 8-cube Birectified 8-cube Rectified 8-cube 8-cube

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.

Rectified 8-orthoplex

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 ${\displaystyle {\sqrt {2}}}$ are all permutations of:

(±1,±1,0,0,0,0,0,0)

Images

orthographic projections
B8 B7
[16] [14]
B6 B5
[12] [10]
B4 B3 B2
[8] [6] [4]
A7 A5 A3
[8] [6] [4]

Birectified 8-orthoplex

Birectified 8-orthoplex
Type uniform 8-polytope
Schläfli symbol t2{3,3,3,3,3,3,4}
Coxeter-Dynkin diagrams
7-faces 272
6-faces 3184
5-faces 16128
4-faces 34048
Cells 36960
Faces 22400
Edges 6720
Vertices 448
Vertex figure {3,3,3,4}x{3}
Coxeter groups C8, [3,3,3,3,3,3,4]
D8, [35,1,1]
Properties convex

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 ${\displaystyle {\sqrt {2}}}$ are all permutations of:

(±1,±1,±1,0,0,0,0,0)

Images

orthographic projections
B8 B7
[16] [14]
B6 B5
[12] [10]
B4 B3 B2
[8] [6] [4]
A7 A5 A3
[8] [6] [4]

Trirectified 8-orthoplex

Trirectified 8-orthoplex
Type uniform 8-polytope
Schläfli symbol t3{3,3,3,3,3,3,4}
Coxeter-Dynkin diagrams
7-faces
6-faces
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure {3,3,4}x{3,3}
Coxeter groups C8, [3,3,3,3,3,3,4]
D8, [35,1,1]
Properties convex

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 ${\displaystyle {\sqrt {2}}}$ are all permutations of:

(±1,±1,±1,±1,0,0,0,0)

Images

orthographic projections
B8 B7
[16] [14]
B6 B5
[12] [10]
B4 B3 B2
[8] [6] [4]
A7 A5 A3
[8] [6] [4]

Notes

1. ^ Klitzing, (o3x3o3o3o3o3o4o - rek)
2. ^ Klitzing, (o3o3x3o3o3o3o4o - bark)
3. ^ Klitzing, (o3o3o3x3o3o3o4o - tark)

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)". o3x3o3o3o3o3o4o - rek, o3o3x3o3o3o3o4o - bark, o3o3o3x3o3o3o4o - tark