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

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Outline

• History of Leonardo Pisano Fibonacci
• What are the Fibonacci numbers?
• Explaining the sequence
• Recursive Definition
• Theorems and Properties
• The Golden Ratio
• Binets Formula
• Fibonacci numbers and Nature

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Leonardo Pisano Fibonacci

• Born in 1170 in the city-state of Pisa
• Books Liber Abaci, Practica Geometriae, Flos,
  and Liber Quadratorum
• Frederick IIs challenge
• Impact on mathematics

http//www-gap.dcs.st-and.ac.uk/history/Mathemati
cians/Fibonacci.html
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What are the Fibonacci Numbers?
1 1 2 3 5 8 13 21 34 55 89
F1 F2 F3 F4 F5 F6 F7 F8 F9 F10 F11 ...

• Recursive Definition F1F21 and, for n gt2,
  FnFn-1 Fn-2
• For example, let n6.
• Thus, F6F6-1 F6-2 F6F5 F4
  F653
• So, F68

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Theorems and Properties

• Telescoping Proof

Theorem For any n ? N, F1 F2 Fn Fn2
- 1
Proof Observe that Fn-2 Fn-1 Fn (n gt2) may
be expressed as Fn-2 Fn Fn-1 (n gt2).
Particularly, F1 F3 F2 F2 F4
F3 F3 F5 F4
Fn-1 Fn1 Fn Fn Fn2 Fn1
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) (Fn1 Fn) (Fn2 Fn1)
Fn2 (F1-F3) Fn2 F2 Fn2 1
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Theorems and Properties

• Proof by Induction

Theorem For any n ? N, F1 F2 Fn Fn2
1.
1) Show P(1) is true. F1 F2 1, F3 2
F1 F12 1 F1 F3 1 F1
2-1 F1 1 Thus, P(1) is true.
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Theorems and Properties

• Let k ? N. Assume P(k) is true.
• Show that P(k 1) is true.
• Assume F1 F2 Fk Fk2 1.
• Examine P(k 1) F1 F2 Fk Fk1
  Fk2 1 Fk1
  Fk3 1
• Thus, P(k 1) holds true.

Therefore, by the Principle of Mathematical
Induction, P(n) is true ?n ? N.
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Theorems and Properties

• Combinatorial Proof
• What is a tiling of an n-board what is fn?
• fnFn1
• How many ways can we tile an 4-board?
• f4F5

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Theorems and Properties
Identity 1 For n ?0, f0 f1 f2 fn
fn2 1.
Question How many tilings of an (n 2)-board
use at least one domino?
Answer 1 There are fn2 tilings of an
(n2)-board. Excluding the all square tiling
gives fn2 1 tilings with at least one domino.
Answer 2 Condition on the location of the last
domino. There are fk tilings where the last
domino covers cells k 1 and k 2. This is
because cells 1 through k can be tiled in fk
ways, cells k 1 and k 2 must be covered by a
domino, and cells k3 through n2 must be covered
by squares. Hence the total number of tilings
with at least one domino is f0 f1 f2
fn (or equivalently fk).
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Combinatorial Proof Diagram
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The Golden Ratio

• What is the Golden Ratio?
• Satisfies the equation
• Positive Root
• Negative Root

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Binets Formula

• What is Binets Formula?
• What is the importance of this formula?
• Direct and Combinatorial Proof
• Lets do an example together where

For any


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Binets Formula
Therefore, when , we find that when
using Binets formula, amazingly equals
832,040.
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Binets Formula

• Combinatorial Method
• Probability
• Proof by Induction
• Telescoping Proof
• Counting Proof
• Convergent Geometric Series
• Together, the above yield Binets Formula

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Fibonacci numbers and Nature

• Pinecones
• Sunflowers
• Pineapples
• Artichokes
• Cauliflower
• Other Flowers

http//www.mcs.surrey.ac.uk/Personal/R.Knott/Fibon
acci/fib.html
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Fibonacci numbers and Nature
http//www.mcs.surrey.ac.uk/Personal/R.Knott/Fibon
acci/fib.html
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Fibonacci numbers and Nature
http//www.mcs.surrey.ac.uk/Personal/R.Knott/Fibon
acci/fib.html
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Fibonacci numbers and Nature
http//www.mcs.surrey.ac.uk/Personal/R.Knott/Fibon
acci/fib.html
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Fibonacci and Phyllotaxis
Tree Number of Turns Number of Leaves Phyllotactic Ratio
Basswood, Elm 1 2 1/2
Beech, Hazel 1 3 1/3
Apricot, Cherry, Oak 2 5 2/5
Pear, Poplar 3 8 3/8
Almond, Willow 5 13 5/13
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Fibonacci and Phyllotaxis

• Thus, we can conclude that approximates

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Further Research Questions

• Looking at Binets Formula in more detail
• Looking at Binets Formula in comparison with
  Lucas Numbers
• Similarities?
• Differences?
• Fibonacci and relationships with other
  mathematical concepts?

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Thank you for listening to my presentation!