Small prismatotetracontoctachoron

The small prismatotetracontoctachoron, or spic, also commonly called the runcinated 24-cell, is a convex uniform polychoron that consists of 48 regular octahedra and 192 triangular prisms. 2 octahedra and 8 triangular prisms join at each vertex. It is the result of expanding the cells of an icositetrachoron outwards.

The small prismatotetracontoctachoron contains the vertices of an octagonal duoprism, small rhombated tesseract, truncated cubic prism, triangular-hexagonal prismantiprismoid and the triangular double prismantiprismoid.

Vertex coordinates
The vertices of a small prismatotetracontoctachoron of edge length 1 are all permutations of:


 * $$\left(±\frac{2+\sqrt2}{2},\,±\frac{\sqrt2}{2},\,0,\,0\right),$$
 * $$\left(±\frac{1+\sqrt2}{2},\,±\frac{1+\sqrt2}{2},\,±\frac12,\,±\frac12\right).$$

The second set of vertices are identical to the vertices of an inscribed small rhombated tesseract.

Representations
The small prismatotetracontoctachoron has the following Coxeter diagrams:


 * x3o4o3x (full symmetry)
 * oxoxoxoxo4oooxxxooo3xxwoqowxx&#xt (BC3 axial, octahedron-first)
 * ox4oo3xx3qo&#zx (BC4 symmetry)
 * qoo3xxx3oqo *b3ooq&#zx (D4 symmetry)
 * Uwqxo oxoxo4oooxx3xxwoq&#zx (BC3×A1 symmetry)
 * ooxxowq4oxoxwoq qowxoxo4qwoxxoo&#zx (BC2×BC2 symmetry)

Variations
The small prismatotetracontoctachoron has a semi-uniform variant under half symmetry called the small disprismatoicositetricositetrachoron.

Related polychora
The small prismatotetracontoctachoron is the colonel of a regiment that includes 20 members, 2 fissaries, and a compound. Among the members, 8 in total have doubled F4 symmetry, including the small prismatotetracontoctachoron itself, the small distetracontoctachoron, and the noble small retrotetracontoctachoron, while the 12 remaining members, and the fissaries and compound, have single F4 symmetry only.

The small prismatotetracontoctachoron can be diminished by removing octahedron atop small rhombicuboctahedron segmentochora. If 8 of these caps are removed, the result is the small rhombated tesseract.

Isogonal derivatives
Substitution by vertices of these following elements will produce these convex isogonal polychora:
 * Octahedron (48): Bitetracontoctachoron
 * Triangular prism (192): Biambotetracontoctachoron
 * Square (288): Tetracontoctachoron
 * Triangle (384): Bitruncatotetracontoctachoron
 * Edge (576): Rectified small prismatotetracontoctachoron