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 Team Number: 020 School Name: Cuba High School Area of Science: Mathematics Project Title: The Fibonacci Sequence
 Abstract Interim Final Report

The Fibonacci Sequence Occurring in Nature

INTRODUCTION
The Fibonacci sequence is a well-known set of numbers starting with 1. This sequence of numbers occurs frequently in nature, from the arrangement of seeds on a flower head to the expansion of branches of a tree. If undisturbed in the wild, rabbits would reproduce in pairs mimicking the Fibonacci sequence. Pinecones grow in numbers of spirals that are also part of the Fibonacci sequence. The objective of this project will be to compose a program that will accurately model the growth of a tree or of a rabbit population in a given amount of time. The equation used to determine this will most likely be f(n)=(1+5)/2-(1-5)/2. The only variables included should be the time period simulated.

RESEARCH
Leonardo Fibonacci, son of Guilielmo Bonacci, discovered the Fibonacci sequence. Leonardo was born in Pisa, Italy around 1170. Leonardo traveled frequently on business with his father and was often around mathematical ideas. Fibonacci traveled widely in the Middle East, where he learned the Hindu Arabic system. Fibonacci was probably the greatest genius of number theory during the 2000 years between Diophantus and Fermat (http://www-groups.dcs.st-and.ac.uk/~history/Mathematics/Fibonacci.html).

The sequence 1, 1, 2, 3, 5, 8, 13, 21, 34, 55... defined by F (1) = 1, F (2) = 1, and F (n) = F (n-1) +F (n-2) for n = 3, 4, 5, ... is named the Fibonacci sequence. Each number after the first two numbers (1,1) equals the sum of the two numbers before it. For any value larger than three, the ratio between any two consecutive numbers in the sequence is approximately 1:1.6. It is exactly (1+5½)/2 and is known as the golden ratio. Since their introduction, mathematicians have studied these numbers (http://www.mcs.surrey.ac.uk/Personal/R.Knott?Fibonacci/Fibmaths.html). Around 1200, Fibonacci started to work on his own mathematical compositions. The five works from this period consist of the Libber abaci (1202, 1228), the Practice geometry (1submissions.shtml1221), which is a letter to Theodore's, Floss (1225). Which is a collection of solutions to problems posed in the presence of Frederick 2, and the Liber quadratorum (1225), a number-theoretic book concerned with the simultaneous solution of quadratic equations with 2 or more variables (http://www-groups.dcs.st-and.ac.uk/~history/Mathematics/Fibonacci.html) ?

The Fibonacci sequence plays a major role in nature. Most everyone knows that a sunflower head is composed of two opposite sets of spirals. Spirals frequently occur in nature. The outsides of pinecones and pineapples all have spirals. Its turns out that the sunflower has 21spirals going in one direction and 34 going in the other (numbers of the Fibonacci sequence). A pinecone has five spirals going in one direction and eight in the other. A pineapple has eight spirals going in 1 direction and 13 in the other. The Fibonacci Sequence can also be used to predict the number of galaxies in the universe. This sequence was used for the number of steps on a staircase in the ancient pyramids. The Fibonacci Sequence can also illustrate the number of stems on a plant. The Fibonacci Sequence can be used to illustrate how a shell will grow, or how a tree can grow. Philosophers and mathematicians have often wondered why mathematics "a construct of the human mind," (Sundaram 176) is so successful in modeling and predicting natural phenomena. Geometrical arrangements in nature appear to be controlled by number sequences.

COMPUTATIONAL PLAN
The method we will use to create this program will include finding or developing formulas that can accurately mimic the Fibonacci Sequence. Inputs will include time and if you want to know a rabbit population or the number of branches on a tree. The program will take these inputs and calculate the number you requested. As of this time, we have obtained several formulas and are in the process of testing them for reliability.

METHOD
To conduct the project we extensively researched Fibonacci's Sequence. A C++ program will display the population of rabbits under controlled circumstances. These circumstances were such that the rabbits could not die or leave a contained area. Each rabbit would produce a pair over a one year time span. This also assumes that each rabbit was able to produce after a month.

ACKNOWLEDGEMENTS
During this time, we would like to thank all the people who have supported and contributed to this project. First, we would like to thank Mr. Moore for being our mentor. We would also like to thank Ms. Drzymalski for her support and pushing us to the limit. In addition, we would like to thank TechNet for giving us Internet access to research our project.

WORKS CONSULTED

Andreasen, Cory. "Fibonacci and Pascal together again: Pattern exploration in". E-Mail to Madlyn Cebada. 22 November 1999.

"The Fibonacci Sequence Page" Online. Available: http://www.mathacademy.com/platonic_realms/encyclop/articles/fibonac.html

Kimberling, Clark. "Fibonacci " Online. Available: http://www.cedar.evansville.edu/~ck6/bstud/fnt.html

"The Mathematics of the Fibonacci sequence." Online. Available: http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fibmaths.html 15 November 1999

Philippou, Andreas. Fibonacci Numbers and Their Applications. Dordrecht, Holland: Reidel Publishing Company, 1984.

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