And indeed I suspect this will happen many times over the years to come.

In modern times it has been almost unheard of for genuinely new science to be presented for the first time in a book that can be read by non-scientists.

And ever since ancient Greek times iterative methods have been used to construct geometrical figures. … (Less formal versions of the idea were also common in ancient Greek times.) … The notion of using definite rules to organize and maneuver formations of soldiers appears to have existed in Babylonian and Assyrian times, and to be well codified by Roman times.

But imposing the constraint that a walk must always avoid anywhere it has been before (as for example in an idealized polymer molecule) leads to correlations over arbitrary times.

For the multiplication rules discussed in the main text both states and rules can immediately be represented by integers, with h = Times , and r = m giving the multiplier. … Typically the issue is whether h[a, b] for large a and b can be found with much less effort than it would take to evaluate h[r, b] about a times. If h = Times , then as discussed in the note above, the most obvious procedure for evaluating h[a, b] would involve about m n operations, where m and n are the numbers of digits in a and b .

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.

Thus for example, however many times one runs a rule 30 cellular automaton, starting with a single black cell, the behavior one gets will always be exactly the same.

The answer, as we have seen many times in this book, is that across a very wide range of programs there is great universality in the behavior that occurs.

The specific value of the entropy will depend on what measurements one makes, but the content of the Second Law is that if one repeats the same measurements at different times, then the entropy deduced from them will tend to increase with time.

For as we have seen many times in this book, there is often a great distance between underlying rules and overall