Rectified 8-orthoplexes

In eight-dimensional geometry, a rectified 8-orthoplex is a convex uniform 8-polytope, being a rectification of the regular 8-orthoplex.


8-orthoplex

Rectified 8-orthoplex

Birectified 8-orthoplex

Trirectified 8-orthoplex

Trirectified 8-cube

Birectified 8-cube

Rectified 8-cube

8-cube
Orthogonal projections in A8 Coxeter plane

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
Typeuniform 8-polytope
Schläfli symbolt1{3,3,3,3,3,3,4}
Coxeter-Dynkin diagrams
7-faces272
6-faces3072
5-faces8960
4-faces12544
Cells10080
Faces4928
Edges1344
Vertices112
Vertex figure6-orthoplex prism
Petrie polygonhexakaidecagon
Coxeter groupsC8, [4,36]
D8, [35,1,1]
Propertiesconvex

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.

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

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
Typeuniform 8-polytope
Schläfli symbolt2{3,3,3,3,3,3,4}
Coxeter-Dynkin diagrams
7-faces272
6-faces3184
5-faces16128
4-faces34048
Cells36960
Faces22400
Edges6720
Vertices448
Vertex figure{3,3,3,4}x{3}
Coxeter groupsC8, [3,3,3,3,3,3,4]
D8, [35,1,1]
Propertiesconvex

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

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
Typeuniform 8-polytope
Schläfli symbolt3{3,3,3,3,3,3,4}
Coxeter-Dynkin diagrams
7-faces16+256
6-faces1024 + 2048 + 112
5-faces1792 + 7168 + 7168 + 448
4-faces1792 + 10752 + 21504 + 14336
Cells8960 + 126880 + 35840
Faces17920 + 35840
Edges17920
Vertices1120
Vertex figure{3,3,4}x{3,3}
Coxeter groupsC8, [3,3,3,3,3,3,4]
D8, [35,1,1]
Propertiesconvex

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

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
      • (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
Family An Bn I2(p) / Dn E6 / E7 / E8 / F4 / G2 Hn
Regular polygon Triangle Square p-gon Hexagon Pentagon
Uniform polyhedron Tetrahedron OctahedronCube Demicube DodecahedronIcosahedron
Uniform polychoron Pentachoron 16-cellTesseract Demitesseract 24-cell 120-cell600-cell
Uniform 5-polytope 5-simplex 5-orthoplex5-cube 5-demicube
Uniform 6-polytope 6-simplex 6-orthoplex6-cube 6-demicube 122221
Uniform 7-polytope 7-simplex 7-orthoplex7-cube 7-demicube 132231321
Uniform 8-polytope 8-simplex 8-orthoplex8-cube 8-demicube 142241421
Uniform 9-polytope 9-simplex 9-orthoplex9-cube 9-demicube
Uniform 10-polytope 10-simplex 10-orthoplex10-cube 10-demicube
Uniform n-polytope n-simplex n-orthoplexn-cube n-demicube 1k22k1k21 n-pentagonal polytope
Topics: Polytope familiesRegular polytopeList of regular polytopes and compounds
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