In the last decades, many researchers concerned their attention on fractals properties of objects. Fractals can be use to describe natural shapes so their applications are various in many fields such as informatics, economics, engineering, medical studies. In this paper we present a way to describe fractal, using the Iterated Function System (IFS). We present the random iterated algorithm implemented in the C++ programming language used to generate selfsimilar fractals.fractal, IFS, random iteration algorithm

Daniela Alexandra CRIŞAN

Justina Lavinia STĂNICĂ

English

Research Papers in Economics

1 
 
THE RANDOM ITERATION ALGORITHM 
 
Daniela Alexandra CRIŞAN, PhD 
Justina Lavinia STĂNICĂ, PhD Candidate 
 
Romanian – American University, Bucharest, Romania 
dacrisan@yahoo.com, lavinia.stanica@gmail.com 
 
ABSTRACT 
In the last decades, many researchers concerned their attention on 
fractals properties of objects. Fractals can be use to describe natural 
shapes so their applications are various in many fields such as 
informatics, economics, engineering, medical studies. In this paper 
we present a way to describe fractal, using the Iterated Function 
System (IFS). We present the random iterated algorithm 
implemented in the C++ programming language used to generate 
selfsimilar fractals. 
Keywords: fractal, IFS, random iteration algorithm 
 
1. INTRODUCTION  
Around the year 1970, the French mathematician Benoit Mandelbrot 
introduced the concept of Fractals in order to describe some dynamic 
systems. Although Mandelbrot named these features as fractals, 
other mathematicians have studied those forms years before: Cantor, 
Sierpinski and Koch were attracted by the strange properties of these 
forms around the beginning of the XIX century.  In 1918, the 
German mathematician introduced the fractional dimension.  
Mandelbrot defined a fractal as a form with self-similarity: a form 
composed by copies transformed of him. Also, as one looks more 
deeply or more closely at a fractal, its inner parts have a similar 
design to that of the whole object. 2 
 
(a) (b)  
Fig. 1. Two famous fractal: (a)Sierpinski Triangle and (b) Barnsley 
Fern, composed by copies of the whole object. 
Later he proposed another definition: a fractal is a form with a 
fractional dimension. In Euclidian terms a form can be one-
dimensional (lines), two-dimensional (filled figures such as squares, 
trapezoids, and circles), and three-dimensional (filled objects such as 
cubes and spheres) objects. Fractals have dimensions that are in 
between these three integer dimensions.  
2. ITERATED FUNCTION SYSTEM (IFS) 
An  Iterated Function System  (IFS) is a finite collection of 
contractions Fi: X X defined on a metric space X. Each extends to 
a mapping (different but denoted by the same letter) Fi: H(X)
H(X), where H(X) is the space whose points are nonempty compact 
subsets of X. When endowed with the Hausdorff metric, H(X) is 
complete if X is. In addition, contractions FiX X remain 
contractions as mappings of H(X). Together, the Fi define another 
contraction F: H(X) H(X), by the following formula: for every A
H(X), F(A) =  Fi(A). M. Barnsley demonstrated that since we 
are working in a complete metric space, F has a fixed point (set!) AF, 
so that F(AF) = AF, and this point can be reached by successive 
approximations from any starting location. This result is known as 
the College Theorem. Fixed points of IFSs are called attractors or 
invariant sets. 
Two problems arise: the direct and the inverse one. The direct 
problem is to find the fixed point of a given IFS and is solved by 
what is known as the deterministic algorithm. Start with a set A0
H(X) and compute successively Ak =  Fi(Ak-1) = F(Ak-1), k > 1. 3 
 
The sequence {Ak) converges to the fixed point AF of {Fi} as k
.  
The inverse problem is: for a given set A H(X), find an IFS {Fi} 
that has A as its fixed point. The mathematics of the second, random 
iteration  algorithm is more complex but implementation is more 
straightforward. Assign positive frequencies pi to the mappings Fi. 
Start with an arbitrary point x0 X. At every step k+1, select xk+1 
from the set {Fi(xk)}. Fj(xk) is selected with the probability pj / pi. 
The sequence {xk} converges to AF. In practical terms, to depict an 
approximation of AF on a computer, the points of the sequence are 
displayed starting with a reasonably large index. The numbers {pi} 
have no effect on the fixed point AF but influence significantly the 
rendering of its approximations. The inverse problem is the base of 
the fractal compression: it is more efficient to store into computer the 
parameters defining the IFS that the whole image.   
3. IFS AND FRACTALS 
Fractal can be represented using iterated function system (IFS). 
Considering that a fractal is a composed by copies of himself, each 
transformation in the IFS describes the affine transformation 
characterizing every copy.  
An affine transformation is a function: 
f(x, y) = (a*x+b*y+e, c*x+d*y+f),  
x, y∈X   
with the parameters a, b, c, d, e, f .  
It can be written also as: 
 


 


+  


 








=
f
e
y
x
d c
b a
y x f ) , ( . 
Consider the famous selfsimilar fractal the Sierpinski Triangle 
composed by tree copies of himself: 4 
 
 






+  


 








=






+  


 








=






+  


 








=
3 2 / 1
2 / 1
*
2 / 1 0
0 2 / 1
) , (
,
0
1
*
2 / 1 0
0 2 / 1
) , (
,
0
0
*
2 / 1 0
0 2 / 1
) , (
3
2
1
y
x
y x f
y
x
y x f
y
x
y x f
 
Fig. 2. The Sierpinski Triangle and the associated IFS 
Two other interesting results are the Barnsley’s Fern and the 
Heighway dragon presented below.  






+  


 








−
=






+  


 







−
=






+  


 







 −
=






+  


 








=
5 . 1
0
*
85 . 0 05 . 0
05 . 0 85 . 0
) , (
,
45 . 0
0
*
25 . 0 25 . 0
3 . 0 15 . 0
) , (
,
5 . 1
0
*
2 . 0 2 . 0
25 . 0 2 . 0
) , (
,
0
0
*
15 . 0 0
0 0
) , (
4
3
2
1
y
x
y x f
y
x
y x f
y
x
y x f
y
x
y x f
 5 
 
 
Fig. 3. The selfsimilar fractal Barnsley’s Fern and the associated IFS 
 






+  


 








=






+  


 








=






+  


 








=
3 2 / 1
2 / 1
*
2 / 1 0
0 2 / 1
) , (
,
0
1
*
2 / 1 0
0 2 / 1
) , (
,
0
0
*
2 / 1 0
0 2 / 3
) , (
3
2
1
y
x
y x f
y
x
y x f
y
x
y x f
 
 
Fig. 4. The Heighway dragon and the associated IFS 
 
4. THE RANDOM ITERATION ALGORITHM  
The random iteration algorithm is a form of the  deterministic 
algorithm. Knowing the IFS we can generate the fractal associated to 
it, using the direct problem and the college theorem discussed above. 
Starting with a single point, we can transform it using the IFS-
transformations for thousands times and finally we obtain the fractal 
we want.  6 
 
The method may be illustrated with the Yuval Fisher’s special 
copyrighter which receives as entry an arbitrary image (may be a 
point) and applies to it the set of affine transformation, generating a 
new image. The image obtained is transmitted, using a feedback 
process, on the entry of the copyrighter and the process is repeated 
for several times. For example, consider that the transformations are 
those which describe the Sierpinski Triangle. If we test the Yuval 
Fisher copyrighter for different initial images we can observe that the 
final image is the same, so it not depends on the initial image but is 
defined by the affine transforms applied to it: 
 
Fig. 5. The Yuval Fisher copyrighter 
The algorithm is: 
P1. Set (x,y) the starting point - may be (0,0); 
P2. Let (x1, y1) be the point obtained by applying an arbitrary 
transformation in the IFS; repeat the second step with (x1,y1) as 
initial point 
In the section below we implemented the algorithm in C++ 
programming  language; consider that PutPixel(x,y) is a function 
which display on screen a point with coordinates (x,y). 
void IFS(double p[][6], int n) 
//n=numar transformari 
//p[k][0]=r 
//p[k][1]=s 
//p[k][2]= α 
//p[k][3]=β 
//p[k][4]=e 
//p[k][5]=f 
{ 
  double a, b, c, d, e, f; 
  int x = 0., y = 0.;//starting point 
  int k;//current transformation 7 
 
  int xx; 
  srand((unsigned)time(NULL)); 
  for( long i=0;i<10000;i++)   
{ 
k=rand()%n; //an arbitrary transformation 
a= p[k][0] *cos(p[k][2]*PI/180); 
b = -p[k][1]*sin(p[k][3]*PI/180); 
c =p[k][0]*sin(p[k][2]*PI/180); 
d = p[k][1]*cos(p[k][3]*PI/180); 
e = p[k][4]; 
f = p[k][5]; 
xx = a*x + b * y + e; 
y = c*x+d*y+f; 
x = xx; 
PutPixel(x, y); 
  } 
} 
CONCLUSIONS 
In 1900’s decades, fractals were regarded as “strange images”. 
Representing a fractal image (although the name “fractal” was 
imagined much later, around 1970) was a difficult task; these forms 
were studied only for mathematical reasons. In 1981 John  
Hutchinson defines the Iterated Function System theory in his work 
„Fractals and selfsimilarity” (Hutchinson, [10]). Later, in 1988 the 
American researcher Michael Barnsley succide to demonstrate the 
collage theorem in his works „Fractals everywhere” (Barnsley, [1]).  
The collage theorem yield to the random iteration algorithm, those 
implementation in C++ language is presented above and which is 
very ingeniously illustrated by Yuval Fisher’s special copyrighter. It 
is one of the most used methods of generating a selfsimilar fractals 
such are Sierpinski Triangle, Barnsley’s Fern and Heighway dragon 
exemplified above.   
As this paper shows, the random iteration algorithm is very simple, it 
is easy to implement in any programming language and the results it 
generates are very spectacular and may be used for any graphical 
goal, not only for mathematical resons! 8 
 
 
BIBLIOGRAPHY 
[1]  Barnsley M., “Fractals Everywhere, 2
nd ed.”, Academic Press, Boston, 
1993 
[2]  Barnsley, M., S.G. Demko, "Iterated Function Systems and the Global 
Construction of Fractals" Proc.Roy.Soc.London, Ser.A 399, 1985, 243-
275 
[3]  Buchnicek M., M. Nezadal, O. Zmeskal, „Numeric calculation of 
fractal dimension”, Nostradamus 2000, Prediction Conference, 2000; 
[4]  Chen S., J.M. Keller and R.M. Crownover, “On the calculation of 
fractal features from images”, IEEE PAMI, 1998 
[5]  Crişan  D.,  "Fractal  spectrum  of  gray-level images", Cercetare 
ştiinţifică.  Sesiunea  cadrelor  didactice,  Universul  Juridic,  Bucureşti, 
2005 
[6]  Dobrescu R.(ed), C. Vasilescu (ed), “Interdisciplinary Applications Of 
Fractal And Chaos Theory”,  Academia Română, Bucureşti, 2004 
[7]  Y., “Fractal image compression with quadtree”, Fractal Compression 
Theory and Application to digital images, Springer Verlag, New York, 
1994 
[8]  Fisher Y., E.W. Jacobs, R. D. Boss, “Fractal image compression using 
iterated transformes”, NOSC Techical Report, Naval Ocean System 
Center, San Diego, 1995 
[9]  Harris J. W.,  H. Stocker, "Hausdorff Dimension", "Scaling Invariance 
and Self-Similarity", "Construction of Self-Similar Objects.", cap. 
4.11.1-4.11.3, Handbook of Mathematics and Computational Science, 
Springer-Verlag, New York, 1998, pg. 113-135 
[10] Hutchinson J., "Fractals and Self-Similarity.",  Indiana Univ. Journal 
Mathematics, 35, 1981, pg. 713-747 
[11] Jaquin A. E., “Image coding based on a fractal theory of iterated 
contractive image transformation”, IEEE Trans. On Image Processing, 
vol I, No 1, January 1992 
[12] Lauwerier H., “Fractals:  Endlessly  ŞRepeated  Geometric  Figures” 
Princeton, NJ: Princeton University Press, 1991, pg. 29-31 
[13] Mandelbrot B. „The fractal geometry of nature”, Freeman, New York, 
1983 
 

Fractal image compression using iterated transformes”, NOSC Techical Report, Naval Ocean System Center,

Fractal image compression with quadtree”, Fractal Compression Theory and Application to digital images,

Fractal spectrum of gray-level images&quot;, Cercetare ştiinţifică. Sesiunea cadrelor didactice,

Fractals and Self-Similarity.&quot;, Indiana Univ.

Fractals: Endlessly ŞRepeated Geometric Figures”

Hausdorff Dimension&quot;, &quot;Scaling Invariance and Self-Similarity&quot;, &quot;Construction of Self-Similar Objects.&quot;,

Image coding based on a fractal theory of iterated contractive image transformation”,

Interdisciplinary Applications Of Fractal And Chaos Theory”, Academia Română, Bucureşti,

Iterated Function Systems and the Global Construction of Fractals&quot;

Numeric calculation of fractal dimension”, Nostradamus 2000, Prediction Conference,

On the calculation of fractal features from images”,

The fractal geometry of nature”,

THE RANDOM ITERATION ALGORITHM

http://www.rebe.rau.ro/RePEc/rau/jisomg/SU08/JISOM-SU08-A28.pdf

THE RANDOM ITERATION ALGORITHM

Abstract

Similar works

Full text

Available Versions

Research Papers in Economics