2. Date: 3 rd Mar, 2011 Time: 11:59:59 Venue: 2302@UST Class: Math 162 Follow Me 1

3. Date: 3 rd Mar, 2011 Time: 11:59:59 Venue: 2302@UST Class: Math 162 Follow Me 2

4. ͵fıb ə ʹna ːʧ ı Sequence 3

5. Fibonacci Sequence & Golden Ratio Chee Ka Ho, Alan Lai Siu Kwan, Justina Wong Wing Yan, Gloria 4

6.  Introduction  Fibonacci Sequence  Golden Ratio  Activities  Conclusion CONTENT 5

7.  named after Leonardo of Pisa (1170~1250)  Italian Mathematician Introduction 6

8. Question Time !!  Fibonacci Sequence is named after Leonardo of Pisa, so why is it called Fibonacci Sequence, but not Leonardo Sequence or Pisa Sequence? A) Because he is a son. B) Because his father is called Bonacci. C) Because this is a short form only. D) All of the above. WHY? 7

9. Question Time !!  Fibonacci Sequence is named after Leonardo of Pisa, so why is it called Fibonacci Sequence, but not Leonardo Sequence or Pisa Sequence? Leonardo is the son of Bonacci. “Son of Bonacci” in Italian is 'filius Bonacci'. To take the short form, people called him Fibonacci. WHY? D) All of the above Oh..IC 8

10. Leonardo of Pisa (1170~1250)  Son of a wealthy Italian Merchant  Traveled with his dad and learnt about Hindu-Arabic numerical system  Wrote 'Book of Calculation'  Fibonacci Sequence is an example in this book 9

11.  He considered the growth of an idealized rabbit population. History of Fibonacci Sequence 10

12. Imagine You are now in a Kingdom of RABBITS : 1. never die. 2. are able to mate at the age of 1 month !!! 3. At the end of the 2 nd month, a female can produce 4. A mating pair always produces one new pair every month. Rabbit population 11

13. Question: How many pairs of rabbits will there be in one year? Rabbit population 1 1 2 3 5 8 12

14.  0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144… for n ≥ 0 and Fibonacci Sequence  Related to nature in many aspects! 13

15.  Number of petals ( 花瓣 )  Spirals in daisy, pinecone…  Arrangements of leaves  … Fibonacci Sequence and nature 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144… 14

16. Number of petals ( 花瓣 ): 1 3 2 5 8 13 21 34 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144… 15

17.  Let’s Go !!!! Spirals in Daisy: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144… 16 http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fibnat.html#petals

18. Spirals in Daisy: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144… 17

19. Spirals in Daisy: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144… 18

20. Spirals in Daisy: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144… 19

21. Spirals in pinecone 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144… 20

22. Spirals in pinecone 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144… 21

23. Spirals in pinecone 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144… 22

24. Spirals in pinecone 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144… 23

25.  Number of paths for going to cell n in a honey comb: Exercise: n01234… Number of paths 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144… 24

26.  Number of paths for going to cell n in a honey comb: Exercise: n01234… Number of paths 123 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144… 25

27. n123456 Fn011235 Fn/Fn-1--121.5 1.66667 Ratios of Fibonacci Numbers n111213141516 Fn5589133233377610 Fn/Fn-1 1.617681.618181.617981.618061.618031.61804 26

28. n123456 Fn011235 Fn/Fn-1--121.5 1.66667 Ratios of Fibonacci Numbers n111213141516 Fn5589133233377610 Fn/Fn-1 1.617681.618181.617981.618061.618031.61804 27

29. Golden Ratio 28

30.  Denoted by Φ = 1.6180339887…  Related to beauty Golden ratio l wl-w 29

31.  Construct a simple square  Draw a line from the midpoint of one side of the square to an opposite corner  Use that line as the radius to draw an arc that defines the height of the rectangle  Complete the golden rectangle. Golden rectangle 30

32. Golden rectangle Φ 1 31

33. Golden Spiral 32

34.  http://www.xgoldensection.com/demos.html http://www.xgoldensection.com/demos.html Golden ratio-nature 33

35. Golden ratio--Architecture Parthenon, Acropolis, Athens 34

36. Golden ratio--Architecture 35

37. Golden ratio--Architecture Golden Rectangle 36

38. Golden ratio--Architecture 37

39.  Da Vinci's Mona Lisa Golden ratio--Paintings 38

40.  Note that not every individual has body dimensions in exact phi proportion but averages across populations tend towards phi and phi proportions are perceived as being the most natural or beautiful. Golden Ratio 39

41. Activity 40

42.  http://www.youtube.com/watch?v=kkGeOWYOFoA&f eature=related http://www.youtube.com/watch?v=kkGeOWYOFoA&f eature=related Conclusion 41

43.  http://britton.disted.camosun.bc.ca/goldslide/jbgoldsli de.htm http://britton.disted.camosun.bc.ca/goldslide/jbgoldsli de.htm  http://en.wikipedia.org/wiki/Fibonacci_number http://en.wikipedia.org/wiki/Fibonacci_number  http://www.goldennumber.net/hand.htm http://www.goldennumber.net/hand.htm  http://britton.disted.camosun.bc.ca/fibslide/jbfibslide. htm http://britton.disted.camosun.bc.ca/fibslide/jbfibslide. htm  http://jwilson.coe.uga.edu/emat6680/parveen/GR_in_ art.htm http://jwilson.coe.uga.edu/emat6680/parveen/GR_in_ art.htm References 42

44. Discussion 43

45.  1) Explain why the exercise in slide 24-25 is related to Fibonacci Sequence.  2) Draw a golden rectangle and derive from the rectangle.  Extra Credit) Prove that for Fibonacci Sequence. Homework 44

46. ~~Thank you~~ 45