Suppose you have a bunch of pennies and want to arrange them on your desk so that they do not overlap but cover as much of the desktop as possible. Honeybees figured out the answer long ago, but a proof that it is the optimal arrangement only came in 1890. Now suppose you have two different sizes of coins and ask the same question. The optimal arrangement depends on the relative sizes of the coins, and lots of nice patterns emerge. For certain sizes some especially nice packings known as "compact packing" are possible. There are only finitely many, and we will attempt to find as many as we can by drawing pictures, arranging discs, using some simple planar geometry and whatever other tools come to mind.