Fibonacci

Fibonacci was the greatest mathematician of his age. He did not simply master the arts of geometry, arithmetic, trigonometry, and algebra, but also made his knowledge useful to all the businesses involving math (by amending some forms of notation and eliminating possible sources of accountancy errors). He eliminated use of complex Roman numerals and made mathematics more accessible to the public because he brought the Hindu-Arabic system (including zero) to Western Europe.

Q. What is your name and its origin?

A. “In actuality my original name was Leonardo, and back then people named each other according to location so I was Leonardo of Pisa. Yes, it is the same city as the famous leaning tower. Anyway, I decided to adopt the more professional name of Fibonacci, or “son of Bonacci”, as Guilielmo Bonacci was my father.

Q. When were you born?

“Sometime around 1175. My memory clouds now that I am around 800 years old; please forgive my vague personal knowledge. I don't even remember my wife's name. Anyway, I was born during a time when the shadow of the Dark Ages was visibly receding and there was a population growth thanks to our farmers using better techniques. This meant a cultural boom as well, and it is important to note that for the first time contacts were being made with Eastern civilizations. I was born as merchants and intellectuals began bringing back news of different mathematical systems. I actually traveled to the East and North Africa and was the one who popularized these new systems and modified them slightly. I grew up with a North African education under the Moors and later traveled extensively around the Mediterranean coast. I remember that my interest in the Arabs and their strange numbers was part of what gave me so much advantage back home.”

Q. Where were you raised and how did this affect you?

A. “I was raised in Pisa, Italy. It was already an independent republic, a small city-state with a pretty large commerce and seaport. My father found work there in a company in the port of Bugia and instructed me a bit in accounting.”

Q. What do you think people see you as?

A. “Most know me as a very serious scholar. I suppose that everyone currently knows about what kind of scholar; namely a mathematician, but I am also very interested in the laws and patterns of nature. Much of my work is highly applicable to the designs by which living things evolve and grow.”

Q. Can you give any examples of how your mathematics are seen in nature?

“The Fibonacci Sequence is 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, etc. The formula basically is a guide to adding the previous two numbers in the Hindu-Arabic system to get a new number ad infinitam. Interestingly enough, this is found everywhere in living things because of the way things grow exponentially in nature. Also, did you notice how much art and music have to do with the sequence? If you look at piano keys or famous works of art you will always see recurring patterns obviously of the "Golden numbers.” We take the ratio of two successive numbers in Fibonacci's series, (1, 1, 2, 3, 5, 8, 13) and we divide each by the number before it, we will find the following series of numbers: = 1, 2/1 = 2, 3/2 = 1.5, 5/3 = 1.666..., 8/5 = 1.6, 13/8 = 1.625, 21/13 = 1.61538... The ratio seems to be settling down to a particular value, which we call the golden ratio or the golden number. It has a value of approximately 1.61804

Here are some pictures Fibonacci gave me which are useful in showing us patterns).

Here you see that there are, for example, 8 or 13 whirls on a pinecone (both fibonacci numbers) depending on which direction you follow. If you also look at a pinecone from the side, each level has a certain number of scales that matches a Fibonacci number. Pinecones are not alone. Just about every plant or animal is governed by Fibonacci numbers! Here's another example.


You can tell how the leaves of the plant (8, another Fibonacci number) go up in spirals so that from the top view, we can see that each leaf gets as much sunlight as possible. The spirals themselves are in Fibonacci increments and proportions.

Q. What are your basic achievements?

A. “I wrote the Liber Abbaci, “the book of calculation,” (one of the problems in this book was the problem about the rabbits in a field which introduced the series 1, 2, 3, 5, 8); Liber Quadratorum, “the book of square numbers,” (this last one demonstrated Fibonacci's ease at solving the problems of Diophantus, the greatest mathematician before him); Practica Geometriae, “the practice of geometry,” Flos (where he solved many problems at the imperial court); and most important was my role in bringing Eastern mathematics into Western mathematics. You may even be familiar with the fact that I introduced the fractional bar because the numbers were otherwise rather confusing in accountant notation.”

Q. What political ties have you made?

A. “The holy emperor Frederick II, who maintained a very culturally diverse court, invited me to his palaces to solve problems posed by the great mathematician Master John of Palermo. I was also awarded an annual honorarium of 20 Pisan pounds, because I worked pro bono and helped further accounting and performed other community services.”

Q. What do you think your greatest contribution has been to the world?

A. “I believe that my book, Liber Abbaci, was the most important thing I put into creation. It seriously aided the introduction of Hindu-Arabic numerals to Western Europe and set a solid example for arithmetic, geometry, and algebra, but more importantly a sturdy foundation for purely theoretical applications of math. The cool thing is that it was noticed by the more common people and actually used. That is the greatest thing a mathematician can hope for- the integration of his work into the systems of the world!”

TRY THIS:

The original problem that Fibonacci investigated (in the year 1202) was about how fast rabbits could breed in ideal circumstances. 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 made was: How many pairs will there be in one year? The number of pairs of rabbits in the field at the start of each month is 1, 1, 2, 3, 5, 8, 13, 21, 34, ...

HERE IS A PROBLEM TO TRY ON YOUR OWN: If you put the Fibonacci numbers in a column, you see that we can actually define a function Fib(N) which, when applied to the number N, gives the Nth Fibonacci number.

PROVE that Fib(5k) it itself always a multiple of 5. Hint: This can be proved by induction. When you prove by induction you establish that the theorem is true for a certain case and then you have to prove that if it is true for all values up to k, it has to be true for k + 1.

Now that you are familiar with the Fibonacci numbers, perhaps you will find yourself looking for them the next time you go outdoors! Scientists today say that Nature uses the spirals and exponential numbers in the Fibonacci sequence to prevent over-packing and create uniform patterns, such as in flower proportions or the bones in your hand. To find out more, click on the websites below or check out these books for some different reading!

Works Cited

http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Fibonacci.html

This website provides a thorough basic account of Fibonacci's origins and a few excerpts of his own, including math problems. However, it does not cover the Fibonacci sequence and other sources should be taken to in that case.

http://www.lib.virginia.edu/science/parshall/fibonacc.html

This website contains a brief biography and some references, especially to his works.

http://pass.maths.org.uk/issue3/fibonacci/index.html

This website is the one I most recommend; it aptly reviews the different functions and especially the Fibonacci sequence in small basic lessons. The graph on the Golden section is a good demonstration of the average.

Kenda, Margaret. Math Wizardry For Kids. Hauppauge: Barrons, 1995.

This is a very elementary book for children and contains many activities, among which are finding the Fibonacci sequence in nature (such as my own pinecone example).

Vorderman, Carol. How Math Works. London: Dorling Kindersley Limited, 1996.

This book is also elementary but delves more deeply into the nuances of fun mathematical problems and gives a bit of history on Fibonacci and the origins of math. I recommend this one to whomsoever is interested in more variations of Fibonacci-related mathematics.