Snub disphenoid

The snub disphenoid also known as the siamese dodecahedron, is one of the 92 Johnson solids (J84). It consists of 4+8 triangles as faces.

It can be constructed from a tetrahedron, seen as a digonal antiprism or disphenoid, by expanding the two halves outward and inserting a set of 8 triangles in between the halves.

Coordinates
Coordinates for a snub disphenoid of unit edge length are given by where r, s and t are given in terms of the unique positive root q ≈ 0.16902 of
 * (±t, 0, –r),
 * (0, ±t, r),
 * (±1/2, 0, s),
 * (0, ±1/2, –s),
 * $$2x^3+11x^2+4x-1$$

as
 * $$r=\frac{\sqrt q}2,\quad s=\sqrt{\frac{1-q}{8q}},\quad t=\sqrt{\frac{1-q}{2}}.$$

With these coordinates, it's possible to calculate the volume of a snub disphenoid with unit edge length as ξ ≈ 0.85949, where ξ is the unique positive root of the polynomial


 * $$5832x^6-1377x^4-2160x^2-4.$$

Related polyhedra
The snub disphenoid can be considered to be the digonal case in the family of snub antiprisms. The snub triangular antiprism is the regular icosahedron, and the snub square antiprism is another Johnson solid. No other convex members of this family can be made to have all regular faces, though nonconvex regular-faced cases do exist.

If each square of a square antiprism are turned into two triangles, the result is a snub disphenoid. If only one square is turned into triangles, the result is a biaugmented triangular prism.

In vertex figures
The snub disphenoid appears as the vertex figure of the nonuniform 13-5 step prism. This vertex figure has an edge length of 1, and has no corealmic realization, because the Johnson snub disphenoid has no circumscribed sphere.