The Fibonacci Numbers

Pine cones. Stock-market quotations. Sunflowers. Classical architecture. Reproduction of bees. Roman poetry. What do they have in common? In one way or another, these and many more creations of nature or works of man all seem to be related to a sequence of numbers named for 13th century Mathematician Leonardo Fibonacci. The earnest mathematics buffs of the California-based Fibonacci Association keep examining the phenomenon. The more they investigate, members insist, the more convinced they become that Fibonacci numbers pervade the world.

The intriguing sequence was first mentioned by Fibonacci in his book Liber Abaci, which was published in Pisa in A.D. 1202. To solve a hypothetical problem about the multiplication of rabbits,-he used the numerical series 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, etc. Each number following the first 1 consisted of the sum of the two previous numbers. Fibonacci attached no great significance to the sequence, and it was generally ignored through the years by all but dedicated mathematicians. Then, in the early 1960s, Brother Alfred Brousseau, who teaches math at St. Mary's College near San Francisco, became interested in the numbers and their applications. "We got a group of people together in 1963," he says, "and just like a bunch of nuts, we started a mathematics magazine." The Fibonacci Quarterly ($6 per year) has survived and--in an academic sort of way--even thrived. It is currently being sent to the 350 members of the Fibonacci Association and has a general circulation of about 900.

Golden Rectangle. There is more than enough Fibonacci lore to fill each issue. "We have a backlog of articles," says Brother Alfred proudly, "and we've been accepted by the mathematical fraternity." Mathematician Verner Hoggatt Jr., editor of the Quarterly, has gone to the extent of establishing the Fibonacci Bibliographical and Research Center at San Jose State College. He tours schools to lecture on Fibonacci numbers, vigorously advocates their use in teaching and has compiled a remarkable dossier on Fibonaccia.

Among the material in his files are research papers showing that there are Fibonacci numerical patterns in the meter of works by Virgil and other Roman poets, and Fibonacci relationships between the different sizes of mosaic patterns in the floors of Greek and Roman ruins. There are studies showing that the ratio between any two successive larger Fibonacci numbers is 1 to 1.618--the same as the ratio between the sides of the "golden rectangle," a form that is traditionally used by artists and architects to produce effects that are most pleasing to the eye.

Mystical Connection. In nature, male bees reproduce Fibonaccially, and the number of spiral floret formations visible in many sunflowers, spiraled scales on a pine cone and segments on the surface of a pineapple have been found to match Fibonacci numbers. The pattern of the branching of many trees, the position of leaves on the branches, and petal formation of many flowers are also described by numbers in the Fibonacci series.

Hoggatt sees Fibonacci everywhere. "The piano octave," he notes with satisfaction, "has eight white keys--five black keys and 13 keys altogether," all Fibonacci numbers. "I always use parking lot eight at school and I watch my Fibonacci numbers on the stock market too." Brother Alfred recently took 40 math teachers on a nature walk, collecting pine cones and counting the number of spirals on each. He has lovingly described other expeditions and his findings in the Quarterly.

Both the association and its magazine may well be bolstered by the recent appearance of a lengthy Fibonacci article in Scientific American, which is widely circulated in the mathematical community. Once they learn about the Fibonacci sequence, says Brother Alfred, "people tend to find an esthetic satisfaction in it. They think that there's some kind of mystical connection between these numbers and the universe."

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