Rectified tesseract Schlegel diagram Type Uniform 4-polytope Schläfli symbol r{4,3,3} = { 4 3 , 3 } {\displaystyle \left\{{\begin{array}{l}4\\3,3\end{array}}\right\}} 1,1 }3 {4,3,3} Coxeter-Dynkin diagrams Cells 24 8 (3.4.3.4 ) (3.3.3 ) Faces 88 64 {3} {4} Edges 96 Vertices 32 Vertex figure Symmetry group B4 [3,3,4], order 3844 [31,1,1 ], order 192 Properties convex , edge-transitive Uniform index 10 12
In geometry , the rectified tesseract , rectified 8-cell is a uniform 4-polytope (4-dimensional polytope ) bounded by 24 cells : 8 cuboctahedra , and 16 tetrahedra . It has half the vertices of a runcinated tesseract , with its runcic tesseract .
Net It has two uniform constructions, as a rectified 8-cell r{4,3,3} and a cantellated demitesseract , rr{3,31,1 }, the second alternating with two types of tetrahedral cells.
E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as tC8 .
Construction edit The rectified tesseract may be constructed from the tesseract by truncating its vertices at the midpoints of its edges.
The Cartesian coordinates of the vertices of the rectified tesseract with edge length 2 is given by all permutations of:
( 0 , ± 2 , ± 2 , ± 2 ) {\displaystyle (0,\ \pm {\sqrt {2}},\ \pm {\sqrt {2}},\ \pm {\sqrt {2}})} Images edit Projections edit In the cuboctahedron-first parallel projection of the rectified tesseract into 3-dimensional space, the image has the following layout:
The projection envelope is a cube . A cuboctahedron is inscribed in this cube, with its vertices lying at the midpoint of the cube's edges. The cuboctahedron is the image of two of the cuboctahedral cells. The remaining 6 cuboctahedral cells are projected to the square faces of the cube. The 8 tetrahedral volumes lying at the triangular faces of the central cuboctahedron are the images of the 16 tetrahedral cells, two cells to each image. Alternative names edit Rit (Jonathan Bowers: for rectified tesseract) Ambotesseract (Neil Sloane & John Horton Conway ) Rectified tesseract/Runcic tesseract (Norman W. Johnson) Runcic 4-hypercube/8-cell/octachoron/4-measure polytope/4-regular orthotope Rectified 4-hypercube/8-cell/octachoron/4-measure polytope/4-regular orthotope Related uniform polytopes edit Runcic cubic polytopes edit Runcic n -cubes n 4 5 6 7 8 [1+ ,4,3n-2 ]n-3,1 ] [1+ ,4,32 ]1,1 ] [1+ ,4,33 ]2,1 ] [1+ ,4,34 ]3,1 ] [1+ ,4,35 ]4,1 ] [1+ ,4,36 ]5,1 ] Runcic Coxeter Schläfli h3 {4,32 } h3 {4,33 } h3 {4,34 } h3 {4,35 } h3 {4,36 }
Tesseract polytopes edit 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
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. (1966) , George Olshevsky. Klitzing, Richard. "4D uniform polytopes (polychora) o4x3o3o - rit".
rectified, tesseract, schlegel, diagramcentered, cuboctahedrontetrahedral, cells, showntype, uniform, polytopeschläfli, symbol, displaystyle, left, begin, array, array, right, coxeter, dynkin, diagrams, cells, faces, edges, 96vertices, 32vertex, figure, elonga. Rectified tesseractSchlegel diagramCentered on cuboctahedrontetrahedral cells shownType Uniform 4 polytopeSchlafli symbol r 4 3 3 43 3 displaystyle left begin array l 4 3 3 end array right 2r 3 31 1 h3 4 3 3 Coxeter Dynkin diagrams Cells 24 8 3 4 3 4 16 3 3 3 Faces 88 64 3 24 4 Edges 96Vertices 32Vertex figure Elongated equilateral triangular prism Symmetry group B4 3 3 4 order 384D4 31 1 1 order 192Properties convex edge transitiveUniform index 10 11 12In geometry the rectified tesseract rectified 8 cell is a uniform 4 polytope 4 dimensional polytope bounded by 24 cells 8 cuboctahedra and 16 tetrahedra It has half the vertices of a runcinated tesseract with its construction called a runcic tesseract NetIt has two uniform constructions as a rectified 8 cell r 4 3 3 and a cantellated demitesseract rr 3 31 1 the second alternating with two types of tetrahedral cells E L Elte identified it in 1912 as a semiregular polytope labeling it as tC8 Contents 1 Construction 2 Images 3 Projections 4 Alternative names 5 Related uniform polytopes 5 1 Runcic cubic polytopes 5 2 Tesseract polytopes 6 ReferencesConstruction editThe rectified tesseract may be constructed from the tesseract by truncating its vertices at the midpoints of its edges The Cartesian coordinates of the vertices of the rectified tesseract with edge length 2 is given by all permutations of 0 2 2 2 displaystyle 0 pm sqrt 2 pm sqrt 2 pm sqrt 2 nbsp Images editorthographic projections Coxeter plane B4 B3 D4 A2 B2 D3Graph nbsp nbsp nbsp Dihedral symmetry 8 6 4 Coxeter plane F4 A3Graph nbsp nbsp Dihedral symmetry 12 3 4 nbsp Wireframe nbsp 16 tetrahedral cellsProjections editIn the cuboctahedron first parallel projection of the rectified tesseract into 3 dimensional space the image has the following layout The projection envelope is a cube A cuboctahedron is inscribed in this cube with its vertices lying at the midpoint of the cube s edges The cuboctahedron is the image of two of the cuboctahedral cells The remaining 6 cuboctahedral cells are projected to the square faces of the cube The 8 tetrahedral volumes lying at the triangular faces of the central cuboctahedron are the images of the 16 tetrahedral cells two cells to each image Alternative names editRit Jonathan Bowers for rectified tesseract Ambotesseract Neil Sloane amp John Horton Conway Rectified tesseract Runcic tesseract Norman W Johnson Runcic 4 hypercube 8 cell octachoron 4 measure polytope 4 regular orthotope Rectified 4 hypercube 8 cell octachoron 4 measure polytope 4 regular orthotopeRelated uniform polytopes editRuncic cubic polytopes edit Runcic n cubesn 4 5 6 7 8 1 4 3n 2 3 3n 3 1 1 4 32 3 31 1 1 4 33 3 32 1 1 4 34 3 33 1 1 4 35 3 34 1 1 4 36 3 35 1 Runcicfigure nbsp nbsp nbsp nbsp nbsp Coxeter nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp Schlafli h3 4 32 h3 4 33 h3 4 34 h3 4 35 h3 4 36 Tesseract polytopes edit B4 symmetry polytopesName tesseract rectifiedtesseract truncatedtesseract cantellatedtesseract runcinatedtesseract bitruncatedtesseract cantitruncatedtesseract runcitruncatedtesseract omnitruncatedtesseractCoxeterdiagram nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp 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 cellCoxeterdiagram nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp 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 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 1966 2 Convex uniform polychora based on the tesseract 8 cell and hexadecachoron 16 cell Model 11 George Olshevsky Klitzing Richard 4D uniform polytopes polychora o4x3o3o rit 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 Rectified tesseract amp oldid 943517959, 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.