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

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

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fbna Sequence

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Fibonacci Sequence & Golden RatioChee Ka Ho, AlanLai Siu Kwan, JustinaWong Wing Yan, Gloria*

IntroductionFibonacci SequenceGolden RatioActivitiesConclusion

CONTENT*

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

Introduction*

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?*

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 aboveOh..IC*

Leonardo of Pisa (1170~1250)Son of a wealthy Italian MerchantTraveled with his dad and learnt about Hindu-Arabic numerical systemWrote 'Book of Calculation'Fibonacci Sequence is an example in this book*

He considered the growth of an idealized rabbit population.

History of Fibonacci Sequence*

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 produce4. A mating pair always produces one new pair every month.Rabbit population*

Question:How many pairs of rabbits will there be in one year?

Rabbit population112358*

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144 for n 0 and Fibonacci SequenceRelated to nature in many aspects!

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Number of petals ()Spirals in daisy, pineconeArrangements of leavesFibonacci Sequence and nature0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144*

Number of petals ():0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144*

Lets Go !!!!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

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

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

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

Spirals in pinecone0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144*

Spirals in pinecone0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144*

Spirals in pinecone0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144*

Spirals in pinecone0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144*

Number of paths for going to cell n in a honey comb:

Exercise:0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144*

n01234Number of paths

Number of paths for going to cell n in a honey comb:

Exercise:1230, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144*

n01234Number of paths

Ratios of Fibonacci Numbers*

n123456Fn011235Fn/Fn-1--121.51.66667

n111213141516Fn5589133233377610Fn/Fn-11.617681.618181.617981.618061.618031.61804

Ratios of Fibonacci Numbers*

n123456Fn011235Fn/Fn-1--121.51.66667

n111213141516Fn5589133233377610Fn/Fn-11.617681.618181.617981.618061.618031.61804

Golden Ratio*

Denoted by = 1.6180339887Related to beautyGolden ratio*

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*

Golden rectangle1*

Golden Spiral*

http://www.xgoldensection.com/demos.html

Golden ratio-nature*

Golden ratio--ArchitectureParthenon, Acropolis, Athens*

Golden ratio--Architecture*

Golden ratio--ArchitectureGolden Rectangle*

Golden ratio--Architecture*

Da Vinci's Mona Lisa

Golden ratio--Paintings*

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*

Activity*

http://www.youtube.com/watch?v=kkGeOWYOFoA&feature=relatedConclusion*

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

References*

Discussion*

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*

~~Thank you~~*

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