Yet as it turns out, few regularities have in fact been found, and often the results that have been established tend only to support the idea that the sequence has many features of randomness. … But in fact with considerable effort it has been proved that all of them are in a sense more random—and eventually cross the axis an infinite number of times, and indeed go any distance up or down.

The procedure starts from a random configuration of squares, and then at each step picks a square at random, then reverses the color of this square whenever doing so reduces the total number of squares that violate the constraint.

Indeed, the main difference is just that in engineering explicit human effort is expended to find an appropriate form for the system, whereas in natural selection an iterative random search process is used instead. … And more often it is probably the result of combinations of large numbers of elements that each produce fairly random behavior.

One might nevertheless imagine that it would be possible to devise a complicated machine, perhaps with an elaborate arrangement of paddles, that would still be able to extract systematic mechanical work even from an apparently random distribution of particles. … The result is that in practice it is never possible to build perpetual motion machines that continually take energy in the form of heat—or randomized particle motions—and convert it into useful mechanical work.

An argument for the Second Law from around 1900, still reproduced in many textbooks, is that if a system is ergodic then it will visit all its possible states, and the vast majority of these will look random. … The argument suffers however from the same difficulties as the ones for chaos theory discussed in Chapter 6 and does not in the end explain in any real way the origins of randomness, or the observed validity of the Second Law. With the Second Law accepted as a general principle, there is confusion about why systems in nature have not all dissipated into complete randomness.

Properties [of network systems] Random behavior seems to occur in a few out of every thousand randomly selected rules of the kind shown here.

And the picture on the facing page then shows how a cellular automaton can retrieve data from a numbered location in what is effectively a random-access memory.

A typical example of the behavior of rule 110 with random initial conditions.

If one imagines a uniform slope with discrete streams of water going randomly in each direction at the top, and then merging whenever they meet, one immediately gets a simple tree structure a little like in the pictures at the top of page 359 . (More complicated models based for example on aggregation, percolation and energy minimization have been proposed in recent years—and perhaps because most random spanning trees are similar, they tend to give similar results.) As emphasized by Benoit Mandelbrot in the 1970s and 1980s, topography and contour lines (notably coastlines) seem to show apparently random structure on a wide range of scales—with definite power laws being measured in quite a few cases.

For the vast majority of rules written down at random, such problems do indeed occur. … But in the other cases, the fluctuations seem instead in many respects random.