delanceyplace.com 8/20/12 - the fabulous fibonacci

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In today's excerpt - in Europe, cumbersome Roman numerals were used until about 1202 AD, when a mathematician trained in north Africa named Leonardo Pisano, a man known as Fibonacci, introduced the Hindu-Arabic numerals (he called them "Indian figures") we still use today. He also brought other dramatic changes in western Europe's use and understanding of mathematics including the amazing sequence of numbers we now know as Fibonacci numbers:

"In a remote section of the Austrian Alps, there is a long-abandoned salt mine entrance with a cornerstone bearing the in­scription 'anno 1180.' It refers to the year that the mine was estab­lished. Clearly there is something wrong with this designation. Scholars have determined that the first published use of the Hindu numerals (our common numerals) in the Western world was in 1202. It was in this year that Leonardo of Pisa (Leonardo Pisano), more commonly known as Fibonacci (pronounced: fee-boh-NACH-ee), published his seminal work Liber Abaci, or 'book of calcula­tion.' He began chapter 1 of his book with:

The nine Indian figures are: 9 8 7 6 5 4 3 2 1.
With these nine figures, and with the sign 0, which the Arabs call zephyr, any number whatsoever is written.

"This constitutes the first formal mention of our base-ten numeral system in the Western world. There is speculation, however, that Arabs already introduced these numerals informally and locally in Spain in the second half of the tenth century.

"Unlike luminaries from the past, whose fame today is largely based on a single work, such as Georges Bizet (1838-1875) for Carmen, Engelbert Humperdinck (1854-1921) for Hansel und Gretel, or J. D. Salinger (b. 1919) for The Catcher in the Rye, we should not think of Fibonacci as a mathematician who is known only for this now-famous sequence of numbers that bears his name. He was one of the greatest mathematical influences in Western cul­ture and unquestionably the leading mathematical mind of his times. Yet it was this sequence of numbers, emanating from a problem on the regeneration of rabbits, that made him famous in today's world.

"Fibonacci was a serious mathematician, who first learned mathematics in his youth in Bugia, a town on the Barbary Coast of Africa, which had been established by merchants from Pisa. He traveled on business throughout the Middle East and along the way met mathematicians with whom he entered into serious discussions. He was familiar with the methods of Euclid (ca. 300 BCE) and used those skills to bring to the European people mathematics in a very usable form. His contributions include introducing a practical nu­meration system, calculating algorithms and algebraic methods, and a new facility with fractions, among others. The result was that the schools in Tuscany soon began to teach Fibonacci's form of calcu­lation. They abandoned the use of the abacus, which involved counting beads on a string and then recording their results with Ro­man numerals. This catapulted the discipline of mathematics for­ward, since algorithms were not possible with these cumbersome symbols. Through his revolutionary book and other subsequent publications, he made dramatic changes in western Europe's use and understanding of mathematics.

"Unfortunately, Fibonacci's popularity today does not encompass these most important innovations. Among the mathematical prob­lems Fibonacci poses in chapter 12 of Liber Abaci, there is one about the regeneration of rabbits. Although its statement is a bit cumbersome, its results have paved the way for a plethora of monumental ideas, which has resulted in his fame today. The prob­lem ... shows the monthly count of rabbits as the following sequence of numbers: 1,1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, . . . , which is known today as the Fibonacci numbers. At first sight one may wonder what makes this sequence of numbers so revered. A quick inspection shows that this sequence of numbers can go on infinitely, as it begins with two 1s and continues to get succeeding terms by adding, each time, the last two numbers to get the next number (i.e., 1 + 1=2, 1+2 = 3, 2 + 3 = 5, and so on). By itself, this is not very impressive. Yet, as you will see, there are no numbers in all of mathematics as ubiquitous as the Fibonacci numbers. They appear in geometry, algebra, number theory, and many other branches of mathematics. However, even more spectacularly, they appear in nature; for example, the number of spirals of bracts on a pinecone is always a Fibonacci number, and, similarly, the number of spirals of bracts on a pineapple is also a Fibonacci number. The appearances in nature seem boundless. The Fibonacci numbers can be found in connection with the arrangement of branches on various species of trees, as well as in the number of ancestors at every generation of the male bee on its family tree. There is practically no end to where these numbers appear."