Schlegel diagram, centered on octahedron, with the octahedra shown.
Perspective projection of the runcinated 24-cell into 3 dimensions, centered on an octahedral cell.
The rotation is only of the 3D image, in order to show its structure, not a rotation in 4-space. Fifteen of the octahedral cells facing the 4D viewpoint are shown here in red. The gaps between them are filled up by a framework of triangular prisms.
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.
Schlegel diagram centered on rhombicuboctahedron only triangular prisms shown
Runcicantic snub 24-cell
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. Like the snub 24-cell, it has symmetry [3+,4,3], order 576. The runcitruncated 24-cell has 192 identical hexagonal faces, while the runcicantic snub 24-cell has 2 constructive sets of 96 hexagons.
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}, . It is not uniform, but it is vertex-transitive and has all regular polygon faces. It is constructed with 24 icosahedra, 24 truncated tetrahedra, 96 triangular prisms, and 96 triangular cupolae in the gaps, for a total of 240 cells, 960 faces, 1008 edges, and 288 vertices. Like the snub 24-cell, it has symmetry [3+,4,3], order 576.[1]
The vertex figure contains one icosahedron, two triangular prisms, one truncated tetrahedron, and 3 triangular cupolae.
The 48 truncated cuboctahedral cells are joined to each other via their octagonal faces. They can be grouped into two groups of 24 each, corresponding with the cells of a 24-cell and its dual. The gaps between them are filled in by a network of 192 hexagonal prisms, joined to each other via alternating square faces in alternating orientation, and to the truncated cuboctahedra via their hexagonal faces and remaining square faces.
Coordinates
The Cartesian coordinates of an omnitruncated 24-cell having edge length 2 are all permutations of coordinates and sign of:
Perspective projection into 3D centered on a truncated cuboctahedron. The nearest great rhombicuboctahedral cell to the 4D viewpoint is shown in red, with the six surrounding great rhombicuboctahedra in yellow. Twelve of the hexagonal prisms sharing a square face with the nearest cell and hexagonal faces with the yellow cells are shown in blue. The remaining cells are shown in green. Cells lying on the far side of the polytope from the 4D viewpoint have been culled for clarity.
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. Its vertex figure is an irregular triangular bipyramid.
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. It has a symmetry of [[3,4,3]+], order 1152.
The uniform snub 24-cell is called a semi-snub 24-cell by John Horton Conway with Coxeter diagram within the F4 family, although it is a full snub or omnisnub within the D4 family, as .
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. Its vertex figure contains 4 tetrahedra, 2 octahedra, and 2 snub cubes. It has 816 cells, 2832 faces, 2592 edges, and 576 vertices.[2]