Rectified 24-cell

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. 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. 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] 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 . 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) 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.