FIBONACCI

1
FIBONACCI
2
Fibonacci

• 1170 1250
• Introduction of Arabic number system into the
  West.
• Root finding almost 300 years before others could
  find such accurate results.
• Problem involving the growth of a populualtion of
  rabbits.

3
The Rabbit Problem
1 1
2 3
5
4
Fibonacci numbers

• 1 1 2 3 5 8 13 21
  34
• Add the last two numbers to get the next.
• F ( i 2 ) F ( i 1 ) F ( i )

5
Lucas

• 1842 1891
• 2 1 3 4 7 11 18 29

6
Pascals Triangle

• 1
• 1 1
• 1 2 1
• 1 3 3 1
• 1 4 6 4 1
• 1 5 10 10 5 1
• 1 6 15 20 15 6
  1

7
Pascals Triangle

• 1
• 1 1
• 1 2 1
• 1 3 3 1
• 1 4 6 4 1
• 1 5 10 10 5 1
• 1 6 15 20 15 6

1
1
1
1
2
2
3
3
5
5
8
8
13
13
8
Number of petals on flowers
1
5
2
3
13
21
8
9
Sneezewort
10
Sneezewort
13 8 5 3 2 1
8 5 3 2 1 1
Branches
Leaves
11
Spirals
12
Spirals
13
Spirals
14
Spirals
15
Turku power station

• Fibonacci Sequence 1 - 55

16
Golden Ratio

• Ratio of two neighbouring Fibonacci numbers.

17

• F ( i 1 ) F ( i 2 ) X
• F ( i ) F ( i 1 )
• F ( i 1 ) F ( i )
• F ( i 1 )
• 1 1/X
• X2 X 1

18

• X2 X 1
• X 1 v5 1.618 Phi
• 2
• Or
• X 1 - v5 - 0.618 phi
• 2
• 1 1.168 equivalent to 0.168 1

19
Phi F

• Golden Rectangle
• Greeks Parthenon
• Credit cards
• KitKat

• Golden Section
• Artists
• Film makers
• Compositions

20
Golden Rectanlge

• Nautilus sea shell

21
(No Transcript)
22
Golden Triangle

• Isosceles triangle
• BC BD AD
• AC BC or AC AD
• BD DC AD DC

23
(No Transcript)