Golden-Section

The Golden Section & Fibonacci Numbers
by David Migliore

The ‘Golden Section’ spirals appearing in my drawings are derived from a series of numbers named after Leonardo Fibonacci, a mathematician who discovered what are now called the ‘Fibonacci Numbers’ or ‘Fibonacci Series’:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55..... (the series goes on forever). These numbers are derived very simply…

Since ‘1’ is the sum of the two digits preceding it (0 and 1), it becomes the next number in the series, so now we have:

The number that comes after ‘1’ is the sum of the two digits preceding it (1 and 1). Since 1 + 1 = 2, the series becomes:

The number that comes after ‘2’ is the sum of the two digits preceding it (1 and 2). Since 1 + 2 = 3, the series now becomes:

The number that comes after ‘3’ is the sum of the two digits preceding it (2 and 3). Since 2 + 3 = 5, the series now becomes:

Using this series of numbers, one can construct the Golden Section spiral using a series of squares. Each square is larger than the next (except for the first two squares, which are the same size) and its size in relation to the last square is a ratio of two consecutive Fibonacci numbers. So if we draw a square with sides equaling 1”, the next square is 1”. The second 1” square is drawn right next to the first 1” square. Right after the second 1” square is a 2” square. Then a 3” square. Then 5”. Then 8” and so on. You get a 1-1-2-3-5-8 Fibonacci sequence of squares:

Within each square, a quarter-circle is drawn. This quarter circle will connect to the next (larger) quarter circle and soon a spiral forms. So the quarter circle from the 1” square meets up with the quarter circle from the next 1” square, which meets up with the quarter circle from the 2” square. Then the quarter circle from the 3” square, then 5”, then 8”:

These Golden Sections further distinguish themselves from randomly-drawn spirals in that they mimic nature. One example is the nautilus shell (or snail shell). The spiral on this shell approximates a Golden Section.

This Golden Section also forms in the growth patterns of plants (number of petals a flower has, the pattern of leaf growth on a plant stem or pine cone) as well as the birth patterns/family trees of animals like rabbits.

Much of the knowledge gained about this topic is from Dr. Ron Knott, Former Lecturer in the Department of Mathematics and Computing Sciences, University of Surrey, Guildford, Surrey, UK. The easiest way to find Dr. Knott’s work is to enter his name in Google. The first hit will be his University of Surrey website, which is a treasure of information on Fibonacci Numbers and the Golden Section.