The cult of the golden ratio

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

Webster's Ninth New Collegiate Dictionary has several definitions of "cult." One is: "great devotion to a person, idea, or thing; esp: such devotion regarded as a literary or intellectual fad." For more than a century a cult, in the sense just defined, has flourished with respect to an arithmetic ratio known variously as the "divine proportion," the "extreme and mean ratio," and the "golden ratio."

Place a point on a line so that it cuts the line into two segments, A and B, such that the length of the line is to A as A is to B. A and B are then in golden ratio to each other.

It is easy to show that this ratio is the root of the equation |x.sup.2~ - x = 1. Applying the quadratic formula gives x a value of 1.61803398+, an irrational number that is half the sum of 1 and the square root of 5. In the U.S. this number is customarily designated by |Phi~, the Greek symbol for phi.

Phi is almost as ubiquitous in mathematical structures as pi or e. The reason, of course, is the simplicity of the equation that produces it. As might be expected, it is also the sum of the simplest possible continued fraction:

|Mathematical Expression Omitted~

Phi is the only positive number that becomes its own reciprocal by subtracting 1. Because 1/|Phi~ is .61803398+, the same infinite sequence of decimal digits, some mathematicians have preferred to call .618 . . . the golden ratio. It doesn't matter because you get the same sequence of decimal digits regardless of whether you divide line segment A by B or vice versa.

Phi is related to a generalized Fibonacci sequence in a startling way. Begin with any two numbers, rational or irrational, add them to get a third number, then add the second and third to get the fourth, and so on, always recursively generating the next number as the sum of the two previous ones. As the sequence goes to infinity, the ratio of two adjacent numbers approaches phi as the limit.

Consider the simplest Fibonacci sequence: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, . . . Divide 89 by 55 and you get 1.618+, or phi correct to three decimals. Continue five more steps to 987/610 and you produce the golden ratio to five decimal places.

Since the Renaissance, an enormous literature has accumulated, most of it nonsense, about the application of the golden ratio to architecture, painting, sculpture, nature, and even to poetry and music. The first great work defending this inflated worship of phi was a 457-page German treatise, Der Goldene Schnitt (The golden cut), published in 1884 by Adolf Zeising. Early works in English, following Zeising's lead, were Nature's Harmonic Unity (1913) by Samuel Coleman, and The Curves of Life (1914), by Theodore Cook. Similar books have been published since, and others are still being written.

My discussion here of what can be called pseudomathematics is based mainly on George Markowsky's sensible paper, "Misconceptions About the Golden Ratio," in the College Mathematics Journal (vol. 23, January 1992, pp. 2-19). Markowsky is a computer scientist at the University of Maine. His paper can be consulted for more details and for its valuable list of 59 references. See also the chapter on phi in my Second Scientific American Book of Mathematical Puzzles and Diversions, where I myself fell for some misconceptions.

The most persistent misconception is the belief that the "golden rectangle," a rectangle with sides in golden ratio, is the most aesthetically pleasing of all rectangles. The first effort to prove this was undertaken by Gustav Fechner (1801-1887), an eccentric German physicist and psychologist who held bizarre opinions. He believed, for example, that the sun, the moon, and all the planets are living organisms and that even a pebble has a "soul." In one experiment he exhibited ten rectangles of varying shapes and asked subjects to select the one they considered most attractive. About three-fourths of their choices were rectangles with side ratios of .57, .62, and .67.

As Markowsky writes, far from establishing the golden rectangle as the most beautiful, Fechner showed that people prefer a rectangle within a fairly large range; not too close and not too far from a square. Fechner made other experiments to prove that phi was the most pleasing of all ratios, but they were crude by today's standards.

Recent research has shown that the golden rectangle does not score any better than rectangles similar to it, such as a 3 x 5, the shape of a file card. To demonstrate how vague such preferences are, Markowsky drew the chart shown in Figure 1, containing 48 rectangles, and asked students to identify the two golden forms. Most of them could not. He then arranged the same rectangles in linear order, their ratios ranging from .4 to 2.5. He found wide variations in choices. Moreover, when asked to pick the most pleasing shape in each chart, subjects rarely chose the same one. The rectangle most often selected was the one in row 3 in column 4 of Figure 1--a rectangle with sides in a 1.83 ratio.