Disdyakis triacontahedron

The disdyakis triacontahedron, also called the small disdyakis triacontahedron or siddykit, is one of the 13 Catalan solids. It has 120 scalene triangles as faces, with 12 order-10, 20 -order-6, and 30 order-4 vertices. It is the dual of the uniform great rhombicosidodecahedron.

It can also be obtained as the convex hull of a dodecahedron, an icosahedron, and an icosidodecahedron. If the dodecahedron has unit edge length, the icosahedron's edge length is $$3\frac{3+\sqrt5}{10} ≈ 1.57082$$ and the icosidodecahedron's edge length is $$3\frac{4+\sqrt5}{22} ≈ 0.85037$$.

Each face of this polyhedron is a scalene triangle. If the shortest edges have unit edge length, the medium edges have length $$3\frac{3+\sqrt5}{10} ≈ 1.57082$$ and the longest edges have length $$\frac{7+\sqrt5}5 ≈ 1.84721$$. These triangles have angles measuring $$\arccos\left(\frac{5-2\sqrt5}{30}\right) ≈ 88.99180°$$, $$\arccos\left(\frac{15-2\sqrt5}{20}\right) ≈ 58.23792°$$, and $$\arccos\left(\frac{9+5\sqrt5}{24}\right) ≈ 32.77028°$$.