Fractal dimensions [of additive cellular automata] The total number of nonzero cells in the first t rows of the pattern generated by the evolution of an additive cellular automaton with k colors and weights w (see page 952 ) from a single initial 1 can be found using g[w_, k_, t_] := Apply[Plus, Sign[NestList[Mod[ ListCorrelate[w, #, {-1, 1}, 0], k] &, {1}, t - 1]], {0, 1}] The fractal dimension of this pattern is then given by the large m limit of Log[k,g[w, k,k m + 1 ]/g[w, k, k m ]] When k is prime it turns out that this can be computed as d[w_, k_:2] := Log[k,Max[Abs[Eigenvalues[With[ {s = Length[w] - 1}, Map[Function[u, Map[Count[u, #] &, #1]], Map[Flatten[Map[Partition[Take[#, k + s - 1], s, 1] &, NestList[Mod[ListConvolve[w, #], k] &, #, k - 1]], 1] &, Map[Flatten[Map[{Table[0, {k - 1}], #} &, Append[#, 0]]] &, #]]] &[Array[IntegerDigits[#, k, s] &, k s - 1]]]]]]] For rule 90 one gets d[{1, 0, 1}] = Log[2, 3] ≃ 1.58 .

One is to toss an object and see which way up or where it lands; the other is to select an object from a collection mixed by shaking.

And in fact it appears that the only parameter values that are not covered are ones where the shell could not easily have been secreted by an animal because its shape is degenerate and leaves little useful room for the animal.

In his 1952 paper Turing used a finite difference approximation to a pair of diffusion equations to show that starting from a random distribution of concentration values dappled regions could develop in which one or the other chemical was dominant.

Mechanistic models [in physics] Until quite recently, it was generally assumed that if one were able to get at the microscopic constituents of the universe they would look essentially like small-scale analogs of objects familiar from everyday life.

One can make arguments that composite particles this small should have masses much larger than are observed—but it is easy to find theories that avoid these.) … Since the 1960s it has been noted in many simplified quantum field theories that there can be a kind of duality in which a soliton or other extended field configuration in one representation becomes what acts like an elementary particle in another representation.

And at one time or another almost every major department of my company has provided help that has been crucial to some aspect of the creation or production of this book. … But over the years what has also been important is that I have interacted personally with a great many individuals, and I have been fortunate that my position in science and technology has brought me into contact at one time or another with the leaders of almost every major technical field.

After essentially one stop, most of these go to the primary visual cortex at the back of the brain, which itself contains more than 100 million nerve cells.

Encodings [of cellular automaton rules] Generalizing the setup in the main text one can say that a cellular automaton i can emulate j if there is some encoding function ϕ that encodes the initial conditions a j for j as initial conditions for i , and which has an inverse that decodes the outcome for i to give the outcome for j .

These include one-of-a-kind beings that have nothing to communicate with, as well as beings with inner intellectual activity but no effect on the outside world.