Runcinated 5-cubes

In five-dimensional geometry, a runcinated 5-cube is a convex uniform 5-polytope that is a runcination (a 3rd order truncation) of the regular 5-cube.

There are 8 unique degrees of runcinations of the 5-cube, along with permutations of truncations and cantellations. Four are more simply constructed relative to the 5-orthoplex.

Alternate names

 * Small prismated penteract (Acronym: span) (Jonathan Bowers)

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


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

Images


Alternate names

 * Runcitruncated penteract
 * Prismatotruncated penteract (Acronym: pattin) (Jonathan Bowers)

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


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

Images


Alternate names

 * Runcicantellated penteract
 * Prismatorhombated penteract (Acronym: prin) (Jonathan Bowers)

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


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

Images


Alternate names

 * Runcicantitruncated penteract
 * Biruncicantitruncated pentacross
 * great prismated penteract (gippin) (Jonathan Bowers)

Coordinates
The Cartesian coordinates of the vertices of a runcicantitruncated 5-cube having an edge length of 2 are given by all permutations of coordinates and sign of:


 * $$\left(1,\ 1+\sqrt{2},\ 1+2\sqrt{2},\ 1+3\sqrt{2},\ 1+3\sqrt{2}\right)$$

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