New Mexico Supercomputing Challenge  



Challenge Team Interim Report
The Fibonacci Sequence Occurring in Nature INTRODUCTION
RESEARCH
The sequence 1, 1, 2, 3, 5, 8, 13, 21, 34, 55... defined by F (1) = 1, F (2) = 1, and F (n) = F (n1) +F (n2) 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 numbertheoretic book concerned with the simultaneous solution of quadratic equations with 2 or more variables (http://wwwgroups.dcs.stand.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
METHOD
ACKNOWLEDGEMENTS
WORKS CONSULTED Andreasen, Cory. "Fibonacci and Pascal together again: Pattern exploration in". EMail 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. Team Members Sponsoring Teachers Project Advisor(s)
