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.

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:


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

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

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:
 * $$\left(0,\ \pm1,\ \pm2,\ \pm2,\ \pm2,\ \pm2 \right)$$

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

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:
 * $$\left(0,\ 0,\ \pm1,\ \pm2,\ \pm2,\ \pm2 \right)$$

Related polytopes
These polytopes are from a set of 63 Uniform 6-polytopes generated from the B6 Coxeter plane, including the regular 6-cube or 6-orthoplex.