Cantellated 6-simplex edit Cantellated 6-simplex Type uniform 6-polytope Schläfli symbol rr{3,3,3,3,3}r { 3 , 3 , 3 , 3 3 } {\displaystyle r\left\{{\begin{array}{l}3,3,3,3\\3\end{array}}\right\}} Coxeter-Dynkin diagrams 5-faces 35 4-faces 210 Cells 560 Faces 805 Edges 525 Vertices 105 Vertex figure 5-cell prism Coxeter group A6 , [35 ], order 5040 Properties convex
Alternate names edit Small rhombated heptapeton (Acronym: sril) (Jonathan Bowers)[1] Coordinates edit The vertices of the cantellated 6-simplex can be most simply positioned in 7-space as permutations of (0,0,0,0,1,1,2). This construction is based on facets of the cantellated 7-orthoplex .
Images edit [2]
Bicantellated 6-simplex edit Bicantellated 6-simplex Type uniform 6-polytope Schläfli symbol 2rr{3,3,3,3,3}r { 3 , 3 , 3 3 , 3 } {\displaystyle r\left\{{\begin{array}{l}3,3,3\\3,3\end{array}}\right\}} Coxeter-Dynkin diagrams 5-faces 49 4-faces 329 Cells 980 Faces 1540 Edges 1050 Vertices 210 Vertex figure Coxeter group A6 , [35 ], order 5040 Properties convex
Alternate names edit Small prismated heptapeton (Acronym: sabril) (Jonathan Bowers)[3] Coordinates edit The vertices of the bicantellated 6-simplex can be most simply positioned in 7-space as permutations of (0,0,0,1,1,2,2). This construction is based on facets of the bicantellated 7-orthoplex .
Images edit Cantitruncated 6-simplex edit cantitruncated 6-simplex Type uniform 6-polytope Schläfli symbol tr{3,3,3,3,3}t { 3 , 3 , 3 , 3 3 } {\displaystyle t\left\{{\begin{array}{l}3,3,3,3\\3\end{array}}\right\}} Coxeter-Dynkin diagrams 5-faces 35 4-faces 210 Cells 560 Faces 805 Edges 630 Vertices 210 Vertex figure Coxeter group A6 , [35 ], order 5040 Properties convex
Alternate names edit Great rhombated heptapeton (Acronym: gril) (Jonathan Bowers)[4] Coordinates edit The vertices of the cantitruncated 6-simplex can be most simply positioned in 7-space as permutations of (0,0,0,0,1,2,3). This construction is based on facets of the cantitruncated 7-orthoplex .
Images edit Bicantitruncated 6-simplex edit bicantitruncated 6-simplex Type uniform 6-polytope Schläfli symbol 2tr{3,3,3,3,3}t { 3 , 3 , 3 3 , 3 } {\displaystyle t\left\{{\begin{array}{l}3,3,3\\3,3\end{array}}\right\}} Coxeter-Dynkin diagrams 5-faces 49 4-faces 329 Cells 980 Faces 1540 Edges 1260 Vertices 420 Vertex figure Coxeter group A6 , [35 ], order 5040 Properties convex
Alternate names edit Great birhombated heptapeton (Acronym: gabril) (Jonathan Bowers)[5] Coordinates edit The vertices of the bicantitruncated 6-simplex can be most simply positioned in 7-space as permutations of (0,0,0,1,2,3,3). This construction is based on facets of the bicantitruncated 7-orthoplex .
Images edit Related uniform 6-polytopes edit The truncated 6-simplex is one of 35 uniform 6-polytopes based on the [3,3,3,3,3] Coxeter group , all shown here in A6 Coxeter plane orthographic projections .
A6 polytopes t0 t1 t2 t0,1 t0,2 t1,2 t0,3 t1,3 t2,3 t0,4 t1,4 t0,5 t0,1,2 t0,1,3 t0,2,3 t1,2,3 t0,1,4 t0,2,4 t1,2,4 t0,3,4 t0,1,5 t0,2,5 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 t0,1,4,5 t0,1,2,3,4 t0,1,2,3,5 t0,1,2,4,5 t0,1,2,3,4,5
Notes edit ^ Klitizing, (x3o3x3o3o3o - sril) ^ Klitzing, (x3o3x3o3o3o - sril) ^ Klitzing, (o3x3o3x3o3o - sabril) ^ Klitzing, (x3x3x3o3o3o - gril) ^ Klitzing, (o3x3x3x3o3o - gabril) 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. "6D uniform polytopes (polypeta)". External links edit Multi-dimensional Glossary
cantellated, simplexes, simplex, cantellated, simplex, bicantellated, simplexbirectified, simplex, cantitruncated, simplex, bicantitruncated, simplexorthogonal, projections, coxeter, planein, dimensional, geometry, cantellated, simplex, convex, uniform, polyto. 6 simplex Cantellated 6 simplex Bicantellated 6 simplexBirectified 6 simplex Cantitruncated 6 simplex Bicantitruncated 6 simplexOrthogonal projections in A6 Coxeter planeIn six dimensional geometry a cantellated 6 simplex is a convex uniform 6 polytope being a cantellation of the regular 6 simplex There are unique 4 degrees of cantellation for the 6 simplex including truncations Contents 1 Cantellated 6 simplex 1 1 Alternate names 1 2 Coordinates 1 3 Images 2 Bicantellated 6 simplex 2 1 Alternate names 2 2 Coordinates 2 3 Images 3 Cantitruncated 6 simplex 3 1 Alternate names 3 2 Coordinates 3 3 Images 4 Bicantitruncated 6 simplex 4 1 Alternate names 4 2 Coordinates 4 3 Images 5 Related uniform 6 polytopes 6 Notes 7 References 8 External linksCantellated 6 simplex editCantellated 6 simplexType uniform 6 polytopeSchlafli symbol rr 3 3 3 3 3 or r 3 3 3 33 displaystyle r left begin array l 3 3 3 3 3 end array right nbsp Coxeter Dynkin diagrams nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp 5 faces 354 faces 210Cells 560Faces 805Edges 525Vertices 105Vertex figure 5 cell prismCoxeter group A6 35 order 5040Properties convexAlternate names edit Small rhombated heptapeton Acronym sril Jonathan Bowers 1 Coordinates edit The vertices of the cantellated 6 simplex can be most simply positioned in 7 space as permutations of 0 0 0 0 1 1 2 This construction is based on facets of the cantellated 7 orthoplex Images edit orthographic projections Ak Coxeter plane A6 A5 A4Graph nbsp nbsp nbsp Dihedral symmetry 7 6 5 Ak Coxeter plane A3 A2Graph nbsp nbsp Dihedral symmetry 4 3 2 Bicantellated 6 simplex editBicantellated 6 simplexType uniform 6 polytopeSchlafli symbol 2rr 3 3 3 3 3 or r 3 3 33 3 displaystyle r left begin array l 3 3 3 3 3 end array right nbsp Coxeter Dynkin diagrams nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp 5 faces 494 faces 329Cells 980Faces 1540Edges 1050Vertices 210Vertex figureCoxeter group A6 35 order 5040Properties convexAlternate names edit Small prismated heptapeton Acronym sabril Jonathan Bowers 3 Coordinates edit The vertices of the bicantellated 6 simplex can be most simply positioned in 7 space as permutations of 0 0 0 1 1 2 2 This construction is based on facets of the bicantellated 7 orthoplex Images edit orthographic projections Ak Coxeter plane A6 A5 A4Graph nbsp nbsp nbsp Dihedral symmetry 7 6 5 Ak Coxeter plane A3 A2Graph nbsp nbsp Dihedral symmetry 4 3 Cantitruncated 6 simplex editcantitruncated 6 simplexType uniform 6 polytopeSchlafli symbol tr 3 3 3 3 3 or t 3 3 3 33 displaystyle t left begin array l 3 3 3 3 3 end array right nbsp Coxeter Dynkin diagrams nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp 5 faces 354 faces 210Cells 560Faces 805Edges 630Vertices 210Vertex figureCoxeter group A6 35 order 5040Properties convexAlternate names edit Great rhombated heptapeton Acronym gril Jonathan Bowers 4 Coordinates edit The vertices of the cantitruncated 6 simplex can be most simply positioned in 7 space as permutations of 0 0 0 0 1 2 3 This construction is based on facets of the cantitruncated 7 orthoplex Images edit orthographic projections Ak Coxeter plane A6 A5 A4Graph nbsp nbsp nbsp Dihedral symmetry 7 6 5 Ak Coxeter plane A3 A2Graph nbsp nbsp Dihedral symmetry 4 3 Bicantitruncated 6 simplex editbicantitruncated 6 simplexType uniform 6 polytopeSchlafli symbol 2tr 3 3 3 3 3 or t 3 3 33 3 displaystyle t left begin array l 3 3 3 3 3 end array right nbsp Coxeter Dynkin diagrams nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp 5 faces 494 faces 329Cells 980Faces 1540Edges 1260Vertices 420Vertex figureCoxeter group A6 35 order 5040Properties convexAlternate names edit Great birhombated heptapeton Acronym gabril Jonathan Bowers 5 Coordinates edit The vertices of the bicantitruncated 6 simplex can be most simply positioned in 7 space as permutations of 0 0 0 1 2 3 3 This construction is based on facets of the bicantitruncated 7 orthoplex Images edit orthographic projections Ak Coxeter plane A6 A5 A4Graph nbsp nbsp nbsp Dihedral symmetry 7 6 5 Ak Coxeter plane A3 A2Graph nbsp nbsp Dihedral symmetry 4 3 Related uniform 6 polytopes editThe truncated 6 simplex is one of 35 uniform 6 polytopes based on the 3 3 3 3 3 Coxeter group all shown here in A6 Coxeter plane orthographic projections A6 polytopes nbsp t0 nbsp t1 nbsp t2 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 t0 5 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 t0 1 5 nbsp t0 2 5 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 t0 1 4 5 nbsp t0 1 2 3 4 nbsp t0 1 2 3 5 nbsp t0 1 2 4 5 nbsp t0 1 2 3 4 5Notes edit Klitizing x3o3x3o3o3o sril Klitzing x3o3x3o3o3o sril Klitzing o3x3o3x3o3o sabril Klitzing x3x3x3o3o3o gril Klitzing o3x3x3x3o3o gabril 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 x3o3x3o3o3o sril o3x3o3x3o3o sabril x3x3x3o3o3o gril o3x3x3x3o3o gabrilExternal 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 Cantellated 6 simplexes amp oldid 1148115464 Cantitruncated 6 simplex, wikipedia, wiki , book, books, library,
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