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?

Also in plants

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+ 1 = = = = = = 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

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

Fibonacci Sequence As the terms increase, the ratio between successive terms approaches This is called the “golden ratio” –Ratio of human leg length to arm length –Ratio of successive layers in a conch shell Reference:

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

Great Wall of China Parthenon, Greece In Art & Architecture.. 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.