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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

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The Sequence of Fibonacci Numbers and How They Relate to Nature

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  1. The Sequence of Fibonacci Numbers and How They Relate to Nature November 30, 2004 Allison Trask

  2. 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

  3. 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

  4. What are the Fibonacci Numbers? • 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

  5. 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

  6. 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.

  7. Theorems and Properties • 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.

  8. 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

  9. 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 fktilings where the last domino covers cells k +1 and k +2. This is because cells 1 through k can be tiled in fkways, 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).

  10. Combinatorial Proof Diagram

  11. The Golden Ratio • What is the Golden Ratio? • Satisfies the equation • Positive Root: • Negative Root:

  12. 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

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

  14. Binet’s Formula • Combinatorial Method • Probability • Proof by Induction • Telescoping Proof • Counting Proof • Convergent Geometric Series • Together, the above yield Binet’s Formula

  15. Fibonacci numbers and Nature • Pinecones • Sunflowers • Pineapples • Artichokes • Cauliflower • Other Flowers http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fib.html

  16. Fibonacci numbers and Nature http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fib.html

  17. Fibonacci numbers and Nature http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fib.html

  18. Fibonacci numbers and Nature http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fib.html

  19. Fibonacci and Phyllotaxis

  20. Fibonacci and Phyllotaxis • Thus, we can conclude that approximates

  21. 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?

  22. Thank you for listening to my presentation!

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