I was always fascinated by the emergence of patterns. One I like most is the synchronization of hundreds or thousands of fireflies. First they flash randomly but after some time and influencing each other, they flash in sync.
The rule behind this is very simple. All fireflies have nearly the same frequency for their flashing, but their phase is shifted. If a firefly receives a flash of a neighbour firefly, it flashes slightly earlier.
This circuit simulates fireflies with small microcontrollers.
Synchronizing Fireflies - [Link]