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

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
Fibonacci numbers and Nature
Leonardo Pisano Fibonacci
 Born in 1170 in the citystate 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?
1
1
2
3
5
8 13 21 34 55 89 …
F1 F2 F3 F4 F5 F6 F7 F8 F9 F10 F11 ...
 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
2)
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 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 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).
n
k 0
Combinatorial Proof Diagram
fn
1
2
3
4
n-2
n-1
n
n+1 n+2
fn-1
1
2
3
4
n-2
n-1
n
n+1 n+2
fn-2
1
2
3
4
n-2
n-1
n
n+1 n+2
f1
1
2
3
4
n-2
n-1
n
n+1 n+2
f0
1
2
3
4
n-2
n-1
n
n+1 n+2
The Golden Ratio
What is the Golden Ratio?
Satisfies the equation
1
x
 x2  x  1  x2  x 1  0
x 1
Positive Root:
Negative Root:
x  the golden ratio   
x  ' 
1 5
 1.61803398875
2
1 5
 0.61803398875
2
Binet’s Formula
What is Binet’s Formula?
n
1 5  1 5 

 

 n  ( ' ) n  2   2 

For any n  Z , Fn 
5
5
What is the importance of this formula?
Direct and Combinatorial Proof
Let’s do an example together where n  30
n
Binet’s Formula
Fn 
 n  ( ' ) n
F30 
F30
F30
F30
 30
5
 ( ' ) 30
5
1 5 




2


30
1 5 




2


5
30
1,860,498

5
 832,040
Therefore, when n  30 , we find that when using Binet’s formula, F30 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.K
nott/Fibonacci/fib.html
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
Fibonacci and Phyllotaxis
Fn
Fn
1
1



Fn  2 Fn  Fn 1 Fn  Fn 1 1  Fn 1
Fn
Fn
  lim
Fn 1
Fn
n
 Thus, we can conclude that
Fn approximates 1
Fn  2
1 
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?
Thank you for listening to my
presentation!