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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.
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