Runcinated 24-cells

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 dual regular skew polyhedron, {8,4|3}, is similarly related to the octagonal faces of the bitruncated 24-cell.

The dual configuration has coordinates generated from all permutations and signs of: A half-symmetry construction of the runcitruncated 24-cell (or runcicantellated 24-cell), as , also called a runcicantic snub 24-cell, as , has an identical geometry, but its triangular faces are further subdivided.

The difference can be seen in the vertex figures: A related 4-polytope is the runcic snub 24-cell or prismatorhombisnub icositetrachoron, s3{3,4,3}, .

The Cartesian coordinates of an omnitruncated 24-cell having edge length 2 are all permutations of coordinates and sign of: Nonuniform variants with [3,4,3] symmetry and two types of truncated cuboctahedra can be doubled by placing the two types of truncated cuboctahedra on each other to produce a nonuniform polychoron with 48 truncated cuboctahedra, 144 octagonal prisms (as ditetragonal trapezoprisms), 192 hexagonal prisms, two kinds of 864 rectangular trapezoprisms (288 with D2d symmetry and 576 with C2v symmetry), and 2304 vertices.

Vertex figure This polychoron can then be alternated to produce another nonuniform polychoron with 48 snub cubes, 144 square antiprisms, 192 octahedra (as triangular antiprisms), three kinds of 2016 tetrahedra (288 tetragonal disphenoids, 576 phyllic disphenoids, and 1152 irregular tetrahedra), and 1152 vertices.

In contrast a full snub 24-cell or omnisnub 24-cell, defined as an alternation of the omnitruncated 24-cell, cannot be made uniform, but it can be given Coxeter diagram , and symmetry [[3,4,3]]+, order 1152, and constructed from 48 snub cubes, 192 octahedrons, and 576 tetrahedrons filling the gaps at the deleted vertices.