The Sequence of Fibonacci Numbers and How They Relate to Nature

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The Sequence of Fibonacci Numbers and How They Relate to Nature. November 30, 2004 Allison Trask. Outline. History of Leonardo Pisano Fibonacci What are the Fibonacci numbers? Explaining the sequence Recursive Definition Theorems and Properties The Golden Ratio Binet’s Formula

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## The Sequence of Fibonacci Numbers and How They Relate to Nature

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### The Sequence of Fibonacci Numbers and How They Relate to Nature

November 30, 2004

Outline
• History of Leonardo Pisano Fibonacci
• What are the Fibonacci numbers?
• Explaining the sequence
• Recursive Definition
• Theorems and Properties
• The Golden Ratio
• Binet’s Formula
• Fibonacci numbers and Nature
Leonardo Pisano Fibonacci
• Born in 1170 in the city-state of Pisa
• Books: Liber Abaci, Practica Geometriae, Flos, and Liber Quadratorum
• Frederick II’s challenge
• Impact on mathematics

http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Fibonacci.html

What are the Fibonacci Numbers?
• Recursive Definition: F1=F2=1 and, for n >2, Fn=Fn-1 + Fn-2
• For example, let n=6.

Thus, F6=F6-1 + F6-2 F6=F5 + F4 F6=5+3

So, F6=8

Theorems and Properties
• Telescoping Proof

Theorem: For any n  N, F1 + F2 + … + Fn = Fn+2 - 1

Proof: Observe that Fn-2 + Fn-1 = Fn(n >2) may be expressed as Fn-2 = Fn – Fn-1 (n >2). Particularly,

F1 = F3 – F2

F2 = F4 – F3

F3 = F5 – F4

Fn-1 = Fn+1 – Fn

Fn = Fn+2 – Fn+1

When we add the above equations and observing that the sum on the right is telescoping, we find that:F1 + F2 + … + Fn =

F1 + (F4 – F3) + (F5 – F4) + … + (Fn+1 – Fn) + (Fn+2 – Fn+1) =

Fn+2 +(F1-F3)=

Fn+2 – F2 =

Fn+2 – 1

Theorems and Properties
• Proof by Induction

Theorem: For any n  N, F1 + F2 + … + Fn = Fn+2 – 1.

1) Show P(1) is true.

F1 = F2 = 1, F3 = 2

F1 = F1+2 – 1

F1 = F3 – 1

F1 = 2-1

F1 = 1

Thus, P(1) is true.

Theorems and Properties
• Let k N. Assume P(k) is true.

Show that P(k +1) is true.

Assume F1 + F2 + … + Fk = Fk+2 – 1.

Examine P(k +1): F1 + F2 + … + Fk+ Fk+1 = Fk+2 – 1 + Fk+1 = Fk+3 – 1

Thus, P(k +1) holds true.

Therefore, by the Principle of Mathematical Induction,

P(n) is true ∀n N.

Theorems and Properties
• Combinatorial Proof
• What is a tiling of an n-board – what is fn?
• fn=Fn+1
• How many ways can we tile an 4-board?
• f4=F5
Theorems and Properties

Identity 1: For n0, f0 + f1 + f2 + … + fn = fn+2 – 1.

Question: How many tilings of an (n +2)-board use at least one domino?

Answer 1: There are fn+2 tilings of an (n+2)-board. Excluding the “all square” tiling gives fn+2 – 1 tilings with at least one domino.

Answer 2: Condition on the location of the last domino. There are fktilings where the last domino covers cells k +1 and k +2. This is because cells 1 through k can be tiled in fkways, cells k +1 and k +2 must be covered by a domino, and cells k+3 through n+2 must be covered by squares. Hence the total number of tilings with at least one domino is

f0 + f1 + f2 + … + fn (or equivalently fk).

The Golden Ratio
• What is the Golden Ratio?
• Satisfies the equation
• Positive Root:
• Negative Root:
Binet’s Formula
• What is Binet’s Formula?
• What is the importance of this formula?
• Direct and Combinatorial Proof
• Let’s do an example together where

For any

Binet’s Formula

Therefore, when , we find that when using Binet’s formula, amazingly equals 832,040.

Binet’s Formula
• Combinatorial Method
• Probability
• Proof by Induction
• Telescoping Proof
• Counting Proof
• Convergent Geometric Series
• Together, the above yield Binet’s Formula
Fibonacci numbers and Nature
• Pinecones
• Sunflowers
• Pineapples
• Artichokes
• Cauliflower
• Other Flowers

http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fib.html

Fibonacci numbers and Nature

http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fib.html

Fibonacci numbers and Nature

http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fib.html

Fibonacci numbers and Nature

http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fib.html

Fibonacci and Phyllotaxis
• Thus, we can conclude that approximates
Further Research Questions
• Looking at Binet’s Formula in more detail
• Looking at Binet’s Formula in comparison with Lucas Numbers
• Similarities?
• Differences?
• Fibonacci and relationships with other mathematical concepts?