Perhaps some interpretation can be made involving eddies existing only in a fractal region, or interacting with each other as well as branching.

Then the rules for the language consisting of balanced runs of parentheses (see page 939 ) can be written as {s[e]  s[e, e], s[e]  s["(", e, ")"], s[e]  s["(",")"]} Different expressions in the language can be obtained by applying different sequences of these rules, say using (this gives so-called leftmost derivations) Fold[# /. rules 〚 #2 〛 &, s[e], list] Given an expression, one can then use the following to find a list of rules that will generate it—if this exists: Parse[rules_, expr_] := Catch[Block[{t = {}}, NestWhile[ ReplaceList[#, MapIndexed[ReverseRule, rules]] &, {{expr, {}}}, (# /. … Given only the rules for a context-free language, it is often very difficult to find out the properties of the language (compare page 944 ).

Elementary ( k = 2 , r = 1 ) cellular automata can be found only up to separations s = 2 . But k = 2 , r = 2 cellular automata can be found for all separations up to 15, as well as 17, 19 and 23. … Of the 4 billion k = 2 , r = 2 cellular automata none turn out to be able to produce for example sequences corresponding to the cubes, powers of 3, Fibonacci numbers, primes, digits of √ 2 , or concatenation sequences.

The idea that as a matter of principle there should be truths in mathematics that can only be reached by some form of inductive reasoning—like in natural science—was discussed by Kurt Gödel in the 1940s and by Gregory Chaitin in the 1970s.

Pictures (a) and (b) below however correspond to the n = 3 multiplication tables {{1, 1, 3}, {3, 3, 2}, {2, 2, 1}} and {{3, 1, 3}, {1, 3, 1}, {3, 1, 2}} . … If the group is Abelian, so that f[i, j]  f[j, i] , then only nested patterns are ever produced (see page 955 ). … The group used is S 3 , which has six elements and multiplication table {{1, 2, 3, 4, 5, 6}, {2, 1, 5, 6, 3, 4}, {3, 4, 1, 2, 6, 5}, {4, 3, 6, 5, 1, 2}, {5, 6, 2, 1, 4, 3}, {6, 5, 4, 3, 2, 1}} The initial condition contains {5, 6} surrounded by 1 's.

But in their usual form, they yield essentially only rather simple repetitive patterns.

In the mid-1960s David Raup used early computer graphics to generate pictures for various ranges of parameters, but perhaps because he considered only specific classes of molluscs there emerged from his work the belief that parameters of shells are greatly constrained—with explanations being proposed based on optimization of such features as strength, relative volume, and stability when falling through water.

The notion that diffusion might be important in embryo development had been suggested in the early 1900s (see page 1004 ), but it was only in 1952 that Alan Turing showed how it could lead to the formation of definite patterns.

Occasionally there have been mechanistic descriptions used—as in the parton and bag models, and various continuum models of high-energy collisions—but they have typically been viewed only as convenient rough approximations.

But it was only in the mid-1800s that there started to be real evidence for the existence of some kind of discrete atoms of matter. … (Actual experiments based on high-energy scattering and precision magnetic moment measurements have shown only that electrons and muons must have sizes smaller than about ℏ c/(10 TeV) ≃ 10 -20 m —or about 10 -5 times the size of a proton.