Truncated 24-cells

In geometry, a truncated 24-cell is a uniform 4-polytope (4-dimensional uniform polytope) formed as the truncation of the regular 24-cell. There are two degrees of truncations, including a bitruncation. The truncated 24-cell or truncated icositetrachoron is a uniform 4-dimensional polytope (or uniform 4-polytope), which is bounded by 48 cells: 24 cubes, and 24 truncated octahedra. Each vertex joins three truncated octahedra and one cube, in an equilateral triangular pyramid vertex figure. The truncated 24-cell can be constructed from polytopes with three symmetry groups: F4 [3,4,3]: A truncation of the 24-cell. B4 [3,3,4]: A cantitruncation of the 16-cell, with two families of truncated octahedral cells. D4 [31,1,1]: An omnitruncation of the demitesseract, with three families of truncated octahedral cells. It is also a zonotope: it can be formed as the Minkowski sum of the six line segments connecting opposite pairs among the twelve permutations of the vector (+1,−1,0,0). The Cartesian coordinates of the vertices of a truncated 24-cell having edge length sqrt(2) are all coordinate permutations and sign combinations of: (0,1,2,3) [4!×23 = 192 vertices] The dual configuration has coordinates at all coordinate permutation and signs of (1,1,1,5) [4×24 = 64 vertices] (1,3,3,3) [4×24 = 64 vertices] (2,2,2,4) [4×24 = 64 vertices] The 24 cubical cells are joined via their square faces to the truncated octahedra; and the 24 truncated octahedra are joined to each other via their hexagonal faces. The parallel projection of the truncated 24-cell into 3-dimensional space, truncated octahedron first, has the following layout: The projection envelope is a truncated cuboctahedron. Two of the truncated octahedra project onto a truncated octahedron lying in the center of the envelope. Six cuboidal volumes join the square faces of this central truncated octahedron to the center of the octagonal faces of the great rhombicuboctahedron. These are the images of 12 of the cubical cells, a pair of cells to each image.