## Systems Based on Constraints

In the course of this book we have looked at many different kinds of systems. But in one respect all these systems have ultimately been set up in the same basic way: they are all based on explicit rules that specify how the system evolves from step to step.

In traditional science, however, it is common to consider systems that are set up in a rather different way: instead of having explicit rules for evolution, the systems are just given constraints to satisfy.

As a simple example, consider a line of cells in which each cell is colored black or white, and in which the arrangement of colors is subject to the constraint that every cell should have exactly one black and one white neighbor. Knowing only this constraint gives no explicit procedure for working out the color of each cell. And in fact it may at first not be clear that there will be any arrangement of colors that can satisfy the constraint. But it turns out that there is—as shown below.

And having seen this picture, one might then imagine that there must be many other patterns that would also satisfy the constraint. After all, the constraint is local to neighboring cells, so one might suppose that parts of the pattern sufficiently far apart should always be independent. But in fact this is not true, and instead the system works a bit like a puzzle in which there is only one way to fit in each piece. And in the end it is only the perfectly repetitive pattern shown below that can satisfy the required constraint at every cell.

Other constraints, however, can allow more freedom. Thus, for example, with the constraint that every cell must have at least one neighbor whose color is different from its own, any of the patterns in the picture at the top of the facing page are allowed, as indeed is any pattern that involves no more than two successive cells of the same color.

A system consisting of a line of black and white cells whose form is defined by the constraint that every cell should have exactly one black and one white neighbor. The pattern shown is the only possible one that satisfies this constraint. The idea of implicitly determining the behavior of a system by giving constraints that it must satisfy is common in traditional science and mathematics.