Background Born 1170, Died 1250 in Pisa (now in Italy). Real name is Leonardo Pisano, Fibonacci is his nickname. Studied in North Africa in mathematics. Wrote many books on Mathematics: Liber abaci, Practica geometriae, Flos, and Liber quadratorum. There are other text of his that were lost.

Fibonacci’s Works  Liber Abaci (The Book of Calculating)  Liber Quadratorum (The Book of Squares)  Practica Geometriae (Book on Geometry)  Flos  Letter to Master Theodorus

Liber Abaci  Written in 1202  Responsible for introduction of Hindu-Arabic system to Western Europe, thus doing away with the Roman numeral system  Taught mathematics and applied it to accounting, money transfer, etc.  Introduced the rabbit problem and consequently the Fibonacci Sequence

Fibonacci’s Question “A man has one pair of rabbits at a certain place entirely surrounded by a wall. We wish to know how many pairs will be bred from it in one year, if the nature of these rabbits is such that they breed every month one other pair and begin to breed in the second month after their birth.” -Liber Abaci, 1202 “A man has one pair of rabbits at a certain place entirely surrounded by a wall. We wish to know how many pairs will be bred from it in one year, if the nature of these rabbits is such that they breed every month one other pair and begin to breed in the second month after their birth.” -Liber Abaci, 1202

Fibonacci’s rabbits How fast can rabbits breed in ideal circumstances?  One male, one female  Rabbits are able to mate at the age of one month  After the second month, a female can produce another pair of rabbits  A female always produces 1 new pair every month following the second month

Fibonacci’s Rabbits Suppose a newly-born pair of rabbits, one male, one female, are put in a field. Rabbits are able to mate at the age of one month so that at the end of its second month a female can produce another pair of rabbits. Suppose that our rabbits never die and that the female always produces one new pair (one male, one female) every month from the second month on. The puzzle that Fibonacci posed was …. How many pairs will there be in one year?

Fibonacci’s Rabbits A A Month 1 A Month 2 AA1A1 Month 3 AA2A2 A1A1 Month 4 AA3A3 A1A1 BA2A2 Month 5

Fibonacci Sequence Month 1 Month 2 Month 6 Month 3 Month 4 Month Can you see the pattern appearing?

Fibonacci Sequence By adding the current month to the previous month you get the next month ? 0 +

Fibonacci Sequence F0F0 F1F1 F2F2 F3F3 F4F4 F5F5 F6F6 F7F7 F8F8 F9F9 F 10 F 11 F 12 F 13 F 14 F F 16 F 17 F 18 F 19 F

Fibonacci Sequences  Begins with zero  Add the last two numbers to get the next  0,1,1,2,3,5,8,13,21,34,55,89,144,……… The recursion formula F 1 =1 F 2 =2 F n =F n-1 + F n-2 If n > or = 3

Fibonacci Squares

Fibonacci Spirals

The Fibonacci Spiral A spiral that grows at the rate of the Fibonacci sequence The spiral consists of quarter-circles inside of squares This concept is closely related to the golden rectangle used in art and architecture

Fibonacci in Nature Honey Bees In a typical hive, there is 1 queen who can lay eggs There are many worker bees who are female, but they cannot lay eggs There are several male bees who do not work. They are drones. Male bees are produced by unfertilized female eggs Females are produced from fertilized female eggs

Fibonacci in Nature Honey Bees Males bees have only 1 parent (unfertilized) Female bees have 2 parents (fertilized) Examine the genealogy ParentsGrand Parents Great Gr Parents Grt Grd Parents Male1235 Female2358

Fibonacci in Nature Plants and seeds

Fibonacci in Nature Human Anatomy 2 hands 5 fingers 3 joints per finger Fibonacci Spirals

Look! It’s a Fibonacci sock!