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.