the sequence of fibonacci numbers and how they relate to nature november 30, 2004 allison trask

The Sequence of Fibonacci Numbers and How They Relate

to Nature

November 30, 2004

Allison Trask

Outline

History of Leonardo Pisano FibonacciWhat are the Fibonacci numbers?

Explaining the sequence Recursive Definition

Theorems and PropertiesThe Golden RatioBinet’s FormulaFibonacci 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?

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 ProofTheorem: 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+1When 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 InductionTheorem: 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 ProofWhat 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.

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

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.

Combinatorial Proof Diagram

n-2 n-1 n n+1 n+21 2 3 4

fn-2

f1

f0

fn-1

fn

1 2 3 4 n-2 n-1 n n+1 n+2

1 2 3 4 n-2 n-1 n n+1 n+2

1 2 3 4 n-2 n-1 n n+1 n+2

n-2 n-1 n n+1 n+21 2 3 4

The Golden Ratio

What is the Golden Ratio?Satisfies the equation

Positive Root:Negative Root:

56180339887.12

51

ratiogoldenthex

56180339887.02

51'

x

0111

1 22

xxxxx

x

Binet’s Formula

What is Binet’s Formula?

What is the importance of this formula?Direct and Combinatorial ProofLet’s do an example together where

For any 5

251

251

5

)'(,

nn

nn

nFZn

30n

Binet’s Formula

30F

040,8325

498,860,15

251

251

5

)'(

5

)'(

30

30

3030

30

3030

30

F

F

F

F

Fnn

n

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

30n

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

PineconesSunflowersPineapplesArtichokesCauliflowerOther 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

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

Thus, we can conclude that approximates

n

F

F

n

n 1limn

n

n

nnnn

n

n

n

FF

FFFFF

F

F

F

1112 1

11

1

1

2n

n

F

F

Further Research Questions

Looking at Binet’s Formula in more detailLooking 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!