In five-dimensional geometry , a truncated 5-orthoplex is a convex uniform 5-polytope , being a truncation of the regular 5-orthoplex .
There are 4 unique truncations of the 5-orthoplex. Vertices of the truncation 5-orthoplex are located as pairs on the edge of the 5-orthoplex. Vertices of the bitruncated 5-orthoplex are located on the triangular faces of the 5-orthoplex. The third and fourth truncations are more easily constructed as second and first truncations of the 5-cube .
Truncated 5-orthoplex Edit Truncated 5-orthoplex Type uniform 5-polytope Schläfli symbol t{3,3,3,4}1,1 } Coxeter-Dynkin diagrams 4-faces 42 10 Cells 240 160 Faces 400 320 Edges 280 240 Vertices 80 Vertex figure ( )v{3,4} Coxeter groups B5 , [3,3,3,4], order 38405 , [32,1,1 ], order 1920 Properties convex
Alternate names Edit Truncated pentacross Truncated triacontaditeron (Acronym: tot) (Jonathan Bowers)[1] Coordinates Edit Cartesian coordinates for the vertices of a truncated 5-orthoplex, centered at the origin, are all 80 vertices are sign (4) and coordinate (20) permutations of
(±2,±1,0,0,0) Images Edit The truncated 5-orthoplex is constructed by a truncation operation applied to the 5-orthoplex . All edges are shortened, and two new vertices are added on each original edge.
Bitruncated 5-orthoplex Edit Bitruncated 5-orthoplex Type uniform 5-polytope Schläfli symbol 2t{3,3,3,4}1,1 } Coxeter-Dynkin diagrams 4-faces 42 10 Cells 280 40 Faces 720 320 Edges 720 480 Vertices 240 Vertex figure { }v{4} Coxeter groups B5 , [3,3,3,4], order 38405 , [32,1,1 ], order 1920 Properties convex
The bitruncated 5-orthoplex can tessellate space in the tritruncated 5-cubic honeycomb .
Alternate names Edit Bitruncated pentacross Bitruncated triacontiditeron (acronym: bittit) (Jonathan Bowers)[2] Coordinates Edit Cartesian coordinates for the vertices of a truncated 5-orthoplex, centered at the origin, are all 80 vertices are sign and coordinate permutations of
(±2,±2,±1,0,0) Images Edit The bitrunacted 5-orthoplex is constructed by a bitruncation operation applied to the 5-orthoplex .
Related polytopes Edit This polytope is one of 31 uniform 5-polytopes generated from the regular 5-cube or 5-orthoplex .
B5 polytopes β5 t1 β5 t2 γ5 t1 γ5 γ5 t0,1 β5 t0,2 β5 t1,2 β5 t0,3 β5 t1,3 γ5 t1,2 γ5 t0,4 γ5 t0,3 γ5 t0,2 γ5 t0,1 γ5 t0,1,2 β5 t0,1,3 β5 t0,2,3 β5 t1,2,3 γ5 t0,1,4 β5 t0,2,4 γ5 t0,2,3 γ5 t0,1,4 γ5 t0,1,3 γ5 t0,1,2 γ5 t0,1,2,3 β5 t0,1,2,4 β5 t0,1,3,4 γ5 t0,1,2,4 γ5 t0,1,2,3 γ5 t0,1,2,3,4 γ5
Notes Edit ^ Klitzing, (x3x3o3o4o - tot) ^ Klitzing, (o3x3x3o4o - bittit) 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. "5D uniform polytopes (polytera)". External links Edit
truncated, orthoplexes, orthoplex, truncated, orthoplex, bitruncated, orthoplex5, cube, truncated, cube, bitruncated, cubeorthogonal, projections, coxeter, planein, five, dimensional, geometry, truncated, orthoplex, convex, uniform, polytope, being, truncation. 5 orthoplex Truncated 5 orthoplex Bitruncated 5 orthoplex5 cube Truncated 5 cube Bitruncated 5 cubeOrthogonal projections in B5 Coxeter planeIn five dimensional geometry a truncated 5 orthoplex is a convex uniform 5 polytope being a truncation of the regular 5 orthoplex There are 4 unique truncations of the 5 orthoplex Vertices of the truncation 5 orthoplex are located as pairs on the edge of the 5 orthoplex Vertices of the bitruncated 5 orthoplex are located on the triangular faces of the 5 orthoplex The third and fourth truncations are more easily constructed as second and first truncations of the 5 cube Contents 1 Truncated 5 orthoplex 1 1 Alternate names 1 2 Coordinates 1 3 Images 2 Bitruncated 5 orthoplex 2 1 Alternate names 2 2 Coordinates 2 3 Images 3 Related polytopes 4 Notes 5 References 6 External linksTruncated 5 orthoplex EditTruncated 5 orthoplexType uniform 5 polytopeSchlafli symbol t 3 3 3 4 t 3 31 1 Coxeter Dynkin diagrams nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp 4 faces 42 10 nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp 32 nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp Cells 240 160 nbsp nbsp nbsp nbsp nbsp nbsp 80 nbsp nbsp nbsp nbsp nbsp nbsp Faces 400 320 nbsp nbsp nbsp nbsp 80 nbsp nbsp nbsp nbsp Edges 280 240 nbsp 40 nbsp Vertices 80Vertex figure nbsp v 3 4 Coxeter groups B5 3 3 3 4 order 3840D5 32 1 1 order 1920Properties convexAlternate names Edit Truncated pentacross Truncated triacontaditeron Acronym tot Jonathan Bowers 1 Coordinates Edit Cartesian coordinates for the vertices of a truncated 5 orthoplex centered at the origin are all 80 vertices are sign 4 and coordinate 20 permutations of 2 1 0 0 0 Images Edit The truncated 5 orthoplex is constructed by a truncation operation applied to the 5 orthoplex All edges are shortened and two new vertices are added on each original edge orthographic projections Coxeter plane B5 B4 D5 B3 D4 A2Graph nbsp nbsp nbsp Dihedral symmetry 10 8 6 Coxeter plane B2 A3Graph nbsp nbsp Dihedral symmetry 4 4 Bitruncated 5 orthoplex EditBitruncated 5 orthoplexType uniform 5 polytopeSchlafli symbol 2t 3 3 3 4 2t 3 31 1 Coxeter Dynkin diagrams nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp 4 faces 42 10 nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp 32 nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp Cells 280 40 nbsp nbsp nbsp nbsp nbsp nbsp 160 nbsp nbsp nbsp nbsp nbsp nbsp 80 nbsp nbsp nbsp nbsp nbsp nbsp Faces 720 320 nbsp nbsp nbsp nbsp 320 nbsp nbsp nbsp nbsp 80 nbsp nbsp nbsp nbsp Edges 720 480 nbsp 240 nbsp Vertices 240Vertex figure nbsp v 4 Coxeter groups B5 3 3 3 4 order 3840D5 32 1 1 order 1920Properties convexThe bitruncated 5 orthoplex can tessellate space in the tritruncated 5 cubic honeycomb Alternate names Edit Bitruncated pentacross Bitruncated triacontiditeron acronym bittit Jonathan Bowers 2 Coordinates Edit Cartesian coordinates for the vertices of a truncated 5 orthoplex centered at the origin are all 80 vertices are sign and coordinate permutations of 2 2 1 0 0 Images Edit The bitrunacted 5 orthoplex is constructed by a bitruncation operation applied to the 5 orthoplex orthographic projections Coxeter plane B5 B4 D5 B3 D4 A2Graph nbsp nbsp nbsp Dihedral symmetry 10 8 6 Coxeter plane B2 A3Graph nbsp nbsp Dihedral symmetry 4 4 Related polytopes EditThis polytope is one of 31 uniform 5 polytopes generated from the regular 5 cube or 5 orthoplex B5 polytopes nbsp b5 nbsp t1b5 nbsp t2g5 nbsp t1g5 nbsp g5 nbsp t0 1b5 nbsp t0 2b5 nbsp t1 2b5 nbsp t0 3b5 nbsp t1 3g5 nbsp t1 2g5 nbsp t0 4g5 nbsp t0 3g5 nbsp t0 2g5 nbsp t0 1g5 nbsp t0 1 2b5 nbsp t0 1 3b5 nbsp t0 2 3b5 nbsp t1 2 3g5 nbsp t0 1 4b5 nbsp t0 2 4g5 nbsp t0 2 3g5 nbsp t0 1 4g5 nbsp t0 1 3g5 nbsp t0 1 2g5 nbsp t0 1 2 3b5 nbsp t0 1 2 4b5 nbsp t0 1 3 4g5 nbsp t0 1 2 4g5 nbsp t0 1 2 3g5 nbsp t0 1 2 3 4g5Notes Edit Klitzing x3x3o3o4o tot Klitzing o3x3x3o4o bittit 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 5D uniform polytopes polytera x3x3o3o4o tot o3x3x3o4o bittitExternal links EditWeisstein Eric W Hypercube MathWorld Polytopes 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 5 orthoplexes amp oldid 1148111844 Truncated 5 orthoplex, wikipedia, wiki , book, books, library,
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