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Truncated 6-cubes

Orthogonal projections in B6 Coxeter plane

In six-dimensional geometry, a truncated 6-cube (or truncated hexeract) is a convex uniform 6-polytope, being a truncation of the regular 6-cube.

There are 5 truncations for the 6-cube. Vertices of the truncated 6-cube are located as pairs on the edge of the 6-cube. Vertices of the bitruncated 6-cube are located on the square faces of the 6-cube. Vertices of the tritruncated 6-cube are located inside the cubic cells of the 6-cube.

Truncated 6-cube

Truncated 6-cube
Type
Class
Schläfli symbol
Coxeter-Dynkin diagrams
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure
Coxeter groups
Properties

Alternate names

  • Truncated hexeract (Acronym: tox) (Jonathan Bowers)

Construction and coordinates

The truncated 6-cube may be constructed by truncating the vertices of the 6-cube at 1/(\sqrt{2}+2) of the edge length. A regular 5-simplex replaces each original vertex.

The Cartesian coordinates of the vertices of a truncated 6-cube having edge length 2 are the permutations of:

:\left(\pm1,\ \pm(1+\sqrt{2}),\ \pm(1+\sqrt{2}),\ \pm(1+\sqrt{2}),\ \pm(1+\sqrt{2}),\ \pm(1+\sqrt{2})\right)

Images

The truncated 6-cube, is fifth in a sequence of truncated hypercubes:

Bitruncated 6-cube

Bitruncated 6-cube
Type
Class
Schläfli symbol
Coxeter-Dynkin diagrams
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure
Coxeter groups
Properties

Alternate names

  • Bitruncated hexeract (Acronym: botox) (Jonathan Bowers)

Construction and coordinates

The Cartesian coordinates of the vertices of a bitruncated 6-cube having edge length 2 are the permutations of: :\left(0,\ \pm1,\ \pm2,\ \pm2,\ \pm2,\ \pm2 \right)

Images

The bitruncated 6-cube is fourth in a sequence of bitruncated hypercubes:

Tritruncated 6-cube

Tritruncated 6-cube
Type
Class
Schläfli symbol
Coxeter-Dynkin diagrams
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure
Coxeter groups
Properties

Alternate names

  • Tritruncated hexeract (Acronym: xog) (Jonathan Bowers)

Construction and coordinates

The Cartesian coordinates of the vertices of a tritruncated 6-cube having edge length 2 are the permutations of: :\left(0,\ 0,\ \pm1,\ \pm2,\ \pm2,\ \pm2 \right)

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.
  • o3o3o3o3x4x - tox, o3o3o3x3x4o - botox, o3o3x3x3o4o - xog

References

  1. "squete".
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