Backtracking [in cellular automata] If one wants to find out which of the 2 n possible initial conditions of width n evolve to yield a specific column of colors in a system like an elementary cellular automaton one can usually do somewhat better than just testing all possibilities. … If one wants to find just a single initial condition that works then one can set up a recursive algorithm that in effect does a depth-first traversal of the tree.

In most cases, however, introducing these kinds of slightly more complicated encodings does not fundamentally seem to expand the set of rules that a given rule can emulate. But often it does allow the emulations to work with smaller blocks. … Convolutional codes (related to sequential cellular automata) are the other major class of codes studied in coding theory, but in their usual form these do not seem especially useful for our present purposes.

Gödel's paper does this first for the statement "this statement is unprovable", and much of the paper is concerned with showing how such a statement can be encoded within arithmetic. Gödel in effect does this by first converting the statement to one about recursive functions and then—by using tricks of number theory such as the beta function of page 1120 —to one purely about arithmetic. … He suggested that his results could be avoided if some form of transfinite hierarchy of formalisms could be used, and appears to have thought that at some level humans and mathematics do this (compare page 1167 ).

This does not seem to happen for rules that involve 4 neighbors, but with 8 neighbors there are cases in which clusters can get fairly large, but end up having no sites where further cells can be added. … With the rule illustrated above, however, those clusters that do successfully grow exhibit complicated and irregular shapes, but nevertheless eventually seem to take on a roughly circular shape, as in the pictures below. … But most generalized aggregation models do not have this property: instead, the form of their internal patterns depends on the sequence of random choices made.

Like essentially all forms of science, however, what I do in this book is done in a rational tradition—with limited relation to the more mystical traditions of Eastern thinking.

And indeed most ways of ensuring that these do not occur are in essence equivalent just to saying that the overall behavior exhibits some specific regularity and is therefore not complex.

But as their names suggest Simplify and FullSimplify were intended to be less predictable—and just to do what they can and then return a result.

But if, as in the past, one tries to do computer experiments on continuous mathematical systems, then the situation can be different.

Fluttering If one releases a stationary piece of paper in air, then unlike a coin, it does not typically maintain the same orientation as it falls.

To do this dynamically is difficult, but in a perfect crystal final patterns of dislocations that remain at the edge of a region affected by fracture can be seen for example by electron diffraction.