1- and 2-connection [network] clusters Clusters with just one or two dangling connections can always in effect be thought of just as adding extra structure to single connections in a network.

And these kinds of expressions often arise in Mathematica when one manipulates functions as a whole before applying them to arguments. ( ∂ xx f[x] for example gives f''[x] which is Derivative[2][f][x] .) (In principle one can imagine representing all objects with forms such as f[x, y] by so-called currying as f[x][y] , and indeed I tried this in the early 1980s in SMP.

History [of recursive sequences] The idea of sequences in which later terms are deduced from earlier ones existed in antiquity, notably in the method of induction and in various approximation schemes (compare page 918 ). … But almost without exception the emphasis was on studying what such functions could in principle do, not on looking at the actual behavior of particular ones.

History [of 2D cellular automata] As indicated on pages 876 – 878 , 2D cellular automata were historically studied more extensively than 1D ones—though rarely with simple initial conditions. The 5-cell neighborhood on page 170 was considered by John von Neumann in 1952; the 9-cell one on page 177 by Edward Moore in 1962.

If there are n nodes in such a network, then if any blocks are excluded, the shortest one of them must be of length less than n . … It is common to see such lengths progressively increase, although in principle they can decrease by as much as 2r from one step to the next.

But the remarkable fact that we see is that just by changing the pattern of Possible networks formed by having one, two or three nodes, with two connections coming out of each node.

And as a result, one can conclude that any of the very wide range of computations that can be performed by practical computers can also be done by cellular automata.

But with a total of 218 out of the 65,536 possible rules, one gets somewhat different behavior, as cases (g) and (h) below show.

Most such patterns look to us quite simple, but the examples shown here were specifically chosen to be ones that look more complicated.

But the table below demonstrates that as soon as one goes beyond the familiar traditions of language and mathematics there are other operators that can also just as well be used as primitives.