Less famous – but equally wonderful – are Ehrhart polynomials, which are what you get when you try to lift Pick’s theorem into higher dimensions. Though geometrically intuitive, they quickly lead into deep mathematical waters. They’re also valued as tools in optimisation problems and in other areas of computer science (I’m told).
This afternoon I gave a – hopefully fairly accessible – talk on these topics. The slides are available here.
(Update: PDF of slides here).