*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.