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Fibonacci, The Golden number, and Spiral Growth in nature

Fibonacci, The Golden number, and Spiral Growth in nature. http://www.youtube.com/watch?v=kkGeOWYOFoA. Fibonacci's Dilemma (year 1202). Original Question: How fast rabbits can rabbits breed in ideal circumstances?

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Fibonacci, The Golden number, and Spiral Growth in nature

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  1. Fibonacci, The Golden number, and Spiral Growth in nature http://www.youtube.com/watch?v=kkGeOWYOFoA

  2. Fibonacci's Dilemma (year 1202) Original Question: How fast rabbits can rabbits breed in ideal circumstances? Suppose a newly-born pair of rabbits, one male, one female, are put in a field. Rabbits are able to mate at the age of one month so that at the end of its second month a female can produce another pair of rabbits. Suppose that our rabbits never die and that the female always produces one new pair (one male, one female) every month from the second month on. The puzzle that Fibonacci posed was... How many pairs will there be in one year?

  3. Fibonacci sequence Each number created by adding the two previous numbers What are the next 5 Fibonacci numbers? 34, 55, 89, 144, 233, … This sequence has fascinated mathematicians for centuries… 1, 1, 2, 3, 5, 8, 13, 21, …

  4. Fibonacci numbers are everywhere! Flower Petals: MOST flowers have petals that occur in Fibonacci numbers (1, 2, 3, 5, 8, …) Very few flowers have petals that do not occur in Fibonacci numbers (4, 6, 7, …) 1 petal

  5. 2 petal 3 petal 5 petal 8 petal

  6. 13 petal 21 petal 34 petal

  7. PINEAPPLE SPIRALS

  8. Pinecone Spirals 8 spirals 13 spirals

  9. Sunflower seed spirals

  10. HUMAN HAND BONE MEASUREMENTS Not to mention, we have 2 hands, each with 5 fingers, each with 3 parts! 5 8 3 2

  11. The Fibonacci Rectangle:The Golden Spiral • A Fibonacci Rectangle (the Golden Rectangle) is created by taking the Fibonacci numbers and arranging them as shown:

  12. Golden spiral • By drawing the curve through the corners of the boxes, we create something called the golden spiral (or sometimes logarithmic spiral)

  13. Golden Spiral:Nautilus shell The most classic example of the golden spiral in nature is the cross section of the chambers of the Nautilus Shell.

  14. Nautilus Shells

  15. The Golden Ratio: If you start dividing the Fibonacci numbers backwards, the quotient gets closer and closer to the number 1.618 2/1=2 3/2=1.5 5/3=1.667 8/5=1.6 13/8=1.625 21/13=1.615 34/21=1.619 55/34=1.618 89/55=1.618 144/89=1.618 We call this number φ It can be pronounced “Fee” or “Fye” Φ=1.618… and is called the Golden Ratio

  16. The Golden Ratio: Phi φ

  17. GOLDEN Ratios & Golden REctangles • The golden ratio is considered to be the most aesthetically pleasing ratio to the human eye. It is used in art, architecture, and advertising. • Any rectangle whose length ÷ width ≈ 1.618 is called a golden rectangle.

  18. Golden Rectangles Apple IPOD dimensions are 1:1.67 and is the closest MP3 player to the golden ratio.

  19. CULT of the golden Ratio Some people are obsessed with finding golden ratios in everything they see. The see the shape of cereal boxes, cigarette packages, and note-cards as a giant conspiracy. Jack Ruby shoots assassin Lee Harvey Oswald in this famous news photo. The area taken up by Ruby: the area taken up by Oswald = 1.618

  20. Fibonacci Falsities? There are just as many sources that say that finding Fibonacci and the Golden Ratio “EVERYWHERE” is garbage. Google “Fibonacci Skeptics” to find much discourse on the subject.

  21. Fibonacci notation • You may see Fibonacci numbers written as Fn F1 = 1 F2 = 1 F3 = 2 F4 = 3 F5 = 5 F6 = 8 etc… 55 What is F10? Recursive Definition of Fibonacci Numbers: FN = FN-1 + FN-2

  22. Explicit Definition of Fibonacci Numbers: Euler improved another mathematician’s theorem to show that: Not a bad estimate for 55! You don’t have to know the 8th and 9th Fibonacci numbers to find it!

  23. http://www.youtube.com/watch?v=kkGeOWYOFoA

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