In four-dimensional geometry , a runcinated 120-cell (or runcinated 600-cell ) is a convex uniform 4-polytope , being a runcination (a 3rd order truncation) of the regular 120-cell .
There are 4 degrees of runcinations of the 120-cell including with permutations truncations and cantellations.
The runcinated 120-cell can be seen as an expansion applied to a regular 4-polytope, the 120-cell or 600-cell.
Runcinated 120-cell edit Net The runcinated 120-cell or small disprismatohexacosihecatonicosachoron is a uniform 4-polytope . It has 2640 cells: 120 dodecahedra , 720 pentagonal prisms , 1200 triangular prisms , and 600 tetrahedra . Its vertex figure is a nonuniform triangular antiprism (equilateral-triangular antipodium): its bases represent a dodecahedron and a tetrahedron, and its flanks represent three triangular prisms and three pentagonal prisms.
Alternate names edit Runcinated 120-cell / Runcinated 600-cell (Norman W. Johnson ) Runcinated hecatonicosachoron / Runcinated dodecacontachoron / Runcinated hexacosichoron / Runcinated polydodecahedron / Runcinated polytetrahedron Small diprismatohexacosihecatonicosachoron (acronym: sidpixhi) (George Olshevsky, Jonathan Bowers)[1] Images edit Polyhedral rings
Runcitruncated 120-cell edit Runcitruncated 600-cell edit Omnitruncated 120-cell edit Omnitruncated 120-cell Type Uniform 4-polytope Uniform index 46 Coxeter diagram Cells 2640 total:4.6.10 4.4.10 4.4.6 4.6.6 Faces 17040 total:{4} , 4800 {6} {10} Edges 28800 Vertices 14400 Vertex figure Schläfli symbol t0,1,2,3 {3,3,5} Symmetry group H4 , [3,3,5], order 14400 Properties convex
The omnitruncated 120-cell or great disprismatohexacosihecatonicosachoron is a convex uniform 4-polytope , composed of 2640 cells : 120 truncated icosidodecahedra , 600 truncated octahedra , 720 decagonal prisms , and 1200 hexagonal prisms . It has 14400 vertices, 28800 edges, and 17040 faces (10800 squares, 4800 hexagons, and 1440 decagons). It is the largest nonprismatic convex uniform 4-polytope .
The vertices and edges form the Cayley graph of the Coxeter group H4 .
Alternate names edit Omnitruncated 120-cell / Omnitruncated 600-cell (Norman W. Johnson ) Omnitruncated hecatonicosachoron / Omnitruncated hexacosichoron / Omnitruncated polydodecahedron / Omnitruncated polytetrahedron Great diprismatohexacosihecatonicosachoron (Acronym gidpixhi) (George Olshevsky, Jonathan Bowers)[4] Images edit Polyhedral rings
Net
Models edit The first complete physical model of a 3D projection of the omnitruncated 120-cell was built by a team led by Daniel Duddy and David Richter on August 9, 2006 using the Zome system in the London Knowledge Lab for the 2006 Bridges Conference.[5]
Full snub 120-cell edit Vertex figure for the omnisnub 120-cell The full snub 120-cell or omnisnub 120-cell , defined as an alternation of the omnitruncated 120-cell, can not be made uniform, but it can be given Coxeter diagram symmetry [5,3,3]+ , and constructed from 1200 octahedrons , 600 icosahedrons , 720 pentagonal antiprisms , 120 snub dodecahedrons , and 7200 tetrahedrons filling the gaps at the deleted vertices. It has 9840 cells, 35040 faces, 32400 edges, and 7200 vertices.[6]
Related polytopes edit These polytopes are a part of a set of 15 uniform 4-polytopes with H4 symmetry:
H4 family polytopes 120-cell rectified truncated cantellated runcinated cantitruncated runcitruncated omnitruncated {5,3,3} r{5,3,3} t{5,3,3} rr{5,3,3} t0,3 tr{5,3,3} t0,1,3 t0,1,2,3 600-cell rectified truncated cantellated bitruncated cantitruncated runcitruncated omnitruncated {3,3,5} r{3,3,5} t{3,3,5} rr{3,3,5} 2t{3,3,5} tr{3,3,5} t0,1,3 t0,1,2,3
Notes edit ^ Klitizing, (x3o3o5x - sidpixhi) ^ Klitizing, (x3o3x5x - prix) ^ Klitizing, (x3x3o5x - prahi) ^ Klitizing, (x3x3x5x - gidpixhi) ^ Photos of Zome model of omnitruncated 120/600-cell ^ "S3s3s5s". References edit 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 (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] J.H. Conway and M.J.T. Guy : Four-Dimensional Archimedean Polytopes , Proceedings of the Colloquium on Convexity at Copenhagen, page 38 und 39, 1965N.W. Johnson : The Theory of Uniform Polytopes and Honeycombs , Ph.D. Dissertation, University of Toronto, 1966Four-dimensional Archimedean Polytopes (German), Marco Möller, 2004 PhD dissertation [1] m55 m62 m60 m64 , George Olshevsky. Klitzing, Richard. "4D uniform polytopes (polychora)". External links edit H4 uniform polytopes with coordinates: t03{5,3,3} t013{3,3,5} t013{5,3,3} t0123{5,3,3}
runcinated, cells, four, runcinations, cell, runcinated, cell, expanded, cell, runcitruncated, cell600, cell, runcitruncated, cell, omnitruncated, cellorthogonal, projections, coxeter, planein, four, dimensional, geometry, runcinated, cell, runcinated, cell, c. Four runcinations 120 cell Runcinated 120 cell Expanded 120 cell Runcitruncated 120 cell600 cell Runcitruncated 600 cell Omnitruncated 120 cellOrthogonal projections in H3 Coxeter planeIn four dimensional geometry a runcinated 120 cell or runcinated 600 cell is a convex uniform 4 polytope being a runcination a 3rd order truncation of the regular 120 cell There are 4 degrees of runcinations of the 120 cell including with permutations truncations and cantellations The runcinated 120 cell can be seen as an expansion applied to a regular 4 polytope the 120 cell or 600 cell Contents 1 Runcinated 120 cell 1 1 Alternate names 1 2 Images 2 Runcitruncated 120 cell 2 1 Alternate names 2 2 Images 3 Runcitruncated 600 cell 3 1 Alternate names 3 2 Images 4 Omnitruncated 120 cell 4 1 Alternate names 4 2 Images 4 3 Models 4 4 Full snub 120 cell 5 Related polytopes 6 Notes 7 References 8 External linksRuncinated 120 cell editRuncinated 120 cellType Uniform 4 polytopeUniform index 38Coxeter diagram nbsp nbsp nbsp nbsp nbsp nbsp nbsp Cells 2640 total 120 5 5 5 nbsp 720 4 4 5 nbsp 1200 4 4 3 nbsp 600 3 3 3 nbsp Faces 7440 2400 3 3600 4 1440 5 Edges 7200Vertices 2400Vertex figure nbsp Equilateral triangular antipodiumSchlafli symbol t0 3 5 3 3 Symmetry group H4 3 3 5 order 14400Properties convex nbsp NetThe runcinated 120 cell or small disprismatohexacosihecatonicosachoron is a uniform 4 polytope It has 2640 cells 120 dodecahedra 720 pentagonal prisms 1200 triangular prisms and 600 tetrahedra Its vertex figure is a nonuniform triangular antiprism equilateral triangular antipodium its bases represent a dodecahedron and a tetrahedron and its flanks represent three triangular prisms and three pentagonal prisms Alternate names edit Runcinated 120 cell Runcinated 600 cell Norman W Johnson Runcinated hecatonicosachoron Runcinated dodecacontachoron Runcinated hexacosichoron Runcinated polydodecahedron Runcinated polytetrahedron Small diprismatohexacosihecatonicosachoron acronym sidpixhi George Olshevsky Jonathan Bowers 1 Images edit Schlegel diagram Only tetrahedral cells shown nbsp Polyhedral rings nbsp Cells on 5 fold axis nbsp Cells on 3 fold axis nbsp Cells on 2 fold axisOrthogonal projections in Coxeter planes nbsp H3 nbsp A2 B3 nbsp A3 B2Runcitruncated 120 cell editRuncitruncated 120 cellType Uniform 4 polytopeUniform index 43Coxeter diagram nbsp nbsp nbsp nbsp nbsp nbsp nbsp Cells 2640 total 120 3 10 10 nbsp 720 4 4 10 nbsp 1200 3 4 4 nbsp 600 3 4 3 4 nbsp Faces 13440 4800 3 7200 4 1440 10 Edges 18000Vertices 7200Vertex figure nbsp Irregular rectangular pyramidSchlafli symbol t0 1 3 5 3 3 Symmetry group H4 3 3 5 order 14400Properties convex nbsp NetThe runcitruncated 120 cell or prismatorhombated hexacosichoron is a uniform 4 polytope It contains 2640 cells 120 truncated dodecahedra 720 decagonal prisms 1200 triangular prisms and 600 cuboctahedra Its vertex figure is an irregular rectangular pyramid with one truncated dodecahedron two decagonal prisms one triangular prism and one cuboctahedron Alternate names edit Runcicantellated 600 cell Norman W Johnson Prismatorhombated hexacosichoron Acronym prix George Olshevsky Jonathan Bowers 2 Images edit Schlegel diagram Only triangular prisms shown nbsp Orthogonal projections in Coxeter planes nbsp H3 nbsp A2 B3 nbsp A3 B2Runcitruncated 600 cell editRuncitruncated 600 cellType Uniform 4 polytopeUniform index 44Coxeter diagram nbsp nbsp nbsp nbsp nbsp nbsp nbsp Cells 2640 total 120 3 4 5 4 nbsp 720 4 4 5 nbsp 1200 4 4 6 nbsp 600 3 6 6 nbsp Faces 13440 2400 3 7200 4 1440 5 2400 6 Edges 18000Vertices 7200Vertex figure nbsp Trapezoidal pyramidSchlafli symbol t0 1 3 3 3 5 Symmetry group H4 3 3 5 order 14400Properties convex nbsp NetThe runcitruncated 600 cell or prismatorhombated hecatonicosachoron is a uniform 4 polytope It is composed of 2640 cells 120 rhombicosidodecahedron 600 truncated tetrahedra 720 pentagonal prisms and 1200 hexagonal prisms It has 7200 vertices 18000 edges and 13440 faces 2400 triangles 7200 squares and 2400 hexagons Alternate names edit Runcicantellated 120 cell Norman W Johnson Prismatorhombated hecatonicosachoron Acronym prahi George Olshevsky Jonathan Bowers 3 Images edit Schlegel diagram nbsp Orthogonal projections in Coxeter planes nbsp H3 nbsp A2 B3 nbsp A3 B2Omnitruncated 120 cell editOmnitruncated 120 cellType Uniform 4 polytopeUniform index 46Coxeter diagram nbsp nbsp nbsp nbsp nbsp nbsp nbsp Cells 2640 total 120 4 6 10 nbsp 720 4 4 10 nbsp 1200 4 4 6 nbsp 600 4 6 6 nbsp Faces 17040 total 10800 4 4800 6 1440 10 Edges 28800Vertices 14400Vertex figure nbsp Chiral scalene tetrahedronSchlafli symbol t0 1 2 3 3 3 5 Symmetry group H4 3 3 5 order 14400Properties convexThe omnitruncated 120 cell or great disprismatohexacosihecatonicosachoron is a convex uniform 4 polytope composed of 2640 cells 120 truncated icosidodecahedra 600 truncated octahedra 720 decagonal prisms and 1200 hexagonal prisms It has 14400 vertices 28800 edges and 17040 faces 10800 squares 4800 hexagons and 1440 decagons It is the largest nonprismatic convex uniform 4 polytope The vertices and edges form the Cayley graph of the Coxeter group H4 Alternate names edit Omnitruncated 120 cell Omnitruncated 600 cell Norman W Johnson Omnitruncated hecatonicosachoron Omnitruncated hexacosichoron Omnitruncated polydodecahedron Omnitruncated polytetrahedron Great diprismatohexacosihecatonicosachoron Acronym gidpixhi George Olshevsky Jonathan Bowers 4 Images edit nbsp nbsp Schlegel diagram centered on truncated icosidodecahedron Orthogonal view centered on decagonal prism cell Stereographic projection centered on truncated icosidodecahedron Orthogonal projections in Coxeter planes nbsp H3 nbsp A2 B3 nbsp A3 B2Polyhedral rings nbsp Cells on 5 fold axis nbsp Cells on 3 fold axis nbsp Cells on 2 fold axisNet nbsp Omnitruncated 120 cell nbsp Dual to omnitruncated 120 cellModels edit The first complete physical model of a 3D projection of the omnitruncated 120 cell was built by a team led by Daniel Duddy and David Richter on August 9 2006 using the Zome system in the London Knowledge Lab for the 2006 Bridges Conference 5 Full snub 120 cell edit nbsp Vertex figure for the omnisnub 120 cellThe full snub 120 cell or omnisnub 120 cell defined as an alternation of the omnitruncated 120 cell can not be made uniform but it can be given Coxeter diagram nbsp nbsp nbsp nbsp nbsp nbsp nbsp and symmetry 5 3 3 and constructed from 1200 octahedrons 600 icosahedrons 720 pentagonal antiprisms 120 snub dodecahedrons and 7200 tetrahedrons filling the gaps at the deleted vertices It has 9840 cells 35040 faces 32400 edges and 7200 vertices 6 Related polytopes editThese polytopes are a part of a set of 15 uniform 4 polytopes with H4 symmetry H4 family polytopes120 cell rectified120 cell truncated120 cell cantellated120 cell runcinated120 cell cantitruncated120 cell runcitruncated120 cell omnitruncated120 cell nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp 5 3 3 r 5 3 3 t 5 3 3 rr 5 3 3 t0 3 5 3 3 tr 5 3 3 t0 1 3 5 3 3 t0 1 2 3 5 3 3 nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp 600 cell rectified600 cell truncated600 cell cantellated600 cell bitruncated600 cell cantitruncated600 cell runcitruncated600 cell omnitruncated600 cell nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp 3 3 5 r 3 3 5 t 3 3 5 rr 3 3 5 2t 3 3 5 tr 3 3 5 t0 1 3 3 3 5 t0 1 2 3 3 3 5 Notes edit Klitizing x3o3o5x sidpixhi Klitizing x3o3x5x prix Klitizing x3x3o5x prahi Klitizing x3x3x5x gidpixhi Photos of Zome model of omnitruncated 120 600 cell S3s3s5s References editKaleidoscopes 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 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 J H Conway and M J T Guy Four Dimensional Archimedean Polytopes Proceedings of the Colloquium on Convexity at Copenhagen page 38 und 39 1965 N W Johnson The Theory of Uniform Polytopes and Honeycombs Ph D Dissertation University of Toronto 1966 Four dimensional Archimedean Polytopes German Marco Moller 2004 PhD dissertation 1 m55 m62 m60 m64 Convex uniform polychora based on the hecatonicosachoron 120 cell and hexacosichoron 600 cell Model 38 44 47 George Olshevsky Klitzing Richard 4D uniform polytopes polychora x3o3o5x sidpixhi x3o3x5x prix x3x3o5x prahi x3x3x5x gidpixhiExternal links editH4 uniform polytopes with coordinates t03 5 3 3 t013 3 3 5 t013 5 3 3 t0123 5 3 3 vteFundamental 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 120 cells amp oldid 1161798938 Runcinated 120 cell, wikipedia, wiki , book, books, library,
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