2. 5.5 Fibonacci's Rabbits 1 Section 5.5 Fibonacci’s Problem

3. 5.5 Fibonacci's Rabbits 2 “Fibonacci” ( Leonardo de Pisa) 1170-1240 And this man’s claim to fame is …?

4. 5.5 Fibonacci's Rabbits 3

5. 4 Rabbit Rules 1.All pairs of rabbits consist of a male and female 2.One pair of newborn rabbits is placed in hutch on January 1 3.When this pair is 2 months old they produce a pair of baby rabbits 4.Every month afterwards they produce another pair 5.All rabbits produce pairs in the same manner 6.Rabbits don’t die

6. 5.5 Fibonacci's Rabbits 5 The Fibonacci Rabbit Problem How many pairs of rabbits will there be 12 months later?

7. 5.5 Fibonacci's Rabbits 6 How many pairs of rabbits will there be on June 1? 1. 5 2. 7 3. 8 4. 11 5. 13

8. 5.5 Fibonacci's Rabbits 7 Jan 1 Feb 1 Mar 1 2 0 1 0

9. 5.5 Fibonacci's Rabbits 8 Apr 1 May 1 Mar 1 2 3 4 0 1 2 1 0 0 0

10. 5.5 Fibonacci's Rabbits 9 Apr 1 May 1 June 1 310 4210 0 5321 1

11. 5.5 Fibonacci's Rabbits 10 Pairs this month Generalize Pairs last month Pairs of newborns =+ Pairs this month Pairs last month Pairs 2 months ago =+

12. 5.5 Fibonacci's Rabbits 11 Pattern of the Sequence 1, 1, 2, 3, 5, 8, 13, 21, 34, … So the answer to the original problem is. Joke Rule: Every term is the of the two preceding terms

13. 5.5 Fibonacci's Rabbits 12 The 16th term of the Fibonacci sequence is 987 and the 17th term is 1597. What is the 19 th term? 1. 2584 2. 4181 3. 6765

14. 5.5 Fibonacci's Rabbits 13 Change Fibonacci’s problem slightly so that each pair of adult rabbits produces 2 pairs per litter. Which recursion formula best describes the rabbit population? 1.This month = Last month + (Two months ago) 2. This month = Last month + 2*(Two months ago) 3. This month = 2*Last month + (Two months ago)

15. 5.5 Fibonacci's Rabbits 14 As in the last problem assume that each pair produces 2 pairs per litter from the second month on. How many pairs will there be in 5 months? 1.8 2.10 3.11

16. 5.5 Fibonacci's Rabbits 15 New Problem: Assume each pair of adult rabbits produces one pair monthly from the 4th month on. Which recursive formula best describes the rabbit population? 1.This month = Last month + (Four months ago) 2.This month = Last month + 3*(Four months ago) 3.This month = Last month + 4*(Four months ago)

17. 5.5 Fibonacci's Rabbits 16 As in last problem, assume each pair of adult rabbits produces one pair monthly from the 4th month on. How many pairs of rabbits will there be in 7 months? 1.5 2.8 3.13 4.21

18. 5.5 Fibonacci's Rabbits 17 Exponential Growth 1 1 2 3 5 8 13 21 34 55 89 144 1 year 2 years 3 years 4 years 46,368 14,930,352 4,807,526,976 Examples

19. 5.5 Fibonacci's Rabbits 18 End of 5.5

20. 5.5 Fibonacci's Rabbits 19 Fibonacci Suite for retuned piano, seven hands Music

21. 5.5 Fibonacci's Rabbits 20 Chromatic Scale Fibonacci numbers?

22. 5.5 Fibonacci's Rabbits 21 5 Spirals Botany

23. 5.5 Fibonacci's Rabbits 22 One-petaled... white calla lily

24. 5.5 Fibonacci's Rabbits 23 Two-petaled flowers… euphorbia

25. 5.5 Fibonacci's Rabbits 24 Three petals… trillium

26. 5.5 Fibonacci's Rabbits 25 Five petals morning glory

27. 5.5 Fibonacci's Rabbits 26 Eight-petaled… bloodroot

28. 5.5 Fibonacci's Rabbits 27 Thirteen... black-eyed susan

29. 5.5 Fibonacci's Rabbits 28 Twenty-one… shasta daisy with 21 petals

30. 5.5 Fibonacci's Rabbits 29 I am sitting quietly, listening for the quiet noises in the darkness, ghostly images flying between the tall pine trees, illusion created by the mind, made by shadows, the brain playing tricks on itself. It sits there, the raven, black as night, looking at me with its dark eyes in the dark night. Inspiration comes. Words form in my head. Evermore. Poetry Jim T. Henriksen

31. 5.5 Fibonacci's Rabbits 30 “Fibs” Six line, 20 syllable poem One Small, Precise, Poetic, Spiraling mixture: Math plus poetry yields the Fib

32. 5.5 Fibonacci's Rabbits 31 Investments Robert Fischer, leader of the Fibonacci approach to trading

33. 5.5 Fibonacci's Rabbits 32 Education

34. 5.5 Fibonacci's Rabbits 33 213413 Algebra 55 …3421 * = *= 441 442 853211

35. 5.5 Fibonacci's Rabbits 34 Fibonacci and the Greeks 8/5 = 1.6 13/8 = 1.625 21/13 = 1.6153846… 34/21= 1.6190476… 55/34 = 1.6176470… 89/55= 1.6181818… 144/89= 1.6179775… 233/144= 1.6180555… 2/1 = 2 3/2 = 1.5 5/3 = 1.666666… 1, 1, 2, 3, 5, 8, 13, 21, 34, … 1/1 = 1

36. 5.5 Fibonacci's Rabbits 35 Golden Number 1.61808…

37. 5.5 Fibonacci's Rabbits 36 Golden Rectangle 1 1.618

38. 5.5 Fibonacci's Rabbits 37 Mona Lisa Classical Art

39. 5.5 Fibonacci's Rabbits 38 Parthenon - Athens

40. 5.5 Fibonacci's Rabbits 39 The Vitruvian Man

41. 5.5 Fibonacci's Rabbits 40 Spiral Generated by Golden Ratio

42. 5.5 Fibonacci's Rabbits 41 The Nautilus Shell

43. 5.5 Fibonacci's Rabbits 42 Modern Art Piet Mondrian

44. 5.5 Fibonacci's Rabbits 43 Diets and Fitness

45. 5.5 Fibonacci's Rabbits 44 Mo and the Boys There are 100 measures in the first movement. The first section, with the theme, has 32 measures, and the last section, with theme variations, has 68 measures. This is a perfect division, using natural numbers, with the golden section. Although there is no physical evidence that Mozart used the Fibonacci sequence in his music, it is still very easy to see the use of perfect divisions.

46. 5.5 Fibonacci's Rabbits 45 Fibonacci Rap

47. 5.5 Fibonacci's Rabbits 46 Fibonacci Waltz

48. 5.5 Fibonacci's Rabbits 47 Lateralus By Tool

49. 5.5 Fibonacci's Rabbits 48 (1)Black, (1) then, (2) white are, (3) all I see, (5) in my infancy, (8) red and yellow then came to be, (5) reaching out to me, (3) lets me see. (2) There is, (1) so, “Lateralus” (1) much, (2) more that (3) beckons me, (5) to look through to these, (8) infinite possibilities. (13) As below so above and beyond I imagine, (8) drawn outside the lines of reason. (5) Push the envelope. (3) Watch it bend.

50. 5.5 Fibonacci's Rabbits 49 Spiral out Keep going Spiral out Keep going Spiral out Keep going Spiral out! Keep going More lyrics

51. 5.5 Fibonacci's Rabbits 50 Facts Keenan begins singing at 1:37 into the song. 1 minute 37 seconds, or 97 seconds, is approximately 1.618 of a full minute. The time signatures of the chorus change from 9/8 to 8/8 to 7/8; as drummer Danny Carey says, "It was originally titled 9-8-7. For the time signatures. Then it turned out that 987 was the 17th number of the Fibonacci sequence. So that was cool.”time signaturesDanny Carey

52. 5.5 Fibonacci's Rabbits 51 Fibonacci Joke How much does a large order of Fibonaccos cost?. The price of a medium The price of a small +

53. 5.5 Fibonacci's Rabbits 52 Meta-Material