This article starts with the penny-packing problem and builds up to discussing the unsolved problem of how to best back tetrahedra in space. (It's the most recent Feature Column at the American Mathematical Society.)
|Is this the best way to pack pentagons? Nobody knows.|
On a related topic, James and I got to talking about putting together polyhedra in nice ways--like to build another one. A cuboctahedron, in particular. That made me think of this paper from Bridges over the summer about making loops out of copies a polyhedron.
And finally, here are some four-dimensional symmetry puzzles. Awesome.