Outer billiards has an unbounded orbit
This just popped up on the arXiv, so I thought I should mention it: Richard Schwartz has a paper up showing that there is a shape for an “outer billiards” table and a starting point whose path gets as far away from the shape as you want. Even better, it’s one of the shapes from the Penrose tiling. Curiouser and curiouser. The first section or so of the paper are very readable, and gives a much better explanation (with pictures!) of outer billiards than I could manage. The proof itself is heavily aided by computer calculations, but seems to be tightly reasoned apart from needing help to handle a lot of cases.
I’m not quite sure how billiards and outer billiards are related. My intuition is that there’s some sort of duality going on, which would exchange lengths of segments in outer billiards with angles in billiards. On the other hand, if there were such a straightforward translation, couldn’t the enormous machinery of billiards have been brought to bear on this problem before now? Do any billiard theorists in the audience know anything about outer billiards?