Chaos rides on the back of a philosophical paradox: how can deterministic mathematical models produce random behavior?

There is a similar surprise regarding symmetry: symmetric causes can have asymmetric effects. This paradox is called symmetry-breaking. In recent years researchers have shown that it plays a major role in pattern formation in physical systems. 

 

Finally, the way in which symmetry and chaos co-exist is a third paradox: symmetry suggests order and regularity while chaos suggests disorder and randomness. The combination leads to a striking series of pictures and to a notion of pattern on average.