There are two Archimedean solids with 62 faces:

They both have 62 faces because of their roots in the dodecahedron and icosahedron. They have a face for each of the faces, vertices, and edges of either of those polyhedra: 12 + 20 + 30 = 62.

The rhomb­icosi­dodeca­hedron shows up in more places than you might expect for a complex polyhedron. The Zometool building kit uses it for hubs, though the squares are adjusted to rectangles:

Recently one also showed up on Kickstarter: Glint, a rhomb­icosi­dodeca­hedron machined in solid brass. I have one. It’s satisfyingly heavy and precise. Here it is with its little Zome sister:

The rhomb­icosi­dodeca­hedron is very pleasing: nearly round, but with simple polygonal faces roughly equal in size. Who’d have thought a number as seemingly uninteresting as 62 could appear in such serendipitous places?