And indeed this phenomenon has been seen in other systems with randomness in their underlying rules. … The idea of this model is to add cells to a cluster one at a time, and to determine where a cell will be added by seeing where a random walk that starts far from the cluster first lands on a square adjacent to the cluster. An example of the behavior obtained in this model is shown below: The lack of smooth overall behavior in this case can perhaps be attributed to the global probing of the cluster that is effectively done by each incoming random walk.

And in the first case, starting from random initial conditions, the system quickly settles down to the all black invariant state. But in the second case, nothing like this happens, and instead the system continues to exhibit complicated and seemingly random behavior forever. … But if one starts these rules from random initial conditions, one typically never gets the pattern of page 211 .

And by the late 1970s it had become popular to believe that the randomness in fluid turbulence was somehow associated with this phenomenon. … And as I argue in the main text, the chaos phenomenon in the end seems quite unlikely to explain most of the randomness we see in turbulence. … Even within the Lorenz equations, however, one can see evidence of intrinsic randomness generation, in which randomness is produced without any need for randomness in initial conditions.

But many are globular, and have at least a core in which the 3D packing of amino acids seems quite random. … But other parts—often including sites important for function—seem more like random walks. … And at some level this is presumably why there seems to be so much randomness in their shapes.

Each panel shows 500 steps of evolution, and rapid randomization is evident.

But to make it work there needs to be some source of appropriate randomness. … I suspect, however, that in fact the most important source of randomness in most cases will instead be the phenomenon of intrinsic randomness generation that I first discovered in systems like the rule 30 cellular automaton. … But with this amount of computation there are many ways to generate random bits.

But what about random initial conditions? … The behavior of rule 110 starting from random initial conditions.

Common approaches are to assume that the surfaces are random with some frequency spectrum, or can be generated as fractals using substitution systems with random parameters.

But if one considers something random then usually one will also consider it meaningless. … Yet there are still cases where things that are presumably quite random are considered meaningful—prices in financial markets being one example.

Central Limit Theorem Averages of large collections of random numbers tend to follow a Gaussian or normal distribution in which the probability of getting value x is Exp[-(x - μ ) 2 /(2 σ 2 )] / (Sqrt[2 π ] σ ) The mean μ and standard deviation σ are determined by properties of the random numbers, but the form of the distribution is always the same. The only conditions are that the random numbers should be statistically independent, and that their distribution should have bounded variance, so that, for example, the probability for very large numbers is rapidly damped.