What is the golden proportion
The golden ratio is a measure often found in Architecture, art and nature. The ratio is related to the fibonacci series of numbers, where each number is the sum of the previous two: 0 – 1 – 1 – 2 – 3 – 5 – 8 – 13 – 21 – etc. With larger numbers, the ratio between two consecutive numbers approximates the golden ratio (Phi=1.618..). (Dividing the smaller of the two by the larger, you get the inverse of phi, 0,618. So for me, the golden ratio can be used either direct as 1.618.. or as the invers, which is 0.618.
Why is it important
Objects and distributions that are set up along the golden ratio are often pleasing to the eye, and for a good basis for composing your bonsai. Is it a proven way used in design (Both in art as well as architecture) to create pleasing objects. Whether it is the ratio between the base of the pyramids and their height, or when composing pictures; The golden ratio is part of it. In photography this is more well-known as the rule of thirds: Place your object at roughly one third from the left/right/top or bottom of your image. And yes.. 1 : 1.618 is roughly a 1 : 2/3 division of your image.
How can I use it
You can use this in different ways. One of the best known, and most universally know ratio’s in bonsai is that “The first branch of the tree is at about 1/3 of the tree height”. Or in other words: “The trunk length is about 1/3 of the total tree height.”.
Give me an example
In John Naka’s Bonsai Techniques 1 he gives a nice example of how one can use this. It directly uses the golden ratio to determine the height / branch position.