Examples Of Golden Ratio

Introducing the Golden Ratio

Refer to the first image. The long blue line is broken into two unequal lengths “A” and “B”, so the total length is “A+B”.

It happens that the ratio of A:B is exactly equal to the ratio (A+B):A. This is the “Golden Ratio”, which also goes by the name “Golden Mean”.

Of course, a line can be broken into any pair of lengths…preferably values from the “positive real” set of numbers. It happens that this is special for any number of reasons.

The numeric value is approximately 1.618, but there is much more to it than that.

The natural world provides examples of this approximate ratio. Consider a person’s height compared to the area in which the waist “cuts” the torso where a person bends at the waist. The head-to-navel length is “B”, the navel-to-toe length is “A”. For most people, the A:B ratio is nearly equal to the (A+B):A ratio. In other words, full height is to toe-to-navel length as toe-to-navel length is to navel-to-scalp length.

Greek architects and sculptors found that this is an aesthetically pleasing way to create buildings and art. In particular, columns and buildings were constructed with a visual break in the height that corresponded to this ratio.

The Greek geometer Euclid was perhaps the first to write about this ratio. His compatriot Plato apparently believed that this number had mystic properties.

Let’s thank William Harris of Middlebury College whose “ The Golden Mean ” gives a very succinct explanation which is re-phrased here.

From the first image, the Golden Ratio equals the ratio of (A+B):A and also A:B. Arbitrarily, we can let the shorter part, B, have the value of ’1′. So then (A+1)/A = A. Multiply both sides by A, so then A+1 = A*A. Then A*A – A – 1 = zero.

The quadratic equation solves “a*(x*x) + b*x + c = zero” with two answers: “x = ( (-b) + square_root(b*b – 4*a*c) ) / (2 * a) as well as  “x = ( (-b) – square_root(b*b – 4*a*c) ) / (2*a).

The explicit values of a,b,c from “A*A – A – 1 = zero” are shown as “1*A*A + (-1)*A – 1″, so a=1, b=-1, c=-1. So one solution is (-(-1) + square_root((-1 * -1) – 4*1*(-1)) ) / 2 = (1 + square_root(1 + 4) ) / 2 = ( square_root(5) + 1 ) / 2.

Nature, Fibonacci Numbers and the Golden Ratio — World Mysteries Blog

Nature, Fibonacci Numbers and the Golden Ratio

The Fibonacci numbers are Nature’s numbering system. They appear everywhere in Nature, from the leaf arrangement in plants, to the pattern of the florets of a flower, the bracts of a pinecone, or the scales of a pineapple. The Fibonacci numbers are therefore applicable to the growth of every living thing, including a single cell, a grain of wheat, a hive of bees, and even all of mankind. Part 1. Golden Ratio & Golden Section, Golden Rectangle, Golden Spiral

The Golden Ratio is a universal law in which is contained the ground-principle of all formative striving for beauty and completeness in the realms of both nature and art, and which permeates, as a paramount spiritual ideal, all structures, forms and proportions, whether cosmic or individual, organic or inorganic, acoustic or optical; which finds its fullest realization, however, in the human form.  –Adolf Zeising