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