Triakis icosahedron

The triakis icosahedron is one of the 13 Catalan solids. It has 60 isosceles triangles as faces, with 12 order-10 and 20 order-3 vertices. It is the dual of the uniform truncated dodecahedron.

It can also be obtained as the convex hull of a dodecahedron and an icosahedron, where the edges of the icosahedron are $$\frac{15+\sqrt5}{10} ≈ 1.72361$$ times the length of those of the dodecahedron. Using an icosahedron that is any number more than $$\frac{1+\sqrt5}2 ≈ 1.61803$$ times the edge length of the dodecahedron gives a fully symmetric variant of this polyhedron.

Each face of this polyhedron is an isosceles triangle with base side length $$\frac{15+\sqrt5}{10} ≈ 1.72361$$ times those of the side edges. These triangles have apex angle $$\arccos\left(-3\frac{1+\sqrt5}{20}\right) ≈ 119.03935°$$ and base angles $$\arccos\left(\frac{15+\sqrt5}{20}\right) ≈ 30.48032°$$.