Spiral Growth in Nature Chapter 9

Fibonacci Numbers 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, .. Is a widely known Fibonacci numbers. They are named after the Italian Leonardo de Pisa, better known by the nickname Fibonacci. The first two numbers stand their own. After the first two, each subsequent number is the sum of the two numbers before it. 2= 1+1, 3 = 2 + 1, 5 = 3+2,…

Fibonacci Numbers Does the list of Fibonacci numbers ever end? No. The list goes on forever, with each new number in the sequence equal to the sum of the previous two. Each Fibonacci number has its place in the Fibonacci sequence. The standard mathematical notation to describe a Fibonacci number is an F followed by a subscript indicating its place in the sequence. For example, F8 stands for the eighth Fibonacci number, which is 21 (F8 = 21).

Fibonacci Numbers Fibonacci numbers that come before FN, are FN-1 and FN-2. The notation to find a Fibonacci number FN from two previous Fibonacci numbers FN-1 and FN-2 is given by: FN = FN-1 + FN-2 Where FN is a generic Fibonacci number, FN-1 is a Fibonacci number right before it and FN-2 is a Fibonacci number two positions before it. We must give the values of F1= 1 and F2 = 1

Fibonacci Numbers (Recursive definition) Seeds: F1= 1 and F2 = 1 Recursive Rules: FN = FN-1 + FN-2 (N >= 3)

Fibonacci Numbers (Recursive definition) The recursive definition gives us a blueprint as to how to calculate any Fibonacci number (E.g., F100), but it is an arduous climb up the hill, one step at a time. Imagine climbing up to F500 or F1000. The practical limitations of the recursive definition lead naturally to the question, Is there a better way? There is.

Fibonacci Numbers (Binet’s formula) FN = 1 + √5 2 N 1 - √5 - √5 Binet’s formula is called an explicit definition of the Fibonacci numbers

Fibonacci Numbers (Binet’s formula) By substituting the constants by letters: FN = (aN – bN)/ c 1 - √5 Where a = 1 + √5, b = , c = √5 2 2

Fibonacci Numbers in Nature The number of petals in certain varieties of flowers: 3 (lily, iris) 5 (buttercup, columbine) 8 (cosmo, rue anemone) 13 (yellow daisy, marigold) 21 (English daisy, aster) 34 (oxeye daisy) 55 (coral Gerber daisy).