Disdyakis dodecahedron

The disdyakis dodecahedron, also called the small disdyakis dodecahedron or siddykid, is one of the 13 Catalan solids. It has 48 scalene triangles as faces, with 6 order-8, 8 order-6, and 12 order-4 vertices. It is the dual of the uniform great rhombicuboctahedron.

It can also be obtained as the convex hull of a cube, an octahedron, and a cuboctahedron. If the cube has unit edge length, the octahedron's edge length is $$3\frac{2+3\sqrt2}{14} ≈ 1.33771$$ and the cuboctahedron's edge length is $$3\frac{1+2\sqrt2}{14} ≈ 0.82038$$.

Each face of this polyhedron is a scalene triangle. If the shortest edges have unit edge length, the medium edges have length $$3\frac{2+3\sqrt2}{14} ≈ 1.33771$$ and the longest edges have length $$\frac{10+\sqrt2}{7} ≈ 1.63060$$. These triangles have angles measuring $$\arccos\left(\frac{2-\sqrt2}{12}\right) ≈ 87.20196°$$, $$\arccos\left(\frac{6-\sqrt2}{8}\right) ≈ 55.02470°$$, and $$\arccos\left(\frac{1+6\sqrt2}{12}\right) ≈ 37.77334°$$.

Vertex coordinates
A disdyakis dodecahedron with dual edge length 1 has vertex coordinates given by all permutations of:
 * $$\left(±3\frac{2+3\sqrt2}{7},\,0,\,0\right),$$
 * $$\left(±3\frac{1+2\sqrt2}{7},\,±3\frac{1+2\sqrt2}{7},\,0\right),$$
 * $$\left(±\sqrt2,\,±\sqrt2,\,±\sqrt2\right).$$