There are 6 unique cantellation for the 5-cube, including truncations. Half of them are more easily constructed from the dual 5-orthoplex
| bgcolor=#e7dcc3 align=center colspan=3 | Cantellated 5-cube | ||
| Type | Uniform 5-polytope | ||
| Schläfli symbol | rr = r\left\{\begin{array}{l}4\\3,3,3\end{array}\right\} | ||
| Coxeter-Dynkin diagram | = | ||
| 4-faces | 122 | 10 80 32 | |
| Cells | 680 | 40 320 160 160 | |
| Faces | 1520 | 80 480 320 640 | |
| Edges | 1280 | 320+960 | |
| Vertices | 320 | ||
| Vertex figure | |||
| Coxeter group | B5 [4,3,3,3] | ||
| Properties | convex, uniform | ||
The Cartesian coordinates of the vertices of a cantellated 5-cube having edge length 2 are all permutations of:
\left(\pm1, \pm1, \pm(1+\sqrt{2}), \pm(1+\sqrt{2}), \pm(1+\sqrt{2})\right)
| bgcolor=#e7dcc3 align=center colspan=3 | Bicantellated 5-cube | ||
| Type | Uniform 5-polytope | ||
| Schläfli symbols | 2rr = r\left\{\begin{array}{l}3,4\\3,3\end{array}\right\} r = r\left\{\begin{array}{l}3,3\ 3\\3\end{array}\right\} | ||
| Coxeter-Dynkin diagrams | = | ||
| 4-faces | 122 | 10 80 32 | |
| Cells | 840 | 40 240 160 320 80 | |
| Faces | 2160 | 240 320 960 320 320 | |
| Edges | 1920 | 960+960 | |
| Vertices | 480 | ||
| Vertex figure | |||
| Coxeter groups | B5, [3,3,3,4] D5, [3<sup>2,1,1</sup>] | ||
| Properties | convex, uniform | ||
The Cartesian coordinates of the vertices of a bicantellated 5-cube having edge length 2 are all permutations of:
(0,1,1,2,2)
| bgcolor=#e7dcc3 align=center colspan=3 | Cantitruncated 5-cube | ||
| Type | Uniform 5-polytope | ||
| Schläfli symbol | tr = t\left\{\begin{array}{l}4\\3,3,3\end{array}\right\} | ||
| Coxeter-Dynkin diagram | = | ||
| 4-faces | 122 | 10 80 32 | |
| Cells | 680 | 40 320 160 160 | |
| Faces | 1520 | 80 480 320 640 | |
| Edges | 1600 | 320+320+960 | |
| Vertices | 640 | ||
| Vertex figure | |||
| Coxeter group | B5 [4,3,3,3] | ||
| Properties | convex, uniform | ||
The Cartesian coordinates of the vertices of a cantitruncated 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+2\sqrt{2}\right)
It is third in a series of cantitruncated hypercubes:
| bgcolor=#e7dcc3 colspan=3 | Bicantitruncated 5-cube | ||
|---|---|---|---|
| Type | uniform 5-polytope | ||
| Schläfli symbol | 2tr = t\left\{\begin{array}{l}3,4\\3,3\end{array}\right\} t = t\left\{\begin{array}{l}3,3\ 3\\3\end{array}\right\} | ||
| Coxeter-Dynkin diagrams | = | ||
| 4-faces | 122 | 10 80 32 | |
| Cells | 840 | 40 240 160 320 80 | |
| Faces | 2160 | 240 320 960 320 320 | |
| Edges | 2400 | 960+480+960 | |
| Vertices | 960 | ||
| Vertex figure | |||
| Coxeter groups | B5, [3,3,3,4] D5, [3<sup>2,1,1</sup>] | ||
| Properties | convex, uniform | ||
Cartesian coordinates for the vertices of a bicantitruncated 5-cube, centered at the origin, are all sign and coordinate permutations of
(±3,±3,±2,±1,0)
These polytopes are from a set of 31 uniform 5-polytopes generated from the regular 5-cube or 5-orthoplex.