Runcinated 5-simplex edit Runcinated 5-simplex Type Uniform 5-polytope Schläfli symbol t0,3 {3,3,3,3} Coxeter-Dynkin diagram 4-faces 47 6 t0,3 {3,3,3} {3}×{3} { }×r{3,3} r{3,3,3} Cells 255 45 {3,3} { }×{3} r{3,3} Faces 420 240 {3} {4} Edges 270 Vertices 60 Vertex figure Coxeter group A5 [3,3,3,3], order 720 Properties convex
Alternate names edit Runcinated hexateron Small prismated hexateron (Acronym: spix) (Jonathan Bowers)[1] Coordinates edit The vertices of the runcinated 5-simplex can be most simply constructed on a hyperplane in 6-space as permutations of (0,0,1,1,1,2) or of (0,1,1,1,2,2), seen as facets of a runcinated 6-orthoplex , or a biruncinated 6-cube respectively.
Images edit Runcitruncated 5-simplex edit Runcitruncated 5-simplex Type Uniform 5-polytope Schläfli symbol t0,1,3 {3,3,3,3} Coxeter-Dynkin diagram 4-faces 47 6 t0,1,3 {3,3,3} {3}×{6} { }×r{3,3} rr{3,3,3} Cells 315 Faces 720 Edges 630 Vertices 180 Vertex figure Coxeter group A5 [3,3,3,3], order 720 Properties convex , isogonal
Alternate names edit Runcitruncated hexateron Prismatotruncated hexateron (Acronym: pattix) (Jonathan Bowers)[2] Coordinates edit The coordinates can be made in 6-space, as 180 permutations of:
(0,0,1,1,2,3) This construction exists as one of 64 orthant facets of the runcitruncated 6-orthoplex .
Images edit Runcicantellated 5-simplex edit Runcicantellated 5-simplex Type Uniform 5-polytope Schläfli symbol t0,2,3 {3,3,3,3} Coxeter-Dynkin diagram 4-faces 47 Cells 255 Faces 570 Edges 540 Vertices 180 Vertex figure Coxeter group A5 [3,3,3,3], order 720 Properties convex , isogonal
Alternate names edit Runcicantellated hexateron Biruncitruncated 5-simplex/hexateron Prismatorhombated hexateron (Acronym: pirx) (Jonathan Bowers)[3] Coordinates edit The coordinates can be made in 6-space, as 180 permutations of:
(0,0,1,2,2,3) This construction exists as one of 64 orthant facets of the runcicantellated 6-orthoplex .
Images edit Runcicantitruncated 5-simplex edit Runcicantitruncated 5-simplex Type Uniform 5-polytope Schläfli symbol t0,1,2,3 {3,3,3,3} Coxeter-Dynkin diagram 4-faces 47 6 t0,1,2,3 {3,3,3} {3}×{6} {}×t{3,3} tr{3,3,3} Cells 315 45 t0,1,2 {3,3} { }×{3} { }×{6} t{3,3} Faces 810 120 {3} Edges 900 Vertices 360 Vertex figure Irregular 5-cell Coxeter group A5 [3,3,3,3], order 720 Properties convex , isogonal
Alternate names edit Runcicantitruncated hexateron Great prismated hexateron (Acronym: gippix) (Jonathan Bowers)[4] Coordinates edit The coordinates can be made in 6-space, as 360 permutations of:
(0,0,1,2,3,4) This construction exists as one of 64 orthant facets of the runcicantitruncated 6-orthoplex .
Images edit Related uniform 5-polytopes edit These polytopes are in a set 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, (x3o3o3x3o - spidtix) ^ Klitizing, (x3x3o3x3o - pattix) ^ Klitizing, (x3o3x3x3o - pirx) ^ Klitizing, (x3x3x3x3o - gippix) 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 Runcinated uniform polytera (spid), Jonathan Bowers Multi-dimensional Glossary
runcinated, simplexes, simplex, runcinated, simplex, runcitruncated, simplexbirectified, simplex, runcicantellated, simplex, runcicantitruncated, simplexorthogonal, projections, coxeter, planein, dimensional, geometry, runcinated, simplex, convex, uniform, pol. 5 simplex Runcinated 5 simplex Runcitruncated 5 simplexBirectified 5 simplex Runcicantellated 5 simplex Runcicantitruncated 5 simplexOrthogonal projections in A5 Coxeter planeIn six dimensional geometry a runcinated 5 simplex is a convex uniform 5 polytope with 3rd order truncations Runcination of the regular 5 simplex There are 4 unique runcinations of the 5 simplex with permutations of truncations and cantellations Contents 1 Runcinated 5 simplex 1 1 Alternate names 1 2 Coordinates 1 3 Images 2 Runcitruncated 5 simplex 2 1 Alternate names 2 2 Coordinates 2 3 Images 3 Runcicantellated 5 simplex 3 1 Alternate names 3 2 Coordinates 3 3 Images 4 Runcicantitruncated 5 simplex 4 1 Alternate names 4 2 Coordinates 4 3 Images 5 Related uniform 5 polytopes 6 Notes 7 References 8 External linksRuncinated 5 simplex editRuncinated 5 simplexType Uniform 5 polytopeSchlafli symbol t0 3 3 3 3 3 Coxeter Dynkin diagram nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp 4 faces 47 6 t0 3 3 3 3 nbsp 20 3 3 15 r 3 3 6 r 3 3 3 nbsp Cells 255 45 3 3 nbsp 180 3 30 r 3 3 nbsp Faces 420 240 3 nbsp 180 4 Edges 270Vertices 60Vertex figure nbsp Coxeter group A5 3 3 3 3 order 720Properties convexAlternate names edit Runcinated hexateron Small prismated hexateron Acronym spix Jonathan Bowers 1 Coordinates edit The vertices of the runcinated 5 simplex can be most simply constructed on a hyperplane in 6 space as permutations of 0 0 1 1 1 2 or of 0 1 1 1 2 2 seen as facets of a runcinated 6 orthoplex or a biruncinated 6 cube respectively Images edit orthographic projections AkCoxeter plane A5 A4Graph nbsp nbsp Dihedral symmetry 6 5 AkCoxeter plane A3 A2Graph nbsp nbsp Dihedral symmetry 4 3 Runcitruncated 5 simplex editRuncitruncated 5 simplexType Uniform 5 polytopeSchlafli symbol t0 1 3 3 3 3 3 Coxeter Dynkin diagram nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp 4 faces 47 6 t0 1 3 3 3 3 20 3 6 15 r 3 3 6 rr 3 3 3 Cells 315Faces 720Edges 630Vertices 180Vertex figure nbsp Coxeter group A5 3 3 3 3 order 720Properties convex isogonalAlternate names edit Runcitruncated hexateron Prismatotruncated hexateron Acronym pattix Jonathan Bowers 2 Coordinates edit The coordinates can be made in 6 space as 180 permutations of 0 0 1 1 2 3 This construction exists as one of 64 orthant facets of the runcitruncated 6 orthoplex Images edit orthographic projections AkCoxeter plane A5 A4Graph nbsp nbsp Dihedral symmetry 6 5 AkCoxeter plane A3 A2Graph nbsp nbsp Dihedral symmetry 4 3 Runcicantellated 5 simplex editRuncicantellated 5 simplexType Uniform 5 polytopeSchlafli symbol t0 2 3 3 3 3 3 Coxeter Dynkin diagram nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp 4 faces 47Cells 255Faces 570Edges 540Vertices 180Vertex figure nbsp Coxeter group A5 3 3 3 3 order 720Properties convex isogonalAlternate names edit Runcicantellated hexateron Biruncitruncated 5 simplex hexateron Prismatorhombated hexateron Acronym pirx Jonathan Bowers 3 Coordinates edit The coordinates can be made in 6 space as 180 permutations of 0 0 1 2 2 3 This construction exists as one of 64 orthant facets of the runcicantellated 6 orthoplex Images edit orthographic projections AkCoxeter plane A5 A4Graph nbsp nbsp Dihedral symmetry 6 5 AkCoxeter plane A3 A2Graph nbsp nbsp Dihedral symmetry 4 3 Runcicantitruncated 5 simplex editRuncicantitruncated 5 simplexType Uniform 5 polytopeSchlafli symbol t0 1 2 3 3 3 3 3 Coxeter Dynkin diagram nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp 4 faces 47 6 t0 1 2 3 3 3 3 20 3 6 15 t 3 3 6 tr 3 3 3 Cells 315 45 t0 1 2 3 3 120 3 120 6 30 t 3 3 Faces 810 120 3 450 4 240 6 Edges 900Vertices 360Vertex figure nbsp Irregular 5 cellCoxeter group A5 3 3 3 3 order 720Properties convex isogonalAlternate names edit Runcicantitruncated hexateron Great prismated hexateron Acronym gippix Jonathan Bowers 4 Coordinates edit The coordinates can be made in 6 space as 360 permutations of 0 0 1 2 3 4 This construction exists as one of 64 orthant facets of the runcicantitruncated 6 orthoplex Images edit orthographic projections AkCoxeter plane A5 A4Graph nbsp nbsp Dihedral symmetry 6 5 AkCoxeter plane A3 A2Graph nbsp nbsp Dihedral symmetry 4 3 Related uniform 5 polytopes editThese polytopes are in a set 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 x3o3o3x3o spidtix Klitizing x3x3o3x3o pattix Klitizing x3o3x3x3o pirx Klitizing x3x3x3x3o gippix 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 x3o3o3x3o spidtix x3x3o3x3o pattix x3o3x3x3o pirx x3x3x3x3o gippixExternal links editGlossary for hyperspace George Olshevsky Polytopes of Various Dimensions Jonathan Bowers Runcinated uniform polytera spid 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 Runcinated 5 simplexes amp oldid 1148111732, 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