6-cubeRectified 6-cubeBirectified 6-cubeBirectified 6-orthoplexRectified 6-orthoplex6-orthoplexOrthogonal projections in B6 Coxeter plane In six-dimensional geometry, a rectified 6-cube is a convex uniform 6-polytope, being a rectification of the regular 6-cube.
There are unique 6 degrees of rectifications, the zeroth being the 6-cube, and the 6th and last being the 6-orthoplex. Vertices of the rectified 6-cube are located at the edge-centers of the 6-cube. Vertices of the birectified 6-cube are located in the square face centers of the 6-cube.
Rectified 6-cube Rectified 6-cubeTypeuniform 6-polytopeSchläfli symbol t1{4,34} or r{4,34}Coxeter-Dynkin diagrams = 5-faces764-faces444Cells1120Faces1520Edges960Vertices192Vertex figure5-cell prismPetrie polygonDodecagonCoxeter groupsB6, [3,3,3,3,4]D6, [33,1,1]Propertiesconvex
Alternate names Rectified hexeract (acronym: rax) (Jonathan Bowers)
Construction The rectified 6-cube may be constructed from the 6-cube by truncating its vertices at the midpoints of its edges.
Coordinates The Cartesian coordinates of the vertices of the rectified 6-cube with edge length are all permutations of:
Images
Birectified 6-cube Birectified 6-cubeTypeuniform 6-polytopeCoxeter symbol 0311Schläfli symbol t2{4,34} or 2r{4,34}Coxeter-Dynkin diagrams = = 5-faces764-faces636Cells2080Faces3200Edges1920Vertices240Vertex figure{4}x{3,3} duoprismCoxeter groupsB6, [3,3,3,3,4]D6, [33,1,1]Propertiesconvex
Alternate names Birectified hexeract (acronym: brox) (Jonathan Bowers) Rectified 6-demicube
Construction The birectified 6-cube may be constructed from the 6-cube by truncating its vertices at the midpoints of its edges.
Coordinates The Cartesian coordinates of the vertices of the rectified 6-cube with edge length are all permutations of:
Images
Related polytopes These polytopes are part of 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. o3o3o3o3x4o - rax, o3o3o3x3o4o - brox
External links
Polytopes of Various Dimensions Multi-dimensional Glossary