1. 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

2. Patterns in Nature

3. 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.

4. Patterns in nature Many different patterns arise in nature. Do we always know what they mean ? Or if it could have been otherwise?

5. Also in plants

7. Mathematics can be fun! There is mathematics at every level of life… …you just need to look! So let’s do just that!

8. 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?

9. 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)

10. 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)

11. Busy Bunnies

13. Fibonacci Series The rule for the sequence? 1, 1, 2, 3, ?, ?, …. The Answer : 1, 1, 2, 3, 5, 8, 13, 24, … 1+ 1 = 21+ 2 = 32+ 3 = 53+ 5 = 85+ 8 = 13 8+ 13 = 24

14. Nature and Fibonacci White calla lily

15. Nature and Fibonacci Euphorbia

16. Nature and Fibonacci trillium

17. Nature and Fibonacci Black eyed susan

18. More Fibonacci Pinecones and pineapples… Count the number of spirals.

19. Spirals in a pine cone: clockwise and anti-clockwise

20. And more

22. Pascal’s Triangle

24. Finding the n-th term of a Fibonacci Sequence

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

26. 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

27. Spirals

29. Golden Ratio Compute the ratio of Fibonacci numbers: 2 ÷ 1 = 3 ÷ 2 = 5 ÷ 3 = 8 ÷ 5 = 13 ÷ 8 = 21 ÷ 13 =

30. 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…

31. The Golden ratio in nature

32. Great Wall of China Parthenon, Greece In Art & Architecture.. The Mona Lisa

33. So why do spirals occur ? Helps in efficient packing of seeds. But flowers and pinecones have to deal with growing.

34. Finding Golden ratio As seeds grow older they are pushed out.