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Rectified 10-cubes

Orthogonal projections in BC10 Coxeter plane

In ten-dimensional geometry, a rectified 10-cube is a convex uniform 10-polytope, being a rectification of the regular 10-cube.

There are 10 rectifications of the 10-cube, with the zeroth being the 10-cube itself. Vertices of the rectified 10-cube are located at the edge-centers of the 10-cube. Vertices of the birectified 10-cube are located in the square face centers of the 10-cube. Vertices of the trirectified 10-cube are located in the cubic cell centers of the 10-cube. The others are more simply constructed relative to the 10-cube dual polytope, the 10-orthoplex.

These polytopes are part of a family of 1023 uniform 10-polytopes with BC10 symmetry.

Rectified 10-cube

Rectified 10-cube
Type
Schläfli symbol
Coxeter-Dynkin diagrams
9-faces
8-faces
7-faces
6-faces
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure
Coxeter groups
Properties

Alternate names

  • Rectified dekeract (Acronym: rade) (Jonathan Bowers)

Cartesian coordinates

Cartesian coordinates for the vertices of a rectified 10-cube, centered at the origin, edge length \sqrt{2} are all permutations of: : (±1,±1,±1,±1,±1,±1,±1,±1,±1,0)

Images

Birectified 10-cube

Birectified 10-cube
Type
Coxeter symbol
Schläfli symbol
Coxeter-Dynkin diagrams
9-faces
8-faces
7-faces
6-faces
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure
Coxeter groups
Properties

Alternate names

  • Birectified dekeract (Acronym: brade) (Jonathan Bowers)

Cartesian coordinates

Cartesian coordinates for the vertices of a birectified 10-cube, centered at the origin, edge length \sqrt{2} are all permutations of: : (±1,±1,±1,±1,±1,±1,±1,±1,0,0)

Images

Trirectified 10-cube

Trirectified 10-cube
Type
Schläfli symbol
Coxeter-Dynkin diagrams
9-faces
8-faces
7-faces
6-faces
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure
Coxeter groups
Properties

Alternate names

  • Trirectified dekeract (Acronym: trade) (Jonathan Bowers)

Cartesian coordinates

Cartesian coordinates for the vertices of a triirectified 10-cube, centered at the origin, edge length \sqrt{2} are all permutations of: : (±1,±1,±1,±1,±1,±1,±1,0,0,0)

Images

Quadrirectified 10-cube

Quadrirectified 10-cube
Type
Schläfli symbol
Coxeter-Dynkin diagrams
9-faces
8-faces
7-faces
6-faces
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure
Coxeter groups
Properties

Alternate names

  • Quadrirectified dekeract
  • Quadrirectified decacross (Acronym: terade) (Jonathan Bowers)

Cartesian coordinates

Cartesian coordinates for the vertices of a quadrirectified 10-cube, centered at the origin, edge length \sqrt{2} are all permutations of: : (±1,±1,±1,±1,±1,±1,0,0,0,0)

Images

Notes

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, wiley.com,
      • (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. (1966)
  • x3o3o3o3o3o3o3o3o4o - ka, o3x3o3o3o3o3o3o3o4o - rake, o3o3x3o3o3o3o3o3o4o - brake, o3o3o3x3o3o3o3o3o4o - trake, o3o3o3o3x3o3o3o3o4o - terake, o3o3o3o3o3x3o3o3o4o - terade, o3o3o3o3o3o3x3o3o4o - trade, o3o3o3o3o3o3o3x3o4o - brade, o3o3o3o3o3o3o3o3x4o - rade, o3o3o3o3o3o3o3o3o4x - deker

References

  1. Klitzing, (o3o3o3o3o3o3x3o3o4o - trade)
  2. Klitzing, (o3o3o3o3o3x3o3o3o4o - terade)
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