But while the first arrangement of colors shown below looks somewhat random, the last two are simple and purely repetitive.

So what about other choices of constraints? We have seen in this book many examples of systems where simple sets of rules give rise to highly complex behavior. But what about systems based on constraints? Are there simple sets of constraints that can force complex patterns?

It turns out that in one-dimensional systems there are not. For in one dimension it is possible to prove that any local set of constraints that can be satisfied at all can always be satisfied by some simple and purely repetitive arrangement of colors.

But what about two dimensions? The proof for one dimension breaks down in two dimensions, and so it becomes at least conceivable that a simple set of constraints could force a complex pattern to occur.

As a first example of a two-dimensional system, consider an array of black and white cells in which the constraint is imposed that every black cell should have exactly one black neighbor, and every white cell should have exactly two white neighbors.

A system consisting of a line of black and white cells whose form is defined by the constraint that every cell should have at least one neighbor whose color is different from its own. There are many possible arrangements of colors that satisfy this constraint. Some, like the first arrangement above, look quite random. But others, like the second two arrangements above, are simple and repetitive. It turns out that in a one-dimensional system no set of local constraints can force arrangements of more complicated types.

A system consisting of a grid of black and white cells defined by the constraint that every black cell should have exactly one black neighbor among its four neighbors, and every white cell should have exactly two white neighbors. The infinite repetitive pattern shown here, together with its rotations and reflections, is the only one that satisfies this constraint. (The picture is assumed to wrap around at each edge.) The pattern can be viewed as a tessellation of 5×5 blocks of cells.