Truncated 5-cubes

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.

Truncated 5-cube

Truncated 5-cube
Type	uniform 5-polytope
Schläfli symbol	t{4,3,3,3}
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{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 [math]\displaystyle{ 1/(\sqrt{2}+2) }[/math] 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:

[math]\displaystyle{ \left(\pm1,\ \pm(1+\sqrt{2}),\ \pm(1+\sqrt{2}),\ \pm(1+\sqrt{2}),\ \pm(1+\sqrt{2})\right) }[/math]

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:

Bitruncated 5-cube

Bitruncated 5-cube
Type	uniform 5-polytope
Schläfli symbol	2t{4,3,3,3}
Coxeter-Dynkin diagrams
4-faces	42	10 32
Cells	280	40 160 25px|link=Truncated tetrahedron 80
Faces	720	80 320 25px|link=Hexagon 320
Edges	800	320 480
Vertices	320
Vertex figure	{ }v{3}
Coxeter groups	B5, [3,3,3,4], order 3840
Properties	convex

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 [math]\displaystyle{ \sqrt{2} }[/math] of the edge length.

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

[math]\displaystyle{ \left(0,\ \pm1,\ \pm2,\ \pm2,\ \pm2\right) }[/math]

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:

Related polytopes

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

References

• H.S.M. Coxeter:
  • H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973
  • Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6 [1]
    • (Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380-407, MR 2,10]
    • (Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559-591]
    • (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
• Norman Johnson Uniform Polytopes, Manuscript (1991)
  • N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D.
• Klitzing, Richard. "5D uniform polytopes (polytera)". https://bendwavy.org/klitzing/dimensions/polytera.htm.  o3o3o3x4x - tan, o3o3x3x4o - bittin

External links

• Polytopes of Various Dimensions
• Multi-dimensional Glossary