One of the oldest math problem states that:
“How many pairs of rabbits will be produced in a year, beginning
with a single pair, if in every month each pair bears a new pair
which becomes productive from the second month on?”
The problem was originated from an Italian businessman who wants to calculate the
amount of rabbits that he could trade. The businessman’s name was Leonardo
Fibonacci of Pisa; he is also mathematician in the Medieval Era.
Taking a look at the problem we see that every in
the first and second month, 1 pair will be
produced. In the third month, 2 pairs will be
produced; one of them is from the original pair and
one from the pair that was born in the first month.
In the fourth month, 3 pairs will be born. And if we
keep on going with the number, there will a
resultant sequence representing the amount of
rabbits born
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233...
This sequence is named after Leonardo Fibonacci. The
rule is that each term of the Fibonacci sequence is
determined by adding the previous two terms together
(except for the first 2 terms that don’t follow the rule).
The sequence is mathematically represented as:
F (n) = F (n – 1) + F (n – 2)
FIBONOCCI AND THE GOLDEN RATIO
When you have a square and add a square of the same size, you
form a new rectangle. If you continue adding squares whose
sides are the length of the longer side of the rectangle; the longer
side will always be a successive Fibonacci number. Eventually the
large rectangle formed will look like a Golden Rectangle - the
longer you continue, the closer it will be.
SIMPLE ENOUGH RIGHT?!
Turns out that the Fibonocci sequence is all around us!
EVERYDAY
ALL THE TIME
In architecture, art, astronomy, nature, and even in your own body.
ART AND ARCHITECTURE
Many architects and artists have proportioned their
works to approximate the golden ratio—especially in
the form of the golden rectangle, in which the ratio of
the longer side to the shorter is the golden ratio—
believing this proportion to be aesthetically pleasing.
ART AND ARCHITECTURE
Its use started as early as with the Egyptians in the design of the pyramids. When the
basic phi relationships are used to create a right triangle, it forms the dimensions
of the pyramids of Egypt.
ART AND ARCHITECTURE
It has also been used in many other well known, and culturally significant buildings:
The Greeks knew it
as the "dividing a
line in the extreme
and mean ratio"
and used it
extensively for
beauty and balance
in the design of the
Parthenon.
The main building of the
Taj Mahal was designed
using the Golden Ratio.
This is why it looks so
perfect. The rectangles
that served as the basic
outline
for the exterior of the
building were all in the
Golden Proportion.
ART AND ARCHITECTURE
It was used it in the design of Notre Dame in
Paris, which was built in the 1163 and
1250.
Its use continues in modern architecture,
as illustrated in the United Nations building:
ART AND ARCHITECTURE
"The Mona Lisa,” Leonardo DaVinci's most famous painting,
is full of Golden Rectangles. If you draw a rectangle
whose base extends from the woman's right wrist to her
left elbow and extend the rectangle vertically until it
reaches the very top of her head, you will have a Golden
Rectangle. Then, if you draw squares inside this Golden
Rectangle you will discover that the edges of these new
squares come to all the important focal points of the
woman: her chin, her eye, her nose, and the upturned
corner of her mysterious mouth. It is believed that
Leonardo, as a mathematician himself, purposefully
made this painting line up with Golden Rectangles in this
fashion in order to further the incorporation of
mathematics into art.
ART AND ARCHITECTURE
We can even HEAR this
sequence! In Mozart’s
Sonata No.1 in C major,
there are 32 measures in
the first section, 68 in the
section section, for a total
of 100 in the first
movement!
But this isn’t an ancient
concept! In Black Star’s
“Astronomy (8th light)” the
sequence is used in the
lyrics of the chorus. A
distant reference, but an
interesting fact
nonetheless!
THE HUMAN BODY
The Human Body is proportioned according to the Golden Ratio, which is found using
Fibonacci Numbers.
The Golden Ratio is a special number approximately equal to 1.618. This number can
be found by dividing a line into two parts so that the longer part divided by the
longer part is equal to the whole length divided by the longer part.
http://www.youtube.com/watch?v=P
NQk_GJuZQo
THE HUMAN BODY
The Fibonocci sequence is also seen in DaVinci's
famous study of the proportions of man,"The Vetruvian
Man" (The Man in Action), is also full of Golden
Rectangles. Unlike the Mona Lisa, where all the lines of
the Golden Rectangle are assumed by the
mathematician, in "The Vetruvian Man", many of the
lines of the rectangles are actually drawn into the
image, at least in part. There are three distinct sets of
Golden Rectangles in this painting:
one set for the head area, one for the torso, and one for
the legs.
NATURE
When observing nature, you may not notice the role Fibonacci Numbers play, but with
a small education on these numbers, and you may find yourself picking up on
certain trends.
The most obvious would be the number of petals on a flower, generally you will find
that this number is a Fibonacci Number, 3, 5, 8, 13, etc.
However, this is just the beginning.
The Fibonacci Sequence can also be
seen when
observing leaves.
NATURE
If you start at the bottom leaf and count up the stem, when you come to the next leaf
that is in line with the initial starting leaf, generally the number will be in line with
the Fibonacci Sequence.
IN CONCLUSION:
MATH AND SCIENCE ARE ALL AROUND US!!
It’s easy to think (especially when students take classes for Gen Ed. Credit) that what
we learn doesn’t actually apply to us in real, everyday life.
BUT IT DOES!