Incommensurable Lengths

Incommensurable line segments have length ratios that are not rational. No square tiles, no matter how small, can completely tile a rectangular room with incommensurable sides! The discovery of incommensurables was a crisis for the Pythagoreans who believed commensurables were sufficient to describe the world. Legend has it Hippasus was thrown overboard by the Pythagoreans for proving the sides and diagonals of squares are incommensurable.