Spinodal decomposition The separation into progressively larger black and white regions seen in the cellular automata shown here is reminiscent of the phenomena that occur for example in the separation of randomly mixed oil and water.

Shapes of [biological] cells Many types of cells are arranged like typical 3D packings of deformable objects (see page 988 )—with considerable apparent randomness in individual shapes and positions, but definite overall statistical properties.

[Spectra of] random block sequences Analytical forms for all but the last spectrum are: 1 , u 2 /(1 + 8u 2 ) , 1/(1 + 8 u 2 ) , u 2 , (1 - 4u 2 ) 2 /(1 - 5u 2 + 8u 4 ) , u 2 /(1 - 5u 2 + 8u 4 ) , u 2 + 1/36 DiracDelta[ ω - 1/3] , where u = Cos[ π ω ] , and ω runs from 0 to 1/2 in each plot. … For a random walk (see page 977 ) in which ± 1 occur with equal probability the spectrum is Csc[ π ω ] 2 /2 , or roughly 1/ ω 2 . The same basic setup also applies to spectra associated with linear filters and ARMA time series processes (see page 1083 ), in which elements in a sequence are generated from external random noise by forming linear combinations of the noise with definite configurations of elements in the sequence.

At least in the case of lightning, there is some evidence that small inhomogeneities in the atmosphere can be important in producing at least some aspects of the apparent randomness that is seen.

Most randomly chosen PDEs appear, however, to have no such conserved quantities.

I originally studied rule 30 in the context of basic science, but I soon realized that it could serve as the basis for practical random sequence generation and cryptography, and I analyzed this extensively in 1985. … Rule 30 has been widely used for random sequence generation, but for a variety of reasons I have not in the past much emphasized its applications in cryptography.

But as soon as one perturbs such initial conditions, one normally seems to get only complicated and seemingly random behavior, as in the top row of pictures in the second image.

Often it is assumed that different parts of a texture are statistically independent, so that the texture can be characterized by probabilities for local patterns, as in a so-called Markov random field or generalized autoregressive moving average (ARMA) process.

First, that starting from random initial conditions, cellular automata could organize themselves to produce complex patterns. … But at the time, I did not study this picture in detail, and I tacitly assumed that whenever I saw randomness it must come from the random initial conditions that I used. … Partly as an application for this computer I then ended up making a detailed study of rule 30 and its randomness—among other things proposing it as a practical random sequence generator and cryptosystem.

A system based on a constraint, in which a complex and largely random pattern is forced to occur.