Download m162-8

Date: 3rd Mar, 2011
Time: 11:59:59
Venue: 2302@UST
Class: Math 162
Follow Me
1
Date: 3rd Mar, 2011
Time: 11:59:59
Venue: 2302@UST
Class: Math 162
Follow Me
2
͵fıbəʹnaːʧı Sequence
3
Fibonacci Sequence
&
Golden Ratio
Chee Ka Ho, Alan
Lai Siu Kwan, Justina
Wong Wing Yan, Gloria
4
CONTENT
 Introduction
 Fibonacci Sequence
 Golden Ratio
 Activities
 Conclusion
5
Introduction
 named after Leonardo of Pisa
(1170~1250)
 Italian Mathematician
6
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.
7
WHY?
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?
Leonardo is the son of Bonacci.
“Son of Bonacci” in Italian is 'filius
Bonacci'. To take the short form,
people called him Fibonacci.
8
Oh..IC
WHY?
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
History of Fibonacci Sequence
 He considered the growth of an
idealized rabbit population.
10
Rabbit population
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 2nd month, a female can produce
4. A mating pair always produces one new pair every month.
11
1
Rabbit population
1
2
Question:
How many pairs of rabbits
will there be in
one year?
3
5
8
12
Fibonacci Sequence
 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144…
for n ≥ 0 and
 Related to nature in many aspects!
13
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
…
14
Number of petals (花瓣):
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144…
8
15
Spirals in Daisy:
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144…
 Let’s Go !!!!
http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fibnat.html#petals
16
Spirals in Daisy:
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144…
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 pinecone
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
Exercise:
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144…
 Number of paths for going to cell n in a honey comb:
n
0
1
2
Number
of paths
24
3
4
…
Exercise:
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144…
 Number of paths for going to cell n in a honey comb:
n
0
1
2
Number
of paths
1
2
3
25
3
4
…
Ratios of Fibonacci Numbers
n
Fn
1
0
2
1
3
1
4
2
5
3
Fn/Fn-1
-
-
1
2
1.5
6
5
1.66667
n
11
12
13
14
15
16
Fn
55
89
133
233
377
610
Fn/Fn-1 1.61768 1.61818 1.61798 1.61806 1.61803 1.61804
26
Ratios of Fibonacci Numbers
n
Fn
1
0
2
1
3
1
4
2
5
3
Fn/Fn-1
-
-
1
2
1.5
6
5
1.66667
n
11
12
13
14
15
16
Fn
55
89
133
233
377
610
Fn/Fn-1 1.61768 1.61818 1.61798 1.61806 1.61803 1.61804
27
Golden Ratio
28
Golden ratio
l
w
l-w
 Denoted by Φ = 1.6180339887…
 Related to beauty
29
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.
30
Golden rectangleΦ
1
31
Golden Spiral
32
Golden ratio-nature
 http://www.xgoldensection.com/demos.html
33
Golden ratio--Architecture
Parthenon, Acropolis,
Athens
34
Golden ratio--Architecture
35
Golden ratio--Architecture
Golden Rectangle
36
Golden ratio--Architecture
37
Golden ratio--Paintings
 Da Vinci's Mona Lisa
38
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.
39
Activity
40
Conclusion
 http://www.youtube.com/watch?v=kkGeOWYOFoA&f
eature=related
41
References
 http://britton.disted.camosun.bc.ca/goldslide/jbgoldsli
de.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
42
Discussion
43
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
Fibonacci Sequence.
for
44
~~Thank you~~
45