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Much has been said about random walks, mostly in the form of bringing up the term and not caring to define precisely what "random" means. Barring those cases when "stochastic" and "random" are used interchangeably, true randomness implies absence of any order definable by the brain (see also psychology's contributions to creative personality and random number generation).

But since any numerical structure is a derivative of an underlying quantity evaluation system (e.g. number system), in mentioning "random numbers" one usually means that which cannot be ordered in any way meaningful to a human brain - or, philosophically, a recognition of some things beyond our cognitive horizons.

Let P be a stochastic/probability space and consider a function (X->Y) such as that of a straight line or a parabola (Copyright Google Images)

In both cases the output (Y-values) of the functions is perfectly predictable from the explanatory (\predictor) variable. Thus, the function shapes the probability space in an easily expressible way. At the same time, if the function's influence on the shape of random variable Y is not ubiquitous (i.e. if after sufficient "zooming in" the shape of probability space is not that determined by the function), the said probability space P has already been transformed by some other function.

Fractals' Matter

Randomness, in the way humans are interested in it, states that and only that there is no further classification allowed of the output given only the fact that it has been transformed by function f, some other function (g) has already transformed the said probability space P.

Thus, all shapes that are not fractal constitute a mix of more than 1 function. Pure randomness, restricted to a relatively narrow discourse, means "no discernible ubiquitous shape has been applied except a set of those defining the said discourse". Given a probability shape defining a particular dscourse, no further transformation is allowed if only random sequences are desired. Hence, a flawless random number generator is discourse-specific and preserves the discourse-defining shape imposition.

Hence, fractals

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