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Spiral Growth in Nature

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.

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Spiral Growth in Nature

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  1. Spiral Growth in Nature Chapter 9

  2. 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,…

  3. 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).

  4. 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

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

  6. 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.

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

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

  9. 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).

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