Using a geometric shape called a Reuleaux triangle, it’s possible to drill square holes. Click through for all the exciting math!
Update: A video of a Reuleaux triangle rotating in a square. (thx, will)
Update: More on the Reuleaux triangle at MathWorld. (thx, nevan)
Update: The Reuleaux triangle is also the basis for the Wankel engine.. (thx, brian & adam)