The golden number: nature seems to have a sense of proportion

what do:

* the arrangements of sunflower seeds;

* the branching of leaves on a stem;

* the flight path of a diving falcon;

* the breeding of rabbits;

* the spiral shapes of nautilus shells and other mollusks;

* the shapes of spiral galaxies; and

* the way black holes change from one "phase" to another all have in common? What shared thread connects the petal arrangement in a red rose with the art of Salvador Dali and the architecture of Frank Lloyd Wright? The answer is, all these phenomena share a close association with a single, extraordinary number. No wonder the number in question has earned the name "golden ratio."

The golden ratio--aka "golden section," "golden number," and even "divine proportion"--is hardly, by itself, a novel concept. The systematizer of Greek geometry, Euclid, who taught in Alexandria around 300 B.C., defined the number in Elements, his famous work on geometry and number theory. But Euclid's definition was entirely geometric and betrayed not the slightest acquaintance with the role of the golden ratio in the natural world. In fact, it was nothing more than a modestly amusing way for geometers to divide a line into two unequal parts. Little did Euclid know that his innocent-looking division would preoccupy mathematicians, physicists, botanists, psychologists, and artists for the next few millennia.

Euclid's number (the name "golden ratio" was applied centuries later) emerges from geometry in the following way: Take any line segment and divide it into two parts, in such a way that the longer part of the line segment is in the same proportion to the shorter part as the entire line segment is to the longer part. The ratio in question is the golden ratio [see diagram below]. (You don't need to follow the mathematics to understand the rest of this article, but for readers who are interested, here's how to figure out the value of Euclid's number: Suppose the length of the shorter part is 1 and the length of the longer part is x. That makes the length of the original line segment equal to x + 1. According to Euclid's definition, then, the value of the golden ratio is x/1, the ratio of the longer part to the shorter part. But that ratio must also be equal to (x + 1)/x, or the ratio of the original line to the longer part. The solution for x is then a straightforward, albeit technical, matter of high school algebra.)

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Turn the crank, and the number that solves the equation for x is equal to the never-ending, never-repeating number 1.6180339887 ..., commonly denoted by the Greek letter phi, or [phi]. Phi is not to be confused with the Greek letter pi, or [pi], which stands for a more familiar never-ending, nonrepeating number also present throughout Euclid's work. Pi, whose decimal value is 3.1415926535 ..., is simply the ratio of the circumference of a circle to its diameter. But pi also makes guest appearances in the most diverse parts of natural science. In that respect phi is like pi: its original definition can be understood by virtually anyone, but it reappears in a remarkable variety of arcane and mysterious guises.

Also like pi, the number phi is an irrational number, one that cannot be expressed as a ratio of two whole numbers, such as 3/1, 3/2, 5/7, or 23/39. In fact, phi is mathematically the "most irrational" number, in the sense that, if you try to approximate it as what is known as a continued fraction (one in which fractions are added in the denominator ad infinitum), you find that the approximation converges on it more slowly than continued-fraction approximations do for any other irrational number.

The number phi would have remained in the relative obscurity of pure mathematics were it not for its propensity to pop up where least expected. Take, for instance, the head of a sunflower. The florets form various clockwise and counterclockwise spiral patterns, intertwined and crisscrossing but otherwise unmistakable to the eye. Each floret arises in the center of the sunflower and gets pushed outward by its successors; the spiral patterning is an outcome of the way the florets are most easily and efficiently packed as they grow. The number of clockwise spirals and the number of counterclockwise spirals vary, depending on the size of the sunflower. Usually you find 55 twisting one way and 34 the other, but you may find 89 and 55, or 144 and 89. Even 233 and 144 has been reported. Amazingly, if you calculate these as ratios (55/34, 89/55, 144/89, 233/144), you find that they get closer and closer to the value of the golden ratio phi!

The patterning of sunflowers is closely related to one of the discoveries made in 1837 by two French brothers, Auguste and Louis Bravais. Auguste, a crystallographer, and Louis, a botanist, observed that as new leaves are put forth from the tip of many growing plants, each new leaf advances by an angle of roughly 137.5 degrees from the preceding leaf, around the circumference of the stem. That angle is what you get if you divide the number of degrees in a complete circle, 360, by the number phi, and then subtract the result from 360.