Figureiv
The Ant
two rules, no plan, and a road
An ant on a grid of black and white squares. On white: turn right, flip the square, step forward. On black: turn left, flip the square, step forward. That is the entire specification. There is nothing else — no goal, no memory, no map, no state beyond a heading and a position.
What this shows
A cellular automaton known as Langton’s ant runs on a grid. For roughly the first five hundred steps the pattern it leaves is small and roughly symmetrical. From about step five hundred to about step ten thousand it is chaotic — a spreading, structureless blot with no discernible order, which is what it looks like to every observer and every statistical test applied to it. Then, at around step ten thousand, without any change in the rules and without anything having been added, the ant begins to build a perfectly regular diagonal corridor one hundred and four steps in period, and it builds that corridor forever. This transition has been proved to occur from every starting configuration ever tested, and nobody can prove it must, and nobody can predict from the rules where the corridor will point.
The road is not in the rules. You can read the rules for a year and not find it. It is not hidden in them, either — it is not a consequence anyone has extracted; it is a thing the rules do, discoverable only by doing them, ten thousand times, and looking.
A pattern with no patterner. It was not designed, it was not intended, and it is completely reliable.