Download power point presentation on fibonacci and the golden ratio!

Who was Fibonacci ?
• Greatest European mathematician of
the middle ages
• Born in Pisa, Italy, the city with the
famous Leaning Tower,~ 1175 AD
• Major contributions in arithmetic,
algebra and number theory
• Decimal system
Patterns in Nature
Studying cool patterns in nature
• Astronomers model the patterns found in giant
spiral galaxies
• Physicists: movements of atoms
• Biologists & Doctors try to understand the random
patterns of growth, spread of disease .
• Geologists study the meandering of rivers,
tortuous coastlines, and awe-inspiring mountain
landscapes.
Patterns in nature
• Many different
patterns arise in
nature.
• Do we always know
what they mean ?
• Or if it could have
been otherwise?
Mathematics can be fun!
• There is mathematics at every level of
life…
• …you just need to look!
• So let’s do just that!
Busy Bunnies
• You have one pair of rabbits on an island
– The rabbits repeat the following:
• Get pregnant one month
• Give birth (to another pair) the next month
– This process repeats indefinitely (no deaths)
– Rabbits get pregnant the month they are born
• How many rabbits are there after 10
months?
Reproducing Rabbits
• First month: 1 pair
– The original pair
• Second month: 1 pair
– The original (and now pregnant) pair
• Third month: 2 pairs
– The child pair (which is pregnant) and the parent pair
(recovering)
• Fourth month: 3 pairs
– “Grandchildren”: Children from the baby pair (now
pregnant)
– Child pair (recovering)
– Parent pair (pregnant)
• Fifth month: 5 pairs
– Both the grandchildren and the parents reproduced
– 3 pairs are pregnant (child and the two new born rabbits)
Reproducing Rabbits
• Sixth month: 8 pairs
– All 3 new rabbit pairs are pregnant, as well as those not
pregnant in the last month (2)
• Seventh month: 13 pairs
– All 5 new rabbit pairs are pregnant, as well as those not
pregnant in the last month (3)
• Eighth month: 21 pairs
– All 8 new rabbit pairs are pregnant, as well as those not
pregnant in the last month (5)
• Ninth month: 34 pairs
– All 13 new rabbit pairs are pregnant, as well as those not
pregnant in the last month (8)
• Tenth month: 55 pairs
– All 21 new rabbit pairs are pregnant, as well as those not
pregnant in the last month (13)
Busy Bunnies
Busy Bunnies
Fibonacci Series
• The rule for the sequence?
1, 1, 2, 3, ?, ?, ….
• The Answer :
1, 1, 2, 3, 5, 8, 13, 24, …
1+ 11+
= 2+
2 = 333+
= 555+
= 88+
= 13
13= 24
Nature and Fibonacci
White calla lily
Nature and Fibonacci
Euphorbia
Nature and Fibonacci
trillium
Nature and Fibonacci
Black eyed susan
More Fibonacci
• Pinecones and pineapples…
• Count the number of spirals.
Spirals in a pine cone: clockwise and anti-clockwise
And more
Pascal’s
Triangle
Finding the n-th term of a
Fibonacci Sequence

1  5   1  5 
F ( n) 
n
5 2
n
n
Summation Formula For the first “n”
numbers in the Fibonacci Sequence
n
F

F

1
 k n2
k 1
Fibonacci Sequence
• As the terms increase, the ratio between
successive terms approaches 1.618
F (n  1)
5 1
lim
 
 1.618933989
n  F ( n)
2
• This is called the “golden ratio”
– Ratio of human leg length to arm length
– Ratio of successive layers in a conch shell
• Reference:
http://en.wikipedia.org/wiki/Golden_ratio
Spirals
Spirals
Golden Ratio
• Compute the ratio of Fibonacci numbers:
2÷1=
3÷2=
5÷3=
8÷5=
13 ÷ 8 =
21 ÷ 13 =
Another fun way to get
Golden ratio
•
•
•
•
Choose any number, say x.
Find x + 1
Find 1/x
Do the above two steps over and over
and over…
The Golden ratio
in nature
In Art & Architecture..
Great Wall of China
Parthenon, Greece
The Mona Lisa
So why do spirals occur ?
• Helps in efficient packing of seeds.
• But flowers and pinecones have to
deal with growing.
Finding Golden ratio
• As seeds grow older they are pushed
out.