Growth rates [of halting times] Some Turing machine can always be found that has halting times that grow at any specified rate. (See page 103 for a symbolic system with halting times that grow like Nest[2 # &, 0, n] .) … The maximum halting times above increase faster than the halting times for any specific Turing machine, and are therefore ultimately not computable by any single Turing machine.

The picture shows the right-hand edge of the pattern; the complete pattern extends about 700 times the width of the page to the left.

Note that in the later cases shown, the head often visits the same position on the grid many times.

Financial Systems During the development of the ideas in this book I have been asked many times whether they might apply to financial systems. … Whether one looks at stocks, bonds, commodities, currencies, derivatives or essentially any other kind of financial instrument, the sequences of prices that one sees at successive times show some overall trends, but also exhibit varying amounts of apparent randomness.

Starting from the node in the middle, one can go around either the left or the right loop in the network any number of times in any order—representing the fact that black and white cells can appear any number of times in any order. … Starting at the right-hand node one can go around the loop to the right any number of times, corresponding to sequences of Four different initial conditions that all lead to the same final state in the rule 4 cellular automaton shown on the previous page .

And indeed what happens is similar to what we have seen many times in this book: the evolution of the cellular automaton generates enough randomness that the effects of the underlying grid tend to be washed out, with the result that the overall behavior produced ends up showing essentially no distinction between different directions in space. … Several times in the past ideas like this have been explored. … For as we have seen many times in this book, if one looks at systems like programs with discrete elements then it immediately becomes much easier for highly complex behavior to emerge.

Note that in practice a coin tossed in the air will typically turn over between ten and twenty times while a die rolled on a table will turn over a few tens of times. A coin spun on a table can rotate several hundred times before falling over and coming to rest.

Note that to generate the pictures that follow requires applying the underlying cellular automaton rule for individual cells a total of about 12 million times.

Note that to generate the pictures that follow requires applying the underlying cellular automaton rule for individual cells a total of about 12 million times.

Note that to generate the pictures that follow requires applying the underlying cellular automaton rule for individual cells a total of about 12 million times.