Many people will have heard of Conway's Game of Life, a 2D simulation based upon a very simple algorithm that shows incredibly complex behaviour. Langton's Ants (wiki) runs along a similar line, however the rules each "Ant" follow are even simpler, and when combined with multiple Ants they create some impressive patterns.
Each Ant, represented in my version by a red square obeys only 2 rules...
1) At a black square, turn 90° right, flip the colour of the square, move forward one unit.
2) At a green square, turn 90° left, flip the colour of the square, move forward one unit.
These 2 rules on their own will exhibit an identical result each time which. Whilst starting off chaotic, it eventually mutates into a structured "Bridge" as pictured below.
The real amazing part comes when you put 2 or more together, the Ants interact, crossing each others paths and creating a crazy trail. You can even lock 2 or more ants together in a higher-order system that shows some seemingly intelligent behaviour.
What I find so amazing about Langton's Ants is there ability to produce organic looking structures from incredibly simple rules, and I think because you can follow 1 Ants progress you have a connection with the actions, unlike Conway's Game of Life where you are slightly disjointed from the sim.
Just noticed that some old-school browsers don't support some of the features used in this demo, like Range controls and Canvas - try it in Google Chrome / Safari.