In six-dimensional geometry , a truncated 6-orthoplex is a convex uniform 6-polytope , being a truncation of the regular 6-orthoplex .
There are 5 degrees of truncation for the 6-orthoplex. Vertices of the truncated 6-orthoplex are located as pairs on the edge of the 6-orthoplex. Vertices of the bitruncated 6-orthoplex are located on the triangular faces of the 6-orthoplex. Vertices of the tritruncated 6-orthoplex are located inside the tetrahedral cells of the 6-orthoplex.
Truncated 6-orthoplex Truncated 6-orthoplex Type uniform 6-polytope Schläfli symbol t{3,3,3,3,4} Coxeter-Dynkin diagrams
5-faces 76 4-faces 576 Cells 1200 Faces 1120 Edges 540 Vertices 120 Vertex figure Coxeter groups B6 , [3,3,3,3,4]6 , [33,1,1 ] Properties convex
Alternate names Truncated hexacross Truncated hexacontatetrapeton (Acronym: tag) (Jonathan Bowers)[1] Construction There are two Coxeter groups associated with the truncated hexacross , one with the C6 or [4,3,3,3,3] Coxeter group, and a lower symmetry with the D6 or [33,1,1 ] Coxeter group.
Coordinates Cartesian coordinates for the vertices of a truncated 6-orthoplex, centered at the origin, are all 120 vertices are sign (4) and coordinate (30) permutations of
(±2,±1,0,0,0,0) Images
Bitruncated 6-orthoplex Bitruncated 6-orthoplex Type uniform 6-polytope Schläfli symbol 2t{3,3,3,3,4} Coxeter-Dynkin diagrams
5-faces 4-faces Cells Faces Edges Vertices Vertex figure Coxeter groups B6 , [3,3,3,3,4]6 , [33,1,1 ] Properties convex
Alternate names Bitruncated hexacross Bitruncated hexacontatetrapeton (Acronym: botag) (Jonathan Bowers)[2] Images
Related polytopes These polytopes are a part of a set of 63 uniform 6-polytopes generated from the B6 Coxeter plane , including the regular 6-cube or 6-orthoplex .
B6 polytopes β6 t1 β6 t2 β6 t2 γ6 t1 γ6 γ6 t0,1 β6 t0,2 β6 t1,2 β6 t0,3 β6 t1,3 β6 t2,3 γ6 t0,4 β6 t1,4 γ6 t1,3 γ6 t1,2 γ6 t0,5 γ6 t0,4 γ6 t0,3 γ6 t0,2 γ6 t0,1 γ6 t0,1,2 β6 t0,1,3 β6 t0,2,3 β6 t1,2,3 β6 t0,1,4 β6 t0,2,4 β6 t1,2,4 β6 t0,3,4 β6 t1,2,4 γ6 t1,2,3 γ6 t0,1,5 β6 t0,2,5 β6 t0,3,4 γ6 t0,2,5 γ6 t0,2,4 γ6 t0,2,3 γ6 t0,1,5 γ6 t0,1,4 γ6 t0,1,3 γ6 t0,1,2 γ6 t0,1,2,3 β6 t0,1,2,4 β6 t0,1,3,4 β6 t0,2,3,4 β6 t1,2,3,4 γ6 t0,1,2,5 β6 t0,1,3,5 β6 t0,2,3,5 γ6 t0,2,3,4 γ6 t0,1,4,5 γ6 t0,1,3,5 γ6 t0,1,3,4 γ6 t0,1,2,5 γ6 t0,1,2,4 γ6 t0,1,2,3 γ6 t0,1,2,3,4 β6 t0,1,2,3,5 β6 t0,1,2,4,5 β6 t0,1,2,4,5 γ6 t0,1,2,3,5 γ6 t0,1,2,3,4 γ6 t0,1,2,3,4,5 γ6
Notes ^ Klitzing, (x3x3o3o3o4o - tag) ^ Klitzing, (o3x3x3o3o4o - botag)
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. "6D uniform polytopes (polypeta)".
External links Multi-dimensional Glossary
truncated, orthoplexes, orthoplex, truncated, orthoplex, bitruncated, orthoplex, tritruncated, cube6, cube, truncated, cube, bitruncated, cubeorthogonal, projections, coxeter, planein, dimensional, geometry, truncated, orthoplex, convex, uniform, polytope, bei. 6 orthoplex Truncated 6 orthoplex Bitruncated 6 orthoplex Tritruncated 6 cube6 cube Truncated 6 cube Bitruncated 6 cubeOrthogonal projections in B6 Coxeter planeIn six dimensional geometry a truncated 6 orthoplex is a convex uniform 6 polytope being a truncation of the regular 6 orthoplex There are 5 degrees of truncation for the 6 orthoplex Vertices of the truncated 6 orthoplex are located as pairs on the edge of the 6 orthoplex Vertices of the bitruncated 6 orthoplex are located on the triangular faces of the 6 orthoplex Vertices of the tritruncated 6 orthoplex are located inside the tetrahedral cells of the 6 orthoplex Contents 1 Truncated 6 orthoplex 1 1 Alternate names 1 2 Construction 1 3 Coordinates 1 4 Images 2 Bitruncated 6 orthoplex 2 1 Alternate names 2 2 Images 3 Related polytopes 4 Notes 5 References 6 External linksTruncated 6 orthoplex EditTruncated 6 orthoplexType uniform 6 polytopeSchlafli symbol t 3 3 3 3 4 Coxeter Dynkin diagrams 5 faces 764 faces 576Cells 1200Faces 1120Edges 540Vertices 120Vertex figure v 3 4 Coxeter groups B6 3 3 3 3 4 D6 33 1 1 Properties convexAlternate names Edit Truncated hexacross Truncated hexacontatetrapeton Acronym tag Jonathan Bowers 1 Construction Edit There are two Coxeter groups associated with the truncated hexacross one with the C6 or 4 3 3 3 3 Coxeter group and a lower symmetry with the D6 or 33 1 1 Coxeter group Coordinates Edit Cartesian coordinates for the vertices of a truncated 6 orthoplex centered at the origin are all 120 vertices are sign 4 and coordinate 30 permutations of 2 1 0 0 0 0 Images Edit orthographic projections Coxeter plane B6 B5 B4Graph Dihedral symmetry 12 10 8 Coxeter plane B3 B2Graph Dihedral symmetry 6 4 Coxeter plane A5 A3Graph Dihedral symmetry 6 4 Bitruncated 6 orthoplex EditBitruncated 6 orthoplexType uniform 6 polytopeSchlafli symbol 2t 3 3 3 3 4 Coxeter Dynkin diagrams 5 faces4 facesCellsFacesEdgesVerticesVertex figure v 3 4 Coxeter groups B6 3 3 3 3 4 D6 33 1 1 Properties convexAlternate names Edit Bitruncated hexacross Bitruncated hexacontatetrapeton Acronym botag Jonathan Bowers 2 Images Edit orthographic projections Coxeter plane B6 B5 B4Graph Dihedral symmetry 12 10 8 Coxeter plane B3 B2Graph Dihedral symmetry 6 4 Coxeter plane A5 A3Graph Dihedral symmetry 6 4 Related polytopes EditThese polytopes are a part of a set of 63 uniform 6 polytopes generated from the B6 Coxeter plane including the regular 6 cube or 6 orthoplex B6 polytopes b6 t1b6 t2b6 t2g6 t1g6 g6 t0 1b6 t0 2b6 t1 2b6 t0 3b6 t1 3b6 t2 3g6 t0 4b6 t1 4g6 t1 3g6 t1 2g6 t0 5g6 t0 4g6 t0 3g6 t0 2g6 t0 1g6 t0 1 2b6 t0 1 3b6 t0 2 3b6 t1 2 3b6 t0 1 4b6 t0 2 4b6 t1 2 4b6 t0 3 4b6 t1 2 4g6 t1 2 3g6 t0 1 5b6 t0 2 5b6 t0 3 4g6 t0 2 5g6 t0 2 4g6 t0 2 3g6 t0 1 5g6 t0 1 4g6 t0 1 3g6 t0 1 2g6 t0 1 2 3b6 t0 1 2 4b6 t0 1 3 4b6 t0 2 3 4b6 t1 2 3 4g6 t0 1 2 5b6 t0 1 3 5b6 t0 2 3 5g6 t0 2 3 4g6 t0 1 4 5g6 t0 1 3 5g6 t0 1 3 4g6 t0 1 2 5g6 t0 1 2 4g6 t0 1 2 3g6 t0 1 2 3 4b6 t0 1 2 3 5b6 t0 1 2 4 5b6 t0 1 2 4 5g6 t0 1 2 3 5g6 t0 1 2 3 4g6 t0 1 2 3 4 5g6Notes Edit Klitzing x3x3o3o3o4o tag Klitzing o3x3x3o3o4o botag 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 6D uniform polytopes polypeta x3x3o3o3o4o tag o3x3x3o3o4o botagExternal 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 6 orthoplexes amp oldid 849657026 Bitruncated 6 orthoplex, wikipedia, wiki , book, books, library,
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