The computer to which I had access at that time was by modern standards a very primitive one. … Yet as it turns out, what I looked at was a particular case of one of the main kinds of systems—cellular automata—that I consider in this book. … What for example is the fundamental origin of the complicated patterns that one sees in turbulent fluids?

One immediate thing that might seem to suggest that elementary particles must somehow be based on simple discrete structures is the fact that their values of quantities like electric charge always seem to be in simple rational ratios. … But in terms of networks one can imagine a much more explicit explanation: that there are just a simple discrete set of possible structures for the cores of particles—each perhaps related in some quite mechanical way by the group of symmetry operations. … So at first, it might seem surprising that one can even set up a particular type of particle to move at different speeds.

If one did not know about the basic phenomenon of universality, then one would most likely assume that by using more complicated rules one would always be able to produce new and different kinds of behavior.

Yet despite all this, we do not in our everyday experience typically have much difficulty telling living systems from non-living ones. … Indeed, following the discoveries in this book I have come to the conclusion that almost any general feature that one might think of as characterizing life will actually occur even in many systems with very simple rules. … If one specifically tries to train an animal to solve

[Sounds based on] musical scores Instead of taking a sequence to correspond directly to the waveform of a sound, one can consider it to give a musical score in which each element represents a note of a certain frequency, played for some specific short time. (One can avoid clicks by arranging the waveform to cross zero at both the beginning and end of each note.) … (One can either determine frequencies of notes directly from the values of elements, or, say, from cumulative sums of such values, or from heights in paths like those on page 892 .)

One can imagine finding the outcome of evolution more efficiently by adding rules that specify what happens to larger blocks of cells after more steps. And as a practical matter, one can look up different blocks using a method like hashing. But much as one would expect from data compression this will only in the end work more efficiently if there are some large blocks that are sufficiently common.

Sequence equations One can ask whether by replacing variables by sequences one can satisfy so-called word or string equations such as Flatten[{x, 0, x, 0, y}]  Flatten[{y, x, 0, y, 1, 0, 1, 0, 0}] (with shortest solution x = {1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0} , y = {1, 0, 1, 0, 0, 1, 0, 1, 0, 0} ). Knowing about PCP and Diophantine equations one might expect that in general this would be undecidable.

At least at a basic level, to compute topological entropy one needs in effect to count every possible sequence that can be generated. But one can potentially get an estimate of measure entropy just by sampling possible sequences. One problem, however, is that even though such sampling may give estimates of probabilities that are unbiased (and have Gaussian errors), a direct computation of measure entropy from them will tend to give a value that is systematically too small.

One can also study directed percolation in which one takes account of the connectivity of cells only in one direction on the lattice.

Olbers' paradox asks why one does not see a bright star in every direction in the night sky. … Focusing a larger and larger distance away, the light one sees was emitted longer and longer ago. And eventually one sees light emitted when the universe was filled with hot opaque gas—now red-shifted to become the 2.7K cosmic microwave background.