Note (d) for Randomness from the Environment…Mechanical randomness It takes only small imperfections in dice or roulette wheels to get substantially non-random results (see page 971 ). … But despite such problems mixing of objects remains by far the most common way to generate randomness when there is a desire for the public to see randomization occur. … There have been a steady stream of inventions for mechanical randomness generation.

Note (e) for The Intrinsic Generation of Randomness…Repeatably random experiments Over the years, I have asked many experimental scientists about repeatability in seemingly random data, and in almost all cases they have told me that they have never looked for such a thing. … Typically there are quite long periods of time where the behavior is rather accurately repeatable—even though it may wiggle tens or hundreds in a seemingly random way—interspersed with jumps of some kind. In most cases the only credible models seem to be ones based on intrinsic randomness generation.

[No text on this page] Examples of the evolution of continuous cellular automata from random initial conditions.

Note (a) for The Intrinsic Generation of Randomness…Card shuffling Another rather poor example of intrinsic randomness generation is perfect card shuffling. … Surprisingly enough, this simple procedure, which can be represented by the function s[list_] := Flatten[ Transpose[Reverse[Partition[list, Length[list]/2]]]] with or without the Reverse , is able to produce orderings which at least in some respects seem quite random.

Note (c) for The Intrinsic Generation of Randomness…Autoplectic processes In the 1985 paper where I introduced intrinsic randomness generation I called processes that show this autoplectic, while I called processes that transcribe randomness from outside homoplectic.

Note (c) for Randomness from the Environment…Physical randomness generators It is almost universally assumed that at some level physical processes must be the best potential sources of true randomness. … But in almost every case I know where detailed analysis has been done substantial deviations from perfect randomness have been found.

In class 3, the behavior is more complicated, and seems in many respects random, although triangles and other small-scale structures are essentially always at some level seen. And finally, as illustrated on the next few pages [ 236 , 237 , 238 , 239 ], class 4 involves a mixture of order and randomness: localized structures are produced which on their own are fairly simple, but these structures move around and interact with each other in very complicated ways.

The Notion of Attractors In this chapter we have seen many examples of patterns that can be produced by starting from random initial conditions and then following the evolution of cellular automata for many steps. … In random initial conditions, absolutely any sequence of black and white cells can be present.

Rule 126 with a typical random initial condition, and with an initial condition that consists of a random sequence of the blocks and .

Note (a) for Randomness in Class 3 Systems