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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)
N
N
1 + √5
2
FN =
-
1 - √5
2
√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
Where a = 1 + √5, b = 1 - √5
2
2
, c = √5
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).