Tridyakis icosahedron

The tridyakis icosahedron is a uniform dual polyhedron. It consists of 120 scalene triangles.

If its dual, the great cubicuboctahedron, has an edge length of 1, then the triangle faces' short edges will be $$5\frac{\sqrt{15}-\sqrt3}{8} ≈ 1.33808$$, the medium edges will be $$\frac{3\sqrt{15}}{4} ≈ 2.90474$$, and the long edges will be $$5\frac{\sqrt{15}+\sqrt3}{8} ≈ 3.50315$$. The triangles have one interior angle of $$\arccos\left(\frac35\right) ≈ 53.13010°$$, one of $$\arccos\left(\frac13+\frac{4\sqrt5}{15}\right) ≈ 21.62463°$$, and one of $$\arccos\left(\frac13-\frac{4\sqrt5}{15}\right) ≈ 105.24526°$$.

Vertex coordinates
A tridyakis icosahedron with dual edge length 1 has vertex coordinates given by all even permutations of:
 * $$\left(±3\frac{5+\sqrt5}{8},\,±\frac{3\sqrt5}{4},\,0\right),$$
 * $$\left(±5\frac{\sqrt5-1}{8},\,±5\frac{1+\sqrt5}{8},\,0\right),$$
 * $$\left(±\frac{3\sqrt5}{4},\,±3\frac{5-\sqrt5}{8},\,0\right),$$
 * $$\left(±\frac54,\,±\frac54,\,±\frac54\right).$$