And to try to derive them all from the kinds of models I have outlined here will certainly take an immense amount of work.

It turns out that using an extension of the argument above it is always possible to take the rules An example of how the color of any square in a nested pattern can be found from its coordinates by a fairly simple mathematical procedure.

The arithmetic system takes the value n that it obtains at each step, computes Mod[n, 30] , and then depending on the result applies to n one of the arithmetic operations specified by the rule above.

(Even if one can do operations on all digits in parallel it still takes of order n steps in a system like a cellular automaton for the effects of different digits to mix together—though see also page 1149 .) … Any iterative procedure (such as FindRoot ) that yields a constant multiple more digits at each step will take about Log[n] steps to get n digits. … The best-known algorithms for evaluating Zeta[1/2 +  x] (see page 918 ) to fixed precision take roughly √ x operations—or 2 n/2 operations if x is an n -digit integer.

In most cases, this search will be done using some iterative scheme such as Newton's method; the result is that the boundaries between regions typically take on an intricate nested form. … But if an iterative scheme for minimization is used, these watersheds are typically no longer sharp, but take on a local nested structure, much as in picture (c) above.

With this setup successive steps in the evolution of the system can be obtained from GMAStep[rules_, {list_, nlist_}] := Module[{a, na}, {a, na} = Transpose[Map[Replace[Take[list, {# - 1, # + 1}], rules]&, nlist]]; {Fold[ReplacePart[#, Last[#2], First[#2]]&, list, Transpose[{nlist, a}]], Union[Flatten[nlist + na]]}]

And thus for example whenever there is a structure containing s identical cells (as on page 462 ), this typically takes about s 2 steps to decay away.

Category theory can be viewed as a formalization of operations on abstract data types in computer languages—though unlike in Mathematica it normally requires that functions take a single bundle of data as an argument.

One can also study directed percolation in which one takes account of the connectivity of cells only in one direction on the lattice.

An example is rule 110, in which repetitive domains form with period 14 in space and 7 in time, but as the second picture below illustrates, the localized structures which separate these domains take a very long time to disappear.