fibonacci-numbers

NATIONAL UNIVERSITY
Fibonacci Number
Munjal Patel
January 30, 2010
Fibonacci numbers are sequence of numbers developed in ancient India and later introduced to
the western European mathematicians by Leonardo of Pisa also known as Fibonacci in 1202.
This paper serves as an introduction to the Fibonacci numbers, their properties, applications
and relation to the mother nature.

Leonardo Pisano Bogollo, (c. 1170 – c. 1250) known as Leonardo of Pisa, or simply
Fibonacci was an Italian mathematician (Anderson, Frazier, & Popendorf, 1999). He is
considered as the most talented mathematician of the middle ages (Eves, 1990). Fibonacci was
first introduced to the number system we currently use with symbols from 0 to 9 along with the
Fibonacci sequence by Indian merchants when he was in northern Africa (Anderson, Frazier, &
Popendorf, 1999). He then introduced the Fibonacci sequence and the number system we
currently use to the western Europe In his book Liber Abaci in 1202 (Singh, Acharya
Hemachandra and the (so called) Fibonacci Numbers, 1986) (Singh, The So‐called Fibonacci
numbers in ancient and medieval India, 1985).
Fibonacci sequence was well known in India and was applied to the metrical sciences
(prosody). Pingala (200 BC), Virahanka (6 th century AD), Gopāla (c.1135 AD), and Hemachandra
(c.1150 AD) are given credits for the development of this sequence (Goonatilake, 1999). In
Sanskrit prosody, long syllables have duration of 2 and short syllables have duration of 1. For
this reason, it is possible to form a pattern of duration by adding a short syllable to the
pattern of duration 1 or a long syllable to the pattern of duration 2. Sanskrit prosodists
proved that the number of patterns of duration is the same as the sum of the previous two
numbers in the sequence. It is believed that this is where the motivation for the Fibonacci
sequence came from. Later, algorithms for finding pattern of duration were developed
and the higher‐order Fibonacci numbers were discovered. This work has been reviewed by
Donald Knuth in The Art of Computer Programming (Knuth, The Art of Computer Programming,
2006) (Hall, 2007).
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In order to study Fibonacci numbers, Fibonacci investigated a problem about growth
rate of rabbit population in a biologically unrealistic situation. The puzzle Fibonacci posed was:
assume that a newly born pair of rabbit, a male and a female is to be mate at the age of one
month to produce another pair of rabbit with a male and a female on the second month.
Assume that rabbits never die and the same cycle repeats every month so that all pairs give
birth to a new pair every month and a newly born pair can start mating from the second month.
How many pairs of rabbit will there be at the end of the year The solution of the problem
looks like the following:
At the end of the first month, they mate, but there is still one only 1 pair.
At the end of the second month the female produces a new pair making 2 pairs of rabbits.
At the end of the third month, the original female produces a second pair, making 3 pairs.
At the end of the fourth month, the original female has produced another new pair, the
female born two months ago produces her first pair as well, making 5 pairs.
If this cycle continues, at the end of n month, number of pairs will be the sum of number of
pairs in the month 2 and 1. This is same as the Fibonacci number (Knott, Fibonacci
Numbers and Nature, 2008).
The Fibonacci numbers are represented by the numbers in the following sequence. By
definition, the first two numbers in the Fibonacci series are 0 and 1, and the rest are the sum of
the previous two numbers.
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, …
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The first 18 Fibonacci numbers denoted by for 0,1,2, … ,17 are the following (Knott, The
Fibonacci numbers, 2005):
0 1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 987 1597
The Fibonacci sequence can be represented by a recurrence relation ,
where seeds 0 and 1. The same sequence can also be extended to negative index .
The sequence with negative index …,8,5,3,2,1,1,0,1,1,2,3,5,8,… can be satisfied by
1 .
One of the interesting properties of Fibonacci numbers is that every third number in the
sequence is even and every element of the sequence is always a multiple of . for this
reason, Fibonacci sequence satisfies a stronger divisibility property described by (Su)
(Ribenboim, 2000) (Renault, 1996):
gcd , ,
This unremarkable recursive sequence has yet another interesting pattern. The sides of
the squares in the figure 1 correspond to the numbers in the Fibonacci sequence and squares
are arranged in the outwardly spiraling pattern. Each rectangle in the image has roughly the
same shape and ratio for their length and width. It is also very interesting to note that as the
sequence of rectangles develops outwards, the ratio of length to width on every step is the
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atio of two successive terms (Platonic Realms). The ratios of these consecutive Fibonacci
numbers form another sequence:
1
1 , 2 1 , 3 2 , 5 3 , 8 5 , 13
8 , 21
13 ,…
In 1753, a Scottish mathematician Robert Simson proved that the ratios of successive Fibonacci
numbers ⁄ converges to a real number known as the golden ratio () as approaches to
infinity (Wells, 1987).
1
lim
→
√
⁄
√
, where is the golden ratio.
√
1.6180339887 … (Sloane)
Figure 1 A tiling with squares whose sides are successive
Fibonacci numbers in length
Figure 2 Approximate and true golden spirals. The green
spiral is made from quarter‐circles tangent to the interior of
each square, while the red spiral is a Golden Spiral, a special
type of logarithmic spiral. Overlapping portions appear
yellow. The length of the side of one square divided by that
of the next smaller square is the golden ratio.
Fibonacci numbers Page 5

Because |1 | ⁄ √5 1⁄ 2for all 0, the closest integer to ⁄ √5 can be obtained
by the rounding as following:
, 0 (Tattersall, 2005)
√
It is possible to test any number whether it belongs to the Fibonacci series of not. As we
know, that the closest integer to is ⁄ √5, the most straightforward and brute‐force test
would be the following identity with is valid if and only if is a Fibonacci number (Posamentier
& Lehmann, 2007).
log √5 1
2
Alternatively, if and only if either 5 4 or 5 4 is a perfect square, where is a positive
integer, than is also a Fibonacci number (Posamentier & Lehmann, 2007). It is possible to
conduct a more sophisticated test by considering the fact that the convergent of the continued
fraction representation of are ratios of successive Fibonacci numbers. The following
inequality is true if and only if ( and are coprime) and and are successive Fibonacci
numbers (Posamentier & Lehmann, 2007).
1
From this, it is possible to derive that is a Fibonacci number if and only if the interval
, contains a positive integer (Möbius, 1998).
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There are many other properties and identities related to Fibonacci numbers and people
have written books about it. Due to the nature of this paper and a limited scope, it is hard to
cover everything there is about for the Fibonacci numbers. For this reason, in this final section
of the paper, I will introduce some real world applications related to Fibonacci numbers and
their relationship with nature.
Fibonacci numbers are important to perform a run‐time analysis of Euclid’s algorithm to
find the greatest common divisor (GCD) of two integers. A pair of two consecutive Fibonacci
numbers makes a worst case input for this algorithm (Knuth, Art of Computer Programming,
Volume 1: Fundamental Algorithms, 1997). Fibonacci numbers have their application in the
polyphase version of the Merge Sort algorithm. This algorithm divides an unsorted list in two
lists such that the length of lists corresponds to two sequential Fibonacci numbers. The ratio of
the lengths of the lists is an approximately same as (Knuth, Art of Computer Programming,
Volume 1: Fundamental Algorithms, 1997). Fibonacci numbers are essential in the analysis of
the Fibonacci heap data structures. A network topology for parallel computing uses a Fibonacci
cube which is an undirected graph with Fibonacci number of nodes. The Fibonacci search
technique is a one‐dimensional optimization method and is developed on the basis of Fibonacci
numbers and their properties (Avriel & Wilde, 1966). IFF 8SVX audio file format in Amiga
computers uses Fibonacci sequence to compand the original audio wave for optional lossy
compression (Addison‐Wesley, 1991). The conversion factor from miles to kilometers
1.609344 . When Fibonacci numbers are replaced by their successors, the sum of the
decomposition of distance in miles into a sum of Fibonacci numbers is approximately same as
kilometer sum. This can be achieved by shifting a radix 2 number register in golden ratio base
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. Shifting the register down the Fibonacci number results into the conversion from kilometers
to miles (Hazewinkel, 2002) (Knott, Using the Fibonacci numbers to represent whole numbers,
2009).
A close observation of our nature revels that Fibonacci numbers are often found in two
consecutive Fibonacci numbers in biological setting (Douady & Couder, 1995), branching in
trees, and arrangement of leaves on a stem, and the fruitlets of a pineapple (Jones & Wilson,
2006) for example. The flowering of artichoke and arrangements of a pine cone is yet another
example (Brousseau, 1969). The Fibonacci numbers have also been observed in the family tree
of honeybees (Thimbleby). H. Vogel in
1979 proposed a model for the pattern of
florets found in the head of a sunflower
(Vogel, 1979) as below:
2 , √
∅ Here, is the index number of the floret
while is a constant scaling factor.
Figure 3 Sunflower head displaying florets in spirals of 34 and 55
around the outside
Therefore, florets lie on Fermat's spiral. The divergence angle is approximately 137.51 ° which is
known as the golden angle because this angle divides a circle in the golden ratio. The reason
sunflower florets pack so efficiently is since this ratio is an irrational number; no floret has a
neighbor at exactly the same angle from the center. The rational approximations to the golden
ratio are of form : 1. For this reason, the nearest possible neighbor of floret number
are those at for some index which is dependent of , the distance from the center.
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It is often observed that sunflowers and similar natural arrangements have 55 spirals in one
direction while 89 in the other (Prusinkiewicz & Lindenmayer, 1991).
Once again I would like to stress that this paper shall only be considered as the
introduction to the Fibonacci numbers and there is still much more to Fibonacci numbers not
covered in this paper.
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