Tetrated dodecahedron

The tetrated dodecahedron is a near-miss Johnson solid. Topologically, its faces are 12 isosceles triangles, 4 triangles and 12 mirror-symmetric pentagons.

To construct it, one needs to inscribe a pentagon in each face of rhombic dodecahedron and fill the remaining holes with triangles.

Variations
There are three variants of the tetrated dodecahedron. The first variant is obtained by making the pentagons 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 pentagons only have mirror symmetry, with the internal angles being approximately 106.20727°, 108.67987°, and 110.22572°. 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 0.65057 and 0.67708 and the pentagons' internal angles are approximately 107.11871° and 111.52516°.