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Original Question: How fast rabbits can rabbits breed in ideal circumstances? Suppose a newly-born pair of.

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Presentation on theme: "Original Question: How fast rabbits can rabbits breed in ideal circumstances? Suppose a newly-born pair of."— Presentation transcript:

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2 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 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, 2, 3, 5, 8, 13, 21, …

4 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

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8 13 spirals 8 spirals

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10 Not to mention, we have 2 hands, each with 5 fingers, each with 3 parts!

11  A Fibonacci Rectangle (the Golden Rectangle) is created by taking the Fibonacci numbers and arranging them as shown:

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

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

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

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17  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 ≈ is called a golden rectangle.

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

19 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 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  You may see Fibonacci numbers written as F n F 1 = 1 F 2 = 1 F 3 = 2 F 4 = 3 F 5 = 5 F 6 = 8 etc… What is F 10 ? 55 Recursive Definition of Fibonacci Numbers: F N = F N-1 + F N-2

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

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