In the pictures below the liquid is divided into cells, with each cell having a temperature from 0 to 1, corresponding exactly to a continuous cellular automaton of the kind discussed on page 155 . … If the temperature of any cell exceeds 1, then only the fractional part is kept, as in the systems on page 158 , representing the consumption of latent heat in the boiling process.

In fact, much as for cellular automata, more explicit experiments have been done on 2D Turing machines than 1D ones. … The specific 4-state rule {s_, c_}  With[{sp = s (2c - 1)  }, {sp, 1 - c, {Re[sp], Im[sp]}}] has been called Langton's ant, and various studies of it were done in the 1990s.

Typically the network topology of a foam continually rearranges itself through cascades of seemingly random T1 processes (rule (b) from page 511 ), with regions that reach zero size disappearing through T2 processes (reversed rule (a)). … Something similar is already visible in the pure T1 pictures in the note above. … But this can again be thought of as a combination of T1 and T2 processes, and in appropriate idealizations can lead to very similar results.)

In case (d), the fluctuation in each f[n] turns out to be essentially just the number of 1's that occur in the base 2 digit sequence for n . And in case (c), the fluctuations are determined by the total number of 1's that occur in the digit sequences of all numbers less than n .

And this implies that in choosing initial conditions for a system like the shift map, one should therefore make no distinction between the exact number 1/2 and numbers that are sufficiently close in size to 1/2 .

For it turns out that an angle between successive elements of about 137.5° is equivalent to a rotation by a number of turns equal to the so-called golden ratio (1+Sqrt[5])/2 ≃ 1.618 which arises in a wide variety of mathematical contexts—notably as the limiting ratio of Fibonacci numbers.

With 0 representing white, 1 gray and 2 black, the rightmost element of the rule gives the result for average color 0, while the element immediately to its left gives the result for average color 1/3—and so on.

Primes have divisors 1 and n only.

For much as on page 943 , one can imagine setting up a 1D cellular automaton with the property that, say, the absence of a particular color of cell throughout the 2D pattern formed by its evolution signifies satisfaction of the constraints. But even starting from a fixed line of cells, the question of whether a given color will ever occur in the evolution of a 1D cellular automaton is in general undecidable, as discussed in the main text.

In such a rule, given a list of how many neighbors around a given cell (out of s possible) make the cell turn black the outer totalistic code for the rule can be obtained from Apply[Plus, 2^Join[2 list, 2 Range[s + 1] - 1]]