The world is one row of cells. In each tile, the top three squares are a cell and its left and right neighbours now; the square below is what that middle cell becomes in the next step. The picture stacks the rows: the top line is the start, and each line below is the next step, so down = time, not space.
Big ideas, shown
Rule 110 knows only 8 cases. Start it from random noise and things appear that move, crash into each other and make new things: particles. Nobody put them in. That is emergence: simple parts, simple rules, complicated whole.
Will a cell far down be black or white? For rule 90 there is a formula: answer at once. For rule 30 nobody has found one. The only way to know is to compute every row on the way. That is computational irreducibility.
In 2004 Matthew Cook proved that rule 110 can compute anything a computer can, if you set up the starting row the right way. Particles carry the data and their crashes work like logic gates. It is Turing complete. And so, as for every real computer, there is no general way to tell in advance whether a pattern will ever stop growing.
Run the same rule twice, the second time with one cell flipped. Red shows where the two runs differ. In rule 90 the change spreads as a neat triangle. In rule 30 it spreads as a mess, and in rule 110 it creeps.
Rule 90 from one cell draws the Sierpinski triangle. It is also Pascal's triangle with the odd numbers black: the same numbers that count the paths on the Galton board. The rule is simply "exactly one of my two neighbours is black".
Rule 184: a black cell is a car, and it moves on if the cell ahead is free. Below half full, everyone drives. Above half, jams appear and slide backwards, just like on the ring road.
All 256 rules
Wolfram sorted the rules into four classes by what they do to a random row. Mirror twins (left and right swapped) and colour twins (black and white swapped) do the same thing, so the 256 rules are only 88 kinds. Click one to run it.