So if one makes a list of all possible axiom systems—say starting with the simplest—where in such a list should one expect to see axiom systems that correspond to traditional areas of mathematics? Most axiom systems as they are given in typical textbooks are sufficiently complicated that they will not show up at all early. … The first argument for each function appears on the left in the picture; the second argument on top.

Each picture can be generated by starting from initial conditions at the top, and then just evolving down the page repeatedly applying the cellular automaton rule.

The top view shows the color of every cell at every step.

But in one respect all these systems have ultimately been set up in the same basic way: they are all based on explicit rules that specify how the system evolves from step to step. … After all, the constraint is local to neighboring cells, so one might suppose that parts of the pattern sufficiently far apart should always be independent. … Thus, for example, with the constraint that every cell must have at least one neighbor whose color is different from its own, any of the patterns in the picture at the top of the facing page are allowed, as indeed is any pattern that involves no more than two successive cells of the same color.

But by the bottom of the first page all that remains of this region is three copies of a rather simple repetitive structure. Yet at the top of the next page the arrival of a diagonal stripe from the left sets off more complicated behavior again. … Will all the structures that are produced eventually annihilate each other, leaving only a very regular pattern?

But soon its rim becomes unstable, and several peaks (often with small drops at the top) appear in a characteristic coronet pattern. … But in any case a peak appears at the center, sometimes with a spherical drop at the top.

In the top rule, the new color of a particular cell is found simply by looking at that cell and its immediate neighbors above and to the right.

But as the table at the top of the next page shows, other square roots have much more complicated digit sequences. In fact, so far as one can tell, all whole numbers other than perfect squares have square roots whose digit sequences appear completely random.

Nevertheless, if one looks at the maximum number of steps needed for any given length of input one finds that this still always increases exactly linearly—just as for the Turing machines that add 1 shown at the top of this page. … All show the same linear growth in maximum number of steps as their size of input increases.

Implementation [of 3/2 system] The evolution for t steps of the system at the top of the page can be computed simply by NestList[If[EvenQ[#], 3#/2, 3(# + 1)/2] &, 1, t]