I just saw the TED talk from Stephen Wolfram below and noticed I'd never written any 1D cellular automata. He briefly mentions rule 30 and the emergence of complexity from a simple set of rules. The Von Neumann automata is very simple to write. Given a single cell, the state of the new cell (either on or off) is determined by the cells in its local neighbourhood. In one dimension this means the state of itself, the cell just to its left, and the cell just to its right. This means there are 8 possible states for local neighbourhoods (8 rules) and therefore 256 possible outcomes of the system.

As Mr Wolfram mentions in the talk, most of the outcomes are not particularly interesting, some are very plain and some are quite pretty. Below is rule 26, it looks nice but beyond that it is quite a simple rule. Self repetition on a number of different scales is pretty much all it is capable of. Just click the image to launch it.

The next two examples are rule 30 and rule 110 respectively. These are extremely interesting. Rule 30 displays what Wolfram calls "Class 3" behaviour, which is a chaotic and seemingly random, whereas Rule 110 displays "Class 4" behaviour, which is neither completely random nor completely repetitive. The interaction of various local structures is used to prove Universality. Again just click the images to launch the programs!

Rule 30 can be used as an excellent random generator:

Notice the life-like region travelling up near the centre in Rule 110:

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