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Truncated 5-cubes

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
Schläfli symbol
Coxeter-Dynkin diagram
4-faces
Cells
Faces
Edges
Vertices
Vertex figure
Coxeter group
Properties

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(\pm1,\ \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.

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

Bitruncated 5-cube

Bitruncated 5-cube
Type
Schläfli symbol
Coxeter-Dynkin diagrams
4-faces
Cells
Faces
Edges
Vertices
Vertex figure
Coxeter groups
Properties

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,\ \pm1,\ \pm2,\ \pm2,\ \pm2\right)

Images

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

Notes

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, http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471010030.html
      • (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.
  • o3o3o3x4x - tan, o3o3x3x4o - bittin
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