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 city-state of Pisa Books: Liber Abaci, Practica Geometriae, Flos, and Liber Quadratorum Frederick II’s challenge Impact on mathematics

What are the Fibonacci Numbers? … F1F1 F2F2 F3F3 F4F4 F5F5 F6F6 F7F7 F8F8 F9F9 F 10 F Recursive Definition: F 1 =F 2 =1 and, for n >2, F n =F n-1 + F n-2 For example, let n=6. Thus, F 6 =F F 6-2 F 6 =F 5 + F 4 F 6 =5+3 So, F 6 =8

Theorems and Properties Telescoping Proof Theorem: For any n  N, F 1 + F 2 + … + F n = F n Proof: Observe that F n-2 + F n-1 = F n (n >2) may be expressed as F n-2 = F n – F n-1 (n >2). Particularly, F 1 = F 3 – F 2 F 2 = F 4 – F 3 F 3 = F 5 – F 4 … F n-1 = F n+1 – F n F n = F n+2 – F n+1 When we add the above equations and observing that the sum on the right is telescoping, we find that: F 1 + F 2 + … + F n = F 1 + (F 4 – F 3 ) + (F 5 – F 4 ) + … + (F n+1 – F n ) + (F n+2 – F n+1 ) = F n+2 +(F 1 -F 3 )= F n+2 – F 2 = F n+2 – 1

Theorems and Properties Proof by Induction Theorem: For any n  N, F 1 + F 2 + … + F n = F n+2 – 1. 1)Show P(1) is true. F 1 = F 2 = 1, F 3 = 2 F 1 = F 1+2 – 1 F 1 = F 3 – 1 F 1 = 2-1 F 1 = 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 F 1 + F 2 + … + F k = F k+2 – 1. Examine P(k +1): F 1 + F 2 + … + F k + F k+1 = F k+2 – 1 + F k+1 = F k+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 f n ?  f n =F n+1 How many ways can we tile an 4-board? f4=F5f4=F5

Theorems and Properties Identity 1: For n  0, f 0 + f 1 + f 2 + … + f n = f n+2 – 1. Answer 2: Condition on the location of the last domino. There are f k tilings where the last domino covers cells k +1 and k +2. This is because cells 1 through k can be tiled in f k 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 f 0 + f 1 + f 2 + … + f n (or equivalently f k ). Question: How many tilings of an (n +2)-board use at least one domino? Answer 1: There are f n+2 tilings of an (n+2)-board. Excluding the “all square” tiling gives f n+2 – 1 tilings with at least one domino.

Combinatorial Proof Diagram

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

Fibonacci numbers and Nature

Fibonacci numbers and Nature

Fibonacci numbers and Nature nott/Fibonacci/fib.html

Fibonacci and Phyllotaxis TreeNumber of Turns Number of Leaves Phyllotactic Ratio Basswood, Elm 121/2 Beech, Hazel131/3 Apricot, Cherry, Oak 252/5 Pear, Poplar383/8 Almond, Willow 5135/13

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?

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