In time there will doubtless also be all sorts of additional material and educational options available. … At first the best thing is probably just to repeat some of the experiments I describe in this book—using the software and resources described at the website, or perhaps just by typing in some of the programs in these notes. … For someone to assimilate all of the new kind of science I describe in this book will take a very significant time.

In the top row of pictures—as well as picture (a)—all one sees is a collection of discrete particles bouncing around.

Material about the programs should be available at the book website—including for example some of the automated tests run to check the programs, as well as annotations about how the programs work.

Will it die out?

At first one might think that one could set up some kind of analog of a cellular automaton and just replace all relevant clusters of nodes at once. … In each pair of pictures in the upper part of the page , the top one shows the form of the network before the replacement, and the bottom one shows the result after doing the replacement—with the cluster of nodes involved in the replacement being highlighted in both cases. In the 3D pictures in the lower part of the page , networks that arise on successive steps are shown stacked one on top of the other, with the nodes involved in each replacement joined by gray lines.

And indeed if space has appropriate curvature one can get all sorts of paths, as in the pictures below. … In each case all the paths shown start parallel, but do not remain so when there is curvature. … Case (b) shows the top of a sphere, which is a surface of positive curvature.

These values can be found using the so-called Lucas-Lehmer test Nest[Mod[# 2 - 2, 2 n - 1] &, 4, n - 2]  0 , and in all cases n itself must be prime. … For odd n up to 500 million the only values near 0 that appear in the curve are {-6, -5, -4, -2, -1, 6, 18, 26, 30, 36} , with, for example, the first 6 occurring at n = 8925 and last 18 occurring at n = 159030135 .

In the top four cases, the pattern produced ultimately has a simple nested form.

Thus, for example, the pictures at the top of the facing page show how the base 10 and base 2 digit sequence representations of π can be used to construct the number π . Digit sequences for cube roots, fourth roots, logarithms and exponentials, given at the top in base 10 and the bottom in base 2.

Already in the early 1980s, however, my studies on simple additive and other cellular automata (see page 26 ) had for example made it rather clear that this is not the case. … This is a k = 8 2D cellular automaton in which toppling of sand above a critical slope is captured by updating an array of relative sand heights s according to the rule SandStep[s_]:= s + ListConvolve[ {{0, 1, 0}, {1, -4, 1}, {0, 1, 0}}, UnitStep[s - 4], 2, 0] Starting from any initial condition, the rule eventually yields a fixed configuration with all values less than 4, as in the picture below. … The pictures at the top of the next page show some of the final fixed configurations, together with the number of steps needed to reach them.