What is the Golden Ratio

Uppercase and lowercase Greek letter phi, the ...

Golden rectangle

In a golden rectangle, it can be divided into a square and a smaller rectangle. The ratio of the width of the small rectangle to the width of the square is the same as that of the width of the square to the entire rectangle. In fact, when you divide a golden rectangle into a square and smaller rectangle, the smaller rectangle is a golden rectangle as well.

Golden ration in Nature

nautilus shell showing the chambe...

The most common example of the golden ratio is the nautilus shell (see image). As it spirals in on itself, the spirals get smaller and smaller in the same proportion to each other as they do to the whole. You can also see the ratio in things like sunflower petals, and the curvature of fern fronds (see image).

Let’s look at the golden ratio, a little bit. You can verify these observations algebraically.

  1. What is one over the golden ratio? 1 / 1.61803398875 . . .=0.61803398875 . . . Does that answer look familiar? It is one less than the golden ratio.
  2. What is the golden ratio squared? (1.61803398875 . . .)2=2.61803398875 . . . Does that answer look familiar? It is one more than the golden ratio.
  3. I was trying to investigate chaos, and I drew the graph of y=x2-1. This parabola intersects the graph of the line y=x in two points. What are the coordinates of those two points? The points are (-0.61803398875 . . .,-0.61803398875 . . .) and (1.61803398875 . . .,1.61803398875 . . .). All four of those numbers should look familiar by now. Of course, the x and y coordinates are identical, for both points, as the points are on the line y=x.

Here is an infinite series which evaluates to the golden ratio:


Notice the alternating signs. Each denominator is the product of two consecutive Fibonacci numbers. Where did I get this series? I deduced it from the fact that the ratio of consecutive Fibonacci numbers approaches the golden ratio, as the numbers get larger. The sequence of ratios of consecutive Fibonacci numbers is 1/1, 2/1, 3/2, 5/3, 8/5, 13/8,… We get the second element of this sequence with this series: 1+1/1·1. The third element is 1+1/1·1-1/1·2. The fourth is 1+1/1·1-1/1·2+1/2·3, and so on. This series was known long before I discovered it.

The golden ratio can be represented as the simplest continued fraction (as shown on the left). This fraction is 1+1/a where a is 1+1/a. You probably evaluate such a continued fraction from the right side, and work toward the left, except that it has infinitely many terms. We can get a sequence from this by starting at the left, and evaluating only part of the continued fraction: 1, 2/1, 3/2, 5/3, 8/5, 13/8… These are the ratios of consecutive Fibonacci numbers.

The golden ratio can be represented as a repeated square root, as in the diagram at the right. Evaluate this from right to left, too.