Rectified truncated icosahedron

The rectified truncated icosahedron is a near-miss Johnson solid. Topologically, its faces are 60 isosceles triangles, 12 pentagons and 20 triambi. It can be constructed by applying a rectification process to the truncated icosahedron.

Variations
There are three variants of the rectified truncated icosahedron. The first variant is obtained by making the hexagons regular, in which the ratio between the two edges is equivalent to 1:$$\frac{3+4\sqrt2-\sqrt5}{6}$$ ≈ 1:1.07013. The second variant is obtained by making all the edges have the same length, but the hexagons only have triangular symmetry, with the internal angles being approximately 115.28055° and 124.71945°. The third variant has the unique property of being canonical and all of its triangles are equilateral. If the canonical variant has a midradius of 1, then the edge lengths are approximately 0.34561 and 0.35956 and the hexagons' internal angles are approximately 118.07524° and 121.92476°.