6-simplexRectified 6-simplexBirectified 6-simplexOrthogonal projections in A6 Coxeter plane In six-dimensional geometry, a rectified 6-simplex is a convex uniform 6-polytope, being a rectification of the regular 6-simplex.
There are three unique degrees of rectifications, including the zeroth, the 6-simplex itself. Vertices of the rectified 6-simplex are located at the edge-centers of the 6-simplex. Vertices of the birectified 6-simplex are located in the triangular face centers of the 6-simplex.
Rectified 6-simplex
E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as S. It is also called 04,1 for its branching Coxeter-Dynkin diagram, shown as .
Alternate names Rectified heptapeton (Acronym: ril) (Jonathan Bowers)
Coordinates The vertices of the rectified 6-simplex can be most simply positioned in 7-space as permutations of (0,0,0,0,0,1,1). This construction is based on facets of the rectified 7-orthoplex.
Images
Birectified 6-simplex Birectified 6-simplexTypeuniform 6-polytopeClassA6 polytopeSchläfli symbol t2{3,3,3,3,3}2r{35} = {33,2}or Coxeter symbol 032Coxeter diagrams5-faces14 total:7 t1{3,3,3,3}7 t2{3,3,3,3}4-faces84Cells245Faces350Edges210Vertices35Vertex figure{3}x{3,3}Petrie polygonHeptagonCoxeter groupsA6, [3,3,3,3,3]Propertiesconvex E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as S. It is also called 03,2 for its branching Coxeter-Dynkin diagram, shown as .
Alternate names Birectified heptapeton (Acronym: bril) (Jonathan Bowers)
Coordinates The vertices of the birectified 6-simplex can be most simply positioned in 7-space as permutations of (0,0,0,0,1,1,1). This construction is based on facets of the birectified 7-orthoplex.
Images
Related uniform 6-polytopes The rectified 6-simplex polytope is the vertex figure of the 7-demicube, and the edge figure of the uniform 241 polytope.
These polytopes are a part of 35 uniform 6-polytopes based on the [3,3,3,3,3] Coxeter group, all shown here in A6 Coxeter plane orthographic projections.
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. o3x3o3o3o3o - ril, o3o3x3o3o3o - bril
External links Polytopes of Various Dimensions Multi-dimensional Glossary