Chamfered icosahedron

The chamfered icosahedron is a modification of the icosahedron that can have one edge length but has irregular faces. It has 20 triangles and 12 hexagons as faces, and 12 order-5 vertices that can be thought of as coming from the icosahedron as well as 60 new order-3 vertices.

The hexagonal faces have angles of $$\arccos\left(\frac{\sqrt5}{5}\right) ≈ 63.43495^\circ$$ on a pair of opposite vertices, and angles of $$\arccos\left(-\sqrt{\frac{5+\sqrt5}{10}}\right) ≈ 148.28253^\circ$$ on the four remaining vertices.

It can be modified such that it has a single inradius, or such that it has a single midradius or "edge radius." The latter version is called the "canonical" version.

It can also be viewed as an order-3-truncated rhombic triacontahedron, or as an icosahedrally-symmetric Goldberg polyhedron.