5-cubeTruncated 5-cubeBitruncated 5-cube5-orthoplexTruncated 5-orthoplexBitruncated 5-orthoplexOrthogonal 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-cubeTypeuniform 5-polytopeSchläfli symbol t{4,3,3,3}Coxeter-Dynkin diagram4-faces4210 32 Cells20040 160 Faces40080 320 Edges40080 320 Vertices160Vertex figure( )v{3,3}Coxeter groupB5, [3,3,3,4], order 3840Propertiesconvex
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 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:
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.
Related polytopes The truncated 5-cube, is fourth in a sequence of truncated hypercubes:
Bitruncated 5-cube Bitruncated 5-cubeTypeuniform 5-polytopeSchläfli symbol 2t{4,3,3,3}Coxeter-Dynkin diagrams4-faces4210 32 Cells28040 160 80 Faces72080 320 320 Edges800320 480 Vertices320Vertex figure{ }v{3}Coxeter groupsB5, [3,3,3,4], order 3840Propertiesconvex
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 of the edge length.
The Cartesian coordinates of the vertices of a bitruncated 5-cube having edge length 2 are all permutations of:
Images
Related polytopes The bitruncated 5-cube is third in a sequence of bitruncated hypercubes:
Related polytopes The truncated 5-cube and bitruncated 5-cube are from the family of 31 uniform 5-polytopes generated from the regular 5-cube or 5-orthoplex.
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 Ivić Weiss, Wiley-Interscience Publication, 1995, wiley.com, (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
External links Polytopes of Various Dimensions Multi-dimensional Glossary