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

Date: 3 rd Mar, 2011 Time: 11:59:59 Venue: 2302@UST Class: Math 162. Follow Me. Date: 3 rd Mar, 2011 Time: 11:59:59 Venue: 2302@UST Class: Math 162. Follow Me. ͵ fıbə ʹ naːʧı Sequence. Fibonacci Sequence & Golden Ratio. Chee Ka Ho, Alan Lai Siu Kwan, Justina

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

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

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

  3. ͵fıbəʹnaːʧı Sequence

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

  5. CONTENT Introduction Fibonacci Sequence Golden Ratio Activities Conclusion

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

  7. Question Time !! • Fibonacci Sequence is named after Leonardo of Pisa, so why is it called Fibonacci Sequence, but not Leonardo Sequence or Pisa Sequence? WHY? 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.

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

  9. 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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

  24. Exercise: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144… 0 2 4 6 8 Start 1 3 5 7 9 Number of paths for going to cell n in a honey comb:

  25. Exercise: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144… 0 2 4 6 8 Start 1 3 5 7 9 1 2 3 Number of paths for going to cell n in a honey comb:

  26. Ratios of Fibonacci Numbers

  27. Ratios of Fibonacci Numbers

  28. Golden Ratio

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

  30. Golden rectangle 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.

  31. Golden rectangle Φ 1

  32. Golden Spiral

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

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

  35. Golden ratio--Architecture

  36. Golden ratio--Architecture Golden Rectangle

  37. Golden ratio--Architecture

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

  39. Golden Ratio 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.

  40. Activity

  41. Conclusion http://www.youtube.com/watch?v=kkGeOWYOFoA&feature=related

  42. References http://britton.disted.camosun.bc.ca/goldslide/jbgoldslide.htm http://en.wikipedia.org/wiki/Fibonacci_number http://www.goldennumber.net/hand.htm http://britton.disted.camosun.bc.ca/fibslide/jbfibslide.htm http://jwilson.coe.uga.edu/emat6680/parveen/GR_in_art.htm

  43. Discussion

  44. Homework • 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.

  45. ~~Thank you~~

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