The pictures at the top of the next page show how averages of successively larger collections of uniformly distributed numbers converge to a Gaussian distribution.

The pictures at the top of the facing page show two very simple examples.

Because of additivity it turns out that one can deduce whether or not some cell a certain number of steps down a given column is black just by seeing whether there are an odd or even number of black cells in certain specific positions in the row at the top.

The top row of pictures start from the repetitive base 2 digit sequence of x=3/5 ; the bottom row of pictures from x=π/4 .

With k = 3 , there are 19,683 possible rules, 1734 of which are fundamentally inequivalent, and many more complicated patterns are seen, as in the pictures at the top of the next page.

On an infinitely long interface, protrusions of cells with one color into a domain of the opposite color get progressively smaller, eventually leaving only a certain pattern of cells in the layer immediately on one side of the interface. 90° corners in an otherwise flat interface effectively act like reflective boundary conditions for the layer of cells on top of the interface.

The fundamental assumption—that in my approach is just a consequence of basic properties of causal networks—is that the photon always goes at the speed of light, so that its path always lies on the surface of light cones like the ones in the top row of pictures.

And for example any of the three pictures at the top of the next page could potentially be referred to as either "quite random" or "quite complex".

The picture at the top of the facing page is one example.

With short initial conditions, the pictures at the top of the next page demonstrate that combinators tend to evolve quickly to simple fixed points.