And in my experience by far the best first step in assessing a model is not to look at numbers or other details, but rather just to use one's eyes, and to compare overall pictures of a system with pictures from the model.

in the majority of cases the best evidence that some particular set of effects are in fact the important ones ultimately comes just from the success of models that are based on these effects.

And finally, as the results in Chapter 7 suggest, for anything beyond the very simplest forms of behavior, iterative random searches rapidly tend to get stuck, and make at best excruciatingly slow progress towards any kind of global optimum.

Almost always, it seems that the best strategy is a simple one: to come up with an appropriate general class of rules, and then just

But in trying to find an ultimate model of space, it seems best to start by considering networks that are somehow as simple as possible in basic structure—and it turns out that the networks of Chapter 5 are somewhat more complicated than is necessary.

And although cellular automata remain some of the very best examples, we will see that a vast range of utterly different systems all in the end turn out to exhibit extremely similar types of behavior.

And so, for example, there are many archeological structures—such as Stonehenge—where it is at best unclear which features were intended to be purposeful.

The best-known system was given by David Hilbert in 1899—and by describing geometrical figures using algebraic equations he showed that it was as consistent as the underlying axioms for numbers.

Linear and nonlinear systems A vast number of different applications of traditional mathematics are ultimately based on linear equations of the form u  m . v where u and v are vectors (lists) and m is a matrix (list of lists), all containing ordinary continuous numbers. … With vectors of length n it generically takes about n 2 steps to compute u given v , and a little less than n 3 steps to compute v given u (the best known algorithms—which are based on matrix multiplication—currently involve about n 2.4 steps). But as soon as the original equation is nonlinear, say u  m 1 . v + m 2 . v 2 , the situation changes dramatically.

One can imagine weighting different network distances differently, but usually I have found that equal weightings work best.