Great rhombated hexateron (Acronym: garx) (Jonathan Bowers)[3]
Coordinates
The vertices of the cantitruncated 5-simplex can be most simply constructed on a hyperplane in 6-space as permutations of (0,0,0,1,2,3) or of (0,1,2,3,3,3). These construction can be seen as facets of the cantitruncated 6-orthoplex or bicantitruncated 6-cube respectively.
The cantellated 5-simplex is one of 19 uniform 5-polytopes based on the [3,3,3,3] Coxeter group, all shown here in A5Coxeter planeorthographic projections. (Vertices are colored by projection overlap order, red, orange, yellow, green, cyan, blue, purple having progressively more vertices)
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, ISBN978-0-471-01003-6[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]
cantellated, simplexes, simplex, cantellated, simplex, bicantellated, simplexbirectified, simplex, cantitruncated, simplex, bicantitruncated, simplexorthogonal, projections, coxeter, planein, five, dimensional, geometry, cantellated, simplex, convex, uniform, . 5 simplex Cantellated 5 simplex Bicantellated 5 simplexBirectified 5 simplex Cantitruncated 5 simplex Bicantitruncated 5 simplexOrthogonal projections in A5 Coxeter planeIn five dimensional geometry a cantellated 5 simplex is a convex uniform 5 polytope being a cantellation of the regular 5 simplex There are unique 4 degrees of cantellation for the 5 simplex including truncations Contents 1 Cantellated 5 simplex 1 1 Alternate names 1 2 Coordinates 1 3 Images 2 Bicantellated 5 simplex 2 1 Alternate names 2 2 Coordinates 2 3 Images 3 Cantitruncated 5 simplex 3 1 Alternate names 3 2 Coordinates 3 3 Images 4 Bicantitruncated 5 simplex 4 1 Alternate names 4 2 Coordinates 4 3 Images 5 Related uniform 5 polytopes 6 Notes 7 References 8 External linksCantellated 5 simplex EditCantellated 5 simplexType Uniform 5 polytopeSchlafli symbol rr 3 3 3 3 r 3 3 3 3 displaystyle r left begin array l 3 3 3 3 end array right Coxeter Dynkin diagram or 4 faces 27 6 r 3 3 3 6 rr 3 3 3 15 x 3 3 Cells 135 30 3 3 30 r 3 3 15 rr 3 3 60 x 3 Faces 290 200 3 90 4 Edges 240Vertices 60Vertex figure Tetrahedral prismCoxeter group A5 3 3 3 3 order 720Properties convexThe cantellated 5 simplex has 60 vertices 240 edges 290 faces 200 triangles and 90 squares 135 cells 30 tetrahedra 30 octahedra 15 cuboctahedra and 60 triangular prisms and 27 4 faces 6 cantellated 5 cell 6 rectified 5 cells and 15 tetrahedral prisms Alternate names Edit Cantellated hexateron Small rhombated hexateron Acronym sarx Jonathan Bowers 1 Coordinates Edit The vertices of the cantellated 5 simplex can be most simply constructed on a hyperplane in 6 space as permutations of 0 0 0 1 1 2 or of 0 1 1 2 2 2 These represent positive orthant facets of the cantellated hexacross and bicantellated hexeract respectively Images Edit orthographic projections AkCoxeter plane A5 A4Graph Dihedral symmetry 6 5 AkCoxeter plane A3 A2Graph Dihedral symmetry 4 3 Bicantellated 5 simplex EditBicantellated 5 simplexType Uniform 5 polytopeSchlafli symbol 2rr 3 3 3 3 r 3 3 3 3 displaystyle r left begin array l 3 3 3 3 end array right Coxeter Dynkin diagram or 4 faces 32 12 t02 3 3 3 20 3 x 3 Cells 180 30 t1 3 3 120 x 3 30 t02 3 3 Faces 420 240 3 180 4 Edges 360Vertices 90Vertex figure Coxeter group A5 2 3 3 3 3 order 1440Properties convex isogonalAlternate names Edit Bicantellated hexateron Small birhombated dodecateron Acronym sibrid Jonathan Bowers 2 Coordinates Edit The coordinates can be made in 6 space as 90 permutations of 0 0 1 1 2 2 This construction exists as one of 64 orthant facets of the bicantellated 6 orthoplex Images Edit orthographic projections AkCoxeter plane A5 A4Graph Dihedral symmetry 6 5 10 AkCoxeter plane A3 A2Graph Dihedral symmetry 4 3 6 Cantitruncated 5 simplex Editcantitruncated 5 simplexType Uniform 5 polytopeSchlafli symbol tr 3 3 3 3 t 3 3 3 3 displaystyle t left begin array l 3 3 3 3 end array right Coxeter Dynkin diagram or 4 faces 27 6 t012 3 3 3 6 t 3 3 3 15 x 3 3 Cells 135 15 t012 3 3 30 t 3 3 60 x 3 30 3 3 Faces 290 120 3 80 6 90 x Edges 300Vertices 120Vertex figure Irr 5 cellCoxeter group A5 3 3 3 3 order 720Properties convexAlternate names Edit Cantitruncated hexateron Great rhombated hexateron Acronym garx Jonathan Bowers 3 Coordinates Edit The vertices of the cantitruncated 5 simplex can be most simply constructed on a hyperplane in 6 space as permutations of 0 0 0 1 2 3 or of 0 1 2 3 3 3 These construction can be seen as facets of the cantitruncated 6 orthoplex or bicantitruncated 6 cube respectively Images Edit orthographic projections AkCoxeter plane A5 A4Graph Dihedral symmetry 6 5 AkCoxeter plane A3 A2Graph Dihedral symmetry 4 3 Bicantitruncated 5 simplex EditBicantitruncated 5 simplexType Uniform 5 polytopeSchlafli symbol 2tr 3 3 3 3 t 3 3 3 3 displaystyle t left begin array l 3 3 3 3 end array right Coxeter Dynkin diagram or 4 faces 32 12 tr 3 3 3 20 3 x 3 Cells 180 30 t 3 3 120 x 3 30 t 3 4 Faces 420 240 3 180 4 Edges 450Vertices 180Vertex figure Coxeter group A5 2 3 3 3 3 order 1440Properties convex isogonalAlternate names Edit Bicantitruncated hexateron Great birhombated dodecateron Acronym gibrid Jonathan Bowers 4 Coordinates Edit The coordinates can be made in 6 space as 180 permutations of 0 0 1 2 3 3 This construction exists as one of 64 orthant facets of the bicantitruncated 6 orthoplex Images Edit orthographic projections AkCoxeter plane A5 A4Graph Dihedral symmetry 6 5 10 AkCoxeter plane A3 A2Graph Dihedral symmetry 4 3 6 Related uniform 5 polytopes EditThe cantellated 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 4Notes Edit Klitizing x3o3x3o3o sarx Klitizing o3x3o3x3o sibrid Klitizing x3x3x3o3o garx Klitizing o3x3x3x3o gibrid 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 x3o3x3o3o sarx o3x3o3x3o sibrid x3x3x3o3o garx o3x3x3x3o gibridExternal links EditGlossary for hyperspace George Olshevsky Polytopes of Various Dimensions Jonathan Bowers 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 Cantellated 5 simplexes amp oldid 783146033 Bicantitruncated 5 simplex, wikipedia, wiki, book, books, library,
