Explicit simple candidates include k = 2 , r = 2 rules with codes 20 and 52, as well as the various k = 3 , r = 1 class 4 rules shown in Chapter 3 .

Scanning the digit sequences from the left, one starts with 0 open parentheses, then adds 1 whenever corresponding digits in the x and y coordinates differ, and subtracts 1 whenever they are the same.

Nand theorems The total number of expressions with n Nand s and s variables is: Binomial[2n, n]s n + 1 /(n + 1) (see page 897 ).

With more colors or more states, the percentage of rules that yield non-repetitive behavior steadily increases, as shown below, roughly like 0.28 (s - 1) (k - 1) .

If the coordinates along a path are given by an expression s (such as {t, 1 + t, t 2 } ) that depends on a parameter t , and the metric at position p is g[p] , then the length of a path turns out to be Integrate[Sqrt[ ∂ t s . g[s] . ∂ t s], {t, t 1 , t 2 }] and geodesics then correspond to paths that extremize this quantity. … If one draws a circle of radius r on a page, then the smaller r is, the more curved the circle will be—and one can define the circle to have a constant curvature equal to 1/r . … One can then compute the Ricci tensor (R ik = R ijk j ) using RicciTensor = Map[Tr, Transpose[Riemann, {1, 3, 2, 4}], {2}] and this has 1/2 d(d + 1) independent components in d > 2 dimensions.

1/ n expansion [and networks] If there are n possible colors for each connection in a network, then for large n it turns out that the vast majority of networks will be planar.

Implementing boundary conditions [in cellular automata] In the bitwise representation discussed on page 865 , 0's outside of a width n can be implemented by applying BitAnd[a, 2 n -1] at each step.

Other uniformly distributed sequences Cases in which Mod[a[n], 1] is uniformly distributed include √ n , n Log[n] , Log[Fibonacci[n]] , Log[n!]

For rule 30, h μ tx < 1.155 , and there is some evidence that its true value may actually be 1. For rule 18 it appears that h μ tx = 1 , while for rule 22, h μ tx and for rule 54 h μ tx .

[Networks generated by] random replacements As indicated in the note above, applying the second rule (T1, shown as (b) on page 511 ) at an appropriate sequence of positions can transform one planar network into any other with the same number of nodes. … (For large n this is approximately λ n with λ = 3/4 ; if 1- and 2-edged regions are allowed then λ = (3 + √ 3 )/6 ≃ 0.79 .) … Another result that has been derived is that the average total number m[n] of edges of all faces around a given face with n edges is 7n + 3 + 9/(n + 1) .