Rectified 5-cubes

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

There are 5 degrees of rectifications of a 5-polytope, the zeroth here being the 5-cube, and the 4th and last being the 5-orthoplex. Vertices of the rectified 5-cube are located at the edge-centers of the 5-cube. Vertices of the birectified 5-cube are located in the square face centers of the 5-cube.

Alternate names

 * Rectified penteract (acronym: rin) (Jonathan Bowers)

Construction
The rectified 5-cube may be constructed from the 5-cube by truncating its vertices at the midpoints of its edges.

Coordinates
The Cartesian coordinates of the vertices of the rectified 5-cube with edge length $$\sqrt{2}$$ is given by all permutations of:
 * $$(0,\ \pm1,\ \pm1,\ \pm1,\ \pm1)$$

Birectified 5-cube
E. L. Elte identified it in 1912 as a semiregular polytope, identifying it as Cr52 as a second rectification of a 5-dimensional cross polytope.

Alternate names

 * Birectified 5-cube/penteract
 * Birectified pentacross/5-orthoplex/triacontiditeron
 * Penteractitriacontiditeron (acronym: nit) (Jonathan Bowers)
 * Rectified 5-demicube/demipenteract

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

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


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

Related polytopes
These polytopes are a part of 31 uniform polytera generated from the regular 5-cube or 5-orthoplex.