Fibonacci Rectangles And Spirals

Fibonacci rectangles are those built to the proportions of consecutive terms in the Fibonacci Series. Because of the nature of the series, any Fib(n):Fib(n+1) rectangle can be divided exactly into the all the previous Fibonacci rectangles. For example, the diagram below shows a 34:21 rectangle within which is contained the 21:13, 13:8, 3:5, 5:3, 3:2, 2:1 and 1:1 rectangles. You will also notice that in doing so we have divided each of our rectangles into perfect squares of length of side Fib(k) for k in the natural numbers, less than or equal to n. (Note that  the numbers inside the squares show the length of the sides).

As n tends to infinity, the proportion of the rectangle will tend to Phi (the golden ratio). This is what we call the golden rectangle: supposedly the most aesthetically pleasing rectangle in existence. It can be approximated by any of the Fibonacci rectangles.

Also of interest is the Fibonacci spiral, constructed using the arcs of demi-semi-circles of radius Fib(n) for a all natural n up to a given point (depending on the required size/complexity of the spiral). The diagram here shows a spiral constructed for all n up to n=8. Fibonacci spirals occur widely in nature.