Additive [cellular automaton] rules Of the 256 elementary cellular automata 8 are additive: {0, 60, 90, 102, 150, 170, 204, 240} . … Of all k k 2r + 1 rules with k colors and range r it turns out that there are always exactly k 2r + 1 additive ones—each obtained by taking the cells in the neighborhood and adding them modulo k with weights between 0 and k - 1 . … Note that each step in the evolution of any additive cellular automaton can be computed as Mod[ListCorrelate[w, list, Ceiling[Length[w]/2]], k] (See page 1087 for a discussion of partial additivity.)

Rule 22 [with simple initial conditions] Randomness is obtained with initial conditions consisting of two black squares 4 m positions apart for any m ≥ 2 . The base 2 digit sequences for 19, 25, 37, 39, 41, 45, 47, 51, 57, 61, ... also give initial conditions that yield randomness. … The overall density of black cells is not 1/2 but is instead approximately 0.35, just as for random initial conditions.

On this page and the ones that follow [ 165 , 166 ] the initial conditions used are u=Exp[-x 2 ] , D[u,t]=0 .

Lorentzian spaces In ordinary Euclidean space distance is given by Sqrt[x 2 + y 2 + z 2 ] . In setting up relativity theory it is convenient (see page 1042 ) to define an analog of distance (so-called proper time) in 4D spacetime by Sqrt[c 2 t 2 - x 2 - y 2 - z 2 ] . … Then one defines a cone of height t whose apex is a given point to be those points whose displacement vector v satisfies 0 > e . g . v > -t (and 0 > v . g. v ).

At each step there is a number x between 0 and 1 that is updated by applying a fixed mapping. … The pictures at the top of the page show the base 2 digit sequences of successive numbers obtained by iterating this mapping, while the pictures in the middle of the page plot the sizes of these numbers. In all cases, the initial conditions consist of the number 1/2—which has a very simple digit sequence.

In case (b), it can move between 0, 1 or 2 positions, while in case (c) it can move any distance between 0 and 1 at each step.

A consequence is that for example 4 steps of evolution can be computed not only as h[r, h[r, h[r, h[r, s]]]] but also as h[h[h[r, r], h[r, r]], s] or u = h[r, r]; h[h[u, u], s] —which requires only 3 applications of h . And in general if h is associative the result Nest[h[r, #]&, s, t] of t steps of evolution can be rewritten for example using the repeated squaring method as h[Fold[If[#2  0, h[#1, #1], h[r, h[#1, #1]]] &, r, Rest[IntegerDigits[t, 2]]], s] which requires only about Log[t] rather than t applications of h . … One can compute the result of 9 steps of evolution as 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 , but a better scheme is to use partial results and compute successively 1 + 1; 2 + 2; 1 + 4; 5 + 5 —which is what the repeated squaring method above does when h = Plus , r = s = 1 .

Gaussian distributions typically arise when measurements involve sums of random quantities; other common distributions are obtained from products or other simple combinations of random quantities, or from the results of simple processes based on random quantities.

[Subset of] elementary rules The examples shown have rule numbers n for which IntegerDigits[n, 2, 8] matches {_, i_, _, j_, i_, _, j_, 0} .

Lengths of [number] representations (a) n , (b) Floor[Log[2, n] + 1] , (c) Tr[FixedPointList[Max[0, Ceiling[Log[2, #]]] &, n + 2]] - n - 3 , (d) 2 Ceiling[Log[3, 2n + 1]] , (e) Floor[Log[GoldenRatio, √ 5 (n + 1/2)]] . Large n approximations: (a) n , (b) Log[2, n] , (c) Log[2, n] + Log[2, Log[2,n ]] + … , (d) 2 Log[3, n] , (e) Log[GoldenRatio, n] .