Stericated 5-cubes

In five-dimensional geometry, a stericated 5-cube is a convex uniform 5-polytope with fourth-order truncations (sterication) of the regular 5-cube.

There are eight degrees of sterication for the 5-cube, including permutations of runcination, cantellation, and truncation. The simple stericated 5-cube is also called an expanded 5-cube, with the first and last nodes ringed, for being constructible by an expansion operation applied to the regular 5-cube. The highest form, the steriruncicantitruncated 5-cube, is more simply called an omnitruncated 5-cube with all of the nodes ringed.

Alternate names

 * Stericated penteract / Stericated 5-orthoplex / Stericated pentacross
 * Expanded penteract / Expanded 5-orthoplex / Expanded pentacross
 * Small cellated penteractitriacontaditeron (Acronym: scant) (Jonathan Bowers)

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


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

Images
The stericated 5-cube is constructed by a sterication operation applied to the 5-cube.

Dissections
The stericated 5-cube can be dissected into two tesseractic cupolae and a runcinated tesseract between them. This dissection can be seen as analogous to the 4D runcinated tesseract being dissected into two cubic cupolae and a central rhombicuboctahedral prism between them, and also the 3D rhombicuboctahedron being dissected into two square cupolae with a central octagonal prism between them.

Alternate names

 * Steritruncated penteract
 * Celliprismated triacontaditeron (Acronym: capt) (Jonathan Bowers)

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


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

Alternate names

 * Stericantellated penteract
 * Stericantellated 5-orthoplex, stericantellated pentacross
 * Cellirhombated penteractitriacontiditeron (Acronym: carnit) (Jonathan Bowers)

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


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

Alternate names

 * Stericantitruncated penteract
 * Steriruncicantellated triacontiditeron / Biruncicantitruncated pentacross
 * Celligreatorhombated penteract (cogrin) (Jonathan Bowers)

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


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

Alternate names

 * Steriruncitruncated penteract / Steriruncitruncated 5-orthoplex / Steriruncitruncated pentacross
 * Celliprismatotruncated penteractitriacontiditeron (captint) (Jonathan Bowers)

Coordinates
The Cartesian coordinates of the vertices of a steriruncitruncated penteract having an edge length of 2 are given by all permutations of coordinates and sign of:


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

Alternate names

 * Steritruncated pentacross
 * Celliprismated penteract (Acronym: cappin) (Jonathan Bowers)

Coordinates
Cartesian coordinates for the vertices of a steritruncated 5-orthoplex, centered at the origin, are all permutations of
 * $$\left(\pm1,\ \pm1,\ \pm1,\ \pm1,\ \pm(1+\sqrt{2})\right)$$

Alternate names

 * Stericantitruncated pentacross
 * Celligreatorhombated triacontaditeron (cogart) (Jonathan Bowers)

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


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

Alternate names

 * Steriruncicantitruncated 5-cube (Full expansion of omnitruncation for 5-polytopes by Johnson)
 * Omnitruncated penteract
 * Omnitruncated triacontiditeron / omnitruncated pentacross
 * Great cellated penteractitriacontiditeron (Jonathan Bowers)

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


 * $$\left(1,\ 1+\sqrt{2},\ 1+2\sqrt{2},\ 1+3\sqrt{2},\ 1+4\sqrt{2}\right)$$

Full snub 5-cube
The full snub 5-cube or omnisnub 5-cube, defined as an alternation of the omnitruncated 5-cube is not uniform, but it can be given Coxeter diagram and symmetry [4,3,3,3]+, and constructed from 10 snub tesseracts, 32 snub 5-cells, 40 snub cubic antiprisms, 80 snub tetrahedral antiprisms, 80 3-4 duoantiprisms, and 1920 irregular 5-cells filling the gaps at the deleted vertices.

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