Golden Mean | Research & Encyclopedia Articles

Golden Mean

The golden mean is a number that appears in a dazzling array of mathematical and natural structures. The golden mean has fascinated both mathematicians and amateurs since the time of Pythagoras, and perhaps even earlier, with the Egyptians. The great astronomer Johannes Kepler called the golden mean one of the "two great treasures" of geometry (the other was the Pythagorean theorem), likening it to a "precious jewel."

There are many ways to define the golden mean, which is usually written phi (). The earliest known appearance of phi among mathematicians is in the Brotherhood of Pythagoras, a group of scholars from the 6th century B.C. In order to recognize its members, the Brotherhood adopted a symbol called the pentagram, which is a pentagon with a five-pointed star inscribed in it. At the center of the star is a smaller pentagon. The golden mean is the ratio of the length of one of the rays of the star to the length of a side of the small pentagon; it is also the ratio of the length of a side of the large pentagon to the length of one of the rays. Historians speculate that the Brotherhood considered the pentagram to be the most beautiful of shapes, and that they were aware that the golden mean also appears in two of the Platonic solids: the dodecahedron and the icosahedron.

A second definition of the golden mean is the one given by Euclid in his Elements, which concerns dividing a line into two parts. If we ask ourselves what is the "most pleasing" way to divide a line into two parts, or a piece of music, or a painting, one way is to divide it into two equal parts; but another interesting way is to divide it so that the ratio of the larger piece to the smaller piece equals the ratio of the whole to the larger piece, a division that art theorists called the principle of "dynamic symmetry." With this division, the ratio of the larger piece to the smaller piece is the golden mean--it is not obvious, but it's true, that this definition gives the same number as the pentagram definition. With this definition, it is not difficult to show that (phi)^2=1+phi, and then the quadratic formula tells us that the value of the golden mean is (1+5)/2 (approximately 1.618034). This expression for phi shows that phi is an irrational number--it is not the ratio of two whole numbers.

The formula (phi)^2=1+phi gives rise to some interesting representations for phi. We can rewrite the formula as phi=(1+phi), and that tells us that wherever we see phi, we can replace it with (1+phi). That means that phi=(1+phi)=(1+(1+phi))=(1+(1+(1+phi)))=(1+(1+(1+(1+...)))). What's more, we can rearrange the formula (phi)^2=1+phi to give the new formula phi=1+1/phi, which means that wherever we see phi we can replace it with 1+1/phi; thus, phi=1+1/phi = 1+1/(1+1/phi) = 1+1/(1+1/(1+1/phi)) = 1+1/(1+1/(1+1/(1+...))). This last expression is called the continued fraction representation of phi.

The golden section also appears in connection with the Fibonacci sequence. That is the sequence of numbers 1, 1, 2, 3, 5, 8, 13, 21, ... in which each term is the sum of the two preceding terms. It can be shown that the ratios of terms, 1/1, 2/1, 3/2, 5/3, 8/5, 13/8, ... get closer and closer to phi, which is their limit.

The above expressions involving phi have been known for centuries; but phi continues to appear in new circumstances, as new mathematics are discovered. Their most recent appearance has been in connection with Penrose tilings. These are tilings of the plane that have the unusual property of being aperiodic--that is, they are not made of copies of a single patterned block. Penrose tilings can be made out of two rhombic tiles, one long and thin, the other more fat. In a Penrose tiling, the ratio of thin tiles to fat tiles is the golden mean.

The golden mean has also been important in the world of art. The Greeks and Egyptians used it in their architecture, and artists have used it through the centuries to create proportions in paintings and sculptures. Recently, music theorists have discovered that the golden ratio also appears as the division between sections in some of Mozart's music. This raises the question, Is the golden ratio the most aesthetically pleasing division? Through the centuries, many people have invested phi with an almost mystical significance, and it is sometimes referred to as the "divine proportion."

The golden mean occurs not just in the works of man, but also in those of nature. In a seed pod, the seeds spiral out from the center, with more and more seeds in each revolution. The ratio of the number of seeds in one revolution to the number of seeds in the previous revolution is the golden mean. The golden mean is also closely related to the logarithmic spiral, which is the shape that appears in seashells.

The golden mean has also been observed, with less precision, as the ratio cut off by the navel in the human body. Some have taken this, together with the other appearances of phi in nature, as evidence of a divine Creator who built an aesthetically pleasing world for humans to inhabit. The appearances of phi in seed pods and seashells may be explained more scientifically, however, in terms of the most efficient arrangements of seeds or shells. And it can also be argued that we find phi aesthetically pleasing because it appears in nature and human anatomy, instead of the other way around.