7-simplexRectified 7-simplexBirectified 7-simplexTrirectified 7-simplexOrthogonal projections in A7 Coxeter plane
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 Rectified 7-simplexTypeuniform 7-polytopeCoxeter symbol 051Schläfli symbol r{36} = {35,1}or Coxeter diagramsor 6-faces165-faces844-faces224Cells350Faces336Edges168Vertices28Vertex figure6-simplex prismPetrie polygonOctagonCoxeter groupA7, , order 40320Propertiesconvex 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 Birectified 7-simplexTypeuniform 7-polytopeCoxeter symbol 042Schläfli symbol 2r{3,3,3,3,3,3} = {34,2}or Coxeter diagramsor 6-faces16:8 r{35} 8 2r{35} 5-faces112:28 {34} 56 r{34} 28 2r{34} 4-faces392:168 {33} (56+168) r{33} Cells770:(420+70) {3,3} 280 {3,4} Faces840:(280+560) {3}Edges420Vertices56Vertex figure{3}x{3,3,3}Coxeter groupA7, , order 40320Propertiesconvex 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 Trirectified 7-simplexTypeuniform 7-polytopeCoxeter symbol 033Schläfli symbol 3r{36} = {33,3}or Coxeter diagramsor 6-faces16 2r{35}5-faces1124-faces448Cells980Faces1120Edges560Vertices70Vertex figure{3,3}x{3,3}Coxeter groupA7×2, [], order 80640Propertiesconvex, isotopic 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
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. o3x3o3o3o3o3o - roc, o3o3x3o3o3o3o - broc, o3o3o3x3o3o3o - he
External links Polytopes of Various Dimensions Multi-dimensional Glossary