6-cubeTruncated 6-cubeBitruncated 6-cubeTritruncated 6-cube6-orthoplexTruncated 6-orthoplexBitruncated 6-orthoplexOrthogonal projections in B6 Coxeter plane In six-dimensional geometry, a truncated 6-cube (or truncated hexeract) is a convex uniform 6-polytope, being a truncation of the regular 6-cube.
There are 5 truncations for the 6-cube. Vertices of the truncated 6-cube are located as pairs on the edge of the 6-cube. Vertices of the bitruncated 6-cube are located on the square faces of the 6-cube. Vertices of the tritruncated 6-cube are located inside the cubic cells of the 6-cube.
Truncated 6-cube Truncated 6-cubeTypeuniform 6-polytopeClassB6 polytopeSchläfli symbol t{4,3,3,3,3}Coxeter-Dynkin diagrams5-faces764-faces464Cells1120Faces1520Edges1152Vertices384Vertex figure( )v{3,3,3}Coxeter groupsB6, [3,3,3,3,4]Propertiesconvex
Alternate names Truncated hexeract (Acronym: tox) (Jonathan Bowers)
Construction and coordinates The truncated 6-cube may be constructed by truncating the vertices of the 6-cube at of the edge length. A regular 5-simplex replaces each original vertex.
The Cartesian coordinates of the vertices of a truncated 6-cube having edge length 2 are the permutations of:
Images
Related polytopes The truncated 6-cube, is fifth in a sequence of truncated hypercubes:
Bitruncated 6-cube Bitruncated 6-cubeTypeuniform 6-polytopeClassB6 polytopeSchläfli symbol 2t{4,3,3,3,3}Coxeter-Dynkin diagrams5-faces4-facesCellsFacesEdgesVerticesVertex figure{ }v{3,3}Coxeter groupsB6, [3,3,3,3,4]Propertiesconvex
Alternate names Bitruncated hexeract (Acronym: botox) (Jonathan Bowers)
Construction and coordinates The Cartesian coordinates of the vertices of a bitruncated 6-cube having edge length 2 are the permutations of:
Images
Related polytopes The bitruncated 6-cube is fourth in a sequence of bitruncated hypercubes:
Tritruncated 6-cube Tritruncated 6-cubeTypeuniform 6-polytopeClassB6 polytopeSchläfli symbol 3t{4,3,3,3,3}Coxeter-Dynkin diagrams5-faces4-facesCellsFacesEdgesVerticesVertex figure{3}v{4} 5D polytope (polyteron)Coxeter groupsB6, [3,3,3,3,4]Propertiesconvex
Alternate names Tritruncated hexeract (Acronym: xog) (Jonathan Bowers)
Construction and coordinates The Cartesian coordinates of the vertices of a tritruncated 6-cube having edge length 2 are the permutations of:
Images
Related polytopes
The table below contains a set of 63 uniform 6-polytopes generated from the B6 Coxeter plane, including the regular 6-cube and 6-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. o3o3o3o3x4x - tox, o3o3o3x3x4o - botox, o3o3x3x3o4o - xog
External links
Polytopes of Various Dimensions Multi-dimensional Glossary