This is the rhombic hexecontahedron. Its faces are 60 rhombi with diagonals in the golden ratio. I made it by starting with a rhombic triacontahedron, then stellating it 26 times. This was done using Stella 4d, a program you can try for free at http://www.software3d.com/Stella.php.
This non-convex polyhedron can also be made using Zome, available at http://www.zometool.com, using all red struts of the same length. This was the reason I made the edges red, and the vertices white (like Zomeballs).
It is not possible to tessellate a plane using only regular pentagons and decagons, but that doesn’t stop me from trying. To get this to work, I had to add rhombi and concave pentagons — but even then, it’s an aperiodic tessellation.
Nothing in this diagram is shown to scale.