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.
| bgcolor=#e7dcc3 colspan=3 | Truncated 5-cube | ||
|---|---|---|---|
| Type | uniform 5-polytope | ||
| Schläfli symbol | t | ||
| Coxeter-Dynkin diagram | |||
| 4-faces | 42 | 10 32 | |
| Cells | 200 | 40 160 | |
| Faces | 400 | 80 320 | |
| Edges | 400 | 80 320 | |
| Vertices | 160 | ||
| Vertex figure | v | ||
| Coxeter group | B5, [3,3,3,4], order 3840 | ||
| Properties | convex | ||
The truncated 5-cube may be constructed by truncating the vertices of the 5-cube at
1/(\sqrt{2}+2)
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)
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.
The truncated 5-cube, is fourth in a sequence of truncated hypercubes:
| bgcolor=#e7dcc3 colspan=3 | Bitruncated 5-cube | ||
|---|---|---|---|
| Type | uniform 5-polytope | ||
| Schläfli symbol | 2t | ||
| Coxeter-Dynkin diagrams | |||
| 4-faces | 42 | 10 32 | |
| Cells | 280 | 40 160 80 | |
| Faces | 720 | 80 320 320 | |
| Edges | 800 | 320 480 | |
| Vertices | 320 | ||
| Vertex figure | v | ||
| Coxeter groups | B5, [3,3,3,4], order 3840 | ||
| Properties | convex | ||
The bitruncated 5-cube may be constructed by bitruncating the vertices of the 5-cube at
\sqrt{2}
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)
The bitruncated 5-cube is third in a sequence of bitruncated hypercubes:
This polytope is one of 31 uniform 5-polytope generated from the regular 5-cube or 5-orthoplex.