Runcinated 24-cells

In four-dimensional geometry, a runcinated 24-cell is a convex uniform 4-polytope, being a runcination (a 3rd order truncation) of the regular 24-cell. There are 3 unique degrees of runcinations of the 24-cell including with permutations truncations and cantellations. In geometry, the runcinated 24-cell or small prismatotetracontoctachoron is a uniform 4-polytope bounded by 48 octahedra and 192 triangular prisms. The octahedral cells correspond with the cells of a 24-cell and its dual. E. L. Elte identified it in 1912 as a semiregular polytope. Runcinated 24-cell (Norman W. Johnson) Runcinated icositetrachoron Runcinated polyoctahedron Small prismatotetracontoctachoron (spic) (Jonathan Bowers) The Cartesian coordinates of the runcinated 24-cell having edge length 2 is given by all permutations of sign and coordinates of: (0, 0, , 2+) (1, 1, 1+, 1+) The permutations of the second set of coordinates coincide with the vertices of an inscribed cantellated tesseract. The regular skew polyhedron, {4,8|3}, exists in 4-space with 8 square around each vertex, in a zig-zagging nonplanar vertex figure. These square faces can be seen on the runcinated 24-cell, using all 576 edges and 288 vertices. The 384 triangular faces of the runcinated 24-cell can be seen as removed. The dual regular skew polyhedron, {8,4|3}, is similarly related to the octagonal faces of the bitruncated 24-cell. The runcitruncated 24-cell or prismatorhombated icositetrachoron is a uniform 4-polytope derived from the 24-cell. It is bounded by 24 truncated octahedra, corresponding with the cells of a 24-cell, 24 rhombicuboctahedra, corresponding with the cells of the dual 24-cell, 96 triangular prisms, and 96 hexagonal prisms. The Cartesian coordinates of an origin-centered runcitruncated 24-cell having edge length 2 are given by all permutations of coordinates and sign of: (0, , 2, 2+3) (1, 1+, 1+2, 1+3) The permutations of the second set of coordinates give the vertices of an inscribed omnitruncated tesseract.