Fibonacci

In this week’s “Eye On Math” spot, we interviewed a man by the name of Leonardo Pisano known much better by his professional name Fibonacci.  He is a renowned mathematician from the late 1100’s through the early to mid 1200’s.  During our interview with Mr. Fibonacci we noticed a very distinct pattern in the wording of his answers, mainly having to do with one of his mathematical masterpieces.  We will leave it to you as the reader to figure out what he “did” with his words. 

Question: Mr. Fibonacci, what year were you born?

1 Answer: 1170

Question: And where were you born in 1170?

1 Answer: Pisa

Question: We understand that you were born to Mr. Guilielmo, and he took you with him in his travels.  Where exactly were you educated and became interested in mathematics?

2 Answer: North Africa

Question: You traveled much throughout your younger years, at what point did you stop and say , “Its time to stop traveling and get on with my life?”

3 Answer: The year 1200.

Question: When you finally settled down, back in Pisa, you wrote a series of texts, of which many still remain today.  Which of these texts do you think makes you who you are in the world today?

5 Answer: Liber abaci, mainly third section.

Question: Putting Liber abaci aside for a moment, for there is much to discuss on this text, might you tell us what you other texts were called?

8 Answer: Practica geometriae, Flos, Liber quadratorium, Di minor guisa.

Question: And as we understand, Liber abaci was your most well known book?

13 Answer: Yes, Liber abaci, brought me to who I am today, principally Fibonacci Numbers.

Question: We researched this book, and found that it was dedicated to a man named Scotus, a man of the court under Emperor

Frederick II in the Holy Roman Empire.  We also found that it is divided into three subdivisions.  Can you tell us about the first subdivision?

21 Answer: In the first section of Liber abaci, I introduced a new system of numbers I had learned while in North Africa.

Question: If I am able to recall, it seemed to be a system of Arabic numerals, also called algorism.  Moving on two the second section, can you tell us anything about the second subdivision?

34 Answer: The second section is focused toward the merchants.  It teaches them how to gain proceeds off of transactions they make, and how to widen their business by converting currency through out the other countries.

Question: Moving right along, we are very excited to discuss with you the third subdivision of Liber abaci, because we know this is where you introduced the prominent Fibonacci Numbers, and  the Fibonacci Sequence.  Can you tell us about them?

55 Answer: The Fibonacci Numbers are a group of numbers that can be found throughout mathematics and even nature.  They are intertwined with the Fibonacci Sequence in that the Fibonacci Numbers take a pattern known as the Fibonacci Sequence.  This is a series of numbers in which each number is the sum of the two proceeding numbers.

Question: Can you give us some examples of where you would find the Fibonacci Sequence? 

89 Answer: The most well known problem in which the Fibonacci Sequence appears is in that of a problem involving the mating of rabbits over twelve months:   
“A certain man put a pair f rabbits in a place surrounded on all sides by walls.  How many pairs of rabbits can be produced from that pair in a year if it supposed that every month each pair begets a new pair which from the second month on becomes productive?”
    I proposed the answer to this problem using the sequence to justify my solution. 

Question: Fascinating! Truly, can you explain to us and to others how you came up with the solution?

144 Answer: First one starts off with one pair of rabbits, male and female, which is able to produce children in the second month.  At the end of the second month the pair produces one pair of children therefore bring the total number of pairs to two.  Since the second pair cannot produce until the end of the fourth month the first pair then produces another pair at the end of the third month, bringing the total number of pairs up to three. At the end of the fourth month the first pair and second each produce a pair bringing the total number of pairs up to five.  This sequence continues until it reaches the twelfth month at which you have a total of 233 pairs of rabbits.  As you can see, each number of pairs is the sum of the two proceeding pairs of rabbits. 

Question: Now we must take into consideration that this is in an ideal very controlled situation in which all the rabbits must stay alive and keep producing.  Are there any other situations or examples that could tie in with the sequence?

233 Answer: Why, of course!  A realistic situation occurring once more with nature is that of in the seeds of a coneflower.  Before I can explain this thoroughly, I must explain about the Fibonacci Rectangles and the Fibonacci Spiral, which is found in the rectangle.  One takes two unit squares and places them next to each other.  To match the sides of the unit squares a square measuring two units is then placed next to them causing the length to now be three units long.  A square measuring three units long is then placed next to the group followed by a five unit square, and then 8,13,21 and so on and so forth.  This causes a spiral shape to form following the order that the squares were placed.  This spiral as well as the sequence occurs in many things such as a seashell and a Coneflower.  On the head of a Coneflower, the seeds are arranged are in accordance with the sequence because of the spirals they form throughout from the center to the outside edge.  This spiral effect and sequence also occurs in pinecones, and even fruits and vegetables.  But for one to grasp the Fibonacci Numbers and Fibonacci Sequence, as well as the Fibonacci Spiral, one must understand that there is a rule behind all Fibonacci Numbers.  This rule or ratio as it could also be considered is a basis for these numbers.

Question: So your saying that there is some sort of Golden Rule/Ratio behind your numbers? 
Answer: One might call it a Golden Rule.  When the sequence is graphed it becomes apparent that there is a ration between each number and the one preceding it, which is almost identical to Phi or approximately 1.6.  As one goes further and further down the sequence the number becomes closer and closer to Phi.  Phi is a number that when squared equals the same number as if someone was to add just 1 to Phi.  These are the basics of the Fibonacci Numbers, Sequence, and Golden Rule.  Moving on to my other works, I also based some of my work upon Euclid, and his famous Elements, and On Divisions.  This text was called Practica geometriae, in which I included geometry and theorems, and mainly geometrical subtleties.  In one of my final works that I finished approximately in the year 1225, at my ripe old age of 55, called Liber quadratorum, I focused on quadratic equations involving two or more variables, while also considering different techniques to find Pythagorean triples.  One very easy idea that I introduced in this book is that of the relationship between odd numbers and square numbers.  Getting the sum of any two odd numbers as I can demonstrate can bring about square numbers.  The formula:
n2 + (2n+1) = (n+1)2
can be used to show the relationship between odd numbers and squares. 
   
This interview was very informative and brings about many interesting questions about occurrences in nature that include the Fibonacci Numbers.  Leonardo Fibonacci died in the year 1250, at a ripe age of 80 years old.  His mathematical genius has advanced the learning of math and also nature from his time to today, and even into the future.  Little is known about his personal life as well as his travels and even some of the texts he wrote are not here today.  His effect on math was forgotten throughout the Middle Ages and then approached again in later centuries.  His math was as like to that of a flower.



Can you figure out what the pattern was in Mr. Fibonacci’s speech?



Works Cited

“Fibonacci.” February 18 2002. http://www.lib.virginia.edu/science/parshall/fibonacc.html

Rating: 1 Star
This site wasn’t so good. It barely gave me half a page of information, and I wouldn’t recommend it.

“Fibonacci.” Mathematicians. February 18, 2002. http://www-groups.dcs.st-and.ac.uk/`history/Mathematicians/Fibonacci.html

Rating: 5 Stars
This site was very good. I got many pages of information out of it. Very useful.

Garland, Trudi Hammel. Fascinating Fibonaccis. Palo Alto, California, Dale Seymour Publications, 1987.

Rating: 5 Stars
This book was actually better than the last. It focuses on every aspect of Fibonacci’s math, not only the Golden Rule and Fibonacci Numbers. I recommend it for everyone.