If a shape has equal sides with 90 degree angles between them then it's a square, right? Well, not quite...

So easy to describe, yet so hard to prove.

Why there are only three regular polygons you can tile a wall with.

Sometimes a piece of maths can be so neat and elegant, it makes you want to shout "eureka!" even if you haven't produced it yourself. One of our favourite examples is the art gallery problem.

Tilings have adorned buildings from ancient Rome to the Islamic world, from Victorian England to colonial Mexico. But while it sometimes seems free from worldly limitations, tiling is a very precise art, where not much can be left to chance. We can push and turn and wiggle, but if the maths is not right, it isn't going to tile. Josefina Alvarez and Cesar L. Garcia investigate.