1. Math around Us: Fibonacci Numbers John Hutchinson March 2005

2. Leonardo Pisano Fibonacci Born: 1170 in (probably) Pisa (now in Italy) Died: 1250 in (possibly) Pisa (now in Italy)

3. What is a Fibonacci Number? Fibonacci numbers are the numbers in the Fibonacci sequence 0, 1, 1, 2, 3, 5, 8, 13, 21,..., each of which, after the second, is the sum of the two previous ones.

4. The Fibonacci numbers can be considered to be a function with domain the positive integers. N12345678910 FNFNFNFN11235813213455 Note that F N+2 = F N+1 + F N

5. Note Every 3rd Fibonacci number is divisible by 2. Every 4th Fibonacci number is divisible by 3. Every 5th Fibonacci number is divisible by 5. Every 6th Fibonacci number is divisible by 8. Every 7th Fibonacci number is divisible by 13. Every 8thFibonacci number is divisible by 21. Every 9th Fibonacci number is divisible by 34.

6. Sums of Fibonacci Numbers 1 + 1 = 2 1 + 1 = 2???? 1 + 1 + 2 = 4 1 + 1 + 2 = 4???? 1 + 1 + 2 + 3 = 7 1 + 1 + 2 + 3 = 7???? 1 + 1 + 2 + 3 + 5 = 12 1 + 1 + 2 + 3 + 5 = 12???? 1 + 1 + 2 + 3 + 5 + 8 = 20 1 + 1 + 2 + 3 + 5 + 8 = 20????

7. Sums of Fibonacci Numbers 1 + 1 = 2 1 + 1 = 2 3 - 1 1 + 1 + 2 = 4 1 + 1 + 2 = 4 5 - 1 1 + 1 + 2 + 3 = 7 1 + 1 + 2 + 3 = 7 8 - 1 1 + 1 + 2 + 3 + 5 = 12 1 + 1 + 2 + 3 + 5 = 12 13 - 1 1 + 1 + 2 + 3 + 5 + 8 = 20 1 + 1 + 2 + 3 + 5 + 8 = 20 21 - 1

8. F 1 + F 2 + F 3 + … + F N = F N+2 -1

9. Sums of Squares 1 2 + 1 2 = 2 1 2 + 1 2 = 2???? 1 2 + 1 2 + 2 2 = 6 1 2 + 1 2 + 2 2 = 6???? 1 2 + 1 2 + 2 2 + 3 2 = 15 1 2 + 1 2 + 2 2 + 3 2 = 15???? 1 2 + 1 2 + 2 2 + 3 2 + 5 2 = 40 1 2 + 1 2 + 2 2 + 3 2 + 5 2 = 40???? 1 2 + 1 2 + 2 2 + 3 2 + 5 2 + 8 2 = 104 1 2 + 1 2 + 2 2 + 3 2 + 5 2 + 8 2 = 104????

10. Sums of Squares 1 2 + 1 2 = 2 1 2 + 1 2 = 2 1 X 2 1 2 + 1 2 + 2 2 = 6 1 2 + 1 2 + 2 2 = 6 2 X 3 1 2 + 1 2 + 2 2 + 3 2 = 15 1 2 + 1 2 + 2 2 + 3 2 = 15 3 X 5 1 2 + 1 2 + 2 2 + 3 2 + 5 2 = 40 1 2 + 1 2 + 2 2 + 3 2 + 5 2 = 40 5 X 8 1 2 + 1 2 + 2 2 + 3 2 + 5 2 + 8 2 = 104 1 2 + 1 2 + 2 2 + 3 2 + 5 2 + 8 2 = 104 8 X 13

11. The Formula F 1 2 + F 2 2 + F 3 2 + …+ F n 2 = F n X F N+1

12. F N+I = F I-1 F N + F I F N+1 Another Formula

13. Pascal’s Triangle

14. Sums of Rows The sum of the numbers in any row is equal to 2 to the nth power or 2 n, when n is the number of the row. For example: 2 0 = 1 2 1 = 1+1 = 2 2 2 = 1+2+1 = 4 2 3 = 1+3+3+1 = 8 2 4 = 1+4+6+4+1 = 16

15. Add Diagonals

16. Pascal’s triangle with odd numbers in red.

17. 1-White Calla Lily

18. 1-Orchid

19. 2-Euphorbia

20. 3-Trillium

21. 3-Douglas Iris

22. 3&5 - Bougainvilla

23. 5-Columbine

24. 5-St. Anthony’s Turnip (buttercup)

25. 5-Unknown

26. 5-Wild Rose

27. 8-Bloodroot

28. 13-Black-eyed Susan

29. 21-Shasta Daisy

30. 34-Field Daisy

31. Dogwood = 4?????

32. Here a sunflower seed illustrates this principal as the number of clockwise spirals is 55 (marked in red, with every tenth one in white) and the number of counterclockwise spirals is 89 (marked in green, with every tenth one in white.)

33. Sweetwart

34. Sweetwart

36. "Start with a pair of rabbits, (one male and one female). Assume that all months are of equal length and that : 1. rabbits begin to produce young two months after their own birth; 2. after reaching the age of two months, each pair produces a mixed pair, (one male, one female), and then another mixed pair each month thereafter; and 3. no rabbit dies. How many pairs of rabbits will there be after each month?"

37. Let’s count rabbits Babies1011235813213445 Adult01123581321345589 Total1123581321345589144

38. Let’s count tokens A token machine dispenses 25-cent tokens. The machine only accepts quarters and half-dollars. How many ways can a person purchase 1 token, 2 tokens, 3 tokens, …?

39. Count them 25C Q 1 50CQQ-H2 75CQQQ-HQ-QH3 100CQQQQ-QQH-QHQ-HQQ-HH5 125CQQQQQ-QQQH-QQHQ-QHQQ-HQQQ- HHQ-HQH-QHH 8

40. 89 Measures Total 55 Measures34 Measures 21 Measures 13 First Movement, Music for Strings, Percussion, and Celeste Bela Bartok Gets loud here Strings remove mutesReplace mutes 21 Theme Texture 138

42. The Keyboard

44. <>

49. The hand

50. Ratios of consecutive 11 22 31.5 51.66666 81.6 131.625 211.615385 341.619048551.617647891.618182 1441.617978 2331.618056 3771.618026 6101.618037 9871.618033 etc1.618034…

51. The golden ratio is approximately 1.610833989… (√5+1)/2 = 2/(√5-1) Or exactly

52. Golden Section S L S/L = L/(S+L) If S = 1 then L= 1.610833989… If L = 1 then S = 1/L =.610833989…

53. Golden Rectangle S L

54. Golden Triangles 5 3 8 5 L S

55. The Parthenon

65. Holy Family, Michelangelo

66. Crucifixion - Raphael

67. Self Portrait - Rembrandt

70. Seurat

72. Fractions 1/1 = 1 1/1 = 1 ½ =.5 ½ =.5 1/3 =.33333 1/3 =.33333 1/5 =.2 1/5 =.2 1/8 =.125 1/8 =.125 … 1/89 = ? 1/89 = ?

73. .011/100.01.0011/1000.011.00022/10000.0112.000033/100000.01123.0000055/1000000.011235.00000088/10000000.0112358.0000001313/100000000.00112393.00000002121/1000000000.0011235951.000000003434/10000000000.00112359544.0000000005555/100000000000.001123595495 1/89 =.00112359550561798 …

74. Are there negative Fibonaccis? F n = F n+2 - F n+1

75. 1 -2 -32 -4-3 -55 -6-8 -713 -8-21 F -n = (-1) n+1 F n

76. For any three Fibonacci Numbers the sum of the cubes of the two biggest minus the cube of the smallest is a Fibonacci number. 8 5 13 125 512 2197 2709 – 125 = 2584 F n+2 3 + F n+1 3 – F n 3 = F 3(n+1)