A review was made of the Fibonacci sequence, its characteristics and applications

Avagyan, Arik

English

Scholarship at Parkland

Parkland CollegeA with Honors Projects Honors Program2010The Fibonacci SequenceArik AvagyanParkland CollegeOpen access to this Article is brought to you by Parkland College's institutional repository, SPARK: Scholarship at Parkland. For more information,please contact spark@parkland.edu.Recommended CitationAvagyan, Arik, "The Fibonacci Sequence" (2010). A with Honors Projects. 9.http://spark.parkland.edu/ah/9AvagyanlArik AvagyanMATH -129Vicki BeitlerFibonacci SequenceIntroduction:Fibonacci sequence is one of the most famous series of numbers in all of the mathematics. Thenumber sequence is named after Leonardo Fibonacci, who first introduced it to westernEuropean mathematics in his 1202 book Liber Abaci, although it may have been previouslyknown by ancient Indian mathematics. In mathematical terms, the sequence Fn of Fibonaccinumbers is defined by the recurrence relationWith seed valuesF0 = 0 and FI = 1As we can see, the first two Fibonacci numbers are 0 and 1, and each subsequent number is thesum of the previous two. The sequence looks like this:0, 1 7 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, . . .AvagyanZSome mathematicians omit zero form the Fibonacci sequence, starting it with two 1's. The firstzero is known as the zeroth Fibonacci number, and has no real practical merit.Origins:The Fibonacci sequence was known in Indian mathematics independently of the West, butscholars differ on the timing of its discovery. First the Fibonacci sequence was noticed byPingala, who introduced it in his book Chandasastra. Later, the number sequence is associatedwith Indian mathematics; Virahanka, Gopala and Hemachadra. According to Parmanand SinghVirahanka was "the first authority who explicitly gave the rule for the formation" of theFibonacci numbers. In contrast, Rachel Hall only mentions Hemachandra among these authors ashaving worked with Fibonacci numbers; she claims that around 1150, Hemachandra noticed thatthe number of possible rhythms followed the Fibonacci sequence.In west the sequence was first introduced by Leonardo Fibonacci, who relates the exponentialgrowth of the pair of rabbits with Fibonacci number. The problem is as follows:Let's say we have two rabbits. These rabbits are able to mate at the age of one month so that atthe end of its second month a female can produce another pair of rabbits. The new pair of rabbitsalso mates at the age of one month, and after another month etc. The rabbits never die, andmating pair always produces a new pair. So how many pairs will be at the end of the nth month?Here is a diagram:Avagyan3time 0(months)pairs 1 1 2 3 5 8As we can see the number of pairs, as time goes up, is the Fibonacci sequence. After 5th monththere are 8 pairs, and after the nth month the number of pairs of rabbits is equal to the number ofnew pairs (which is the number of pairs in month n-2) plus the number of pairs alive last month.This is the nth Fibonacci number.The mathematics of Fibonacci numbers:Besides the definition shown above the Fibonacci numbers can be represented in manyother ways. Here are a few examples of its sum identities:i - 0thThe sum of the first n Fibonacci numbers is equal to the (n+2) numberminus 1.f = nFn+2 - Fn+3 + 2i=0Avagyan4ny Fi = FnFn+ii=o The sum of the squares of the first n Fibonacci numbers is the multiple of the«thand(n+l)th Fibonacci numbers.Furthermore, Fibonacci numbers obey many mathematical formulas and relations:The Fibonacci numbers obey the negation formulathe addition formula\ f-n + Lm Fn ).where ^ is a Lucas number, where L* = Ln-\ if,-: with *-i ~~ 'and ^ = -\e subtractionformula• i i1.. IV fjr / - / r \ •) \ / \ m *-« *-»r • fl /ithe fundamental identityconjugation relationsuccessor relationdouble-angle formula• 2 n "~ * n *-f i •multiple-angle recurrence<n-li ~ (~ \)multiple-angle formulasAvagyanS> : :t k,=u:*•:1=1)fort wenfor A: odd0• •For only n >l the extensionproduct expansionsFmFB = -Jandsquare expansion,and power expansionI ^ik\a-:,,,, for/t odd— I V '"*" }" » _ j f t , fort even.Here are given the general relations,fn-m FM_i Fm -f Fn Fm+ifa*i)n = Fn_i F^n + F,, Ft,,,iFn = F/Fn-/*i +F/- i FB_/.In the case / = M - / + 1, then / = (« + 1 )/2 and for n odd,Similarly, for n even,Avagyan6Avagyan?Furthermore, Fibonacci numbers have many properties. Some of them are fascinating,For example, every 3rd Fibonacci number is an even number, because 3rd Fibonacci number is 2.To put it more generally, every A>th Fibonacci number is a multiple of F(k). Moreover, every realpositive number n can be represented by the sum of Fibonacci numbers, using one number atmost once.Relation to the Golden Ratio:In mathematics and arts, two quantities are in the golden ratio if the ratio between thesum of those quantities and the larger one is the same as the ratio between the larger one and thesmaller. The golden ration is referred to with the letter cp and approximated as1.6180339887...The best way to understand this is by using line segment:BBIf we arrange A and B such that - = <p then (A + B}/B = <p too.There are many ways in which the Fibonacci sequence is related to the golden ratio. The mainrelation is that the further you look at the terms of the sequence, the ratio of the two successivevalues of terms of Fibonacci sequence becomes closer and closer to golden ratio. The tablebelow shows this.First number Second number _ . Second numberIvculU First numberAvagyanS11235813213455891235813213455891441.00002.00001.50001.66671.60001.62501.61541.61901.61761.61821.6180As we can see the ratio of the twelfth and eleventh terms is equal to the golden ratio by fourdecimal places. As we go further, the ration limit of two successive terms approaches the goldenratio (f:ton _, ,»—<» F(n)Furthermore, we can describe this graphically by using squares. Let's take a 1 on 1 square, thenadd another square to it with the same side. This will create a rectangle with sides 2 and 1. Thenwe add another square with the side equal to 2. Subsequently, we add more squares whose sidesare equal to the longest sides of the resulting rectangles. Here is the picture of the first 6 squares.1 1Avagyan98We see that the side of the next square is equal to the sum of the sides of previous two squares.The sequence of the length of the sides of successive squares is Fibonacci number. Furthermore,if we draw a quarter circle in each square we get a Fibonacci spiral like this:Occurrence in Nature:The greatest mystery revolving around Fibonacci sequence is that it occurs in manyplaces in Nature. This is the main reason why it is so famous. Fibonacci numbers appear innature in different ways. A good example would be the curved section of sea shells. Here is thesketch on the x and y axis of Nautilus shell.AvagyanlOWe can see that the spiral part crosses at 1 2 5 on the positive axis, and 0 1 3 on the negativeaxis. If we look more closely, we notice the oscillatory part of the shell crosses at 0 1 1 2 3 5 onthe positive axis, which are the first 6 numbers of Fibonacci sequence! Furthermore, as the spiralgrows, it approximately becomes a Fibonacci spiral.Another example is the petals of flowers. The number of petals on a flower that still has all ofits petals intact and has not lost any, for many flowers is a Fibonacci number. Although there arespecies that have very precise number of petals, most of the flowers have petals, whose numbersare very close to those above, with the average being a Fibonacci number. For example, theflowers that have two or four petals are not common, while there are thousands of species thathave one, three or five petals. Here is a list of some common flowers:Avagyanll• 3 petals: lily, iris• 5 petals: buttercup, wild rose, larkspur, columbine (aquilegia)• 8 petals: delphiniums• 13 petals: ragwort, corn marigold, cineraria,• 21 petals: aster, black-eyed susan, chicory• 34 petals: plantain, pyrethrum• 55, 89 petals: michaelmas daisies, the asteraceae familyThe last two examples are generalized, as because of the likelihood of under-development and over-development those flowers may end up with a few less or more petals.In addition, Fibonacci numbers can be seen in the patterns of florets in the head ofsunflowers. The sunflower displays florets in spirals in clockwise and counter-clockwisedirections.As shown in the picture, there are 21 florets in clockwise direction, and 34 in counter-clockwisedirection. The numbers 34 and 55 are 9th and 10th numbers in the Fibonacci sequence.^/.vl^*.f&M&'>* »o-<»A<£•*.Avagyanl2In addition, Fibonacci numbers occur in our human bodies. Let's take human hand forexample. Every human has two hands, each one of these has five fingers, each finger has threeparts which are separated by two knuckles. All of these numbers fit into the sequence.In the picture the length of each part of the finger is a Fibonacci number. This examplemay be a coincidence, considering how close each of the lengths to the corresponding Fibonaccinumber is. However, we can safely assume that most of the human hands are very close to thisone.The more astonishing example would be the human DNA. DNA is a nucleic acid thatcontains the genetic instructions used in the development and functioning of all known livingorganisms, with the exception of some viruses. The main role of DNA molecules is the long-term storage of information. Our bodies grow according to this information stored in the DNA ofour every cell. Here is the picture of the cut section of human DNA:AvagyanlSAs we see, the DNA looks like the chain of bounded small circles. It measures 34angstroms long by 21 angstroms wide for each full cycle of its double helix spiral. 34 and 21, ofcourse, are numbers in the Fibonacci series.Of course, these are only some of the many examples of Fibonacci numbers in nature.With the advancement of technology, scientists are able to view different galaxies, and theastonishing thing is that one category of those galaxies, which are called spiral galaxies,resembles Fibonacci spiral. Here is the picture of M51 galaxies, which is also known as TheWhirlpool Galaxy:Avagyanl4If we look closely, we can notice two spiral patterns, and each one of them isapproximately a Fibonacci spiral!The Fibonacci sequence retains a mystique, partly because excellent approximations of itturn up in many unexpected places in nature. The Fibonacci numbers appear in many otherunexpected contexts in mathematics, and they continue to spark interest in the scientificcommunity.

The Fibonacci Sequence

SPARK: Scholarship at Parkland | Parkland College

Parkland College
A with Honors Projects Honors Program
2010
The Fibonacci Sequence
Arik Avagyan
Parkland College
Open access to this Article is brought to you by Parkland College's institutional repository, SPARK: Scholarship at Parkland. For more information,
please contact spark@parkland.edu.
Recommended Citation
Avagyan, Arik, "The Fibonacci Sequence" (2010). A with Honors Projects. 9.
http://spark.parkland.edu/ah/9
Avagyanl
Arik Avagyan
MATH -129
Vicki Beitler
Fibonacci Sequence
Introduction:
Fibonacci sequence is one of the most famous series of numbers in all of the mathematics. The
number sequence is named after Leonardo Fibonacci, who first introduced it to western
European mathematics in his 1202 book Liber Abaci, although it may have been previously
known by ancient Indian mathematics. In mathematical terms, the sequence Fn of Fibonacci
numbers is defined by the recurrence relation
With seed values
F0 = 0 and FI = 1
As we can see, the first two Fibonacci numbers are 0 and 1, and each subsequent number is the
sum of the previous two. The sequence looks like this:
0, 1 7 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, . . .
AvagyanZ
Some mathematicians omit zero form the Fibonacci sequence, starting it with two 1's. The first
zero is known as the zeroth Fibonacci number, and has no real practical merit.
Origins:
The Fibonacci sequence was known in Indian mathematics independently of the West, but
scholars differ on the timing of its discovery. First the Fibonacci sequence was noticed by
Pingala, who introduced it in his book Chandasastra. Later, the number sequence is associated
with Indian mathematics; Virahanka, Gopala and Hemachadra. According to Parmanand Singh
Virahanka was "the first authority who explicitly gave the rule for the formation" of the
Fibonacci numbers. In contrast, Rachel Hall only mentions Hemachandra among these authors as
having worked with Fibonacci numbers; she claims that around 1150, Hemachandra noticed that
the number of possible rhythms followed the Fibonacci sequence.
In west the sequence was first introduced by Leonardo Fibonacci, who relates the exponential
growth of the pair of rabbits with Fibonacci number. The problem is as follows:
Let's say we have two rabbits. These rabbits are able to mate at the age of one month so that at
the end of its second month a female can produce another pair of rabbits. The new pair of rabbits
also mates at the age of one month, and after another month etc. The rabbits never die, and
mating pair always produces a new pair. So how many pairs will be at the end of the nth month?
Here is a diagram:
Avagyan3
time 0
(months)
pairs 1 1 2 3 5 8
As we can see the number of pairs, as time goes up, is the Fibonacci sequence. After 5th month
there are 8 pairs, and after the nth month the number of pairs of rabbits is equal to the number of
new pairs (which is the number of pairs in month n-2) plus the number of pairs alive last month.
This is the nth Fibonacci number.
The mathematics of Fibonacci numbers:
Besides the definition shown above the Fibonacci numbers can be represented in many
other ways. Here are a few examples of its sum identities:
i - 0
thThe sum of the first n Fibonacci numbers is equal to the (n+2) number
minus 1.
f = nFn+2 - Fn+3 + 2
i=0
Avagyan4
n
y Fi = FnFn+i
i=o The sum of the squares of the first n Fibonacci numbers is the multiple of the
«thand(n+l)th Fibonacci numbers.
Furthermore, Fibonacci numbers obey many mathematical formulas and relations:
The Fibonacci numbers obey the negation formula
the addition formula
\ f-n + Lm Fn ).
where ^ is a Lucas number, where L* = Ln-\ if,-: with *-i ~~ 'and ^ = -\e subtraction
formula
• i i1.. IV fjr / - / r \ •) \ / \ m *-« *-»r • fl /i
the fundamental identity
conjugation relation
successor relation
double-angle formula
• 2 n "~ * n *-f i •
multiple-angle recurrence
<n-li ~ (~ \)
multiple-angle formulas
AvagyanS
> : :
t k
,=u
:*•:
1=1)
fort wen
for A: odd
0
• •
For only n >l the extension
product expansions
FmFB = -J
and
square expansion,
and power expansion
I ^ik\a-:,,,, for/t odd
— I V '"*" }
" » _ j f t , fort even.
Here are given the general relations,
fn-m FM_i Fm -f Fn Fm+i
fa*i)n = Fn_i F^n + F,, Ft,,,i
Fn = F/Fn-/*i +F/- i FB_/.
In the case / = M - / + 1, then / = (« + 1 )/2 and for n odd,
Similarly, for n even,
Avagyan6
Avagyan?
Furthermore, Fibonacci numbers have many properties. Some of them are fascinating,
For example, every 3rd Fibonacci number is an even number, because 3rd Fibonacci number is 2.
To put it more generally, every A>th Fibonacci number is a multiple of F(k). Moreover, every real
positive number n can be represented by the sum of Fibonacci numbers, using one number at
most once.
Relation to the Golden Ratio:
In mathematics and arts, two quantities are in the golden ratio if the ratio between the
sum of those quantities and the larger one is the same as the ratio between the larger one and the
smaller. The golden ration is referred to with the letter cp and approximated as
1.6180339887...
The best way to understand this is by using line segment:
B
BIf we arrange A and B such that - = <p then (A + B}/B = <p too.
There are many ways in which the Fibonacci sequence is related to the golden ratio. The main
relation is that the further you look at the terms of the sequence, the ratio of the two successive
values of terms of Fibonacci sequence becomes closer and closer to golden ratio. The table
below shows this.
First number Second number _ . Second numberIvculU First number
AvagyanS
1
1
2
3
5
8
13
21
34
55
89
1
2
3
5
8
13
21
34
55
89
144
1.0000
2.0000
1.5000
1.6667
1.6000
1.6250
1.6154
1.6190
1.6176
1.6182
1.6180
As we can see the ratio of the twelfth and eleventh terms is equal to the golden ratio by four
decimal places. As we go further, the ration limit of two successive terms approaches the golden
ratio (f:
ton _, ,
»—<» F(n)
Furthermore, we can describe this graphically by using squares. Let's take a 1 on 1 square, then
add another square to it with the same side. This will create a rectangle with sides 2 and 1. Then
we add another square with the side equal to 2. Subsequently, we add more squares whose sides
are equal to the longest sides of the resulting rectangles. Here is the picture of the first 6 squares.
1 1
Avagyan9
8
We see that the side of the next square is equal to the sum of the sides of previous two squares.
The sequence of the length of the sides of successive squares is Fibonacci number. Furthermore,
if we draw a quarter circle in each square we get a Fibonacci spiral like this:
Occurrence in Nature:
The greatest mystery revolving around Fibonacci sequence is that it occurs in many
places in Nature. This is the main reason why it is so famous. Fibonacci numbers appear in
nature in different ways. A good example would be the curved section of sea shells. Here is the
sketch on the x and y axis of Nautilus shell.
AvagyanlO
We can see that the spiral part crosses at 1 2 5 on the positive axis, and 0 1 3 on the negative
axis. If we look more closely, we notice the oscillatory part of the shell crosses at 0 1 1 2 3 5 on
the positive axis, which are the first 6 numbers of Fibonacci sequence! Furthermore, as the spiral
grows, it approximately becomes a Fibonacci spiral.
Another example is the petals of flowers. The number of petals on a flower that still has all of
its petals intact and has not lost any, for many flowers is a Fibonacci number. Although there are
species that have very precise number of petals, most of the flowers have petals, whose numbers
are very close to those above, with the average being a Fibonacci number. For example, the
flowers that have two or four petals are not common, while there are thousands of species that
have one, three or five petals. Here is a list of some common flowers:
Avagyanll
• 3 petals: lily, iris
• 5 petals: buttercup, wild rose, larkspur, columbine (aquilegia)
• 8 petals: delphiniums
• 13 petals: ragwort, corn marigold, cineraria,
• 21 petals: aster, black-eyed susan, chicory
• 34 petals: plantain, pyrethrum
• 55, 89 petals: michaelmas daisies, the asteraceae family
The last two examples are generalized, as because of the likelihood of under-
development and over-development those flowers may end up with a few less or more petals.
In addition, Fibonacci numbers can be seen in the patterns of florets in the head of
sunflowers. The sunflower displays florets in spirals in clockwise and counter-clockwise
directions.
As shown in the picture, there are 21 florets in clockwise direction, and 34 in counter-clockwise
direction. The numbers 34 and 55 are 9th and 10th numbers in the Fibonacci sequence.
^/.vl^*
.f&M&'>
* »
o
-<»
A
<£•*.
Avagyanl2
In addition, Fibonacci numbers occur in our human bodies. Let's take human hand for
example. Every human has two hands, each one of these has five fingers, each finger has three
parts which are separated by two knuckles. All of these numbers fit into the sequence.
In the picture the length of each part of the finger is a Fibonacci number. This example
may be a coincidence, considering how close each of the lengths to the corresponding Fibonacci
number is. However, we can safely assume that most of the human hands are very close to this
one.
The more astonishing example would be the human DNA. DNA is a nucleic acid that
contains the genetic instructions used in the development and functioning of all known living
organisms, with the exception of some viruses. The main role of DNA molecules is the long-
term storage of information. Our bodies grow according to this information stored in the DNA of
our every cell. Here is the picture of the cut section of human DNA:
AvagyanlS
As we see, the DNA looks like the chain of bounded small circles. It measures 34
angstroms long by 21 angstroms wide for each full cycle of its double helix spiral. 34 and 21, of
course, are numbers in the Fibonacci series.
Of course, these are only some of the many examples of Fibonacci numbers in nature.
With the advancement of technology, scientists are able to view different galaxies, and the
astonishing thing is that one category of those galaxies, which are called spiral galaxies,
resembles Fibonacci spiral. Here is the picture of M51 galaxies, which is also known as The
Whirlpool Galaxy:
Avagyanl4
If we look closely, we can notice two spiral patterns, and each one of them is
approximately a Fibonacci spiral!
The Fibonacci sequence retains a mystique, partly because excellent approximations of it
turn up in many unexpected places in nature. The Fibonacci numbers appear in many other
unexpected contexts in mathematics, and they continue to spark interest in the scientific
community.


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The Fibonacci Sequence

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