truncation
I've been thinking a lot about truncating the Platonic solids lately. If you snip the corners off at different lengths you create different solids, including most of the Archimedean solids. It turns out that 11 of the 13 Archimedean solids can be made from truncating the Platonic solids, excluding the snub cube and snub dodecahedron.
truncating the cube/octahedron
Here are some pictures drawn by me and my students showing the truncations of the cube and octahedron that give Archimedean solids.
Notice how truncations of the cube and octahedron give the same shapes (not surprising considering their duality). The cuboctahedron is right in the middle. The truncated octahedron is particularly exciting because it is one of the only shapes that fills 3D space! Here are some of these shapes made using origami.
truncating the dodeca and icosahedron
The picture below shows successive truncations of the icosahedron and dodecahedron.
In the middle is the icosidodecahedron, which is like the cuboctahedron above in that it is 'in the middle' of two Platonic solids. It is one of the most beautiful Archimedean solids, perhaps because it is so spherical. It has 6 'equators' that run around its centre, a fact which I exploited when making the model on the right.
The truncated icosahedron is often called the 'buckyball' after the architect Buckminster Fuller, and more commonly used as a football. It forms the basis of geodesic domes and is a special case of a goldberg polyhedron. There are some notes and videos on this on the Buckyball at the RI website here. Here are some images of my students exploring truncation with Polydron. 
