Rectified 24-cell Schlegel diagram Type Uniform 4-polytope Schläfli symbols r{3,4,3} = { 3 4 , 3 } {\displaystyle \left\{{\begin{array}{l}3\\4,3\end{array}}\right\}} r { 3 3 , 4 } {\displaystyle r\left\{{\begin{array}{l}3\\3,4\end{array}}\right\}} 1,1,1 } = r { 3 3 3 } {\displaystyle r\left\{{\begin{array}{l}3\\3\\3\end{array}}\right\}} Coxeter diagrams Cells 48 24 3.4.3.4 4.4.4 Faces 240 96 {3} {4} Edges 288 Vertices 96 Vertex figure Triangular prism Symmetry groups F4 [3,4,3], order 11524 [3,3,4], order 3844 [31,1,1 ], order 192 Properties convex , edge-transitive Uniform index 22 24
In geometry , the rectified 24-cell or rectified icositetrachoron is a uniform 4-dimensional polytope (or uniform 4-polytope ), which is bounded by 48 cells : 24 cubes , and 24 cuboctahedra . It can be obtained by rectification of the 24-cell, reducing its octahedral cells to cubes and cuboctahedra.
Net E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as tC24 .
It can also be considered a cantellated 16-cell with the lower symmetries B4 = [3,3,4]. B4 would lead to a bicoloring of the cuboctahedral cells into 8 and 16 each. It is also called a runcicantellated demitesseract in a D4 symmetry, giving 3 colors of cells, 8 for each.
Construction edit The rectified 24-cell can be derived from the 24-cell by the process of rectification : the 24-cell is truncated at the midpoints. The vertices become cubes , while the octahedra become cuboctahedra .
Cartesian coordinates edit A rectified 24-cell having an edge length of √2 has vertices given by all permutations and sign permutations of the following Cartesian coordinates :
(0,1,1,2) [4!/2!×23 = 96 vertices] The dual configuration with edge length 2 has all coordinate and sign permutations of:
(0,2,2,2) [4×23 = 32 vertices] (1,1,1,3) [4×24 = 64 vertices] Images edit Symmetry constructions edit There are three different symmetry constructions of this polytope. The lowest D 4 {\displaystyle {D}_{4}} C 4 {\displaystyle {C}_{4}} D 4 {\displaystyle {D}_{4}} F 4 {\displaystyle {F}_{4}}
The vertex figure is a triangular prism , containing two cubes and three cuboctahedra. The three symmetries can be seen with 3 colored cuboctahedra in the lowest D 4 {\displaystyle {D}_{4}} C 4 {\displaystyle {C}_{4}} F 4 {\displaystyle {F}_{4}}
Coxeter group F 4 {\displaystyle {F}_{4}} C 4 {\displaystyle {C}_{4}} D 4 {\displaystyle {D}_{4}} 1,1 ] Order 1152 384 192 Full [3,4,3] [4,3,3] <[3,31,1 ]> = [4,3,3]1,1,1 ]] = [3,4,3] Coxeter diagram Facets 3: 2: 2,2: 2: 1,1,1: 2: Vertex figure
Alternate names edit Rectified 24-cell, Cantellated 16-cell (Norman Johnson ) Rectified icositetrachoron (Acronym rico) (George Olshevsky, Jonathan Bowers) Cantellated hexadecachoron Disicositetrachoron Amboicositetrachoron (Neil Sloane & John Horton Conway ) Related polytopes edit The convex hull of the rectified 24-cell and its dual (assuming that they are congruent) is a nonuniform polychoron composed of 192 cells: 48 cubes , 144 square antiprisms , and 192 vertices. Its vertex figure is a triangular bifrustum .
Related uniform polytopes edit D4 uniform polychora {3,31,1 } 2r{3,31,1 } 3 {4,3,3} t{3,31,1 } 2 {4,3,3} 2t{3,31,1 } 2,3 {4,3,3} r{3,31,1 } 1,1,1 }={3,4,3} rr{3,31,1 } 1,1,1 }=r{3,4,3} tr{3,31,1 } 1,1,1 }=t{3,4,3} sr{3,31,1 } 1,1,1 }=s{3,4,3}
24-cell family polytopes Name 24-cell truncated 24-cell snub 24-cell rectified 24-cell cantellated 24-cell bitruncated 24-cell cantitruncated 24-cell runcinated 24-cell runcitruncated 24-cell omnitruncated 24-cell Schläfli {3,4,3} t0,1 s{3,4,3} t1 t0,2 t1,2 t0,1,2 t0,3 t0,1,3 t0,1,2,3 Coxeter Schlegel F4 B4 B3 (a) B3 (b) B2
The rectified 24-cell can also be derived as a cantellated 16-cell :
B4 symmetry polytopes Name tesseract rectified truncated cantellated runcinated bitruncated cantitruncated runcitruncated omnitruncated Coxeter Schläfli {4,3,3} t1 t0,1 t0,2 t0,3 t1,2 t0,1,2 t0,1,3 t0,1,2,3 Schlegel B4 Name 16-cell rectified truncated cantellated runcinated bitruncated cantitruncated runcitruncated omnitruncated Coxeter Schläfli {3,3,4} t1 t0,1 t0,2 t0,3 t1,2 t0,1,2 t0,1,3 t0,1,2,3 Schlegel B4
Citations edit References edit T. Gosset : On the Regular and Semi-Regular Figures in Space of n Dimensions , Messenger of Mathematics, Macmillan, 1900Coxeter, H.S.M. (1973) [1948]. Regular Polytopes John H. Conway , Heidi Burgiel, Chaim Goodman-Strauss, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 (Chapter 26. pp. 409: Hemicubes: 1n1 )Norman Johnson Uniform Polytopes , Manuscript (1991) N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs , Ph.D. (1966) , George Olshevsky. , George Olshevsky. , George Olshevsky. Klitzing, Richard. "4D uniform polytopes (polychora) o3x4o3o - rico".
rectified, cell, schlegel, diagram8, cuboctahedral, cells, shown, type, uniform, polytope, schläfli, symbols, displaystyle, left, begin, array, array, right, displaystyle, left, begin, array, array, right, displaystyle, left, begin, array, array, right, coxete. Rectified 24 cell Schlegel diagram8 of 24 cuboctahedral cells shown Type Uniform 4 polytope Schlafli symbols r 3 4 3 3 4 3 displaystyle left begin array l 3 4 3 end array right rr 3 3 4 r 3 3 4 displaystyle r left begin array l 3 3 4 end array right r 31 1 1 r 3 3 3 displaystyle r left begin array l 3 3 3 end array right Coxeter diagrams or Cells 48 24 3 4 3 4 24 4 4 4 Faces 240 96 3 144 4 Edges 288 Vertices 96 Vertex figure Triangular prism Symmetry groups F4 3 4 3 order 1152B4 3 3 4 order 384D4 31 1 1 order 192 Properties convex edge transitive Uniform index 22 23 24 In geometry the rectified 24 cell or rectified icositetrachoron is a uniform 4 dimensional polytope or uniform 4 polytope which is bounded by 48 cells 24 cubes and 24 cuboctahedra It can be obtained by rectification of the 24 cell reducing its octahedral cells to cubes and cuboctahedra 1 Net E L Elte identified it in 1912 as a semiregular polytope labeling it as tC24 It can also be considered a cantellated 16 cell with the lower symmetries B4 3 3 4 B4 would lead to a bicoloring of the cuboctahedral cells into 8 and 16 each It is also called a runcicantellated demitesseract in a D4 symmetry giving 3 colors of cells 8 for each Contents 1 Construction 2 Cartesian coordinates 3 Images 4 Symmetry constructions 5 Alternate names 6 Related polytopes 7 Related uniform polytopes 8 Citations 9 ReferencesConstruction editThe rectified 24 cell can be derived from the 24 cell by the process of rectification the 24 cell is truncated at the midpoints The vertices become cubes while the octahedra become cuboctahedra Cartesian coordinates editA rectified 24 cell having an edge length of 2 has vertices given by all permutations and sign permutations of the following Cartesian coordinates 0 1 1 2 4 2 23 96 vertices The dual configuration with edge length 2 has all coordinate and sign permutations of 0 2 2 2 4 23 32 vertices 1 1 1 3 4 24 64 vertices Images editorthographic projections Coxeter plane F4 Graph nbsp Dihedral symmetry 12 Coxeter plane B3 A2 a B3 A2 b Graph nbsp nbsp Dihedral symmetry 6 6 Coxeter plane B4 B2 A3 Graph nbsp nbsp Dihedral symmetry 8 4 Stereographic projection nbsp Center of stereographic projectionwith 96 triangular faces blueSymmetry constructions editThere are three different symmetry constructions of this polytope The lowest D 4 displaystyle D 4 nbsp construction can be doubled into C 4 displaystyle C 4 nbsp by adding a mirror that maps the bifurcating nodes onto each other D 4 displaystyle D 4 nbsp can be mapped up to F 4 displaystyle F 4 nbsp symmetry by adding two mirror that map all three end nodes together The vertex figure is a triangular prism containing two cubes and three cuboctahedra The three symmetries can be seen with 3 colored cuboctahedra in the lowest D 4 displaystyle D 4 nbsp construction and two colors 1 2 ratio in C 4 displaystyle C 4 nbsp and all identical cuboctahedra in F 4 displaystyle F 4 nbsp Coxeter group F 4 displaystyle F 4 nbsp 3 4 3 C 4 displaystyle C 4 nbsp 4 3 3 D 4 displaystyle D 4 nbsp 3 31 1 Order 1152 384 192 Fullsymmetrygroup 3 4 3 4 3 3 lt 3 31 1 gt 4 3 3 3 31 1 1 3 4 3 Coxeter diagram nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp Facets 3 nbsp nbsp nbsp nbsp nbsp 2 nbsp nbsp nbsp nbsp nbsp 2 2 nbsp nbsp nbsp nbsp nbsp 2 nbsp nbsp nbsp nbsp nbsp 1 1 1 nbsp nbsp nbsp nbsp nbsp 2 nbsp nbsp nbsp nbsp nbsp Vertex figure nbsp nbsp nbsp Alternate names editRectified 24 cell Cantellated 16 cell Norman Johnson Rectified icositetrachoron Acronym rico George Olshevsky Jonathan Bowers Cantellated hexadecachoron Disicositetrachoron Amboicositetrachoron Neil Sloane amp John Horton Conway Related polytopes editThe convex hull of the rectified 24 cell and its dual assuming that they are congruent is a nonuniform polychoron composed of 192 cells 48 cubes 144 square antiprisms and 192 vertices Its vertex figure is a triangular bifrustum Related uniform polytopes editD4 uniform polychora 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 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 31 1 h 4 3 3 2r 3 31 1 h3 4 3 3 t 3 31 1 h2 4 3 3 2t 3 31 1 h2 3 4 3 3 r 3 31 1 31 1 1 3 4 3 rr 3 31 1 r 31 1 1 r 3 4 3 tr 3 31 1 t 31 1 1 t 3 4 3 sr 3 31 1 s 31 1 1 s 3 4 3 24 cell family polytopes Name 24 cell truncated 24 cell snub 24 cell rectified 24 cell cantellated 24 cell bitruncated 24 cell cantitruncated 24 cell runcinated 24 cell runcitruncated 24 cell omnitruncated 24 cell Schlaflisymbol 3 4 3 t0 1 3 4 3 t 3 4 3 s 3 4 3 t1 3 4 3 r 3 4 3 t0 2 3 4 3 rr 3 4 3 t1 2 3 4 3 2t 3 4 3 t0 1 2 3 4 3 tr 3 4 3 t0 3 3 4 3 t0 1 3 3 4 3 t0 1 2 3 3 4 3 Coxeterdiagram 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 nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp Schlegeldiagram nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp F4 nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp B4 nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp B3 a nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp B3 b nbsp nbsp nbsp nbsp nbsp nbsp B2 nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp The rectified 24 cell can also be derived as a cantellated 16 cell B4 symmetry polytopes Name tesseract rectifiedtesseract truncatedtesseract cantellatedtesseract runcinatedtesseract bitruncatedtesseract cantitruncatedtesseract runcitruncatedtesseract omnitruncatedtesseract Coxeterdiagram 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 nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp Schlaflisymbol 4 3 3 t1 4 3 3 r 4 3 3 t0 1 4 3 3 t 4 3 3 t0 2 4 3 3 rr 4 3 3 t0 3 4 3 3 t1 2 4 3 3 2t 4 3 3 t0 1 2 4 3 3 tr 4 3 3 t0 1 3 4 3 3 t0 1 2 3 4 3 3 Schlegeldiagram nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp B4 nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp Name 16 cell rectified16 cell truncated16 cell cantellated16 cell runcinated16 cell bitruncated16 cell cantitruncated16 cell runcitruncated16 cell omnitruncated16 cell Coxeterdiagram 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 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 Schlaflisymbol 3 3 4 t1 3 3 4 r 3 3 4 t0 1 3 3 4 t 3 3 4 t0 2 3 3 4 rr 3 3 4 t0 3 3 3 4 t1 2 3 3 4 2t 3 3 4 t0 1 2 3 3 4 tr 3 3 4 t0 1 3 3 3 4 t0 1 2 3 3 3 4 Schlegeldiagram nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp B4 nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp Citations edit Coxeter 1973 p 154 8 4 References editT Gosset On the Regular and Semi Regular Figures in Space of n Dimensions Messenger of Mathematics Macmillan 1900 Coxeter H S M 1973 1948 Regular Polytopes 3rd ed New York Dover John H Conway Heidi Burgiel Chaim Goodman Strauss The Symmetries of Things 2008 ISBN 978 1 56881 220 5 Chapter 26 pp 409 Hemicubes 1n1 Norman Johnson Uniform Polytopes Manuscript 1991 N W Johnson The Theory of Uniform Polytopes and Honeycombs Ph D 1966 2 Convex uniform polychora based on the tesseract 8 cell and hexadecachoron 16 cell Model 23 George Olshevsky 3 Convex uniform polychora based on the icositetrachoron 24 cell Model 23 George Olshevsky 7 Uniform polychora derived from glomeric tetrahedron B4 Model 23 George Olshevsky Klitzing Richard 4D uniform polytopes polychora o3x4o3o rico vteFundamental convex regular and uniform polytopes in dimensions 2 10 Family An Bn I2 p Dn E6 E7 E8 F4 G2 Hn Regular polygon Triangle Square p gon Hexagon Pentagon Uniform polyhedron Tetrahedron Octahedron Cube Demicube Dodecahedron Icosahedron Uniform polychoron Pentachoron 16 cell Tesseract Demitesseract 24 cell 120 cell 600 cell Uniform 5 polytope 5 simplex 5 orthoplex 5 cube 5 demicube Uniform 6 polytope 6 simplex 6 orthoplex 6 cube 6 demicube 122 221 Uniform 7 polytope 7 simplex 7 orthoplex 7 cube 7 demicube 132 231 321 Uniform 8 polytope 8 simplex 8 orthoplex 8 cube 8 demicube 142 241 421 Uniform 9 polytope 9 simplex 9 orthoplex 9 cube 9 demicube Uniform 10 polytope 10 simplex 10 orthoplex 10 cube 10 demicube Uniform n polytope n simplex n orthoplex n cube n demicube 1k2 2k1 k21 n pentagonal polytope Topics Polytope families Regular polytope List of regular polytopes and compounds Retrieved from https en wikipedia org w index php title Rectified 24 cell amp oldid 1189602161, wikipedia, wiki , book, books, library,
article , read, download, free, free download, mp3, video, mp4, 3gp, jpg, jpeg, gif, png, picture, music, song, movie, book, game, games.