Reference articles on history, science, culture and more
Surprise meSurprise meSubscribe
Encyclopedia

Truncated 5-cubes

Wikipedia contributorsAdapted for Fathomly · View authors
2 min read
Source updated May 17, 2026
Image credit is listed at the end of this article.

5-cube
Truncated 5-cube
Bitruncated 5-cube

5-orthoplex
Truncated 5-orthoplex
Bitruncated 5-orthoplex
Orthogonal projections in B5 Coxeter plane

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.

01Truncated 5-cube

Truncated 5-cube
Typeuniform 5-polytope
Schläfli symbolt{4,3,3,3}
Coxeter-Dynkin diagram
4-faces4210
32
Cells20040
160
Faces40080
320
Edges40080
320
Vertices160
Vertex figure
( )v{3,3}
Coxeter groupB5, [3,3,3,4], order 3840
Propertiesconvex

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:

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

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.

Orthographic projections
Coxeter plane B5 B4 / D5 B3 / D4 / A2
Graph
Dihedral symmetry [10] [8] [6]
Coxeter plane B2 A3
Graph
Dihedral symmetry [4] [4]

Related polytopes

The truncated 5-cube, is fourth in a sequence of truncated hypercubes:

Truncated hypercubes
Image ...
Name Octagon Truncated cube Truncated tesseract Truncated 5-cube Truncated 6-cube Truncated 7-cube Truncated 8-cube
Coxeter diagram
Vertex figure ( )v( )
( )v{ }
( )v{3}
( )v{3,3}		 ( )v{3,3,3} ( )v{3,3,3,3} ( )v{3,3,3,3,3}

02Bitruncated 5-cube

Bitruncated 5-cube
Typeuniform 5-polytope
Schläfli symbol2t{4,3,3,3}
Coxeter-Dynkin diagrams
4-faces4210
32
Cells28040
160
80
Faces72080
320
320
Edges800320
480
Vertices320
Vertex figure
{ }v{3}
Coxeter groupsB5, [3,3,3,4], order 3840
Propertiesconvex

Alternate names

  • Bitruncated penteract (Acronym: bittin) (Jonathan Bowers)

Construction and coordinates

The bitruncated 5-cube may be constructed by bitruncating the vertices of the 5-cube at {\sqrt {2}} of the edge length.

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

\left(0,\ \pm 1,\ \pm 2,\ \pm 2,\ \pm 2\right)

Images

Orthographic projections
Coxeter plane B5 B4 / D5 B3 / D4 / A2
Graph
Dihedral symmetry [10] [8] [6]
Coxeter plane B2 A3
Graph
Dihedral symmetry [4] [4]

Related polytopes

The bitruncated 5-cube is third in a sequence of bitruncated hypercubes:

Bitruncated hypercubes
Image ...
Name Bitruncated cube Bitruncated tesseract Bitruncated 5-cube Bitruncated 6-cube Bitruncated 7-cube Bitruncated 8-cube
Coxeter
Vertex figure
( )v{ }
{ }v{ }
{ }v{3}
{ }v{3,3}		 { }v{3,3,3} { }v{3,3,3,3}
Watch videos about Truncated 5-cubesExplainers and documentaries on YouTube (opens in a new tab)

Sources and credits

This article is adapted from the Wikipedia article “Truncated 5-cubes”, written by its contributors and licensed under CC BY-SA 4.0. Fathomly has changed the layout, removed citation markers, navigation and maintenance notices, and adjusted punctuation. This adapted version is shared under the same license. For references, see the original article.

Images, from Wikimedia Commons:

Fathomly is not affiliated with or endorsed by the Wikimedia Foundation. Spotted a problem? Tell us.