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 In 1202, Leonardo of Pisa completed and published Liber Abaci, or Book of the Abacus. At the time, the book was generally disregarded because it used Arabic numbers, which were way too outlandish for most Western Europeans. However, one problem in this book is still referred to today: "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?"      The first time you read the problem, it sounds a little intimidating. However, through reasoning the problem can be worked out. In the first month, there is one pair of rabbits. In the second month, there is still only one pair of rabbits, which reach maturity. In the third month, there are two pairs of rabbits. In the fourth month, there are five pairs. This creates a sequence: 1, 1, 2, 3, 5, 8, 13, 21... In more mathematical terms, the sequence is: Numn = Numn-1 + Numn-2      That means that the sequence is a recursively enumerable set; it is generated from a pair of axioms based on repeated rules of inference. To put that in simpler terms, each sucessive term is the sum of the two preceding terms--1+1=2, 2+1=3, 3+2=5...Basically, it's a mathematical snowball. All sequences like this are called Lucas Sequences.      Whenever you take two subsequent numbers from any Lucas sequence and divide them, you get that magical value of Ø, or the Golden Mean. The higher the numbers you use are, the closer to you get. Since the numbers go on forever, Ø does too. It's the most irrational number.      However, the Fibonacci sequence isn't special just because it gives us the Golden Mean. The Fibonacci sequence shows up in all sorts of weird places...and it gives us the Golden Ratio. | home | introduction | constructions | biology | aesthetics | games | about | contact | Created by Andi, Mel, and Shuj for Thinkquest Internet Challenge 2000