The truncated icosahedron is familiar to many people as the soccer ball, or the C60 molecule. This polyhedron often is confused with it — for, at first glance, it is easy to confuse them.
However, for a true truncated icosahedron, every vertex is identical, with two hexagons meeting a pentagon. In this polyhedron, by contrast, some vertices have three hexagons meeting there — which is enough to prove that these hexagons are irregular, for, if regular, they would be coplanar.
What is this thing? Well, among other things, it is the shape of the C80 molecule. It has the twelve pentagons of the truncated icosahedron. However, that polyhedron has twenty hexagonal faces, all regular. In this polyhedron, by contrast, there are thirty hexagons, with regularity sacrificed.
Software credit: see the programs you can try for free at http://www.software3d.com/stella.php.