The pictures below show which cells occur in more than 10% of 1000 randomly grown clusters.

Successive steps in the iterative procedure used on this page are given by Move[list_] := (If[Cost[#] < Cost[list], #, list] &)[ MapAt[1 - # &, list, Random[Integer, {1, Length[list]}]]] while those in the procedure on page 347 have ≤ in place of < .

On such a grid it is possible to randomize microscopic particle motions, but nevertheless conserve overall momenta.

In addition, my 1984 identification of rule 30 as a randomness generator was the result of a small-scale systematic search. … (Plant and animal breeding can be viewed as a simple form of randomized search done since the dawn of civilization.)

The Riemann Hypothesis then states that all r[i] satisfy Re[r[i]]  1/2 , which implies a certain randomness in the distribution of prime numbers, and a bound of order √ n Log[n] on PrimePi[n] - LogIntegral[n] . … The statistical distribution of zeros was studied by Andrew Odlyzko and others starting in the late 1970s (following ideas of David Hilbert and George Pólya in the early 1900s), and it was found that to a good approximation, the spacings between zeros are distributed like the spacings between eigenvalues of random unitary matrices (see page 977 ).

(Already in the late 1940s John von Neumann had suggested using x  4x (1 - x) as a random number generator, commenting on its extraction of initial condition digits, as mentioned on page 921 .) … Various features of randomness such as uniform distribution were established, and connections to smooth iterated maps emerged after the development of symbolic dynamics in the late 1930s.

The digit is 0 when the number is even and 1 when it is odd, and, as shown, the digits alternate in a seemingly random way.

And indeed the plots on the facing page show various features of this sequence which indicate that it is in many respects quite random.

And thus, for example, essentially as a consequence of randomness generation, a wide range of cellular automata show the simple density diffusion law on page 464 —whether or not their underlying rules happen to be simple.

But whenever there is significant randomness produced by the evolution of the multiway system, this does not typically appear to occur.