6-orthoplexRectified 6-orthoplexBirectified 6-orthoplexBirectified 6-cubeRectified 6-cube6-cubeOrthogonal projections in B6 Coxeter plane In six-dimensional geometry, a rectified 6-orthoplex is a convex uniform 6-polytope, being a rectification of the regular 6-orthoplex.
There are unique 6 degrees of rectifications, the zeroth being the 6-orthoplex, and the 6th and last being the 6-cube. Vertices of the rectified 6-orthoplex are located at the edge-centers of the 6-orthoplex. Vertices of the birectified 6-orthoplex are located in the triangular face centers of the 6-orthoplex.
Rectified 6-orthoplex Rectified hexacrossTypeuniform 6-polytopeSchläfli symbols t1{34,4} or r{34,4}r{3,3,3,31,1}Coxeter-Dynkin diagrams = = 5-faces76 total:64 rectified 5-simplex12 5-orthoplex4-faces576 total:192 rectified 5-cell384 5-cellCells1200 total:240 octahedron960 tetrahedronFaces1120 total:160 and 960 trianglesEdges480Vertices60Vertex figure16-cell prismPetrie polygonDodecagonCoxeter groupsB6, [3,3,3,3,4]D6, [33,1,1]Propertiesconvex
The rectified 6-orthoplex is the vertex figure for the demihexeractic honeycomb. or
Alternate names Rectified hexacross Rectified hexacontatetrapeton (acronym: rag) (Jonathan Bowers)
Construction There are two Coxeter groups associated with the rectified hexacross, one with the C6 or [4,3,3,3,3] Coxeter group, and a lower symmetry with two copies of pentacross facets, alternating, with the D6 or [33,1,1] Coxeter group.
Cartesian coordinates Cartesian coordinates for the vertices of a rectified hexacross, centered at the origin, edge length are all permutations of: (±1,±1,0,0,0,0)
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Root vectors The 60 vertices represent the root vectors of the simple Lie group D6. The vertices can be seen in 3 hyperplanes, with the 15 vertices rectified 5-simplices cells on opposite sides, and 30 vertices of an expanded 5-simplex passing through the center. When combined with the 12 vertices of the 6-orthoplex, these vertices represent the 72 root vectors of the B6 and C6 simple Lie groups.
The 60 roots of D6 can be geometrically folded into H3 (Icosahedral symmetry), as to , creating 2 copies of 30-vertex icosidodecahedra, with the Golden ratio between their radii:Icosidodecahedron from D6 John Baez, January 1, 2015 Rectified 6-orthoplex2 icosidodecahedra3D (H3 projection)A4/B5/D6 Coxeter planeH2 Coxeter plane
Birectified 6-orthoplex Birectified 6-orthoplex Typeuniform 6-polytopeSchläfli symbols t2{34,4} or 2r{34,4}t2{3,3,3,31,1}Coxeter-Dynkin diagrams = = 5-faces764-faces636Cells2160Faces2880Edges1440Vertices160Vertex figure{3}×{3,4} duoprismPetrie polygonDodecagonCoxeter groupsB6, [3,3,3,3,4]D6, [33,1,1]Propertiesconvex The birectified 6-orthoplex can tessellation space in the trirectified 6-cubic honeycomb.
Alternate names Birectified hexacross Birectified hexacontatetrapeton (acronym: brag) (Jonathan Bowers)
Cartesian coordinates Cartesian coordinates for the vertices of a rectified hexacross, centered at the origin, edge length are all permutations of: (±1,±1,±1,0,0,0)
Images
It can also be projected into 3D-dimensions as → , a dodecahedron envelope.
Related polytopes These polytopes are in a family 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. o3x3o3o3o4o - rag, o3o3x3o3o4o - brag
External links Polytopes of Various Dimensions Multi-dimensional Glossary