Truncated 5-cubes

Orthogonal projections in B₅ Coxeter plane
5-cube Template:CDD	Truncated 5-cube Template:CDD	Bitruncated 5-cube Template:CDD
5-orthoplex Template:CDD	Truncated 5-orthoplex Template:CDD	Bitruncated 5-orthoplex Template:CDD

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.

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Truncated 5-cube

Truncated 5-cube
Type	uniform 5-polytope
Schläfli symbol	t{4,3,3,3}
Coxeter-Dynkin diagram	Template:CDD
4-faces	42	10 Template:CDD 32 Template:CDD
Cells	200	40 Template:CDD 160 Template:CDD
Faces	400	80 Template:CDD 320 Template:CDD
Edges	400	80 Template:CDD 320 Template:CDD
Vertices	160
Vertex figure	( )v{3,3}
Coxeter group	B₅, [3,3,3,4], order 3840
Properties	convex

Alternate names

Truncated penteract (Acronym: tan) (Jonathan Bowers)

Construction and coordinates

The truncated 5-cube may be constructed by truncating the vertices of the 5-cube at $1 / (\sqrt{2} + 2)$ of the edge length. A regular 5-cell is formed at each truncated vertex.

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

(\pm 1, \pm (1 + \sqrt{2}), \pm (1 + \sqrt{2}), \pm (1 + \sqrt{2}), \pm (1 + \sqrt{2}))

Images

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.

Template:5-cube Coxeter plane graphs

Related polytopes

The truncated 5-cube, is fourth in a sequence of truncated hypercubes: Template:Truncated hypercube polytopes

Bitruncated 5-cube

Bitruncated 5-cube
Type	uniform 5-polytope
Schläfli symbol	2t{4,3,3,3}
Coxeter-Dynkin diagrams	Template:CDD Template:CDD
4-faces	42	10 Template:CDD 32 Template:CDD
Cells	280	40 Template:CDD 160 Template:CDD 80 Template:CDD
Faces	720	80 Template:CDD 320 Template:CDD 320 Template:CDD
Edges	800	320 Template:CDD 480 Template:CDD
Vertices	320
Vertex figure	{ }v{3}
Coxeter groups	B₅, [3,3,3,4], order 3840
Properties	convex