To make this, I started with the rhombic hexecontahedron (shown in this post), and then I augmented its twelve indentations with rhombic triacontahedra. I did this using Stella 4d, which you can try for free right here.
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.
The orange rhombi have interior angles which measure 30 and 150 degrees. For the blue rhombi, 60 and 120 degrees are the measures of the interior angles.
RobertLovesPi.net 