In five-dimensional geometry , a truncated 5-simplex is a convex uniform 5-polytope , being a truncation of the regular 5-simplex .
There are unique 2 degrees of truncation. Vertices of the truncation 5-simplex are located as pairs on the edge of the 5-simplex. Vertices of the bitruncation 5-simplex are located on the triangular faces of the 5-simplex.
Truncated 5-simplex edit Truncated 5-simplex Type Uniform 5-polytope Schläfli symbol t{3,3,3,3} Coxeter-Dynkin diagram 4-faces 12 6 {3,3,3} t{3,3,3} Cells 45 30 {3,3} t{3,3} Faces 80 60 {3} {6} Edges 75 Vertices 30 Vertex figure ( )v{3,3} Coxeter group A5 [3,3,3,3], order 720 Properties convex
The truncated 5-simplex has 30 vertices , 75 edges , 80 triangular faces , 45 cells (15 tetrahedral , and 30 truncated tetrahedron ), and 12 4-faces (6 5-cell and 6 truncated 5-cells ).
Alternate names edit Truncated hexateron (Acronym: tix) (Jonathan Bowers)[1] Coordinates edit The vertices of the truncated 5-simplex can be most simply constructed on a hyperplane in 6-space as permutations of (0,0,0,0,1,2) or of (0,1,2,2,2,2). These coordinates come from facets of the truncated 6-orthoplex and bitruncated 6-cube respectively.
Images edit Bitruncated 5-simplex edit bitruncated 5-simplex Type Uniform 5-polytope Schläfli symbol 2t{3,3,3,3} Coxeter-Dynkin diagram 4-faces 12 6 2t{3,3,3} t{3,3,3} Cells 60 45 {3,3} t{3,3} Faces 140 80 {3} {6} Edges 150 Vertices 60 Vertex figure { }v{3} Coxeter group A5 [3,3,3,3], order 720 Properties convex
Alternate names edit Bitruncated hexateron (Acronym: bittix) (Jonathan Bowers)[2] Coordinates edit The vertices of the bitruncated 5-simplex can be most simply constructed on a hyperplane in 6-space as permutations of (0,0,0,1,2,2) or of (0,0,1,2,2,2). These represent positive orthant facets of the bitruncated 6-orthoplex , and the tritruncated 6-cube respectively.
Images edit Related uniform 5-polytopes edit The truncated 5-simplex is one of 19 uniform 5-polytopes based on the [3,3,3,3] Coxeter group , all shown here in A5 Coxeter plane orthographic projections . (Vertices are colored by projection overlap order, red, orange, yellow, green, cyan, blue, purple having progressively more vertices)
A5 polytopes t0 t1 t2 t0,1 t0,2 t1,2 t0,3 t1,3 t0,4 t0,1,2 t0,1,3 t0,2,3 t1,2,3 t0,1,4 t0,2,4 t0,1,2,3 t0,1,2,4 t0,1,3,4 t0,1,2,3,4
Notes edit ^ Klitizing, (x3x3o3o3o - tix) ^ Klitizing, (o3x3x3o3o - bittix) 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 , George Olshevsky. Polytopes of Various Dimensions, Jonathan Bowers Truncated uniform polytera (tix), Jonathan Bowers Multi-dimensional Glossary
truncated, simplexes, simplex, truncated, simplex, bitruncated, simplex, orthogonal, projections, coxeter, plane, five, dimensional, geometry, truncated, simplex, convex, uniform, polytope, being, truncation, regular, simplex, there, unique, degrees, truncatio. 5 simplex Truncated 5 simplex Bitruncated 5 simplex Orthogonal projections in A5 Coxeter plane In five dimensional geometry a truncated 5 simplex is a convex uniform 5 polytope being a truncation of the regular 5 simplex There are unique 2 degrees of truncation Vertices of the truncation 5 simplex are located as pairs on the edge of the 5 simplex Vertices of the bitruncation 5 simplex are located on the triangular faces of the 5 simplex Contents 1 Truncated 5 simplex 1 1 Alternate names 1 2 Coordinates 1 3 Images 2 Bitruncated 5 simplex 2 1 Alternate names 2 2 Coordinates 2 3 Images 3 Related uniform 5 polytopes 4 Notes 5 References 6 External linksTruncated 5 simplex editTruncated 5 simplex Type Uniform 5 polytope Schlafli symbol t 3 3 3 3 Coxeter Dynkin diagram nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp 4 faces 12 6 3 3 3 nbsp 6 t 3 3 3 nbsp Cells 45 30 3 3 nbsp 15 t 3 3 nbsp Faces 80 60 3 20 6 Edges 75 Vertices 30 Vertex figure nbsp v 3 3 Coxeter group A5 3 3 3 3 order 720 Properties convex The truncated 5 simplex has 30 vertices 75 edges 80 triangular faces 45 cells 15 tetrahedral and 30 truncated tetrahedron and 12 4 faces 6 5 cell and 6 truncated 5 cells Alternate names edit Truncated hexateron Acronym tix Jonathan Bowers 1 Coordinates edit The vertices of the truncated 5 simplex can be most simply constructed on a hyperplane in 6 space as permutations of 0 0 0 0 1 2 or of 0 1 2 2 2 2 These coordinates come from facets of the truncated 6 orthoplex and bitruncated 6 cube respectively Images edit orthographic projections AkCoxeter plane A5 A4 Graph nbsp nbsp Dihedral symmetry 6 5 AkCoxeter plane A3 A2 Graph nbsp nbsp Dihedral symmetry 4 3 Bitruncated 5 simplex editbitruncated 5 simplex Type Uniform 5 polytope Schlafli symbol 2t 3 3 3 3 Coxeter Dynkin diagram nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp 4 faces 12 6 2t 3 3 3 nbsp 6 t 3 3 3 nbsp Cells 60 45 3 3 nbsp 15 t 3 3 nbsp Faces 140 80 3 nbsp 60 6 nbsp Edges 150 Vertices 60 Vertex figure nbsp v 3 Coxeter group A5 3 3 3 3 order 720 Properties convex Alternate names edit Bitruncated hexateron Acronym bittix Jonathan Bowers 2 Coordinates edit The vertices of the bitruncated 5 simplex can be most simply constructed on a hyperplane in 6 space as permutations of 0 0 0 1 2 2 or of 0 0 1 2 2 2 These represent positive orthant facets of the bitruncated 6 orthoplex and the tritruncated 6 cube respectively Images edit orthographic projections AkCoxeter plane A5 A4 Graph nbsp nbsp Dihedral symmetry 6 5 AkCoxeter plane A3 A2 Graph nbsp nbsp Dihedral symmetry 4 3 Related uniform 5 polytopes editThe truncated 5 simplex is one of 19 uniform 5 polytopes based on the 3 3 3 3 Coxeter group all shown here in A5 Coxeter plane orthographic projections Vertices are colored by projection overlap order red orange yellow green cyan blue purple having progressively more vertices A5 polytopes nbsp t0 nbsp t1 nbsp t2 nbsp t0 1 nbsp t0 2 nbsp t1 2 nbsp t0 3 nbsp t1 3 nbsp t0 4 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 t0 1 2 3 nbsp t0 1 2 4 nbsp t0 1 3 4 nbsp t0 1 2 3 4Notes edit Klitizing x3x3o3o3o tix Klitizing o3x3x3o3o bittix 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 x3x3o3o3o tix o3x3x3o3o bittixExternal links editGlossary for hyperspace George Olshevsky Polytopes of Various Dimensions Jonathan Bowers Truncated uniform polytera tix Jonathan Bowers Multi dimensional Glossary vteFundamental 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 polychoron Pentachoron 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 Retrieved from https en wikipedia org w index php title Truncated 5 simplexes amp oldid 1148110839 Bitruncated 5 simplex, wikipedia, wiki , book, books, library,
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