Consequences of an ultimate model [of physics] Even if one knows an ultimate model for the universe, there will inevitably be irreducible difficulty in working out all its consequences.

This history makes it seem more plausible that one might be able to come up with an ultimate model of physics on largely aesthetic grounds, rather than mainly by working from detailed experimental observations.

In the late 1800s it was noted in recreational mathematics that one could find the value of π by looking at randomly dropped needles. … (In the 1960s it had been noted that one can factor polynomials by filling in random integers for variables and factoring the resulting numbers.) … And the same is true when one picks unique IDs, say to keep track of repeat web transactions with a low probability of collisions.

An initial condition consisting of n white cells with one black cell in the middle can then be obtained with the function (see below for comments on this and other Mathematica functions) CenterList[n_Integer] := ReplacePart[Table[0, {n}], 1, Ceiling[n/2]] For cellular automata of the kind discussed in this chapter, the rule can also be represented by a list. … But since the C program explicitly updates values sequentially from left to right, the left-hand neighbor of a particular cell will already have been given its new value when one tries to updates the cell itself. … Since in a practical computer one can use only a finite array of cells, one must decide how the cellular automaton rule is to be applied to the cells at each end of the array.

Fourier transforms In a typical Fourier transform, one uses basic forms such as Exp[  π r x/n] with r running from 1 to n . … Applying BitReverseOrder to this matrix yields a matrix which has an essentially nested form, and for size n = 2 s can be obtained from Nest[With[{c = BitReverseOrder[Range[0, Length[#] - 1]/ Length[#]]}, Flatten2D[MapIndexed[#1 {{1, 1}, {1, -1} (-1)^c 〚 Last[#2] 〛 } &, #, {2}]]] &, {{1}}, s] Using this structure, one obtains the so-called fast Fourier transform which operates in n Log[n] steps and is given by With[{n = Length[data]}, Fold[Flatten[Map[With[ {k = Length[#]/2}, {{1, 1}, {1, -1}} .

If one ignores the first component of the spectrum the remainder is flat for a constant sequence, or for a random sequence in the limit of infinite length. … By adding a suitable constant to each element one can then arrange in such cases for the whole spectrum to be flat.

In the most general case one can imagine directly exchanging a representation of a program, that is run on the computer that receives it, and induces whatever effect one wants.

If there was no merging, then if a typical state yielded more than one new state, then inevitably the total number of states would increase exponentially. … With most rules, states that appear at one step can disappear at later steps.

Models of crystal growth There are two common types of models for crystal growth: ones based on the physics of individual atoms, and ones based on continuum descriptions of large collections of atoms.

For any given rule one can define the neighborhood size s to be the largest block of cells that is ever needed to determine the color of a single new cell. … There are only 8 rules (the inequivalent ones being 16740555 and 3327051468) where s > s , and in each case s = 6 while s = 5 .