Golden Ratio

Is there an underlying order to all of nature? This is an ancient question with many answers from “cause and effect” to Einstein’s 4-dimensional space-time. One of the most popular, and still somewhat mysterious relationships hiding in basic mathematics and natural forms all around us is the Golden Ratio. Also known as “Phi” (a Greek letter) or “the Divine Proportion”, the Golden Ratio has been attracting interest for nearly 2,500 years.

It was the Greek Euclid who first defined the Golden Ratio clearly in writing, but his predecessors Proclus, Plato, and Pythagoras probably knew about it, too. This excites people because the Golden Ratio pops up in the construction in many geometrical figures, as if indeed a “divine” principle behind all geometric forms. Countless people claim to have found the Golden Ratio throughout human architecture, biology, and art, sparking the idea that it is in some sense the most beautiful proportion to the human eye. Others claim that half the spottings of Phi in human culture are delusional; you can always find what you’re looking for!

You have probably seen this illustration of Phi before:

Phi is the ratio between the long and short legs of any of these rectangles. This illustration should remove any mystery about its essential nature:

Phi is the ratio between a + b and a, and the ratio between a and b; that is the key to its elegance – dividing a line so that the ratio between the whole and its longer piece is the same as the ratio between its two pieces. You can use the construction shown here to derive and solve an n equation for Phi, which then turns out to be 1.6180339 . . . ad infinitum; it continues on forever without repeating (we think!), one of many irrational numbers, like the more famous Pi.

Phi also crops up in the construction of many other basic shapes, such as a five-pointed star:

In this case, Phi happens to be the ratio between the red line and the green line, or between the green line and the small pink line. Once again, we see Phi’s self-recursive nature; natural shapes are often composed of parts with the same pattern as the whole, repeating again and again at smaller scales, such as this example of Phi at work:

Plants display the best-confirmed examples of Phi in the natural world. The same man who discovered this also found Phi in skeletons, veins, and crystals. Many people claim to see Phi in other natural objects, such as spiral galaxies and Nautilus shells, however, these identifications are contested. There are other spirals, such as the Nautilus shells logarithmic spiral which look similar to the Golden Spiral, but they are not, at least according to this mathematician.

Phi is also “the limit” of the ratios between numbers in the Fibonacci sequence, a similarly natural construction. The Fibonacci sequence is simply the list of numbers when you start with 1 and then keep making new numbers by adding the two previous ones:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, etc.

As you continue, the ratio of any two adjacent numbers gets closer and closer to Phi: ½, 2/3, 3/5, 5/8, 8/13, etc. But really, this isn’t so shocking when you realize that the rule for generating the Fibonacci sequence is analogous to the rule for dividing a line segment into the Golden Ratio, but in reverse.

Many people are intrigued by Phi, having heard about its supposed occurrence in thousands of works of art, architecture, and in the human body. Apparently, this fame may be undeserved. It’s easy to draw a golden rectangle around numerous human body parts, doorways, windows, and other things, especially if you’re free to choose where to put the lines and how thick to make them. Nevertheless, Phi is certainly in hundreds of natural objects – flowers, trees, starfish, the behavior of electrons, and more. Maybe you can spot an example yourself. After all, it could be a good excuse to stop and look at the flowers.

Quotes

“Geometry has two great treasures: one is the Theorem of Pythagoras; the other, the division of a line into extreme and mean ratio. The first we may compare to a measure of gold; the second we may name a precious jewel.” – Johannes Kepler

“Without mathematics there is no art.” – Luca Pacioli