Against the Gods. Bernstein Peter L.

was serving as Pisan consul. While Fibonacci was there, an Arab mathematician revealed to him the wonders of the Hindu-Arabic numbering system that Arab mathematicians had introduced to the West during the Crusades to the Holy Land. When Fibonacci saw all the calculations that this system made possible – calculations that could not possibly be managed with Roman letter-numerals – he set about learning everything he could about it. To study with the leading Arab mathematicians living around the Mediterranean, he set off on a trip that took him to Egypt, Syria, Greece, Sicily, and Provence.

      The result was a book that is extraordinary by any standard. Liber Abaci made people aware of a whole new world in which numbers could be substituted for the Hebrew, Greek, and Roman systems that used letters for counting and calculating. The book rapidly attracted a following among mathematicians, both in Italy and across Europe.

      Liber Abaci is far more than a primer for reading and writing with the new numerals. Fibonacci begins with instructions on how to determine from the number of digits in a numeral whether it is a unit, or a multiple of ten, or a multiple of 100, and so on. Later chapters exhibit a higher level of sophistication. There we find calculations using whole numbers and fractions, rules of proportion, extraction of square roots and roots of higher orders, and even solutions for linear and quadratic equations.

      Ingenious and original as Fibonacci’s exercises were, if the book had dealt only with theory it would probably not have attracted much attention beyond a small circle of mathematical cognoscenti. It commanded an enthusiastic following, however, because Fibonacci filled it with practical applications. For example, he described and illustrated many innovations that the new numbers made possible in commercial bookkeeping, such as figuring profit margins, money-changing, conversions of weights and measures, and – though usury was still prohibited in many places – he even included calculations of interest payments.

      Liber Abaci provided just the kind of stimulation that a man as brilliant and creative as the Emperor Frederick would be sure to enjoy. Though Frederick, who ruled from 1211 to 1250, exhibited cruelty and an obsession with earthly power, he was genuinely interested in science, the arts, and the philosophy of government. In Sicily, he destroyed all the private garrisons and feudal castles, taxed the clergy, and banned them from civil office. He also set up an expert bureaucracy, abolished internal tolls, removed all regulations inhibiting imports, and shut down the state monopolies.

      Frederick tolerated no rivals. Unlike his grandfather, Frederick Barbarossa, who was humbled by the Pope at the Battle of Legnano in 1176, this Frederick reveled in his endless battles with the papacy. His intransigence brought him not just one excommunication, but two. On the second occasion, Pope Gregory IX called for Frederick to be deposed, characterizing him as a heretic, rake, and anti-Christ. Frederick responded with a savage attack on papal territory; meanwhile his fleet captured a large delegation of prelates on their way to Rome to join the synod that had been called to remove him from power.

      Frederick surrounded himself with the leading intellectuals of his age, inviting many of them to join him in Palermo. He built some of Sicily’s most beautiful castles, and in 1224 he founded a university to train public servants – the first European university to enjoy a royal charter.

      Frederick was fascinated with Liber Abaci. Some time in the 1220s, while on a visit to Pisa, he invited Fibonacci to appear before him. In the course of the interview, Fibonacci solved problems in algebra and cubic equations put to him by one of Frederick’s many scientists-in-residence. Fibonacci subsequently wrote a book prompted by this meeting, Liber Quadratorum, or The Book of Squares, which he dedicated to the Emperor.

      Fibonacci is best known for a short passage in Liber Abaci that led to something of a mathematical miracle. The passage concerns the problem of how many rabbits will be born in the course of a year from an original pair of rabbits, assuming that every month each pair produces another pair and that rabbits begin to breed when they are two months old. Fibonacci discovered that the original pair of rabbits would have spawned a total of 233 pairs of offspring in the course of a year.

      He discovered something else, much more interesting. He had assumed that the original pair would not breed until the second month and then would produce another pair every month. By the fourth month, their first two offspring would begin breeding. After the process got started, the total number of pairs of rabbits at the end of each month would be as follows: 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233. Each successive number is-the sum of the two preceding numbers. If the rabbits kept going for a hundred months, the total number pairs would be 354,224,848,179,261,915,075.

      The Fibonacci series is a lot more than a source of amusement. Divide any of the Fibonacci numbers by the next higher number. After 3, the answer is always 0.625. After 89, the answer is always 0.618; after higher numbers, more decimal places can be filled in.24 Divide any number by its preceding number. After 2, the answer is always 1.6. After 144, the answer is always 1.618.

      The Greeks knew this proportion and called it “the golden mean.” The golden mean defines the proportions of the Parthenon, the shape of playing cards and credit cards, and the proportions of the General Assembly Building at the United Nations in New York. The horizontal member of most Christian crosses separates the vertical member by just about the same ratio: the length above the crosspiece is 61.8 % of the length below it. The golden mean also appears throughout nature – in flower patterns, the leaves of an artichoke, and the leaf stubs on a palm tree. It is also the ratio of the length of the human body above the navel to its length below the navel (in normally proportioned people, that is). The length of each successive bone in our fingers, from tip to hand, also bears this ratio.25

      In one of its more romantic manifestations, the Fibonacci ratio defines the proportions and shape of a beautiful spiral. The accompanying illustrations demonstrate how the spiral develops from a series of squares whose successive relative dimensions are determined by the Fibonacci series. The process begins with two small squares of equal size. It then progresses to an adjacent square twice the size of the first two, then to a square three times the size of the first two, then to five times, and so on. Note that the sequence produces a series of rectangles with the proportions of the golden mean. Then quarter-circle arcs connect the opposite corners of the squares, starting with the smallest squares and proceeding in sequence.

       Construction of an equiangular spiral using Fibonacci proportions.

      Begin with a 1-unit square, attach another 1-unit square, then a 2-unit square then a 2-unit square where it fits, followed by a 3-unit square where it fits and, continuing in the same direction, attach squares of 5, 8, 13, 21, and 34 units and so on.

      (Reproduced with permission from Fascinating Fibonaccis, by Trudy Hammel Garland; copyright 1987 by Dale Seymour Publications, P.O. Box 10888, Palo Alto, CA 94303.)

      This familiar-looking spiral appears in the shape of certain galaxies, in a ram’s horn, in many seashells, and in the coil of the ocean waves that surfers ride. The structure maintains its form without change as it is made larger and larger and regardless of the size of the initial square with which the process is launched: form is independent of growth. The journalist William Hoffer has remarked, “The great golden spiral seems to be nature’s way of building quantity without sacrificing quality.”26

      Some people believe that the Fibonacci numbers can be used to make a wide variety of predictions, especially predictions about the stock market; such predictions work just often enough to keep the enthusiasm going. The Fibonacci sequence is so fascinating that there is even an American Fibonacci Association, located at Santa Clara University in California, which has published thousands of pages of research on the subject since 1962.

      Fibonacci’s Liber Abaci was a spectacular first step in making measurement the key factor in the taming of risk. But society was not yet prepared to attach numbers to risk. In Fibonacci’s day, most people still thought that risk stemmed from the capriciousness of nature. People would have to learn to recognize man-made risks and acquire the courage to do battle with the fates before they would accept the

24

One of chose odd quirks that numbers can produce reveals that you can derive 0.618 if you take the square root of 5, which is 2.24, subtract 1, and then divide by 2; this result is the algebraic proof of Fibonacci’s sequence of numbers.

In technical terms, the formula for the Fibonacci ratio is as follows: the ratio of the smaller part to the larger part equals the ratio of the larger part to the whole.

The Arabic term survives even in Russian, where it appears as tsifra, which is the word for number.

25

In technical terms, the formula for the Fibonacci ratio is as follows: the ratio of the smaller part to the larger part equals the ratio of the larger part to the whole.

The Arabic term survives even in Russian, where it appears as tsifra, which is the word for number.

26

Two stimulating commentaries on the Fibonacci numbers are Garland, 1987, and Hoffer, 1975. The examples here are drawn from those two sources.