The Fibonacci Numbers in Nature Include Those Found in: the spirals of sunflower heads and pine cones; the genealogy of the male bee; the related logarithmic spiral in snail shells; the arrangement of leaf buds on a stem and animal horns.
Sunflowers Have a Golden Spiral of Seed Arrangements and can Contain the Number 89, or even 144
Scientists Find Clues to the Formation of Fibonacci Spirals in Nature
Logarithmic Spirals (NASA May 17, 2008)
The Fibonacci Rabbits: Leonardo posed the problem of "How Many Pairs of Rabbits Are Created by One Pair in One Year?" in his Book of Calculation. At the end of the first month, they mate, but there is still only 1 pair. At the end of the second month the female produces a new pair, so now there are 2 pairs of rabbits in the field. At the end of the third month, the original female produces a second pair, making 3 pairs in all in the field. At the end of the fourth month, the original female has produced yet another new pair, the female born two months ago produces her first pair also, making 5 pairs.
"A certain man put a pair of rabbits in a place surrounded on all sides by a wall. How many pairs of rabbits can be produced from that pair in a year
if it is supposed that every month each pair begets a new pair which from the second month on becomes productive?"
This problem gave rise to the Fibonacci sequence 1.1.2.3.5.8.13.21.34 55 and so on, in which each number is the sum of the two preceding numbers, was the first recursive number sequence developed in Europe. It was also the first in which the relation between two or more successive terms can be expressed by a formula. Although he is known as the creator of the Fibonacci Sequence, he most likely included this problem that had already been developed by Indian scholars. In his book he says that he included the information to "make their use more commonly understood in his native Italy." Leonardo's father, Gugliemo, was a customs official and engaged in commerce representing Pisa at Bougie on the north coast of Africa. Young Leonardo consequently received a Moorish education as well as the traditional European education and was introduced to Hindu-Arabic numbers. Later on he traveled about the Mediterranean visiting Egypt, Syria, Greece, Sicily and Provence, meeting with scholars and becoming acquainted with the various arithmetical system used by the merchants.
Chimney of Turku Energia, Turku, Finland Featuring Fibonacci Sequence in 2m High Neon Lights
By Italian Artist Mario Merz for an Environmental Art Project (1994)
Europe was still using the Roman and Greek alphabet for numbers consequently the abacus had to be used for calculations. By helping to introduce Hindu-Arabic numbers, Fibonacci freed arithmetic from this need for the abacus.
The Abacus
Though not the first to write about the Hindu-Arabic number system- 1, 2, 3, 4, 5, 6, 7, 9, 0 – he showed by examples the superiority of this system over the Roman system. To Fibonacci also goes the credit for first using the bar in a fraction i.e. 5/6 separating the numerator and denominator. In reading Liber Abaci, it is helpful to understand Fibonacci's notation for rational numbers, a notation that is intermediate in form between the Egyptian fractions commonly used until that time and the vulgar fractions still in use today.
Leonardo Fibonacci (aka Leonardo Pisano and Leonardo de Piza)
Liber Abaci (1202)
In the "Liber Abaci" Fibonacci introduces the so-called "Modus Indorum" (method of the Indians), today known as Arabic numerals.The book advocated numeration with the digits 0–9 and place value. The book showed the practical importance of the new numeral system, using lattice multiplication and Egyptian fractions, by applying it to commercial bookkeeping, conversion of weights and measures, the calculation of interest, money-changing, and other applications. The book was well received throughout educated Europe and had a profound impact on European thought. In the Fibonacci sequence of numbers, each number is the sum of the previous two numbers, starting with 0 and 1. This sequence begins 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987 and so on. The higher up in the sequence, the closer two consecutive "Fibonacci Numbers" of the sequence divided by each other will approach the golden ratio (approximately 1 : 1.618 or 0.618 : 1).
