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Mysterious Mathematics: The Fibonacci Sequence

The Fibonacci sequence is a series of numbers created in 1202 by Leonardo Fibonacci. Fibonacci numbers are generated by the equation F0=0, F1=1, followed by the recursive formula Fn=Fn-1+Fn-2. It follows the rule that any number is the sum of two numbers before it. The sequence is given as 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, and 233 and so on. Two preceding numbers are added together to equal the Fibonacci number, for example, 1 + 2 equals 3, 2 + 3 equals 5, 3 + 5 equals 8. The Fibonacci sequence is named after Leonardo Fibonacci (also known as Leonardo Pisano), an Italian mathematician who lived from 1170 to 1250. He used the sequence arithmetic to solve a problem based on a pair of rabbits: "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 solution can be found numerically using the Fibonacci sequence. Pingala, a Sanskrit grammarian, is also credited for one of the first mentions of the sequence from fifth century B.C. to third century A.D. Fibonacci numbers are found in nature, art and music. They can be found in the number of branches on a plant or from the spirals on a pineapple. Fibonacci numbers are commonly studied as a part of the number theory. It also has applications in the counting of mathematical objects, such as sequences, sets and permutations to computer science. A Fibonacci spiral approximates the golden ratio. Unlike the ratio that uses a mathematical constant of 1.6180339887, the Fibonacci ratio uses squares of Fibonacci-numbers. This can be seen in shell spirals, pine cones and petals. For example, the number of petals in most plants is a Fibonacci number. Buttercups have 5 petals, irises have 3 petals, marigolds have 13 petals and daises can be found with 34, 55 or 89 petals. Fibonacci ratios are also used in other areas, such as in stock market analysis. The method known as the “Elliot Wave Theory” is used to predict natural patterns of behavior of traders as reflected in a stock chart. The significance of the Fibonacci sequence numbers remains an important branch in number theory with more followers being drawn into the complex mind of Fibonacci.

University of Surrey – Who Was Fibonacci?
A brief biography of Leonardo of Pisa (now known as Fibonacci) and his mathematical achievements.

University of Evansville – The Fibonacci Sequence for Visual Layout
Page describing the Fibonacci sequence in composition and how the numerical convention can be translated into various forms.

Bromeliad Encyclopedia – The Fibonacci Sequence and Pineapples
A look at how the sequence of numbers in the Fibonacci sequence can be related to any plant, such as a pineapple.

Temple University – The Fibonacci Sequence, Spirals and the Golden Mean
Brief history of Leonardo Pisa and a solution to one of his well-known brain-teasers.

University of Illinois – Fibonacci Sequence
Description of the sequence of numbers known as the arithmetic series and the geometric series and how to calculate numbers in the Fibonacci sequence.

Math Academy – The Fibonacci Sequence
Explanation of the Fibonacci sequence and the solution to a problem posed by Leonardo Pisano in his treatise ‘Liber Abaci’ (pub. 1202).

Oracle Think Quest – The Fibonacci Series
Information, applications, biographies of important mathematicians, quizzes, flash demonstrations and other interesting resources relating to the Fibonacci series.

University of Arkansas – The Fibonacci Sequence and the Golden Mean
Details on the Fibonacci sequence as well as illustrations on how the sequence works to solve mathematical puzzles.

Camosun College – Fibonacci Numbers in Nature
Set of slides on Fibonacci numbers and how they relate to nature through flowers, fruits and trees.

Lock Haven University – Fibonacci Film-Flam
Fascinating claims made on the Fibonacci series, golden spiral hype and the Fibonacci relation to the golden ratio.

University of New Jersey – Fibonacci Sequence Calculator
Calculator that computes the nth term of the Fibonacci sequence and will also compute the previous and next terms.

Ward Melville High School – Fibonacci
Questions and answers about the life and work of the world’s greatest mathematician, Fibonacci.

Archimedes Lab – Fibonacci Number Properties
Calculations of how Fibonacci number sequences work as well as a Fibonacci calculator and applications.

Middlebury College – Fibonacci Numbers
Background of the life of Leonardo of Pisa and a detailed description of the Fibonacci series and how it unfolds.

James Madison University – The Fibonacci Sequence and More
Article containing several sections of information, including the Fibonacci sequence, the musical octave, Pascal’s triangle, phi, the golden ratio and rectangle and the pentagon and pentagram.