In most traditional computer languages it is usually the case that most programs chosen at random will just crash if run, often as a result of trying to write to memory outside the arrays they have allocated. In Mathematica, there is almost no similar issue, and programs chosen at random tend instead just to return unchanged. … For the kinds of systems like cellular automata that I have discussed in this book programs chosen at random do very often produce some sort of non-trivial behavior.

And in the first million steps in this sequence, for example, it never repeats, and indeed none of the tests I have ever done on it show any meaningful deviation at all from perfect randomness. In a sense, however, there is a certain simplicity to such perfect randomness.

Indeed, for the first few hundred steps there is a region that seems essentially random. … Or will more and more structures appear until the whole pattern becomes quite random?

And still others have leaves with complex and seemingly somewhat random boundaries. … And there are patterns with intricate and seemingly somewhat random shapes that look like sycamore or grape leaves.

As one example, I discovered the classification scheme for cellular automata with random initial conditions described at the beginning of Chapter 6 when I first looked at large numbers of different cellular automata together on high-resolution graphics displays. Similarly, I discovered the randomness of rule 30 (page 27 ) when I was in the process of setting up large simulations for an early parallel-processing computer.

Determinism in brains Early investigations of internal functioning in the brain tended to suggest considerable randomness—say in the sequence of electrical pulses from a nerve cell.

Computer printouts The printouts show a series of elementary cellular automata started from random initial conditions (see page 232 ).

It is not too difficult to make various statements about details of the particular arrangements of creases that can occur, but nothing seems to be known about the origin of the overall randomness that is almost universally seen.

The sequence expands by at least one digit every two steps; more rapid expansion is typically correlated with increased randomness. … If one always includes one new digit on the left at every step, even when it is 0, then a rather random pattern is produced.

And from the results shown one sees that in some cases random-looking ground states should occur. Note that a rather different way to get a somewhat similar ground state is to consider a spin glass, in which the standard Ising model energy function is used, but multiplied by -1 or +1 at random for each spin.