In five-dimensional geometry , a runcinated 5-cube is a convex uniform 5-polytope that is a runcination (a 3rd order truncation) of the regular 5-cube .
There are 8 unique degrees of runcinations of the 5-cube, along with permutations of truncations and cantellations. Four are more simply constructed relative to the 5-orthoplex .
Runcinated 5-cube Alternate names Small prismated penteract (Acronym: span) (Jonathan Bowers) Coordinates The Cartesian coordinates of the vertices of a runcinated 5-cube having edge length 2 are all permutations of:
( ± 1 , ± 1 , ± 1 , ± ( 1 + 2 ) , ± ( 1 + 2 ) ) {\displaystyle \left(\pm 1,\ \pm 1,\ \pm 1,\ \pm (1+{\sqrt {2}}),\ \pm (1+{\sqrt {2}})\right)} Images
Runcitruncated 5-cube Alternate names Runcitruncated penteract Prismatotruncated penteract (Acronym: pattin) (Jonathan Bowers) Construction and coordinates The Cartesian coordinates of the vertices of a runcitruncated 5-cube having edge length 2 are all permutations of:
( ± 1 , ± ( 1 + 2 ) , ± ( 1 + 2 ) , ± ( 1 + 2 2 ) , ± ( 1 + 2 2 ) ) {\displaystyle \left(\pm 1,\ \pm (1+{\sqrt {2}}),\ \pm (1+{\sqrt {2}}),\ \pm (1+2{\sqrt {2}}),\ \pm (1+2{\sqrt {2}})\right)} Images
Runcicantellated 5-cube Alternate names Runcicantellated penteract Prismatorhombated penteract (Acronym: prin) (Jonathan Bowers) Coordinates The Cartesian coordinates of the vertices of a runcicantellated 5-cube having edge length 2 are all permutations of:
( ± 1 , ± 1 , ± ( 1 + 2 ) , ± ( 1 + 2 2 ) , ± ( 1 + 2 2 ) ) {\displaystyle \left(\pm 1,\ \pm 1,\ \pm (1+{\sqrt {2}}),\ \pm (1+2{\sqrt {2}}),\ \pm (1+2{\sqrt {2}})\right)} Images
Runcicantitruncated 5-cube Type Uniform 5-polytope Schläfli symbol t0,1,2,3 {4,3,3,3} Coxeter-Dynkin 4-faces 202 Cells 1560 Faces 4240 Edges 4800 Vertices 1920 Vertex figure Irregular 5-cell Coxeter group B5 [4,3,3,3] Properties convex , isogonal
Alternate names Runcicantitruncated penteract Biruncicantitruncated pentacross great prismated penteract (gippin) (Jonathan Bowers) Coordinates The Cartesian coordinates of the vertices of a runcicantitruncated 5-cube having an edge length of 2 are given by all permutations of coordinates and sign of:
( 1 , 1 + 2 , 1 + 2 2 , 1 + 3 2 , 1 + 3 2 ) {\displaystyle \left(1,\ 1+{\sqrt {2}},\ 1+2{\sqrt {2}},\ 1+3{\sqrt {2}},\ 1+3{\sqrt {2}}\right)} Images
Related polytopes These polytopes are a part of a set of 31 uniform polytera 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
References 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 , George Olshevsky. Polytopes of Various Dimensions, Jonathan Bowers Runcinated uniform polytera (spid), Jonathan Bowers Multi-dimensional Glossary
runcinated, cubes, cube, runcinated, cube, runcinated, orthoplexruncitruncated, cube, runcicantellated, cube, runcicantitruncated, cuberuncitruncated, orthoplex, runcicantellated, orthoplex, runcicantitruncated, orthoplexorthogonal, projections, coxeter, plane. 5 cube Runcinated 5 cube Runcinated 5 orthoplexRuncitruncated 5 cube Runcicantellated 5 cube Runcicantitruncated 5 cubeRuncitruncated 5 orthoplex Runcicantellated 5 orthoplex Runcicantitruncated 5 orthoplexOrthogonal projections in B5 Coxeter planeIn five dimensional geometry a runcinated 5 cube is a convex uniform 5 polytope that is a runcination a 3rd order truncation of the regular 5 cube There are 8 unique degrees of runcinations of the 5 cube along with permutations of truncations and cantellations Four are more simply constructed relative to the 5 orthoplex Contents 1 Runcinated 5 cube 1 1 Alternate names 1 2 Coordinates 1 3 Images 2 Runcitruncated 5 cube 2 1 Alternate names 2 2 Construction and coordinates 2 3 Images 3 Runcicantellated 5 cube 3 1 Alternate names 3 2 Coordinates 3 3 Images 4 Runcicantitruncated 5 cube 4 1 Alternate names 4 2 Coordinates 4 3 Images 5 Related polytopes 6 References 7 External linksRuncinated 5 cube EditRuncinated 5 cubeType Uniform 5 polytopeSchlafli symbol t0 3 4 3 3 3 Coxeter diagram 4 faces 202 10 80 80 32 Cells 1240 40 240 320 160 320 160 Faces 2160 240 960 640 320 Edges 1440 480 960Vertices 320Vertex figure Coxeter group B5 4 3 3 3 Properties convexAlternate names Edit Small prismated penteract Acronym span Jonathan Bowers Coordinates Edit The Cartesian coordinates of the vertices of a runcinated 5 cube having edge length 2 are all permutations of 1 1 1 1 2 1 2 displaystyle left pm 1 pm 1 pm 1 pm 1 sqrt 2 pm 1 sqrt 2 right Images Edit orthographic projections Coxeter plane B5 B4 D5 B3 D4 A2Graph Dihedral symmetry 10 8 6 Coxeter plane B2 A3Graph Dihedral symmetry 4 4 Runcitruncated 5 cube EditRuncitruncated 5 cubeType Uniform 5 polytopeSchlafli symbol t0 1 3 4 3 3 3 Coxeter Dynkin diagrams 4 faces 202 10 80 80 32 Cells 1560 40 240 320 320 160 320 160 Faces 3760 240 960 320 960 640 640 Edges 3360 480 960 1920Vertices 960Vertex figure Coxeter group B5 3 3 3 4 Properties convexAlternate names Edit Runcitruncated penteract Prismatotruncated penteract Acronym pattin Jonathan Bowers Construction and coordinates Edit The Cartesian coordinates of the vertices of a runcitruncated 5 cube having edge length 2 are all permutations of 1 1 2 1 2 1 2 2 1 2 2 displaystyle left pm 1 pm 1 sqrt 2 pm 1 sqrt 2 pm 1 2 sqrt 2 pm 1 2 sqrt 2 right Images Edit orthographic projections Coxeter plane B5 B4 D5 B3 D4 A2Graph Dihedral symmetry 10 8 6 Coxeter plane B2 A3Graph Dihedral symmetry 4 4 Runcicantellated 5 cube EditRuncicantellated 5 cubeType Uniform 5 polytopeSchlafli symbol t0 2 3 4 3 3 3 Coxeter Dynkin diagram 4 faces 202 10 80 80 32 Cells 1240 40 240 320 320 160 160 Faces 2960 240 480 960 320 640 320 Edges 2880 960 960 960Vertices 960Vertex figure Coxeter group B5 4 3 3 3 Properties convexAlternate names Edit Runcicantellated penteract Prismatorhombated penteract Acronym prin Jonathan Bowers Coordinates Edit The Cartesian coordinates of the vertices of a runcicantellated 5 cube having edge length 2 are all permutations of 1 1 1 2 1 2 2 1 2 2 displaystyle left pm 1 pm 1 pm 1 sqrt 2 pm 1 2 sqrt 2 pm 1 2 sqrt 2 right Images Edit orthographic projections Coxeter plane B5 B4 D5 B3 D4 A2Graph Dihedral symmetry 10 8 6 Coxeter plane B2 A3Graph Dihedral symmetry 4 4 Runcicantitruncated 5 cube EditRuncicantitruncated 5 cubeType Uniform 5 polytopeSchlafli symbol t0 1 2 3 4 3 3 3 Coxeter Dynkindiagram 4 faces 202Cells 1560Faces 4240Edges 4800Vertices 1920Vertex figure Irregular 5 cellCoxeter group B5 4 3 3 3 Properties convex isogonalAlternate names Edit Runcicantitruncated penteract Biruncicantitruncated pentacross great prismated penteract gippin Jonathan Bowers Coordinates Edit The Cartesian coordinates of the vertices of a runcicantitruncated 5 cube having an edge length of 2 are given by all permutations of coordinates and sign of 1 1 2 1 2 2 1 3 2 1 3 2 displaystyle left 1 1 sqrt 2 1 2 sqrt 2 1 3 sqrt 2 1 3 sqrt 2 right Images Edit orthographic projections Coxeter plane B5 B4 D5 B3 D4 A2Graph Dihedral symmetry 10 8 6 Coxeter plane B2 A3Graph Dihedral symmetry 4 4 Related polytopes EditThese polytopes are a part of a set of 31 uniform polytera generated from the regular 5 cube or 5 orthoplex B5 polytopes b5 t1b5 t2g5 t1g5 g5 t0 1b5 t0 2b5 t1 2b5 t0 3b5 t1 3g5 t1 2g5 t0 4g5 t0 3g5 t0 2g5 t0 1g5 t0 1 2b5 t0 1 3b5 t0 2 3b5 t1 2 3g5 t0 1 4b5 t0 2 4g5 t0 2 3g5 t0 1 4g5 t0 1 3g5 t0 1 2g5 t0 1 2 3b5 t0 1 2 4b5 t0 1 3 4g5 t0 1 2 4g5 t0 1 2 3g5 t0 1 2 3 4g5References 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 o3x3o3o4x span o3x3o3x4x pattin o3x3x3o4x prin o3x3x3x4x gippinExternal 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 cubes amp oldid 1133802466 Runcicantitruncated 5 cube, wikipedia, wiki , book, books, library,
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