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Notice that the rectangles which result at each stage are all roughly the same shape, that is, that the ratio of length to width seems to settle down as we build the pattern outward. We may discover this number by exploiting the recursive definition of the Fibonacci sequence in the following way.

A rectangle whose sides had this proportion was called the Golden Rectangle. And not just the rate of growth, but the pattern of growth.

The Fibonacci sequence makes its appearance in other ways within mathematics as well.

 

The Elliott wave theory is the basis of a technical analysis technique for predicting the behavior and market trends in the stock market, invented by Ralph Nelson Elliott in 1939. It is based on the belief that markets exhibit well-defined wave patterns that can be used to predict market direction: specifically that stock prices are governed by cycles which adhere to the Fibonacci sequence 0, 1, 1, 2, 3, 5, 8, 13, 21, etc. It claims that the stock market, acting as a meter for mob or crowd psychology, displays many of the same geometric features as other organic structures. Proponents of the Elliott wave theory claim that the pattern is exhibited repeatedly in past market price patterns, and that the fractal nature of such patterns creates a repetition of them on varying levels of order and magnitude.


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