Snub icosahedron

The snub icosahedron, si, or first compound of ten octahedra is a uniform polyhedron compound. It consists of 20+60 triangles, with 4 triangles joining at each vertex.

Each octahedral component has triangular antiprismatic symmetry. If each component is rotated by 60°, the great snub icosahedron, the other uniform compound of ten octahedra, is produced.

Its quotient prismatic equivalent is the small triangular antiprismatic decayottoorthowedge, which is twelve-dimensional.

Vertex coordinates
The vertices of a snub icosahedron of edge length 1 are given by all even permutations of:
 * $$\left(0,\,\pm\frac{2-\sqrt2+2\sqrt5+\sqrt{10}}{12},\,\pm\frac{2+\sqrt2-2\sqrt5+\sqrt{10}}{12}\right),$$
 * $$\left(\pm\frac{6-4\sqrt2+2\sqrt5}{24},\,\pm\frac{\sqrt2+\sqrt5}{6},\,\pm\frac{3+2\sqrt2-\sqrt5}{12}\right),$$
 * $$\left(\pm\frac{1+\sqrt2+\sqrt5-\sqrt{10}}{6},\,\pm\frac{-1+\sqrt2+\sqrt5+\sqrt{10}}{12},\,\pm\frac12\right).$$