On The Application Of Fractal Geometry in Coding

On The Application Of Fractal Geometry in Coding Information Adil Al-Rammahi 

Abstract— In this paper a modified fractal method of coding Procedure ( Pickover's ana code) : information was introduced. Pickover introduces the ana sequence and fractal , two self-referential constructions arising from the use 1) let a → ana , n → ann of language. In his work, the code is represented by fractal figure instead of formula In this paper , the fractal code is 2) take a = black , and n = white , so the stages of ana code represented by contraction maps. are:

Keywords— fractals, coding information, I. INTRODUCTION

P

ICOVER in [5] considers a broad class of verbal sequences, but for simplicity, he focuses on the most typical sequence, which in [ 5 ] referred to as the ana sequence . This sequence of words ( or strings ) is defined in stages by induction . In [ 4 ] , Pickover's questions on the relative composition of sequence terms and the dimension of the fractal was solved. Also, it presents a beautiful variant of the ana constructions involving the golden ratio. In [4 ] and [5] , the code has no the formula of function . Huthinson [3] introduced the concept of fractal geometry by using the iterated function systems ( IFS ) .He proved that the fractal set satisfies two properties . The first is self similarity and the second is the Hausdorff fractional dimension . In this paper , the code was introduced as contraction maps of iterated function systems which satisfying the self similarity and fractional dimension .

The third stage of ana code be anaannana.

II. CUBIC SPLINE INTERPOLATING FUNCTIONS III. HUTCHINSON FRACTAL SETS In Pickover's method the first four terms of the ana sequence are: a ana anaannana anaannanaanaannannanaannana : To see how the ana sequence arises from a verbal context , observe that a letter is replaced in the next stage by its description using the indefinite articles a and an , appropriately, a is described by "ana " and n by " ann ".

Adil L-Rammahi, Kufa University, Faculty of Mathematics and Computer Science, Department of Mathematics, Njaf, IRQ (phone:+964(0)33219195; B.O. 21 Kufa, e-mail: adilm.hasan@ uokufa.edu.iq).

In this section , the construction

of fractal sets which

introduced by Hutchinson [ 3 ] is summarized as follows : Definition (3.1) (Metric Space):Let

X be a non empty set

.A real valued function d is defined on ordered pairs of elements in function on

X  X , such that

X are called a metric or distance

X iff it satisfies , for every x, y, z  X , the

following axioms :

1.d ( x, y )  0, andd ( x, x)  0 2.d ( x, y )  d ( y, x) 3.d ( x, z )  d ( x, y )  d ( y, z )

The real number

d ( x, y ) is called the metric distance from Definition (3.5) (Cantor Set) The Cantor set C is define by

x and y

c  c1  c2  c3  ...

Definition (3.2) (Contraction Function): A continuous function f : X  X is contraction if there is an   (0,1)

Clearly that { 0 , 1 } ε C. In fact, one can find the endpoint, and all the endpoints are in C.

such that

d ( f ( x), f ( y ))  d ( x, y )x, y  X .

Definition (3.3) (Iterated Function Systems) : Let X be a complete metric space, and let f i : X  X be

i  1,2,  , n .,with Constants 1 ,  2 ,…,  n . Let α = max  i . clearly all f i are contraction

contraction maps

for

maps with parameter α . The functions

f i are called Iterated

IV. CONTRACTION CODE In this section the proposition method of contraction code is introduced as in the following procedure . Procedure ( contraction code ) 1 ) Let a=dot ,and n=dash

Function System with parameter α. 2 ) Applying the two contraction functions on L = [ 0 , 1 ]. Definition (3.4) (Attractor) An attractor for the iterated function system { f i } is a non-empty compact set such that :

E  f1 E   f 2 E ... f N E 

w1 ( x)  ax w2 ( x)  ax  b 3)Let D be the dimension of the fractal curve ( attractor ) ,so

Theorem ( 3.1 ,[ 2 ] ) A complete metric space with a finite number of Contraction maps has a unique attractor E.

1 Ln  a D= ,and that it is a fractal [ 1 ].where the number 1 Ln  b

Example (3.1) ( Cantor Set ) The Cantor set is a subset of the real line. We’ll consider subsets of [0, 1]:

of iterated functions system be 2 and the scaling factor be a .

[0] ----------------------------------------------------------[1]

Example (4.1) One can take the parameters a and b are

Will look at a sequence of approximations to the Cantor set.

1 2 a  , b  .So the code in first three stages are : 3 3 a , ana , anannnana And it is a similar result of Pickover's method when the axioms be

Let C0 = [0 , 1]. Remove the middle third, and get C1 = [0 ,

a  ana n  nnn

1 2 ] [ ,1 ] 3 3

And in dot –dash form is .-.---.-. Example (4.2 ) One can take the parameters a and b are

It is a closed interval, keep endpoints [ 0 ]--------------[

1 ] 3

[

a

2 ]-------------[ 1 ] 3

Continue the process. Next is C2 = [ 0 ,

1 2 1 2 7 8 ]  [ , ]  [ , ]  [ ,1] 9 9 3 3 9 9

In general, C n is the union of 2

n

closed intervals, each of

n

size 3 . clearly C0

 C 1

For explain the proposed method , the following examples are given :

C 2

C3 …



1 3 , b  So the code in first three stages are : 4 4

a , anna , annannnnnnnnanna Ad it is a similar result of Pickover's method when the axioms be

a  ana n  nnnn

And in dot –dash form is

.--.--------.--.

V.CONCLUSION: Pickover`s code has no the formula of code function . It deals with figure only .The contraction functions and iterated function system were used here for constructing fractal code instead of the fractal figure . Hutchinson fractal set was used in this proposed method .To generate a population , a practical procedure was introduced and used in finite number of stages . REFERENCES

[1] Alfonseca , M . & Ortega , A.; " Determination of Fractal Dimensions From Equivalent L System ; IBM J. RES & DEV VOL.45; NO.6 , November ; 2001 . [2] Barnsley , M. ; Press) ; 1988 .

Fractals Everywhere ; (Academic

[3] Hutchinson J " Fractal and Self Similarity " , Indian University Mathematics Journal .vol.30 , pp. 713740 , 1981 . [4] Joseph , L. PE ; ANA'S Golden Fractal ; Fractals, Vol. 11, No. 4 pp. 309-313 ; 2003 . [5] Pickover,C.,Wonders of Numbers;Oxford University press;2001. Adil Al-Ramahi was born on 1963 in Najaf, Iraq. He studied Applied Mathematics at University of Technology, Baghdad, Iraq. From the same university, he obtained his M. Sc in stability. The title of Assistant professor was awarded to him in 2002. He was awarded the degree of PhD in Fractals in 2005. He has supervised several M.Sc. dissertations. He has headed the Mathematics Department for three years from 2008-2011. His area of research is Fractals, Numerical Analysis, Cryptography and Image Processing. He published more than 25 papers and one book. He was selected as an editor, reviewer and a scientific committee member in many journals and conferences.