Several years ago, retired military officer Guan Baihua invented what may be the ultimate geometry geek bicycle. The front wheel is a pentagonal curve, and the back wheel is a reuleaux triangle (a triangular curve). It seems like it shouldn't work, but it does — and beautifully!
It's called the Reuleaux Bicycle. I'm not sure why it hasn't caught on yet, given the lust for artisanal bikes among urbanites.