that no fixed point is reached, and instead there is exponential growth in total size—with apparently rather random internal behavior.

The apparent randomness of these patterns reflects the likely difficulty of the problem.

The method of averaging to correct for what were assumed to be random errors of observation began to be used, primarily in astronomy, in the mid-1700s, while least squares fitting and the notion of probability distributions became established around 1800. Probabilistic models based on random variations between individuals began to be used in biology in the mid-1800s, and many of the classical methods now used for statistical analysis were developed in the late 1800s and early 1900s in the context of agricultural research.

Constructing patterns in which templates occur with definite densities is also difficult, although randomized iterative schemes allow some approximation to be obtained.

And with surprising regularity it is assumed that random variations in such data follow a Gaussian distribution (see page 976 ).

If a large amount of numerical data has been made up by a person this can be detectable through statistical deviations from expected randomness—particularly in structural details such as frequency of digits.

In the late 1970s, particularly after the work of Fischer Black and Myron Scholes on options pricing, new models of markets based on methods from statistical physics began to be used, but in these models randomness was taken purely as an assumption. In another direction, it was noticed that dynamic versions of game theory could yield iterated maps and ordinary differential equations which would lead to chaotic behavior in prices, but connections with randomness in actual markets were not established.

Despite the simplicity of its definition, the digit sequence of π seems for practical purposes completely random.

In fact, so far as one can tell, all whole numbers other than perfect squares have square roots whose digit sequences appear completely random.

And indeed even when such growth leads to a certain amount of apparent randomness it typically in the end seems to exhibit some form of repetition.