This is the great rhombicosidodecahedron, one of the thirteen Archimedean solids.

Here’s its dual, the disdyakis triacontahedron.

I use a program called Stella 4d to make these .gifs and manipulate polyhedra, and one of Stella‘s functions is “try to make faces regular.” I performed this function on the disdyakis triacontahedron, which has ten triangles meeting at some vertices — so 600 degrees’ worth of triangle-angles tried to squeeze in around those points when the faces were made to be regular. This forces the polyhedron to become non-convex — to the point of looking wrinkled.

“That’s weird looking,” I thought. “I wonder what its dual looks like?” With Stella, I could find out with one mouse-click, and I was most surprised by the result.

In this polyhedron, there are thirty orange rectangles, twelve light blue 10/4-gons, and twenty violet 6/2-gons. None of them are regular. Here are what the faces look like in isolation, starting with an orange rectangle, then a light blue 10/4-gon, and lastly a violet 6/2-gon.

If you’d like to try Stella for yourself, there is a free trial download available at http://www.software3d.com/Stella.php.