The rectified truncated icosahedron is a surprising new polyhedron discovered by Craig S. Kaplan. It has a total of 60 triangles, 12 pentagons and 20 hexagons as faces.
This image by intocontinuum show how you can take 8 perfectly rigid regular tetrahedra and connect them along their edges to form a ring that you can turn inside-out.
Here Greg Egan has drawn two regular dodecahedra, in red and blue. They share 8 corners—and these are the corners of a cube, shown in green. Adrian Ocneanu calls these twin dodecahedra, and has proved some fascinating results about them.
This is the small cubicuboctahedron, as drawn by Robert Webb’s Great Stella software. It looks simple enough, but it conceals some interesting mathematics.
The icosidodecahedron can be built by truncating either an icosahedron or a dodecahedron. It has 30 vertices. It is a beautiful, highly symmetrical shape. But it’s just a shadow of an even more symmetrical shape with twice as many vertices in twice as many dimensions!
This picture by Toby Hudson shows the densest known packing of the regular pentagon.
This picture by Toby Hudson shows the densest known packing of the regular heptagon. Of all convex shapes, the regular heptagon is believed to have the lowest maximal packing density.
This is the densest packing of regular octagons in the plane, drawn by Graeme McRae. It is interesting because it is a counterexample to the 2-dimensional analogue of a conjecture made in 3 dimensions by Stanislaw Ulam.