The cantellated 24-cell or small rhombated icositetrachoron is a uniform 4-polytope.
The boundary of the cantellated 24-cell is composed of 24 truncated octahedral cells, 24 cuboctahedral cells and 96 triangular prisms. Together they have 288 triangular faces, 432 square faces, 864 edges, and 288 vertices.
Constructionedit
When the cantellation process is applied to 24-cell, each of the 24 octahedra becomes a small rhombicuboctahedron. In addition however, since each octahedra's edge was previously shared with two other octahedra, the separating edges form the three parallel edges of a triangular prism - 96 triangular prisms, since the 24-cell contains 96 edges. Further, since each vertex was previously shared with 12 faces, the vertex would split into 12 (24*12=288) new vertices. Each group of 12 new vertices forms a cuboctahedron.
Coordinatesedit
The Cartesian coordinates of the vertices of the cantellated 24-cell having edge length 2 are all permutations of coordinates and sign of:
(0, √2, √2, 2+2√2)
(1, 1+√2, 1+√2, 1+2√2)
The permutations of the second set of coordinates coincide with the vertices of an inscribed runcitruncated tesseract.
The dual configuration has all permutations and signs of:
(0,2,2+√2,2+√2)
(1,1,1+√2,3+√2)
Structureedit
The 24 small rhombicuboctahedra are joined to each other via their triangular faces, to the cuboctahedra via their axial square faces, and to the triangular prisms via their off-axial square faces. The cuboctahedra are joined to the triangular prisms via their triangular faces. Each triangular prism is joined to two cuboctahedra at its two ends.
Cantic snub 24-celledit
A half-symmetry construction of the cantellated 24-cell, also called a cantic snub 24-cell, as , has an identical geometry, but its triangular faces are further subdivided. The cantellated 24-cell has 2 positions of triangular faces in ratio of 96 and 192, while the cantic snub 24-cell has 3 positions of 96 triangles.
The difference can be seen in the vertex figures, with edges representing faces in the 4-polytope:
The convex hull of two cantellated 24-cells in opposite positions is a nonuniform polychoron composed of 864 cells: 48 cuboctahedra, 144 square antiprisms, 384 octahedra (as triangular antipodiums), 288 tetrahedra (as tetragonal disphenoids), and 576 vertices. Its vertex figure is a shape topologically equivalent to a cube with a triangular prism attached to one of its square faces.
The cantitruncated 24-cell or great rhombated icositetrachoron is a uniform 4-polytope derived from the 24-cell. It is bounded by 24 truncated cuboctahedra corresponding with the cells of a 24-cell, 24 truncated cubes corresponding with the cells of the dual 24-cell, and 96 triangular prisms corresponding with the edges of the first 24-cell.
Coordinatesedit
The Cartesian coordinates of a cantitruncated 24-cell having edge length 2 are all permutations of coordinates and sign of:
(1,1+√2,1+2√2,3+3√2)
(0,2+√2,2+2√2,2+3√2)
The dual configuration has coordinates as all permutations and signs of:
Coxeter, Regular Polytopes, (3rd edition, 1973), Dover edition, ISBN0-486-61480-8, p.296, Table I (iii): Regular Polytopes, three regular polytopes in n-dimensions (n≥5)
H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973, p.296, Table I (iii): Regular Polytopes, three regular polytopes in n-dimensions (n≥5)
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, ISBN978-0-471-01003-6[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]
John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, The Symmetries of Things 2008, ISBN978-1-56881-220-5 (Chapter 26. pp. 409: Hemicubes: 1n1)
cantellated, cells, cell, cantellated, cell, cantitruncated, cell, orthogonal, projections, coxeter, plane, four, dimensional, geometry, cantellated, cell, convex, uniform, polytope, being, cantellation, order, truncation, regular, cell, there, unique, degrees. 24 cell Cantellated 24 cell Cantitruncated 24 cell Orthogonal projections in F4 Coxeter plane In four dimensional geometry a cantellated 24 cell is a convex uniform 4 polytope being a cantellation a 2nd order truncation of the regular 24 cell There are 2 unique degrees of cantellations of the 24 cell including permutations with truncations Contents 1 Cantellated 24 cell 1 1 Construction 1 2 Coordinates 1 3 Structure 1 4 Cantic snub 24 cell 1 5 Images 1 6 Related polytopes 2 Cantitruncated 24 cell 2 1 Coordinates 2 2 Projections 3 Related polytopes 4 ReferencesCantellated 24 cell editCantellated 24 cell Type Uniform 4 polytope Schlafli symbol rr 3 4 3 s2 3 4 3 Coxeter diagram nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp Cells 144 24 nbsp 3 4 4 4 24 nbsp 3 4 3 4 96 nbsp 3 4 4 Faces 720 288 triangles432 squares Edges 864 Vertices 288 Vertex figure nbsp Wedge Symmetry group F4 3 4 3 order 1152 Properties convex Uniform index 24 25 26 nbsp Net The cantellated 24 cell or small rhombated icositetrachoron is a uniform 4 polytope The boundary of the cantellated 24 cell is composed of 24 truncated octahedral cells 24 cuboctahedral cells and 96 triangular prisms Together they have 288 triangular faces 432 square faces 864 edges and 288 vertices Construction edit When the cantellation process is applied to 24 cell each of the 24 octahedra becomes a small rhombicuboctahedron In addition however since each octahedra s edge was previously shared with two other octahedra the separating edges form the three parallel edges of a triangular prism 96 triangular prisms since the 24 cell contains 96 edges Further since each vertex was previously shared with 12 faces the vertex would split into 12 24 12 288 new vertices Each group of 12 new vertices forms a cuboctahedron Coordinates edit The Cartesian coordinates of the vertices of the cantellated 24 cell having edge length 2 are all permutations of coordinates and sign of 0 2 2 2 2 2 1 1 2 1 2 1 2 2 The permutations of the second set of coordinates coincide with the vertices of an inscribed runcitruncated tesseract The dual configuration has all permutations and signs of 0 2 2 2 2 2 1 1 1 2 3 2 Structure edit The 24 small rhombicuboctahedra are joined to each other via their triangular faces to the cuboctahedra via their axial square faces and to the triangular prisms via their off axial square faces The cuboctahedra are joined to the triangular prisms via their triangular faces Each triangular prism is joined to two cuboctahedra at its two ends Cantic snub 24 cell edit A half symmetry construction of the cantellated 24 cell also called a cantic snub 24 cell as nbsp nbsp nbsp nbsp nbsp nbsp nbsp has an identical geometry but its triangular faces are further subdivided The cantellated 24 cell has 2 positions of triangular faces in ratio of 96 and 192 while the cantic snub 24 cell has 3 positions of 96 triangles The difference can be seen in the vertex figures with edges representing faces in the 4 polytope nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp Images edit orthographic 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 Schlegel diagrams nbsp Schlegel diagram nbsp Showing 24 cuboctahedra nbsp Showing 96 triangular prisms Related polytopes edit The convex hull of two cantellated 24 cells in opposite positions is a nonuniform polychoron composed of 864 cells 48 cuboctahedra 144 square antiprisms 384 octahedra as triangular antipodiums 288 tetrahedra as tetragonal disphenoids and 576 vertices Its vertex figure is a shape topologically equivalent to a cube with a triangular prism attached to one of its square faces Cantitruncated 24 cell editCantitruncated 24 cell nbsp Schlegel diagram centered on truncated cuboctahedron Type Uniform 4 polytope Schlafli symbol tr 3 4 3 Coxeter diagram nbsp nbsp nbsp nbsp nbsp nbsp nbsp Cells 144 24 4 6 8 nbsp 96 4 4 3 nbsp 24 3 8 8 nbsp Faces 720 192 3 288 4 96 6 144 8 Edges 1152 Vertices 576 Vertex figure nbsp sphenoid Symmetry group F4 3 4 3 order 1152 Properties convex Uniform index 27 28 29 nbsp Net The cantitruncated 24 cell or great rhombated icositetrachoron is a uniform 4 polytope derived from the 24 cell It is bounded by 24 truncated cuboctahedra corresponding with the cells of a 24 cell 24 truncated cubes corresponding with the cells of the dual 24 cell and 96 triangular prisms corresponding with the edges of the first 24 cell Coordinates edit The Cartesian coordinates of a cantitruncated 24 cell having edge length 2 are all permutations of coordinates and sign of 1 1 2 1 2 2 3 3 2 0 2 2 2 2 2 2 3 2 The dual configuration has coordinates as all permutations and signs of 1 1 2 1 2 5 2 2 1 3 2 3 2 3 2 2 2 2 2 2 2 4 2 2 Projections edit orthographic 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 Related polytopes edit24 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 References editT Gosset On the Regular and Semi Regular Figures in Space of n Dimensions Messenger of Mathematics Macmillan 1900 H S M Coxeter Coxeter Regular Polytopes 3rd edition 1973 Dover edition ISBN 0 486 61480 8 p 296 Table I iii Regular Polytopes three regular polytopes in n dimensions n 5 H S M Coxeter Regular Polytopes 3rd Edition Dover New York 1973 p 296 Table I iii Regular Polytopes three regular polytopes in n dimensions n 5 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 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 3 Convex uniform polychora based on the icositetrachoron 24 cell Model 24 25 George Olshevsky Klitzing Richard 4D uniform polytopes polychora x3o4x3o srico o3x4x3o grico 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 Cantellated 24 cells amp oldid 1189601959 Cantellated 24 cell, wikipedia, wiki, book, books, library,
