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Rectified 7-orthoplexes

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7-orthoplex
Rectified 7-orthoplex
Birectified 7-orthoplex
Trirectified 7-orthoplex

Birectified 7-cube
Rectified 7-cube
7-cube
Orthogonal projections in B7 Coxeter plane

In seven-dimensional geometry, a rectified 7-orthoplex is a convex uniform 7-polytope, being a rectification of the regular 7-orthoplex.

There are unique 7 degrees of rectifications, the zeroth being the 7-orthoplex, and the 6th and last being the 7-cube. Vertices of the rectified 7-orthoplex are located at the edge-centers of the 7-orthoplex. Vertices of the birectified 7-orthoplex are located in the triangular face centers of the 7-orthoplex. Vertices of the trirectified 7-orthoplex are located in the tetrahedral cell centers of the 7-orthoplex.

Rectified 7-orthoplex

Rectified 7-orthoplex
Type uniform 7-polytope
Schläfli symbol r{3,3,3,3,3,4}
Coxeter-Dynkin diagrams
6-faces 142
5-faces 1344
4-faces 3360
Cells 3920
Faces 2520
Edges 840
Vertices 84
Vertex figure 5-orthoplex prism
Coxeter groups B7,
D7,
Properties convex

The rectified 7-orthoplex is the vertex figure for the demihepteractic honeycomb. The rectified 7-orthoplex's 84 vertices represent the kissing number of a sphere-packing constructed from this honeycomb.

or

Alternate names

  • rectified heptacross
  • rectified hecatonicosoctaexon (Acronym rez) (Jonathan Bowers) - rectified 128-faceted polyexon

Images

orthographic projections
Coxeter plane B7 / A6 B6 / D7 B5 / D6 / A4
Graph
Dihedral symmetry
Coxeter plane B4 / D5 B3 / D4 / A2 B2 / D3
Graph
Dihedral symmetry
Coxeter plane A5 A3
Graph
Dihedral symmetry

Construction

There are two Coxeter groups associated with the rectified heptacross, one with the C7 or Coxeter group, and a lower symmetry with two copies of pentacross facets, alternating, with the D7 or Coxeter group.

Cartesian coordinates

Cartesian coordinates for the vertices of a rectified heptacross, centered at the origin, edge length 2   {\displaystyle {\sqrt {2}}\ } are all permutations of:

(±1,±1,0,0,0,0,0)

Root vectors

Its 84 vertices represent the root vectors of the simple Lie group D7. The vertices can be seen in 3 hyperplanes, with the 21 vertices rectified 6-simplexs cells on opposite sides, and 42 vertices of an expanded 6-simplex passing through the center. When combined with the 14 vertices of the 7-orthoplex, these vertices represent the 98 root vectors of the B7 and C7 simple Lie groups.

Birectified 7-orthoplex

Birectified 7-orthoplex
Type uniform 7-polytope
Schläfli symbol 2r{3,3,3,3,3,4}
Coxeter-Dynkin diagrams
6-faces 142
5-faces 1428
4-faces 6048
Cells 10640
Faces 8960
Edges 3360
Vertices 280
Vertex figure {3}×{3,3,4}
Coxeter groups B7,
D7,
Properties convex

Alternate names

  • Birectified heptacross
  • Birectified hecatonicosoctaexon (Acronym barz) (Jonathan Bowers) - birectified 128-faceted polyexon

Images

orthographic projections
Coxeter plane B7 / A6 B6 / D7 B5 / D6 / A4
Graph
Dihedral symmetry
Coxeter plane B4 / D5 B3 / D4 / A2 B2 / D3
Graph
Dihedral symmetry
Coxeter plane A5 A3
Graph
Dihedral symmetry

Cartesian coordinates

Cartesian coordinates for the vertices of a birectified 7-orthoplex, centered at the origin, edge length 2   {\displaystyle {\sqrt {2}}\ } are all permutations of:

(±1,±1,±1,0,0,0,0)

Trirectified 7-orthoplex

A trirectified 7-orthoplex is the same as a trirectified 7-cube.

Notes

  1. Klitzing, (o3o3x3o3o3o4o - rez)
  2. Klitzing, (o3o3x3o3o3o4o - barz)

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,
      • (Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II,
      • (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III,
  • Norman Johnson Uniform Polytopes, Manuscript (1991)
    • N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D.
  • Klitzing, Richard. "7D uniform polytopes (polyexa)". o3x3o3o3o3o4o - rez, o3o3x3o3o3o4o - barz

External links

Fundamental convex regular and uniform polytopes in dimensions 2–10
Family An Bn I2(p) / Dn E6 / E7 / E8 / F4 / G2 Hn
Regular polygon Triangle Square p-gon Hexagon Pentagon
Uniform polyhedron Tetrahedron Octahedron • Cube Demicube Dodecahedron • Icosahedron
Uniform polychoron Pentachoron 16-cell • Tesseract Demitesseract 24-cell 120-cell • 600-cell
Uniform 5-polytope 5-simplex 5-orthoplex • 5-cube 5-demicube
Uniform 6-polytope 6-simplex 6-orthoplex • 6-cube 6-demicube 122 • 221
Uniform 7-polytope 7-simplex 7-orthoplex • 7-cube 7-demicube 132 • 231 • 321
Uniform 8-polytope 8-simplex 8-orthoplex • 8-cube 8-demicube 142 • 241 • 421
Uniform 9-polytope 9-simplex 9-orthoplex • 9-cube 9-demicube
Uniform 10-polytope 10-simplex 10-orthoplex • 10-cube 10-demicube
Uniform n-polytope n-simplex n-orthoplex • n-cube n-demicube 1k2 • 2k1 • k21 n-pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds
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