secretes new shell material faster on one side than the other, causing the shell to grow in a spiral. The rates at which shell material is secreted at different points around the opening are presumably determined by details of the anatomy of the animal. … Case (a) is typical of a nautilus shell, (b) of a cone shell and (c) of one-half of a clam shell.

And in the case shown here the slow growth of the region visited by the active cell in the underlying evolution is reflected in rapid growth of the corresponding causal network. … But with different mobile automaton rules one can still already get tremendous diversity. … A one-dimensional mobile automaton which yields a causal network that in effect grows exponentially with time.

The actual numbers of functions which require 0, 1, 2, ... terms is for n = 2 : {1, 9, 6} ; for n = 3 : {1, 27, 130, 88, 10} , and for n = 4 : {1, 81, 1804, 13472, 28904, 17032, 3704, 512, 26} . … The reason for this is essentially that these functions are the ones that make the coloring of the Boolean hypercube maximally fragmented.

Projections from 3D [cellular automata] Looking from above, with closer cells shown darker, the following show patterns generated after 30 steps, by (a) the rule at the top of page 183 , (b) the rule at the bottom of page 183 , (c) the rule where a cell becomes black if exactly 3 out of 26 neighbors were black and (d) the same as (c), but with a 3×3×1 rather than a 3×1×1 initial block of black cells:

But experiments—the most direct of which are based on looking for quantization in the measured decay times of very short-lived particles—have only demonstrated continuity on scales longer than about 10 -26 seconds, and there is nothing to say that on shorter scales time is not in fact discrete.

An example suggested by Stanislaw Ulam around 1960 (in a peculiar attempt to get a 1D analog of a 2D cellular automaton; see pages 877 and 928 ) starts with {1, 2} , then successively appends the smallest number that is the sum of two previous numbers in just one way, yielding {1, 2, 3, 4, 6, 8, 11, 13, 16, 18, 26, 28, 36, 38, 47, 48, 53, 57, …} With this initial condition, the sequence is known to go on forever.

The last two pictures in each row above give the distribution of points whose coordinates in two and three dimensions are obtained by taking successive numbers from the linear congruential generator. If the output from the generator was perfectly random, then in each case these points would be uniformly distributed.

How can one avoid this? … But the details of how one does this tend to have a great effect on the results one gets. … With this procedure, evolution from any initial condition can visit every possible configuration, so that the configurations which satisfy the constraints will at least eventually be reached.

In the third picture, however, the points where the curve crosses the axis come in two regularly spaced families. And as the pictures on the facing page indicate, for any curve like Sin[x] + Sin[ α x] the relative arrangements of these crossing points turn out to be related to the output of a generalized substitution system in which the rule at each step is obtained from a term in the continued fraction representation of ( α – 1)/( α + 1) .

One way this can happen, illustrated in the first set of pictures below, is for the system to conserve some quantity—such as total density of black—and for this quantity to end up being spread uniformly throughout the system by its evolution. … And in general, in any system with definite rules that only ever visits a limited number of states, it is With each cell at each step having a gray level that is the average of its predecessor and its two neighbors the total amount of black is conserved, but eventually becomes spread uniformly throughout the system. … Such discrete transitions are somewhat less common in one dimension than elsewhere.