Rectified 8-orthoplexes

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8-cube t7.svg
8-orthoplex
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
8-cube t6.svg
Rectified 8-orthoplex
CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
8-cube t5.svg
Birectified 8-orthoplex
CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
8-cube t4.svg
Trirectified 8-orthoplex
CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
8-cube t3.svg
Trirectified 8-cube
CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
8-cube t2.svg
Birectified 8-cube
CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
8-cube t1.svg
Rectified 8-cube
CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.png
8-cube t0.svg
8-cube
CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node 1.png
Orthogonal projections in A8 Coxeter plane

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[edit]

Rectified 8-orthoplex
Type uniform 8-polytope
Schläfli symbol t1{3,3,3,3,3,3,4}
Coxeter-Dynkin diagrams CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.png
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[edit]

The rectified 8-orthoplex is the vertex figure for the demiocteractic honeycomb.

CDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.png or CDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png

Alternate names[edit]

  • rectified octacross
  • rectified diacosipentacontahexazetton (Acronym: rek) (Jonathan Bowers)[1]

Construction[edit]

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[edit]

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[edit]

orthographic projections
B8 B7
8-cube t6.svg 8-cube t6 B7.svg
[16] [14]
B6 B5
8-cube t6 B6.svg 8-cube t6 B5.svg
[12] [10]
B4 B3 B2
8-cube t6 B4.svg 8-cube t6 B3.svg 8-cube t6 B2.svg
[8] [6] [4]
A7 A5 A3
8-cube t6 A7.svg 8-cube t6 A5.svg 8-cube t6 A3.svg
[8] [6] [4]

Birectified 8-orthoplex[edit]

Birectified 8-orthoplex
Type uniform 8-polytope
Schläfli symbol t2{3,3,3,3,3,3,4}
Coxeter-Dynkin diagrams CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.png
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[edit]

  • birectified octacross
  • birectified diacosipentacontahexazetton (Acronym: bark) (Jonathan Bowers)[2]

Cartesian coordinates[edit]

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[edit]

orthographic projections
B8 B7
8-cube t5.svg 8-cube t5 B7.svg
[16] [14]
B6 B5
8-cube t5 B6.svg 8-cube t5 B5.svg
[12] [10]
B4 B3 B2
8-cube t5 B4.svg 8-cube t5 B3.svg 8-cube t5 B2.svg
[8] [6] [4]
A7 A5 A3
8-cube t5 A7.svg 8-cube t5 A5.svg 8-cube t5 A3.svg
[8] [6] [4]

Trirectified 8-orthoplex[edit]

Trirectified 8-orthoplex
Type uniform 8-polytope
Schläfli symbol t3{3,3,3,3,3,3,4}
Coxeter-Dynkin diagrams CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.png
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[edit]

  • trirectified octacross
  • trirectified diacosipentacontahexazetton (acronym: tark) (Jonathan Bowers)[3]

Cartesian coordinates[edit]

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[edit]

orthographic projections
B8 B7
8-cube t4.svg 8-cube t4 B7.svg
[16] [14]
B6 B5
8-cube t4 B6.svg 8-cube t4 B5.svg
[12] [10]
B4 B3 B2
8-cube t4 B4.svg 8-cube t4 B3.svg 8-cube t4 B2.svg
[8] [6] [4]
A7 A5 A3
8-cube t4 A7.svg 8-cube t4 A5.svg 8-cube t4 A3.svg
[8] [6] [4]

Notes[edit]

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

References[edit]

  • 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

External links[edit]

Fundamental convex regular and uniform polytopes in dimensions 2–10
Family An Bn I2(p) / Dn E6 / E7 / E8 / E9 / E10 / F4 / G2 Hn
Regular polygon Triangle Square p-gon Hexagon Pentagon
Uniform polyhedron Tetrahedron OctahedronCube Demicube DodecahedronIcosahedron
Uniform 4-polytope 5-cell 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