Space-filling curves One can conveniently scan a finite 2D grid just by going along each successive row in turn. One can scan a quadrant of an infinite grid using the σ function on page 1127 , or one can scan a whole grid by for example going in a square spiral that at step t reaches position (1/2(-1) # ({1, -1}(Abs[# 2 - t] - #) + # 2 - t- Mod[#, 2]) &)[ Round[ √ t ]]

And even if one can already see the result of an experiment in a picture in this book, it has been my consistent observation that one internalizes results of experiments much better if one gets them by running a program oneself than if one just sees them printed in a book. … With the appropriate setup, one can immediately run a program to find out. Often one will have some kind of guess about what the answer should be.

Some Conclusions In the chapter before this one, we discovered the remarkable fact that even though their underlying rules are extremely simple, certain cellular automata can nevertheless produce behavior of great complexity. … And one might think that features like these could be crucial in making it possible to produce complex behavior from simple underlying rules. … But to get complexity in the overall behavior of a system one needs to go beyond some threshold in the complexity of its underlying rules.

But as the picture at the bottom of the facing page shows, one can equally well use other bases. … But if one looks not at these overall sizes, but rather at digit sequences, then what one sees is considerably more complicated.

But now, with the discovery that simple programs can capture the essential mechanisms for all sorts of complex behavior in nature, one can imagine just sampling such programs to explore generalizations of the forms we see in nature. Traditional scientific intuition—and early computer art—might lead one to assume that simple programs would always produce pictures too simple and rigid to be of artistic interest. … But what the discoveries in this book now show is that by using the types of rules embodied in simple programs one can capture many of the essential mechanisms of nature.

But in every other case the constraints can be satisfied, though typically by just one or sometimes two simple infinite repetitive patterns. … But among the 3,527,988,252 that remain, it turns out that every single one can be satisfied by a simple repetitive pattern. In fact the number of different repetitive patterns that are ever needed is quite small: if a particular constraint can be satisfied by any pattern, then one of the set of 171 repetitive patterns on the next two pages [ 214 , 215 ] is always sufficient.

But if one is not constrained by the need for such analysis, then as we saw in the cellular automaton example above, remarkably simple rules can successfully generate highly random behavior. … For if the only way for intrinsic randomness generation to occur was through very complicated sets of rules, then one would expect that this mechanism would be seen in practice only in a few very special cases. … And as a result, one can expect that the mechanism will be found often in nature.

But when one is dealing with the basic structure of organisms, the vast majority of programs sampled at random will no doubt have immediate disastrous consequences. … And insofar as this is the case the results of this book should allow one to develop some fairly general characterizations of what can happen. … Yet what we now see is that in fact such analogies may be quite direct—and that many of the most obvious features of actual biological organisms may in effect be direct reflections of typical behavior that one sees in simple programs.

For purposes of display the ring of active material is unrolled into a line, and successive states of this line are shown one on top of each other going up the page. … So what happens if one changes the details of the model? … But in the majority of cases one sees rather rapid convergence to almost precisely 137.5°.

But what happens if one has two particles that are moving with different velocities? What will the events associated with the second particle look like if one takes slices through the causal network so that the first particle appears to be at rest? … For in effect any node that was associated with the particle on either one slice or the next must be updated—and the more the particle moves, the less these will overlap.