The Golden Mean

The Golden Mean (or Golden Section), represented by the Greek letter phi, is one of those mysterious natural numbers, like e or pi, that seem to arise out of the basic structure of our cosmos. Unlike those abstract numbers, however, phi appears clearly and regularly in the realm of things that grow and unfold in steps, and that includes living things.The decimal representation of phi is

Number Series If you start with the numbers 0 and 1, and make a list in which each new number is the sum of the previous two, you get a list like this:0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144,... to infinity--> This is called a 'Fibonacci series'.

The Golden Section (or Golden Ratio)is formed when the larger of two consecutive Fibonacci is divided by the smaller giving the number as approximately 1.6e.g. 13 / 8 = / 13 = / 21 = 1.619Many things in nature and art (e.g. the Mona Lisa) are formed in the ratio of 1:1.6 and it is believed that shapes which are ” pleasing to the eye ” are formed according to this ratio.The ratio is also frequently found in music, architecture, art, biology etc.

The Golden Mean Pure Math Proportion is the relationship of the size of two things. Arithmetic proportion exists when a quantity is changed by adding some amount. Geometric proportion exists when a quantity is changed by multiplying by some amount.

Many books on oil painting and water colour in your local library will point out that it is better to position objects not in the centre of the picture but to one side or "about one-third" of the way across, and to use lines which divide the picture into thirds. This seems to make the picture design more pleasing to the eye and relies again on the idea of the golden section being "ideal".

The Annunciation by Leonardo Da Vinci is a picture that looks like it is in a frame of 1:sqrt(5) shape (a root-5 rectangle). Print it and measure it - is it a root-5 rectangle? Divide it into a square on the left and another on the right. (If it is a root-5 rectangle, these lines mark out two golden-section rectangles as the parts remaining after a square has been removed). Also mark in the lines across the picture which are 0·618 of the way up and 0·618 of the way down it. Also mark in the vertical lines which are 0·618 of the way along from both ends. You will see that these lines mark out significant parts of the picture or go through important objects. You can then try marking lines that divide these parts into their golden sections too.