In seven-dimensional geometry, a rectified 7-simplex is a convex uniform 7-polytope, being a rectification of the regular 7-simplex. There are four unique degrees of rectifications, including the zeroth, the 7-simplex itself. Vertices of the rectified 7-simplex are located at the edge-centers of the 7-simplex. Vertices of the birectified 7-simplex are located in the triangular face centers of the 7-simplex. Vertices of the trirectified 7-simplex are located in the tetrahedral cell centers of the 7-simplex. Rectified 7-simplex The rectified 7-simplex is the edge figure of the 251 honeycomb. It is called 05,1 for its branching Coxeter-Dynkin diagram, shown as . E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as S. Alternate names Rectified octaexon (Acronym: roc) (Jonathan Bowers) Coordinates The vertices of the rectified 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,0,0,0,1,1). This construction is based on facets of the rectified 8-orthoplex. Images Birectified 7-simplex E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as S. It is also called 04,2 for its branching Coxeter-Dynkin diagram, shown as . Alternate names Birectified octaexon (Acronym: broc) (Jonathan Bowers) Coordinates The vertices of the birectified 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,0,0,1,1,1). This construction is based on facets of the birectified 8-orthoplex. Images Trirectified 7-simplex The trirectified 7-simplex is the intersection of two regular 7-simplexes in dual configuration. E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as S. This polytope is the vertex figure of the 133 honeycomb. It is called 03,3 for its branching Coxeter-Dynkin diagram, shown as . Alternate names Hexadecaexon (Acronym: he) (Jonathan Bowers) Coordinates The vertices of the trirectified 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,0,1,1,1,1). This construction is based on facets of the trirectified 8-orthoplex. The trirectified 7-simplex is the intersection of two regular 7-simplices in dual configuration. This characterization yields simple coordinates for the vertices of a trirectified 7-simplex in 8-space: the 70 distinct permutations of (1,1,1,1,−1,−1,−1,-1). Images Related polytopes Related polytopes These polytopes are three of 71 uniform 7-polytopes with A7 symmetry. See also List of A7 polytopes 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 Ivic Weiss, Wiley-Interscience Publication, 1995, (Paper 22) H.S.M. Coxeter, Regular