Who was Fibonacci Greatest European mathematician of the

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

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 = 33+ 3= 55+ 5 = 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 anticlockwise

And more

Pascal’s Triangle

Finding the n-th term of a Fibonacci Sequence

Summation Formula For the first “n” numbers in the Fibonacci Sequence

Fibonacci Sequence • As the terms increase, the ratio between successive terms approaches 1. 618 • 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

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…

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.