The fixed points of this procedure are the perfect numbers (see above ).

The paths are geodesics which go the minimum distance on the surface to get to all the points they reach.

Lagrange points and resonances often lead to simple geometrical patterns of orbiting bodies.

For strings the analogous problem is straightforward, since in a string of length n one can ultimately just try each of the n possible starting points for the substring and see for which of them a match occurs.

[Models involving] non-local processes It follows from the fact that any path in a finite network must always eventually return to a node where it has been before that any Markov process must be fundamentally local, in the sense that the probabilities it implies for what happens at a given point in a sequence must be independent of those for points sufficiently far away.

And in fact in general finding such packings is an NP-complete problem: it is equivalent to the problem of finding the maximum clique (completely connected set) in the graph whose vertices are joined whenever they correspond to grid points on which non-overlapping circles could be centered.

In practice, such computations are most often done by requiring explicit synchronization of all elements at appropriate points, and implementing this using a mechanism that is outside of the computation.

Generating textures As discussed on page 217 , it is in general difficult to find 2D patterns which at all points match some definite set of templates.

Indeed in many ways the only real difference is that instead of The digit sequences of positions of points on successive steps in the two examples of kneading processes at the bottom of the previous page .

In the center of the cellular automaton is then a cell whose possible colors correspond to possible points in the program for the register machine.