In the one specific case shown at the top of the facing page it turns out to be fairly easy. … So to determine whether a particular square in the pattern is black or white, all one need do is to compute the corresponding binomial coefficient, and see whether or not it is an odd number. … Much as in the top picture numbers can be assigned to each square, but now these numbers are computed by successively adding together triples rather Algebraic representations of the patterns on the facing page .

The pictures at the top of the page show the base 2 digit sequences of successive numbers obtained by iterating this mapping, while the pictures in the middle of the page plot the sizes of these numbers. In all cases, the initial conditions consist of the number 1/2—which has a very simple digit sequence.

The pictures at the top of the next page show one possible approach based on always arranging the nodes in each network in a line across the page. … As in all the other networks shown, each node here is identical, and has just two connections coming out of it.

But if each picture were extended sufficiently far to the left, then all these lines would eventually be seen to start from the top.

But in fact what we will find in this chapter is that many systems spontaneously tend to organize themselves, so that even with completely random initial conditions they end up producing behavior that has many features that are not at all random. The picture at the top of the next page shows as a simple first example a cellular automaton which starts from a typical random initial condition, then evolves down the page according to the very simple rule that a cell becomes black if either of its neighbors are black. What the picture then shows is that every region of white that exists in the initial conditions progressively gets filled in with black, so that in the end all that remains is a uniform state with every cell black.

The pictures below show several examples, and in all cases the procedures are fairly straightforward. … In the top row of examples, the initial condition for the substitution system is a single black square, and the start state for the finite automaton is also its black state.

From the point of view of statistical analysis, a sequence is perfectly random if it is somehow consistent with a model in which all possible sequences occur with equal probability. … The pictures at the top of the next page show the results of computing the frequencies of different blocks in various sequences, and in each case each successive row shows results for all possible blocks of a given length. The gray levels on every row are set up so that the average of all possible sequences corresponds to the pattern of uniform gray shown below.

As the pictures at the top of the facing page demonstrate, the overall patterns produced in all cases tend to look complex, and in many respects random. … So despite their apparent complexity, all the patterns on the facing page can in effect be described by simple traditional mathematical formulas. … Is it possible that in the end all such patterns could just be described by simple mathematical formulas?

But k = 2 , r = 2 cellular automata can be found for all separations up to 15, as well as 17, 19 and 23. … If one looks not just at specific sequences, but instead at all 2 n possible sequences of length n , one can ask how many cellular automaton rules (say with k = 2 , r = 2 ) one has to go through in order to generate every one of these. … (Note that the sequence is the first one that cannot be generated by any of the 256 elementary cellular automata; the first sequence that cannot be generated by any k = 2 , r = 2 cellular automata is probably of length 26.)

Polynomial value sets Closely related to issues of solving Diophantine equations is the question of what set of positive values a polynomial can achieve when fed all possible positive integer values for its variables. … Nevertheless, from the representation for PrimeQ in the note above it has been shown that the positive values of a particular polynomial with 26 variables, 891 terms and total degree 97 are exactly the primes.