Truncated 5-cubes

In five-dimensional geometry, a truncated 5-cube is a convex uniform 5-polytope, being a truncation of the regular 5-cube.

There are four unique truncations of the 5-cube. Vertices of the truncated 5-cube are located as pairs on the edge of the 5-cube. Vertices of the bitruncated 5-cube are located on the square faces of the 5-cube. The third and fourth truncations are more easily constructed as second and first truncations of the 5-orthoplex.

Alternate names

 * Truncated penteract (Acronym: tan) (Jonathan Bowers)

Construction and coordinates
The truncated 5-cube may be constructed by truncating the vertices of the 5-cube at $$1/(\sqrt{2}+2)$$ of the edge length. A regular 5-cell is formed at each truncated vertex.

The Cartesian coordinates of the vertices of a truncated 5-cube having edge length 2 are all permutations of:


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

Images
The truncated 5-cube is constructed by a truncation applied to the 5-cube. All edges are shortened, and two new vertices are added on each original edge.

Related polytopes
The truncated 5-cube, is fourth in a sequence of truncated hypercubes:

Alternate names

 * Bitruncated penteract (Acronym: bittin) (Jonathan Bowers)

Construction and coordinates
The bitruncated 5-cube may be constructed by bitruncating the vertices of the 5-cube at $$\sqrt{2}$$ of the edge length.

The Cartesian coordinates of the vertices of a bitruncated 5-cube having edge length 2 are all permutations of:


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

Related polytopes
The bitruncated 5-cube is third in a sequence of bitruncated hypercubes:

Related polytopes
This polytope is one of 31 uniform 5-polytope generated from the regular 5-cube or 5-orthoplex.