In the top picture of each pair every individual update is indicated by a black dot. … Note that in this representation, effects can propagate all the way across the system in a single step. … The top row shows evolution with the boundary forced to be white; the bottom row shows cyclic boundary conditions.

So this means that the complete evolution can be represented as in the picture at the top of the facing page , with each sequence shown explicitly only once, and any sequence generated more than once indicated just by an arrow going back to its first occurrence.

The picture at the top left shows the action of a single Nand function.

The overall structure of space that emerges may be complicated, and there may be objects that end up moving at all sorts of speeds. … So from this one can use essentially standard arguments to derive all the various phenomena familiar from ordinary relativity theory. … The black line is always assumed to be moving at the speed of light—so that it always lies on the surface of a light cone, as indicated in the top row of pictures.

The region begins much like region (a), except that the four localized structures at the top are more narrowly spaced. Starting around the middle of the region, however, the behavior becomes quite different from region (a): while region (a) yields an object that allows information to pass through, region (g) yields one that stops all information, as shown in regions (h) and (i). … The pictures on the last few pages [ 683 , 684 , 685 , 686 ] were all made for a cyclic tag system with a specific underlying rule.

But while there are a few other initial conditions for which differences can die out after several steps most forms of averaging will say that the majority of initial conditions lead to growing patterns of differences.

The pictures at the top of the facing page show what happens if one uses as the initial conditions for this system two numbers whose sizes differ by just one part in a billion billion. … For as the pictures at the top of the facing page show, the fact that the numbers which are used as initial conditions differ only by a very small amount in size just means that their first several digits are the same.

The network built up by the evolution of the multiway system from the top of the page .

And as examples of these, the pictures at the top of the facing page show networks that are specifically constructed to correspond to ordinary one-, two- and three-dimensional space. … The first line shows all possible networks with up to four nodes.

But if one tries to keep all edges the same length the surface will inevitably become curved—like a soccer ball or a geodesic dome. … If every region in the network is in effect a hexagon—as in the picture at the top of the page—then the network will behave as if it is flat.