Short description: Convex uniform 7-polytope in seven-dimensional geometry 7-simplex Rectified 7-simplex | Birectified 7-simplex Trirectified 7-simplex Orthogonal 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. ## Contents * 1 Rectified 7-simplex * 1.1 Alternate names * 1.2 Coordinates * 1.3 Images * 2 Birectified 7-simplex * 2.1 Alternate names * 2.2 Coordinates * 2.3 Images * 3 Trirectified 7-simplex * 3.1 Alternate names * 3.2 Coordinates * 3.3 Images * 3.4 Related polytopes * 4 Related polytopes * 5 See also * 6 References * 7 External links ## Rectified 7-simplex Rectified 7-simplex Type | uniform 7-polytope Coxeter symbol | 051 Schläfli symbol | r{36} = {35,1} or [math]\displaystyle{ \left\\{\begin{array}{l}3, 3, 3, 3, 3\\\3\end{array}\right\\} }[/math] Coxeter diagrams | 6-faces | 16 5-faces | 84 4-faces | 224 Cells | 350 Faces | 336 Edges | 168 Vertices | 28 Vertex figure | 6-simplex prism Petrie polygon | Octagon Coxeter group | A7, [36], order 40320 Properties | convex 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 S17. ### 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 orthographic projections Ak Coxeter plane | A7 | A6 | A5 | | | Graph | | | Dihedral symmetry | [8] | [7] | [6] Ak Coxeter plane | A4 | A3 | A2 Graph | | | Dihedral symmetry | [5] | [4] | [3] ## Birectified 7-simplex Birectified 7-simplex Type | uniform 7-polytope Coxeter symbol | 042 Schläfli symbol | 2r{3,3,3,3,3,3} = {34,2} or [math]\displaystyle{ \left\\{\begin{array}{l}3, 3, 3, 3\\\3, 3\end{array}\right\\} }[/math] Coxeter diagrams | 6-faces | 16: 8 r{35} 8 2r{35} 5-faces | 112: 28 {34} 56 r{34} 25px 28 2r{34} 4-faces | 392: 168 {33} (56+168) r{33} Cells | 770: (420+70) {3,3} 280 {3,4} Faces | 840: (280+560) {3} Edges | 420 Vertices | 56 Vertex figure | {3}x{3,3,3} Coxeter group | A7, [36], order 40320 Properties | convex E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as S27. 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 orthographic projections Ak Coxeter plane | A7 | A6 | A5 | | | Graph | | | Dihedral symmetry | [8] | [7] | [6] Ak Coxeter plane | A4 | A3 | A2 Graph | | | Dihedral symmetry | [5] | [4] | [3] ## Trirectified 7-simplex Trirectified 7-simplex Type | uniform 7-polytope Coxeter symbol | 033 Schläfli symbol | 3r{36} = {33,3} or [math]\displaystyle{ \left\\{\begin{array}{l}3, 3, 3\\\3, 3, 3\end{array}\right\\} }[/math] Coxeter diagrams | 6-faces | 16 2r{35} 5-faces | 112 4-faces | 448 Cells | 980 Faces | 1120 Edges | 560 Vertices | 70 Vertex figure | {3,3}x{3,3} Coxeter group | A7×2, 36, order 80640 Properties | convex, 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 S37. 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, ISBN 978-0-471-01003-6 [1] * (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. * Klitzing, Richard. "7D uniform polytopes (polyexa)". https://bendwavy.org/klitzing/dimensions/polyexa.htm. o3o3x3o3o3o3o - broc, o3x3o3o3o3o3o - roc, o3o3x3o3o3o3o - he ## External links * Polytopes of Various Dimensions * Multi-dimensional Glossary * v * t * e Fundamental convex regular and uniform polytopes in dimensions 2–10 Family | An | Bn | I2(p) / Dn | E6 / E7 / E8 / F4 / G2 | Hn Regular polygon | Triangle | Square | p-gon | Hexagon | Pentagon Uniform polyhedron | Tetrahedron | Octahedron • Cube | Demicube | | Dodecahedron • Icosahedron Uniform 4-polytope | 5-cell | 16-cell • Tesseract | Demitesseract | 24-cell | 120-cell • 600-cell Uniform 5-polytope | 5-simplex | 5-orthoplex • 5-cube | 5-demicube | | Uniform 6-polytope | 6-simplex | 6-orthoplex • 6-cube | 6-demicube | 122 • 221 | Uniform 7-polytope | 7-simplex | 7-orthoplex • 7-cube | 7-demicube | 132 • 231 • 321 | Uniform 8-polytope | 8-simplex | 8-orthoplex • 8-cube | 8-demicube | 142 • 241 • 421 | Uniform 9-polytope | 9-simplex | 9-orthoplex • 9-cube | 9-demicube | | Uniform 10-polytope | 10-simplex | 10-orthoplex • 10-cube | 10-demicube | | Uniform n-polytope | n-simplex | n-orthoplex • n-cube | n-demicube | 1k2 • 2k1 • k21 | n-pentagonal polytope Topics: Polytope families • Regular polytope • List of regular polytopes and compounds 0.00 (0 votes) Original source: https://en.wikipedia.org/wiki/Rectified 7-simplexes. 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