Fibonacci, also known as Leonard of Pisa or Leonardo Pisano, was one of the most talented mathematicians of the Medieval period. He is largely remembered today for introducing the Hindu-Arabic numeral system into Europe, causing a quiet revolution in European mathematics, and for the sequence of numbers which came to be named after him. Here we explore the man and some of his most important contributions using simple Flash animations.
LIFE OF FIBONACCI
Leonardo Pisano (also known as Leonard of Pisa, or by the nickname Fibonacci) was born around 1170, the son of Guglielmo Bonacci, a wealthy merchant in the Italian city-state of Pisa. His father also directed a trading post in Bugia (modern day Bejaia in Algeria) and Leonardo often travelled with him to North Africa. It was there that he first encountered the Hindu-Arabic numeral system, and realized that it would make arithmetic much simpler and more efficient than the Roman numerals which were used at the time (see the side-bar at right). After spending some further years travelling through the Mediterranean world to study under the leading Arab mathematicians of the time, he returnd to Pisa in about 1200 and he published his "Liber Abaci" ("Book of Calculation") in 1202. The book was well received throughout educated Europe and had a profound impact on European thought. It was in the "Liber Abaci" that he first introduced to Europe the Hindu-Arabic numerals and the concept of place value ("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 may be written"). Among other things, the horizontal fraction bar notation was also first used in this work. The "Liber Abaci" also introduced to Europe a particular number sequence, which has since become known as Fibonacci Numbers or the Fibonacci Sequence. Fibonacci discovered the sequence - the first recursive number sequence known in Europe - through his consideration of a problem involving the growth of a hypothetical population of rabbits based on idealized assumptions. Many of the implications and relationships of the sequence, however, were not discovered until several centuries after his death. In later years, Leonardo published other books on mathematics, including "Practica Geometriae" in 1220, "Flos" in 1225 and "Liber quadratorum". He later became a long-term guest of the mathematically inclined Holy Roman Emperor Frederick II, and was offiicially honoured by the Republic of Pisa in 1240. It is thought that he died around 1250.
The Fibonacci Sequence (or Fibonacci Numbers) is the sequence of numbers, starting with 0 and 1, where each number is the sum of the previous two numbers. The series begins 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, etc, and continues to infinity. In mathematical terms, Fn = Fn-1 + Fn-2 with seed values F0= 0 and F1 = 1. The sequence itself had been known to Indian mathematicians since the 6th Century. Although Fibonacci recognized the sequence in his "Liber Abaci" of 1202, it was not expressed in algebraic notation until the 17th Century, and the phrase "Fibonacci Sequence" was not coined until the 1870s. The sequence has many interesting mathematical properties which are outside the scope of this web page (for example, every fourth number in the sequence is even, consecutive Fibonacci numbers have no common divisor, prime number terms are themselves prime numbers, the sum of any ten consecutive Fibonacci numbers is divisible by 11, etc). The Fibonacci numbers are also used in the financial markets (in trading algorithms, applications and strategies) as well as in numerous other mathematical and scientific applications.
FIBONACCI'S RABBITS Fibonacci actually discovered his famous sequence of numbers while considering a problem involving the growth of a hypothetical population of rabbits based on idealized assumptions. He noted that, after each monthly generation, the number of pairs of rabbits increased from 1 to 2 to 3 to 5 to 8 to 13, etc. He identified how the sequence progressed by adding the previous two terms, and noted that such a series may extend indefinitely.
GOLDEN RATIO, GOLDEN RECTANGLE & GOLDEN SPIRAL
The Fibonacci Sequence has connections to other mathematical concepts which Fibonacci himself never dreamed of, and which were not identified until the 18th Century. The ratio of each term in the sequence to the previous term approaches, with ever greater accuracy the higher the terms, a ratio of approximately 1 : 1.618 (it is actually an irrational number equal to (1 + √5)/2 which has been calculated to thousands of decimal places). This value is referred to as the Golden Ratio, also known as the Golden Mean, Golden Section, Divine Proportion, etc, and is usually denoted by the Greek letter phi (φ or sometimes the capital letter Phi Φ). Phi has many unique properties, such as those involving its reciprocal and its square (see the animation at right), but essentially two quantities are in the Golden Ratio if the ratio of the sum of the quantities to the larger quantity is equal to the ratio of the larger quantity to the smaller one. A rectangle with sides in the ratio of 1 : φ is known as a Golden Rectangle. An arc connecting opposite points of ever smaller nested Golden Rectangles forms a logarithmic spiral, known as a Golden Spiral (see the animation at right for a more detailed explanation of Golden Rectangles and Golden Spirals).
GOLDEN RATIO IN ART & NATURE
The Golden Ratio, and the related Golden Rectangle and Golden Spiral, has long been considered to be particularly aesthetically pleasing. Many artists and architects throughout history have proportioned their works approximately using the Golden Ratio and Golden Rectangles. The use of the ratio dates back to ancient Egypt and Greece, but it was particularly popular in the Renaissance art of Leonardo da Vinci and his contemporaries. The same ratio can also be found in a surprising number of instances in Nature, from shells to flowers to animal horns to human bodies to storm systems to complete galaxies (see the animation at right), although these are often approximations and one should be wary of reading too much into such natural occurrences.