The Pentakis Dodecahedron
The pentakis dodecahedron is a 3D Catalan solid bounded by 60 isosceles triangles, 90 edges, and 32 vertices. It is the dual of the truncated icosahedron.
The faces are transitive isosceles triangles with 2 short edges and 1 long edge. The edge length ratio is (12+3φ)/19 : 1 (approximately 0.887 : 1), where φ=(1+√5)/2 is the Golden Ratio.
The 32 vertices are of two kinds: 20 vertices where 5 short edges meet, corresponding to an inscribed icosahedron, and 12 vertices where 6 edges meet (3 long, 3 short), corresponding to an inscribed dodecahedron.
Projections
The following are images of the pentakis dodecahedron from various viewpoints:
Projection  Description 

Projection centered on an order5 vertex. 

Centered on an order6 vertex. 

Centered on a long edge. 

Centered on a short edge. 
Animation
Here's an animation of a pentakis dodecahedron rotating around the vertical axis:
Coordinates
The Cartesian coordinates for the pentakis dodecahedron are all permutations of coordinate and all changes of sign of:
 (1, 1, 1)
along with even permutations of coordinate and all changes of sign of:
 (0, A, Aφ)
 (0, φ, 1/φ)
where φ=(1+√5)/2 is the Golden Ratio, and A=(12+3φ)/19 is the length ratio of short edges to long edges.
The edge length of the corresponding dual truncated icosahedron is 2/(3φ^{2}).