Rectified 24-cellSchlegel diagram8 of 24 cuboctahedral cells shownTypeUniform 4-polytopeSchläfli symbolsr{3,4,3} = rr{3,3,4}=r{31,1,1} = Coxeter diagrams or Cells4824 3.4.3.4 24 4.4.4 Faces24096 {3}144 {4}Edges288Vertices96Vertex figureTriangular prismSymmetry groupsF4 [3,4,3], order 1152B4 [3,3,4], order 384D4 [31,1,1], order 192Propertiesconvex, edge-transitiveUniform index22 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.
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
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
A rectified 24-cell having an edge length of 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
Stereographic projectionCenter of stereographic projectionwith 96 triangular faces blue
Symmetry constructions
There are three different symmetry constructions of this polytope. The lowest construction can be doubled into by adding a mirror that maps the bifurcating nodes onto each other. can be mapped up to 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 construction, and two colors (1:2 ratio) in , and all identical cuboctahedra in .
Coxeter group = [3,4,3] = [4,3,3] = [3,31,1]Order1152384192Fullsymmetrygroup[3,4,3][4,3,3]<[3,31,1]> = [4,3,3][3[31,1,1]] = [3,4,3]Coxeter diagramFacets3: 2: 2,2: 2: 1,1,1: 2: Vertex figure
Alternate names 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 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
The rectified 24-cell can also be derived as a cantellated 16-cell:
Citations
References T. Gosset: On the Regular and Semi-Regular Figures in Space of n Dimensions, Messenger of Mathematics, Macmillan, 1900
John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, The Symmetries of Things 2008, (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)