In seven-dimensional geometry , a truncated 7-simplex is a convex uniform 7-polytope , being a truncation of the regular 7-simplex .
There are unique 3 degrees of truncation. Vertices of the truncation 7-simplex are located as pairs on the edge of the 7-simplex. Vertices of the bitruncated 7-simplex are located on the triangular faces of the 7-simplex. Vertices of the tritruncated 7-simplex are located inside the tetrahedral cells of the 7-simplex.
Truncated 7-simplex Edit
Truncated 7-simplex Type uniform 7-polytope Schläfli symbol t{3,3,3,3,3,3} Coxeter-Dynkin diagrams 6-faces 16 5-faces 4-faces Cells 350 Faces 336 Edges 196 Vertices 56 Vertex figure ( )v{3,3,3,3} Coxeter groups A7 , [3,3,3,3,3,3] Properties convex , Vertex-transitive
In seven-dimensional geometry , a truncated 7-simplex is a convex uniform 7-polytope , being a truncation of the regular 7-simplex .
Alternate names Edit Truncated octaexon (Acronym: toc) (Jonathan Bowers)[1] Coordinates Edit The vertices of the truncated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,0,0,0,1,2). This construction is based on facets of the truncated 8-orthoplex .
Images Edit Bitruncated 7-simplex Edit Bitruncated 7-simplex Type uniform 7-polytope Schläfli symbol 2t{3,3,3,3,3,3} Coxeter-Dynkin diagrams 6-faces 5-faces 4-faces Cells Faces Edges 588 Vertices 168 Vertex figure { }v{3,3,3} Coxeter groups A7 , [3,3,3,3,3,3] Properties convex , Vertex-transitive
Alternate names Edit Bitruncated octaexon (acronym: bittoc) (Jonathan Bowers)[2] Coordinates Edit The vertices of the bitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,0,0,1,2,2). This construction is based on facets of the bitruncated 8-orthoplex .
Images Edit Tritruncated 7-simplex Edit Tritruncated 7-simplex Type uniform 7-polytope Schläfli symbol 3t{3,3,3,3,3,3} Coxeter-Dynkin diagrams 6-faces 5-faces 4-faces Cells Faces Edges 980 Vertices 280 Vertex figure {3}v{3,3} Coxeter groups A7 , [3,3,3,3,3,3] Properties convex , Vertex-transitive
Alternate names Edit Tritruncated octaexon (acronym: tattoc) (Jonathan Bowers)[3] Coordinates Edit The vertices of the tritruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,0,1,2,2,2). This construction is based on facets of the tritruncated 8-orthoplex .
Images Edit Related polytopes Edit These three polytopes are from a set of 71 uniform 7-polytopes with A7 symmetry.
A7 polytopes t0 t1 t2 t3 t0,1 t0,2 t1,2 t0,3 t1,3 t2,3 t0,4 t1,4 t2,4 t0,5 t1,5 t0,6 t0,1,2 t0,1,3 t0,2,3 t1,2,3 t0,1,4 t0,2,4 t1,2,4 t0,3,4 t1,3,4 t2,3,4 t0,1,5 t0,2,5 t1,2,5 t0,3,5 t1,3,5 t0,4,5 t0,1,6 t0,2,6 t0,3,6 t0,1,2,3 t0,1,2,4 t0,1,3,4 t0,2,3,4 t1,2,3,4 t0,1,2,5 t0,1,3,5 t0,2,3,5 t1,2,3,5 t0,1,4,5 t0,2,4,5 t1,2,4,5 t0,3,4,5 t0,1,2,6 t0,1,3,6 t0,2,3,6 t0,1,4,6 t0,2,4,6 t0,1,5,6 t0,1,2,3,4 t0,1,2,3,5 t0,1,2,4,5 t0,1,3,4,5 t0,2,3,4,5 t1,2,3,4,5 t0,1,2,3,6 t0,1,2,4,6 t0,1,3,4,6 t0,2,3,4,6 t0,1,2,5,6 t0,1,3,5,6 t0,1,2,3,4,5 t0,1,2,3,4,6 t0,1,2,3,5,6 t0,1,2,4,5,6 t0,1,2,3,4,5,6
See also Edit Notes Edit ^ Klitizing, (x3x3o3o3o3o3o - toc) ^ Klitizing, (o3x3x3o3o3o3o - roc) ^ Klitizing, (o3o3x3x3o3o3o - tattoc) References Edit 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)". External links Edit Multi-dimensional Glossary
truncated, simplexes, simplex, truncated, simplexbitruncated, simplex, tritruncated, simplexorthogonal, projections, coxeter, planein, seven, dimensional, geometry, truncated, simplex, convex, uniform, polytope, being, truncation, regular, simplex, there, uniq. 7 simplex Truncated 7 simplexBitruncated 7 simplex Tritruncated 7 simplexOrthogonal projections in A7 Coxeter planeIn seven dimensional geometry a truncated 7 simplex is a convex uniform 7 polytope being a truncation of the regular 7 simplex There are unique 3 degrees of truncation Vertices of the truncation 7 simplex are located as pairs on the edge of the 7 simplex Vertices of the bitruncated 7 simplex are located on the triangular faces of the 7 simplex Vertices of the tritruncated 7 simplex are located inside the tetrahedral cells of the 7 simplex Contents 1 Truncated 7 simplex 1 1 Alternate names 1 2 Coordinates 1 3 Images 2 Bitruncated 7 simplex 2 1 Alternate names 2 2 Coordinates 2 3 Images 3 Tritruncated 7 simplex 3 1 Alternate names 3 2 Coordinates 3 3 Images 4 Related polytopes 5 See also 6 Notes 7 References 8 External linksTruncated 7 simplex EditTruncated 7 simplexType uniform 7 polytopeSchlafli symbol t 3 3 3 3 3 3 Coxeter Dynkin diagrams nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp 6 faces 165 faces4 facesCells 350Faces 336Edges 196Vertices 56Vertex figure v 3 3 3 3 Coxeter groups A7 3 3 3 3 3 3 Properties convex Vertex transitiveIn seven dimensional geometry a truncated 7 simplex is a convex uniform 7 polytope being a truncation of the regular 7 simplex Alternate names Edit Truncated octaexon Acronym toc Jonathan Bowers 1 Coordinates Edit The vertices of the truncated 7 simplex can be most simply positioned in 8 space as permutations of 0 0 0 0 0 0 1 2 This construction is based on facets of the truncated 8 orthoplex Images Edit orthographic projections Ak Coxeter plane A7 A6 A5Graph nbsp nbsp nbsp Dihedral symmetry 8 7 6 Ak Coxeter plane A4 A3 A2Graph nbsp nbsp nbsp Dihedral symmetry 5 4 3 Bitruncated 7 simplex EditBitruncated 7 simplexType uniform 7 polytopeSchlafli symbol 2t 3 3 3 3 3 3 Coxeter Dynkin diagrams nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp 6 faces5 faces4 facesCellsFacesEdges 588Vertices 168Vertex figure v 3 3 3 Coxeter groups A7 3 3 3 3 3 3 Properties convex Vertex transitiveAlternate names Edit Bitruncated octaexon acronym bittoc Jonathan Bowers 2 Coordinates Edit The vertices of the bitruncated 7 simplex can be most simply positioned in 8 space as permutations of 0 0 0 0 0 1 2 2 This construction is based on facets of the bitruncated 8 orthoplex Images Edit orthographic projections Ak Coxeter plane A7 A6 A5Graph nbsp nbsp nbsp Dihedral symmetry 8 7 6 Ak Coxeter plane A4 A3 A2Graph nbsp nbsp nbsp Dihedral symmetry 5 4 3 Tritruncated 7 simplex EditTritruncated 7 simplexType uniform 7 polytopeSchlafli symbol 3t 3 3 3 3 3 3 Coxeter Dynkin diagrams nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp 6 faces5 faces4 facesCellsFacesEdges 980Vertices 280Vertex figure 3 v 3 3 Coxeter groups A7 3 3 3 3 3 3 Properties convex Vertex transitiveAlternate names Edit Tritruncated octaexon acronym tattoc Jonathan Bowers 3 Coordinates Edit The vertices of the tritruncated 7 simplex can be most simply positioned in 8 space as permutations of 0 0 0 0 1 2 2 2 This construction is based on facets of the tritruncated 8 orthoplex Images Edit orthographic projections Ak Coxeter plane A7 A6 A5Graph nbsp nbsp nbsp Dihedral symmetry 8 7 6 Ak Coxeter plane A4 A3 A2Graph nbsp nbsp nbsp Dihedral symmetry 5 4 3 Related polytopes EditThese three polytopes are from a set of 71 uniform 7 polytopes with A7 symmetry A7 polytopes nbsp t0 nbsp t1 nbsp t2 nbsp t3 nbsp t0 1 nbsp t0 2 nbsp t1 2 nbsp t0 3 nbsp t1 3 nbsp t2 3 nbsp t0 4 nbsp t1 4 nbsp t2 4 nbsp t0 5 nbsp t1 5 nbsp t0 6 nbsp t0 1 2 nbsp t0 1 3 nbsp t0 2 3 nbsp t1 2 3 nbsp t0 1 4 nbsp t0 2 4 nbsp t1 2 4 nbsp t0 3 4 nbsp t1 3 4 nbsp t2 3 4 nbsp t0 1 5 nbsp t0 2 5 nbsp t1 2 5 nbsp t0 3 5 nbsp t1 3 5 nbsp t0 4 5 nbsp t0 1 6 nbsp t0 2 6 nbsp t0 3 6 nbsp t0 1 2 3 nbsp t0 1 2 4 nbsp t0 1 3 4 nbsp t0 2 3 4 nbsp t1 2 3 4 nbsp t0 1 2 5 nbsp t0 1 3 5 nbsp t0 2 3 5 nbsp t1 2 3 5 nbsp t0 1 4 5 nbsp t0 2 4 5 nbsp t1 2 4 5 nbsp t0 3 4 5 nbsp t0 1 2 6 nbsp t0 1 3 6 nbsp t0 2 3 6 nbsp t0 1 4 6 nbsp t0 2 4 6 nbsp t0 1 5 6 nbsp t0 1 2 3 4 nbsp t0 1 2 3 5 nbsp t0 1 2 4 5 nbsp t0 1 3 4 5 nbsp t0 2 3 4 5 nbsp t1 2 3 4 5 nbsp t0 1 2 3 6 nbsp t0 1 2 4 6 nbsp t0 1 3 4 6 nbsp t0 2 3 4 6 nbsp t0 1 2 5 6 nbsp t0 1 3 5 6 nbsp t0 1 2 3 4 5 nbsp t0 1 2 3 4 6 nbsp t0 1 2 3 5 6 nbsp t0 1 2 4 5 6 nbsp t0 1 2 3 4 5 6See also EditList of A7 polytopesNotes Edit Klitizing x3x3o3o3o3o3o toc Klitizing o3x3x3o3o3o3o roc Klitizing o3o3x3x3o3o3o tattoc References EditH 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 x3x3o3o3o3o3o toc o3x3x3o3o3o3o roc o3o3x3x3o3o3o tattocExternal links EditPolytopes of Various Dimensions Multi dimensional GlossaryvteFundamental convex regular and uniform polytopes in dimensions 2 10Family An Bn I2 p Dn E6 E7 E8 F4 G2 HnRegular polygon Triangle Square p gon Hexagon PentagonUniform polyhedron Tetrahedron Octahedron Cube Demicube Dodecahedron IcosahedronUniform polychoron Pentachoron 16 cell Tesseract Demitesseract 24 cell 120 cell 600 cellUniform 5 polytope 5 simplex 5 orthoplex 5 cube 5 demicubeUniform 6 polytope 6 simplex 6 orthoplex 6 cube 6 demicube 122 221Uniform 7 polytope 7 simplex 7 orthoplex 7 cube 7 demicube 132 231 321Uniform 8 polytope 8 simplex 8 orthoplex 8 cube 8 demicube 142 241 421Uniform 9 polytope 9 simplex 9 orthoplex 9 cube 9 demicubeUniform 10 polytope 10 simplex 10 orthoplex 10 cube 10 demicubeUniform n polytope n simplex n orthoplex n cube n demicube 1k2 2k1 k21 n pentagonal polytopeTopics Polytope families Regular polytope List of regular polytopes and compounds Retrieved from https en wikipedia org w index php title Truncated 7 simplexes amp oldid 849657035 Truncated 7 simplex, wikipedia, wiki , book, books, library,
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