[Biological] species One feature of biological organisms is that they normally occur in discrete species, with distinct differences between different species.

Directed network systems If one adds directionality to the connections in a network it becomes particularly easy to set up rules for clusters of nodes that cannot overlap.

In rule 154R, each diagonal stripe is followed by at least one 0; otherwise, the positions of the stripes appear to be quite random, with a density around 0.44.

The minimal solutions can be large; the largest ones for successive coefficient sizes are given below. … If one can find a bound on solutions—say by Baker's method—then one can also try to show that no values below this bound are actually solutions. … But as one continues the enumeration there are increasingly a few equations that seem more and more difficult to handle.

The straight lines in the diagrams represent electrons; the wavy ones photons. … To work out the total probability for a process from Feynman diagrams, what one does is to find the expression corresponding to each diagram, then one adds these up, and squares the result. … Since for QED α ≃ 1/137 , one might expect that this would give quite an accurate result—and indeed experiments suggest that it does.

The facing page and the one that follows show as an example the cellular automaton that we first discussed on page 32 .

In the third picture, however, one no longer sees such regularity, and instead there is behavior that seems in many respects random.

I know of one class of examples where this happens, illustrated in the pictures on the next page .

A similar scheme to the one for mobile automata can also be used for Turing machines, as illustrated in the pictures below.

The key unifying idea that has allowed me to formulate the Principle of Computational Equivalence is a simple but immensely powerful one: that all processes, whether they are produced by human effort or occur spontaneously in nature, can be viewed as computations.