Truncated 6-cubes

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 hexeract (Acronym: tox) (Jonathan Bowers)

The truncated 6-cube may be constructed by truncating the vertices of the 6-cube at ${\displaystyle 1/({\sqrt {2}}+2)}$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:

${\displaystyle \left(\pm 1,\ \pm (1+{\sqrt {2}}),\ \pm (1+{\sqrt {2}}),\ \pm (1+{\sqrt {2}}),\ \pm (1+{\sqrt {2}}),\ \pm (1+{\sqrt {2}})\right)}$

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

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

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

${\displaystyle \left(0,\ \pm 1,\ \pm 2,\ \pm 2,\ \pm 2,\ \pm 2\right)}$

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

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

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

${\displaystyle \left(0,\ 0,\ \pm 1,\ \pm 2,\ \pm 2,\ \pm 2\right)}$

The table below contains a set of 63 Uniform 6-polytopes generated from the B₆ Coxeter plane, including the regular 6-cube or 6-orthoplex.

• 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,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. "6D uniform polytopes (polypeta) with acronyms". o3o3o3o3x4x - tox, o3o3o3x3x4o - botox, o3o3x3x3o4o - xog

• Weisstein, Eric W. "Hypercube". MathWorld.
• Polytopes of Various Dimensions
• Multi-dimensional Glossary