Fibonacci

Leonardo of Pisa, also known as Fibonacci (son of Bonacci) was born around 1170 in Pisa, Italy. he was given the name Fibonacci after his death. It meant "son of Bonacci", which was the nickname of his father.
Fibonacci was a mathematician that many consider "the most talented mathematician of the Middle Ages." His most notable contributions are introducing the Hindu-Arabic numeral system to Europe and a number sequence known as the Fibonacci numbers.
Fibonacci's father directed a trading post in Bugia, a port east of Algiers in North Africa. Young Leonardo often traveled there to help his father and began to pick up the Hindu-Arabic number system. He recognized that doing mathematical functions would be a lot easier with these numbers rather than with Roman numerals. He began traveling around the Mediterrean, learning as much as he could about the Hindu-Arabic system. He returned to Pisa around 1200 and in 1202 he published Liber Abaci (Book of Abacus or Book of Calculation.) Thus the system was introduced to the Western world.
Liber Abaci advocated using the numerals 0-9 and place value. He referred to this as modus Indorum (method of the Indians). He showed the practicality of it by applying it commercial bookkeeping, conversions of weight, calculation of interest, money-changing and other applications. The book was well-received and had huge impact on European thinking.
Liber Abaci also introduced to the West a mathematical sequence now known as the Fibonacci sequence. The sequence was known to Indian mathematicians as early as the sixth century.
Fibonacci died around 1250.
In the 19th century a statue of Fibonacci was erected in Pisa. It is located in the western gallery of the Camposanto, a historical cemetery on the Piazza dei Miracoli.
Fibonacci Sequence
The Fibonacci numbers are 0, 1, 1, 2, 3, 5, 8,13,21,34,55,89,.... By definition the first two numbers are 0 and 1 ( although some some omit the 0 and begin with 1,1) and each remaining number is the sum of the previous two. In Liber Abaci Fibonacci posed the following problem:
The following paragraphs are from
www.mathacademy.com/pr/prime/articles/fibonac/index.asp
How many pairs of rabbits will be produced in a year, beginning with a single pair, if in every month each pair bears a new pair which becomes productive from the second month on?
It is easy to see that 1 pair will be produced the first month, and 1 pair also in the second month (since the new pair produced in the first month is not yet mature), and in the third month 2 pairs will be produced, one by the original pair and one by the pair which was produced in the first month. In the fourth month 3 pairs will be produced, and in the fifth month 5 pairs. After this things expand rapidly, and we get the following sequence of numbers:
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, ...
This is an example of a recursive sequence, obeying the simple rule that to calculate the next term one simply sums the preceding two:

F(1) = 1
F(2) = 1
F(n) = F(n – 1) + F(n – 2)
Thus 1 and 1 are 2, 1 and 2 are 3, 2 and 3 are 5, and so on.
This simple, seemingly unremarkable recursive sequence has fascinated mathematicians for centuries. Its properties illuminate an array of surprising topics, from the aesthetic doctrines of the ancient Greeks to the growth patterns of plants (not to mention populations of rabbits!).
(I do not want to get bogged down in mathematics. To read the whole article use the above link)
The Greeks believe that the equation they refer to as f(phi) was the most most pleasing, indeed the aesthetically perfect proportion, and all of their artwork, sculpture, and especially architecture made use of this proportion. A rectangle whose sides had this proportion was called the Golden Rectangle.
Whether or not you agree with the Greeks’ aesthetic judgment, it's a safe bet that Nature herself does:

The growth of this nautilus shell, like the growth of populations and many other kinds of natural “growing,” are somehow governed by mathematical properties exhibited in the Fibonacci sequence. And not just the rate of growth, but the pattern of growth. Examine the crisscrossing spiral seed pattern in the head of a sunflower, for instance, and you will discover that the number of spirals in each direction are invariably two consecutive Fibonacci numbers.
The Fibonacci sequence also occurs in music.