Introducing the Golden Ratio

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What is the Golden Ratio?

“Golden Ratio AB” Image by Mike DeHaan

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 History of the Golden Ratio

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.

“Golden Ratio Body” Image by Mike DeHaan

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.

The Algebra Behind the Geometry of the Golden Ratio

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.

The other solution would have been ( 1 – square_root(5) ) / 2. This is a negative number, which does not help when dealing with lengths of lines.

Therefore, the Golden Mean’s exact value is “one-half of ( ( the square root of 5 ) plus 1 )”.

The Golden Ratio and Fibonacci Numbers

“Fibonacci Series and Golden Ratio” Image by Mike DeHaan

The Fibonacci sequence is {0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144…}. Starting with the numbers zero and one, the next number is the sum of the previous two numbers. The “nth” Fibonacci number is written as x[n], where x[n] = x + x.

There is a surprising relationship between the Fibonacci sequence and the Golden Ratio. The result of dividing x/x[n] becomes very close to the Golden Ratio as “n” becomes larger. In fact, as early as 13/8, the result is less than 1% off from the Golden Ratio. For 89/55, the error drops under 1% of 1%. This is an incredibly “easy” way to calculate the Golden Ratio without determining the square root of 5.

Yet Another Golden Surprise

Earlier, we had noted (A+1)/A = A. Solving the equation another way, we start by distributing the division on the left hand side. Then “(A/A + 1/A) = A”, or “1 + 1/A = A”. Reading from right to left, it says that the number A is equal to its own reciprocal, plus 1.

Sure enough, 1/1.618 = 0.618. This is a fun exercise on a scientific calculator that has a “1/x” function.

As a matter of notation, the lower-case Greek letter “phi” is drawn to indicate the Golden Ratio of 1.618; the capitalized PHI indicates the inverse, 0.618.

“Golden Ratio Chart” Image by Mike DeHaan

A Series of Golden Ratios

This chart shows a sequence of vertical lines. Each line is 1.618 times taller than its neighbor to the left.

 

References:

William Harris (Professor Emeritus), Middlebury College, “THE GOLDEN MEAN”, referenced May 30, 2011.

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