THE GOLDEN RATIO

The harmonious qualities of the golden ratio—phi—are pleasingly scanned in this history of the number, and, by extension, a historical tour of numbers in general.

Phi—1.6180339887…—is never-ending, never-repeating, irrational, incommensurable, one of those special numbers like pi that confound and delight in the same breath. It has been called the divine proportion for its visual effectiveness and Livio, head of the Science Division/Hubble Space Telescope Institute, is willing to concur with this view, although he is also willing to accept that beauty is in the eye of the beholder and the golden ratio (or golden number, golden section, golden this, golden that) may not be primary in its aesthetic appeal. What he is more concerned with here is the frequency of its occurrence in nature, from the petal arrangements on flowers and leaves on stems (phyllotaxis) to the spiral shells of mollusks (“nature loves logarithmic spirals,” from unicellular foraminifers to the arms of galaxies) to the apple's pentagram, which simply knows no end to its mysterious implications (mystery and surprise are, Livio notes, much of the joy of mathematics). He traces the history of the number, starting with the mists, proceeding through Euclid, the founder of geometry (it threw the Pythagoreans, who liked tidy numbers, into a fit), Francesca, Leonardo, Dürer, Kepler, to Le Corbusier and contemporary mathematicians. He tackles the grander instances where enthusiasts of phi say the number can be found: the pyramids, the Mona Lisa, the Parthenon. What he finds is that, through juggling the numbers, in almost any work of human creation can be found a golden ratio. The nature of the number itself—and others like the Fibonacci series, in which the ratio of successive numbers converges on the golden ratio—beguiles Livio, a keystone to the very meaning of mathematics, concluding that it was both discovered and invented, “a symbolic counterpart of the universe we perceive.” Those with math anxiety beware: this portrait of a number would be adrift without its healthy, if accessible, dose of algebra and geometry.

A shining example of the aesthetics of mathematics. (Illustrations)