7. Fibonacci’s Rabbits Problem: 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 the rabbits never die and that the female always produces one new pair (one male, one female) every month from the second month on. How many pairs will there be in one year?
8. Pairs 1 pair At the end of the first month there is still only one pair
9. Pairs 1 pair 1 pair 2 pairs End first month… only one pair At the end of the second month the female produces a new pair, so now there are 2 pairs of rabbits
10. Pairs 1 pair 1 pair 2 pairs 3 pairs End second month… 2 pairs of rabbits At the end of the third month, the original female produces a second pair, making 3 pairs in all in the field. End first month… only one pair
11. Pairs 1 pair 1 pair 2 pairs 3 pairs End third month… 3 pairs 5 pairs End first month… only one pair End second month… 2 pairs of rabbits At the end of the fourth month, the first pair produces yet another new pair, and the female born two months ago produces her first pair of rabbits also, making 5 pairs.
12.
13. So 144 Pairs will be there at the end of One Year….
26. Note that 8 and 13 are Consecutive Fibonacci numbers
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30. The Fibonacci numbers can be found in pineapples and bananas. Bananas have 3 or 5 flat sides, Pineapple scales have Fibonacci spirals in sets of 8, 13, 21
39. One interesting thing about Phi is its reciprocal 1/ φ = 1/1.618 = 0.618 . It is highly unusual for the decimal integers of a number and its reciprocal to be exactly the same.
40. A golden rectangle is a rectangle where the ratio of its length to width is the golden ratio. That is whose sides are in the ratio 1:1.618
41. The golden rectangle has the property that it can be further subdivided in to two portions a square and a golden rectangle This smaller rectangle can similarly be subdivided in to another set of smaller golden rectangle and smaller square. And this process can be done repeatedly to produce smaller versions of squares and golden rectangles
46. Aha! Notice that as we continue down the sequence, the ratios seem to be converging upon one number (from both sides of the number)! 2/1 = 2.0 (bigger) 3/2 = 1.5 (smaller) 5/3 = 1.67 (bigger) 8/5 = 1.6 (smaller) 13/8 = 1.625 (bigger) 21/13 = 1.615 (smaller) 34/21 = 1.619 (bigger) 55/34 = 1.618 (smaller) 89/55 = 1.618 The Fibonacci sequence is 1,1,2,3,5,8,13,21,34,55,….
47. If we continue to look at the ratios as the numbers in the sequence get larger and larger the ratio will eventually become the same number, and that number is the Golden Ratio !
51. Golden ratio in Art Many artists who lived after Phidias have used this proportion. Leonardo Da Vinci called it the "divine proportion" and featured it in many of his paintings