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. But more theoretical investigations of formal concurrent systems, temporal logics, dataflow systems, Petri nets and so on have led to ideas about distributed computing that are somewhat closer to the ones I discuss here for the universe.

Moire patterns The pictures below show moire patterns formed by superimposing grids of points at different angles. … The second row of pictures illustrates what happens if points closer than distance 1/ √ 2 are joined.

The first six give basic properties of betweenness of points and congruence of line segments. … The axioms given can prove most of the results in an elementary geometry textbook—indeed all results that are about geometrical figures such as triangles and circles specified by a fixed finite number of points, but which do not involve concepts like area.

When I have discussed models like the ones in this chapter with other scientists I have however often encountered great confusion about such issues. … Thus, for example, a cellular automaton can readily be set up to represent the effect of an inhibition on growth at points on the surface of a snowflake where new material has recently been added. … But the whole point of a model is to have a simplified representation of a system, from which those features in which one is interested can readily be deduced or understood.

At first one might think that it must be a consequence of nature somehow intrinsically following mathematics. … But one of the starting points for the science in this book is that when it comes to more complex behavior mathematics has never in fact done well at explaining most of what we see every day in nature. … And indeed if one looks at a presentation of almost any piece of modern mathematics it will tend to seem quite complex.

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. … One-dimensional cellular automata are especially convenient generators of distinctive textures.

. • (a) At step t , the only new string produced is the one containing t black elements. • (b) All strings of length n containing exactly one black cell are produced—after at most 2n - 1 steps. • (c) All strings containing even-length runs of white cells are produced. • (d) The set of strings produced is complicated. The last length 4 string produced is , after 16 steps; the last length 6 one is , after 26 steps. • (e) All strings that begin with a black element are produced. • (f) All strings that end with a white element but contain at least one black element, or consist of all white elements ending with black, are produced. … Those of length n appear after at most 3n - 3 steps. • (l) The same strings as in (k) are produced, taking now at most 2n + 1 steps. • (m) All strings beginning with a black element are produced, after at most 3n + 1 steps. • (n) The set of strings produced is complicated, and seems to include many but not all that do not end with . • (o) All strings that do not end in are produced. • (p) All strings are produced, except ones in which every element after the first is white.

And for the particular mobile automaton rule used here, the network one gets ends up being highly regular, as illustrated in pictures (h) and (i). … And the connections between nodes in the network can then be thought of as defining the pattern of neighbors for points in spacetime. But unlike in the space networks that we discussed two sections ago , the connections in the causal networks we consider here always go only one way: each connection corresponds to a causal relationship in which one event leads to another, but not the other way around.

And so when clusters of nodes that are nearby with respect to connections on the network get updated, they can potentially propagate effects to what might be considered distant points in space. … And particularly in a causal network as regular as the one on page 518 one can then immediately view each connection in the causal network as corresponding to an effect propagating a certain distance in space during a certain interval in time. … One might imagine that its connections could perhaps represent varying distances in space and varying intervals in time.

In general the density for an arrangement of white squares with offsets v is given in s dimensions by (no simple closed formula seems to exist except for the 1 × 1 case) Product[With[{p = Prime[n]}, 1 - Length[Union[Mod[v, p]]]/p s ], {n, ∞ }] White squares correspond to lattice points that are directly visible from the origin at the top left of the picture, so that lines to them do not pass through any other integer points.