Download Final Paper: All About Fibonacci Numbers

Final Paper: All About Fibonacci Numbers
All About Fibonacci Numbers
Miesha Wilson
MAT-135
Professor Aldo Maldonado
10/9/16
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Introduction
Born from humble beginnings, Leonardo of Pisa used his unique educational background and
knowledge to inform and bring forth a better future and new perspective of mathematics to the
world. Without the spread of this ancient Indian knowledge, the western world would have little
understanding of the mathematical link between science, math, and nature. In addition to the lack
of understanding, we as a civilization would have continued to use ancient European
mathematical concepts, achieving little to no expansion on theoretical and numerical ideas and
theories.
Before Fibonacci numbers were introduced performing simple mathematics was a very timely
chore. The introduction of a place value and ‘0’ proved to be extremely helpful. The invention of
zero meant that they could make numbers infinitely as large or small as they wanted. The
Fibonacci sequence even introduced the concept of using a single digit to represent a numerical
value. Such as 1,2,3 and 4. Instead of the previous roman numerals like I which represented one
and V which represented four.
As we seek to grasp an understanding of mathematical patterns and their rules, we can better
comprehend the natural patterns found in nature and the blueprint used to create this planet and
every living thing in it.
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History of Leonardo Bonacci
Leonardo Bonacci was born in Italy in 1170 to Alessandra and Guglielmo Bonaccio. While his
father worked as a customs officer, Leonardo traveled with him across North Africa. There, the
Moors taught Leonardo extensively about mathematical studies, concepts, and ideas. He was
learning much more about the vastness of mathematics and the power of numbers than he could
have ever hoped to learn in Italy.
At the age of 32, he wrote and published the book titled Liber Abaci, which is translated to mean
“the book of calculations.” This book consisted of the mathematical knowledge that he acquired
by the Moors while stationed in North Africa. After the publication of Liber Abaci, Leonardo
was later given the nickname “Fibonacci” due to a mistranslation in the title thus giving birth to
his new surname. No matter what he’s called, Fibonacci’s spread of the decimal system to west
Europe has greatly influenced our current society and the way we understand the relationship
between numbers.
Mathematical concepts in Liber Abaci
While reading the example that Leonardo gave on the mating rabbits, I realized that there seems
to be a structured pattern in almost every aspect of nature as well as science and math. What
seems to be a chaotic random occurrence in nature, has a definitively structured pattern. With the
example of the mating rabbits, Leonardo dared to ask the question “How fast will rabbits
multiply in an ideal circumstance of events?” The ideal scenario would be if no rabbits ever died
and every rabbit mated when they reached puberty (1 month). He used a formula to help
simplify this phenomenon. The formula was Xn+1=Xn+Xn-1.
This formula simplifies the idea that if the original pair of rabbits continues to produce offspring
each month and the offspring produce more offspring after they hit the stage of puberty then an
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“x” amount of rabbits would be produced. By adding 2 to the first number to get the next
number. For instance, with the numeric set “2,4,6,8… “by increasing the previous number by
two the next possible number becomes apparent.
Fibonacci in real-life occurrences
On the website plus math.org, the author gives another example to help the reader better
understand Fibonacci numbers in the natural world.
“Honeybees provide an example. In a colony of honeybees, there is one special female called the
queen.” (Knott, 2013) If only a queen bee can produce a female bee, but the male bee is
produced by both the female (queen) bee and a male bee, how many parents does the male bee
have? The sequence to help explain this occurrence is 1,2,3,5,8 with the one being the original
queen mother and the eight being the three times great grandparents.
The most fantastic aspect of Fibonacci numbers is that it occurs in every aspect of nature!
“Flowers often have a Fibonacci number of petals; daisies can have 34, 55 or even as many as 89
petals!” (Knott,2013) From flower petals to leaves and seeds, Fibonacci numbers are running
rampant.
In the given illustration on the website, patterns in the number of seeds growing in the direction
left and right in a single flower head follow a sequence. In one of the illustrated flowers, there
were 34 seeds going left and 21 going right. The sequence of seeds packed inside one head is the
perfect amount so that the flower head isn’t too dense nor very sparse. From the number of
seeds present inside of a sunflower to the number of leaves along a flower stem, Fibonacci
patterns will always be detectable.
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Fibonacci Patterns Explained
Leonardo Fibonacci’s most respected contribution to the mathematic world was the spread of
Fibonacci numbers. Fibonacci numbers are a sequence or pattern of numbers that follow a single
rule.
In the example of the reproduction of bees, the male is reproduced with only one parent bee
(male) while two parents are needed to produce a female bee. If the number of bees from each
generation is summed up it will reflect the infamous Fibonacci sequence
0,1,1,2,3,5,8,13,21,34,35 and so on. Another great example of a Fibonacci sequence is the
pattern of spirals in a flower head or pinecones. To count the spirals in both a flower head and
pine cone, first start by counting the spirals going in one direction, next count the spirals going in
the opposite direction. The number of spirals occurring in any given direction follows a single
rule.
Rules of the Game
No matter the sequence, all Fibonacci numbers have a given rule. To reveal the rule of a
Fibonacci sequence, you must first discover the hidden pattern. In the example of the Fibonacci
pattern 0,1,1,2,3,5,8,13 there is a “rule” to uncover. For this pattern the rule is xn = xn-1 + xn-2.
(Knott,2013)
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In this mathematical equation, xn represents the unknown number or the number that will follow
the previous numbers. The xn-1 represents the previous number, and xn-2 represents the term
before that. To simplify this equation, I’ll use the number 2 in the previously given equation. To
get the number “2”, we added, “1” (xn-1; previous number) +”1” (xn-2; number before that).
(Knott,2013)
Finite and infinite sequence
When dealing with finite sequences, it is important to keep in mind that both finite and infinite
sequences both have a beginning but only one sequence has a “fixed” ending. This numbering
pattern has come to be known as a finite sequence. Because this series of numbers have an
ending point, it doesn't continue forever.
Infinite sequences are a little trickier. Like a finite series, it has a beginning, but an infinite
sequence has no barriers or boundaries; it keeps continuing forever. In a number sequence of all
even numbers in a numeric system, there is no stopping point. Therefore, resulting in a countless
series of numbers, continuing for eternity. A good way to explain this phenomenon of countless
even numerical values is 2,4,6,8,10,12…∞
Examples of infinite sequences are N = (0, 1, 2, 3, ...) and S = (1, 1/2, 1/4, 1/8, ..., 1/2 n , ...).
Infinity is represented by the three dots located after the last value in the series. (Rouse,2005)
Infinite groups of numbers can go in either direction, both positive and negative. A clear
representation of this would be. (…-5,-4-3-2-1 0 1,2.3,4,5…)
Golden Ratio
According to Wolfram math world, the golden ration is the divine proportion, golden mean, or
golden section. It is a number often encountered when taking the ratios of distances in simple
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geometric figures such as the pentagon, pentagram, decagon, and dodecahedron”. (Golden Ratio,
2016)
The golden ratio represents aesthetically pleasing measurements, or what we know as perfection!
German researcher Adolf Zeising found that the relation between a man’s upper and lower body
has proportions that can be represented by the fraction 13/8 which equals to 1.625. This
numerical value compared to that of a women’s body (1.6) is closer to the golden ratio. “If the
numbers of the segments on a human body (arms, legs, fingers and toes) are written as a series of
numbers, it would be Fibonacci numbers.
Real World Applications
Leonardo Fibonacci’s most respected contribution to the mathematic world was the spread of
Fibonacci numbers. Fibonacci numbers are a sequence or pattern of numbers that follow a single
rule. No matter the Fibonacci sequence every pattern has one.
Both finite and infinite patterns have a beginning but only one contains an ending. A good
example of a finite sequence is a series of numbers with limitations. These limitations could
consist of even numbers ranging from 2 through 20. {2,4,6,8,10,12,14,16,18,20}
Infinite numbers are limitless and boundless. An example of this occurrence would be a
sequence of all even numbers. {2,4,6,8,10…∞}
As explained in the first strand with the example of honey bees, in a colony of honeybees, there
is one special female called the queen.” (Knott, 2013) If only a queen bee can produce a female
bee, but the male bee is produced by both the female (queen) bee and a male bee, how many
parents does the male bee have? The sequence to help explain this occurrence is 1,2,3,5,8 with
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the one being the original queen mother and the eight being the three times great grandparents.
Real world examples of Fibonacci numbers can be seen in multiple aspects of the natural world.
Flowers come in various colors, shades, and even number patterns! The petals on a flower can
have a natural occurrence of the Fibonacci sequence 0, 1, 1, 2, 3, 5, 8, 13, 21.
“Buttercups have 5 petals; lilies and iris have 3 petals; some delphiniums have 8; corn marigolds
have 13 petals; some asters have 21 whereas daisies can be found with 34, 55 or even 89 petals.”
(Knott,2016)
According to the article titled Fibonacci in Nature by Nikhart Praveen “Phyllotaxis is the study
of the ordered position of leaves on a stem.” The leaves on a plant grow in the most efficient way
to maximize sun exposure. The number of leaves present on the stem of a plant are an excellent
example of Fibonacci numbers.
In the previous strand exercise, the occurrence of the Fibonacci sequence in flowers was
explored. The sequence of seeds packed inside one head is the perfect amount so that the flower
head isn’t too dense nor very sparse. From the number of seeds present inside of a sunflower to
the number of leaves along a flower stem, Fibonacci patterns will always be detectable.
Fibonacci numbers can even be found in the depths of the deep blue sea.
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Sea stars and sand dollars have five points while octopus and squid have eight arms. “The
distance between the tips of a starfish’s arms compared to the distance from tip to tip across the
entire body is very close to the golden ratio, and the eye, fins and tail of dolphins all fall at points
along the dolphin’s body that correspond to the ratio.” (Haelle.2014)
“The Milky Way has several spiral arms, each of them a logarithmic spiral of about 12 degrees.”
(Dvorsky,2013) With the prsecence of Fibonacci patterns found in the cosmos, it makes you
wonder about the possible existence of a creator. Some mathematicians even use the natural
prescence of Fibonacci sequences, to develop the argument that God does indeed exist.
Scientist and mathematical geniuses have all wondered if the repetitive occurrence of
Fibonacci numbers throughout the world is credible proof of a creator. No one knows why this
phenomenal existence occurs on earth, or whether it exists on other planets or in another
unidentified life form. As scientist make new discoveries, more unanswered questions will
bubble to the service. Maybe one day we will have a scientific answer to explain the presence of
the Fibonacci sequence or maybe some questions will never be answered by science alone. As I
dove into Fibonacci numbers, sequences, and the golden ratios, I have learned that math isn’t just
about solving equations, plotting points on a graph or finding the value for ‘x’, Mathematics has
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a place in numerous aspects of life that I personally find interesting, like science, art and music.
The Fibonacci sequence has yet to unlock all of the earth’s mysteries or answer all of humanity’s
most challenging questions about the universe, space and time. I believe that one day, we will be
able to uncover these mysteries. In the future I trust that as people become more aware of the
connections and complexities of this universe, we as humans will gain a higher level of respect
for the planet and solar system we reside in. I look forward to diving deeper into the pool of
knowledge that is the heart of mathematics.
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citations
Dr R Knott: fibandphi (AT) ronknott.com. (n.d.). Fibonacci Numbers and Nature. Retrieved
October 01, 2016, from http://www.maths.surrey.ac.uk/hostedsites/R.Knott/Fibonacci/fibnat.html
Dividing, B. (n.d.). Fibonacci in Nature. Retrieved October 01, 2016, from
http://jwilson.coe.uga.edu/emat6680/parveen/fib_nature.htm
The life and numbers of Fibonacci. (n.d.). Retrieved September 25, 2016, from
https://plus.maths.org/content/life-and-numbers-fibonacci
Golden Ratio. (n.d.). Retrieved September 25, 2016, from
https://www.mathsisfun.com/numbers/golden-ratio.html
S. (2011). Finite Sequence and Infinite Sequence - Types of Sequences & Series. Retrieved
September 25, 2016, from https://www.youtube.com/watch?v=LjF8dGAOKnI
A. (2015). What is Golden Ratio - easy explanation. Retrieved September 25, 2016, from
https://www.youtube.com/watch?v=wdk37T8TltM
[IMA Videos]. (2011, January 19). Finite sequence and infinite sequence-types of sequences and
series. [video file] Retrieved from https://www.youtube.com/watch?v=LjF8dGAOKnI
[Assignment Expert. (2015, March 26). What is Golden Ratio? [video file]. Retrieved from
https://www.youtube.com/watch?v=wdk37T8TltM
The life and numbers of Fibonacci. (n.d.). Retrieved September 11, 2016, from
https://plus.maths.org/content/life-and-numbers-fibonacci
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Space and fibonacci numbers - Google Search. (n.d.). Retrieved October 10, 2016, from
https://www.google.com/search?q=space and fibonacci numbers
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