Transcript Fibonacci numbers - University of Georgia

FIBONACCI NUMBERS
The Golden Ratio and Fibonacci Numbers in Nature
By: Mary Catherine Clark
HISTORY
Leonardo Fibonacci was the most outstanding
mathematician of the European Middle Ages.
 He was known by other names including Leonardo
Pisano or Leonard of Pisa.
 Little was know about his life except for the few
facts given in his mathematical writings.
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HISTORY CONT.
Fibonacci was born around 1170.
 Received his early education from a Muslim
schoolmaster.
 His first book was published in 1202 called
Liber Abaci. This book is devoted to arithmetic
and elementary algebra.
 His next book, Practica Geometriae, he wrote in
1220. This book presents geometry and
trigonometry with Euclidean originality.

HISTORY CONT.
He employed algebra to solve geometric
problems and geometry to solve algebraic
problems. This was radical in Europe at the
time.
 He wrote two other books. One of which
included Liber Quadratorum. This earned him
the reputation as a major number theorist.

FIBONACCI TODAY
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He was one of the first
people to introduce the
Hindu-Arabic number system
to Europe- the system used
today.
Liber Abaci remained the
European standard for more
than two centuries replacing
the Roman numeration
system.
Today a statue of Fibonacci
stands in a garden across
the Arno River, near the
Leaning Tower of Pisa.
FIBONACCI NUMBERS
The sequence in which each number is the sum of
the two preceding numbers.
 This sequence is defined by the linear recurrence
equation.

and by the definition above
 The Fibonacci numbers for n=1,2 are 1, 1, 2, 3, 5,
8, 13, 21,…
 Before Fibonacci wrote his work, these numbers
had already been investigated by Indian Scholars
interested in rhythmic patterns of syllables.
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FIBONACCI’S RABBITS
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Fibonacci’s book Liber Abcai asked a question involving the reproduction of
a single pair of rabbits which is the basis of the Fibonacci sequence.
It was posed: Suppose a newly born pair of rabbits (a male and female) are
put in a field. The rabbits are able to mate at the age of one month so that
at the end of the second month a female can produce another pair of
rabbits. Assuming that the rabbits never die and the female always
produces a new pair every month from the second month on. How many
pairs will there be in one year?
Answer: 144 rabbits
THE GOLDEN RATIO
The golden ratio is an irrational number defined to
be
 This has a value of 1.61803 and is sometimes
denoted by φ after the mathematician Phidias who
studied its properties.
 Its unique properties were first considered in the
idea of dividing a line into two segments such that
the ratio of the total length to the length of the
longer segment is equal to the ratio of the length
of the longer segment to the length of the shorter
segment.

THE GOLDEN RECTANGLE
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The golden rectangle can be constructed from these line
segment so that the length to width ratio is φ.
The golden rectangle may be divided into a square and
a smaller golden rectangle.
The ancient Greeks believe that a rectangle constructed
in this manner was the most aesthetically pleasing of all
rectangles and they incorporated this shape into a lot of
their art and architectural designs.
WHAT DOES THIS HAVE TO DO WITH FIBONACCI
NUMBERS?
The ratio of successive Fibonacci numbers is
something you might be surprised by!
 As n increases, the ratio of
approaches
the golden ratio and is expressed as
=
 This is the fundamental property of both the
Fibonacci sequence and the golden ratio.
 Both of these ratios converge at the same limit
and are the positive root of the quadratic
equation

FIBONACCI NUMBERS AND THE GOLDEN
RECTANGLE
If the two smallest squares have a width and
height of 1, then the box to their left has a
measurement of 2 and the other boxes measure
3, 5, 8, and 13.
 The golden ratio is expressed in spiraling shells.
 There is a quarter of a circle in each square going
from one corner to the opposite.
 This is not a true mathematical spiral.
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FIBONACCI NUMBERS IN NATURE
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Look at any seed head, and you will notice what
look like spiral patterns curving out form the
center left and right.
If you count these
spirals you will find a
Fibonacci number. If
you look at the
spirals to the left and
then the right you will
notices these are two
consecutive
Fibonacci numbers.
FIBONACCI NUMBERS IN NATURE
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These can also be seen in pinecones,
pineapples, cauliflower, and much more!
MORE FIBONACCI NUMBERS IN NATURE
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Most of the time, the number of pedals on a flower
is a Fibonacci number!
1 pedal-calla lily
3 pedals-trillium
8 pedalsbloodroot
2 pedals-euphorbia
5 pedalscolumbine
13 pedals-black
eyed susan
WORKS CITED
Dunlap, Richard A. The Golden Ratio and
Fibonacci Numbers. Singapore: World
Scientific, 1997.
 Koshy, Thomas. Fibonacci and Lucas Numbers
with Applications. New York: John Wiley & Sons,
inc., 2001.
 Vorobiev, Nicolai N. Fibonacci Numbers. 6th.
Basel: Birkhauser Verlag, 1992.
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WORKS CITED
Http://www.world-mysteries.com/sci_17.htm
 http://science.howstuffworks.com/evolution/fi
bonacci-nature1.htm
 http://mathworld.wolfram.com/FibonacciNumb
er.html
 http://www.mcs.surrey.ac.uk/Personal/R.Knott
/Fibonacci/fibnat.html
 http://www.branta.connectfree.co.uk/fibonacci
.htm
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FOXTROT BY BILL AMEND