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.

Fibonacci numbers

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 (6th 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).

Fibonacci numbers

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,…

Fibonacci numbers

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

Fibonacci numbers

ratio 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).

lim→

1

√

⁄

√, 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

Because |1 | √5⁄ 1 2⁄ for 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 √51

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).

Fibonacci numbers

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

Fibonacci numbers

. 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.

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.

Figure 3 Sunflower head displaying florets in spirals of 34 and 55 around the outside

Fibonacci numbers

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.

Fibonacci numbers

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Fibonacci Numbers

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