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August 2013. Vol. 4, No. 2 ISSN2305-8269

International Journal of Engineering and Applied Sciences © 2012-2013 EAAS & ARF. All rights reserved www.eaas-journal.org

49

NUMERICAL TRANSFORMATIONS AND FRACTAL

CHARACTERIZATION OF SOME SELECTED

IFS-BASED FRACTALS

Salau T.A.O. 1 and Ajide O.O.

2

1, 2 Department of Mechanical Engineering, University of Ibadan, Nigeria

2 Corresponding Author: [email protected]

ABSTRACT

This study investigated the non-zero arbitrary positive power transformation and fractal characterization by

optimum disk counted of some selected IFS-based fractals (IFS-1 to IFS-10) on a complex plane. In addition, visual

evaluation (quest motivated by information available in literatures) was obtained for the transformed fractals

associated with IFS-1, IFS-5 and IFS-6 at some selected power values: golden ratio (1.618), quadratic (2.0),

exponent (2.718), phi (3.142) and 5.678. The results of this study correspondingly show the existence of both

quantitative and aesthetically appealing qualitative differences between the untransformed (standard) and the

transformed fractals. The range of measured absolute percentage difference is (0.1-25.5) in the estimated disk

dimension for all fractals and their corresponding different transformations powers studied. These differences have

dependence on both the fractal and transformation power used. There can be application of results not only in

fractals beautification and exhibition but including potential in robots application industries.

Keywords: Transformed Fractals, IFS-based Fractals, Selected Power Values, Optimum Disk Counted and

Estimated Disk Dimension

1. INTRODCUTION

The geometry of fractals and mathematics of fractal

dimension has been reported by [13] to provide tools

for a variety of scientific disciplines (especially the

study of chaos). Fractal dimension has been reported

by many researchers in this field as an important

estimator in fractal characterization of nonlinear

systems. The determination of fractal dimension is an

approach of characterizing the complexity of a shape

[4]. In the author’s view, the use of box-counting

algorithm, fractal spectrum and local fractal

dimension can be satisfactorily used in the study of

fractal characteristics of binary and grey-level

images. The paper concluded that the local dimension

is an alternative way to characterize the binary

images and equally useful for multi-structures. A

fractal characterization of velocity components and

turbulent shear stress in open channel has been

studied by [1]. An efficient algorithm was developed

to construct fractal interpolation functions (FIF) and

was used to estimate fractal dimension. The fractal

dimensions of turbulent data were accurately

obtained by applying only 500 data points. Findings

revealed that some differences exist among the fractal

dimension of Reynolds shear stress process. This

paper has shown one of the key applications of fractal

characterization in nonlinear dynamics. [2] research

paper bothers on fractal image compression and

texture analysis. Their paper adequately describes a

new method for building object models for the

purpose of overlapped object recognition. The

technique relies on local fragments of the boundary

which helps in deriving a set of autoregressive

parameters that serve to detect similar boundary

fragments. The authors’ paper has introduced several

researchers into a new concept of fractal image. The

Iterated Function System (IFS) has attracted more of



50

researchers’ interests in the recent time. This is due to

its conceptual simplicity, computational efficiency

and enormous ability to reproduce natural formations

and complex phenomena [15]. This motivated [15] to

develop a new MATLAB program called “IFS

MATLAB Generator”. This program was found

useful in generating and rendering of IFS Fractals as

well as launching interested researchers to the

mathematical basis of IFS. This work has been

exploited as a platform by many researchers in this

field for fractal characterization.The importance of

equation transformations in nonlinear problem

analysis cannot be overemphasized. It has been found

useful in simplifying complex equations to forms that

can easily be analyzed. [11] work bothers on

numerical transformation of nonlinear dynamic

systems with reference to jerky motion and its

application to minimal chaotic flows. The situation

whereby the three dimensional autonomous

dynamical systems exhibits at least one equivalent

jerky dynamics has been examined. A simple

criterion that excludes chaotic dynamics for some

classes of jerky dynamics has been provided as a

result of this transformation. The author was able to

show that numerical transformation of equations

plays significant roles in the characterization of

nonlinear dynamical systems. The dynamics of

fractionally damped Duffing equation has been

studied by [9]. The fractionally damped Duffing

equation was numerically transformed into a set of

fractional integral equations solved by a predictor-

correction method. The authors adopted Largest

Lyapunov exponents for verifying the occurrence and

nature of chaotic attractors. Findings obtained

showed that fractional order of damping has a

significant effect on the dynamic behaviour of the

motion. This paper has to some extent demonstrated

the benefit of equations transformation in the study of

nonlinear dynamics. As part of efforts of making

analysis of nonlinear system dynamics easier, [10]

applied Lie group transformations to nonlinear

dynamic systems .Their paper was able to explain the

theory of Lie group operators in a form appropriate in

the field of applied dynamics. The authors adopted

Hausdorff formula which is found to be implicitly

reproduced in most averaging techniques during the

transformation process of the equations of motion.

This technique is employed in examining nonlinear

modal interaction in a double pendulum. This

transformation of equations has made the authors to

successfully analyze the dynamic behaviour of

nonlinear systems under various conditions. [14]

have equally demonstrated the benefit of equations

transformation in nonlinear dynamics study. The

authors adopted the popular Homotopy perturbation

transform technique for nonlinear system of

equations using He’s polynomials. This approach has

been found very useful in the analysis of nonlinear

problems. In the very recent time, [16] investigated

the possibility of finding different cases for

equivalence dimensional autonomous dynamical and

hyperjerk dynamics. The authors were able to

transform a 4-D dynamical system to hyperjerk form.

Although the transformation between the 4-D

dynamic systems to the hyperjerk form was rigorous,

the outcome of the authors’ study has launched

numerous interested researchers to new ways of

characterizing hyperjerk chaotic dynamics. From the

foregoing, there is no iota of doubt that quite a

number of works exist in the literature on the

relevance of equation transformations for nonlinear

dynamics characterization. Notwithstanding, there is

still an obvious lacuna.

In the quest of the authors of the present paper to

further enrich the volume of relevant literature in the

application of numerical transformations that bothers

on nonlinear dynamics characterization, an obvious

lacuna in the relevant of this in the field of fractal

characterization was examined. The aim of this paper

is the investigation of the non-zero arbitrary positive

power transformation and fractal characterization by

optimum disk counted of some selected IFS-based

fractals (IFS-1 to IFS-10) on a complex plane.

2. METHODOLOGY

[3], [6] and [7] provided the simplified presentation

of how to utilize iterated function systems (IFS) and a

contractive transformation mapping to provide a

framework for the generation of fractals images.

According to mathematical definitions given by [8] ,

a transform :f X X on a metric space ( , )X dis called a contractive mapping if there is a constant



51

0 1s such that equation (1) holds while an

iterated function systems (IFS) consists of a finite set

of contractive mappings , 1,2,..., ,iw for i n

denoted as 1 2{ ; , ,..., }nW X w w w and captured

by equation (2) for 2R metric space. Generalization

of MÖbius transformation called analytic

transformations by [3] concentrated on the behaviour

of only quadratic transformations; however the

present study focuses both on simulation and fractal

characterization of selected fractal images by their

respective IFS-functions iterated to steady state by

what is termed in literature as ‘Chaos Game’. The

corresponding resulting fractal image is subjected to

only arbitrary positive (non zero) power

transformations including the quadratic and then

characterized assuming that the fractal image fit on a

complex plane. The zero power will not results in

interesting fractals while negative powers can

potentially leads to diverge fractals. The relevant

equations for the transformation are (3), (4) and (5)

( ( ), ( )) . ( , ) , ,d f x f y s d x y x y X (1)

x a b x ew

y c d y f

(2)

The appropriate use of prescribed finite number of

affine functions as in equation (2) produce a definite

fractal image, the coordinate points on the image are

captured by ( , )x y . The complex representation of an

arbitrary point ( , )x y is given by equation (3) with

the corresponding polar form provided by equation

(4): see [12].

Z x iy (3)

iZ re (4)

In equation (4), the polar length (r) is 2 2r x y

and the polar angle ( ) relative to x-axis is

arctan( )y

x for non zero x . The positive non-

zero n-order power transformation is given by

equation (5) applicable to all the coordinate points of

the corresponding fractal on a one to one basis. Thus

the transformation takes old coordinate to new

coordinate according to equation (6).

n n inZ r e (5)

, cos( ), sin( )n n

old newx y r n r n (6)

Similarly, equations (7) and (8) provided the

respective equivalent of equation (6) for

transformation effected by multiplication and

division by an arbitrary selected non-zero

(magnitude) complex number ( s s sZ x iy ).

However , it is to be noted that2 2

s s sr x y ,

arctan( )ss

s

y

x ,

2 2( ) ( )ss s s s sr xx yy yx xy and

arctan( )s sss

s s

yx xy

xx yy

.

, ( ) cos( ( )),( ) sin( ( ))n n

s s s sold newx y rr n rr n

(7)

2 2, ( ) cos( ),( ) sin( )n n

ss s ss ss s ssold newx y r r n r r n

(8)

2.1 IFS-Based Fractals Parameters

The details of the prescribed affine functions ( iw )

used for the generation of the ten selected fractal

images as obtained from literatures are provided in

table 1. The reference tags used are IFS-1 to IFS-10.

Table 1: IFS-Parameters of Selected fractal images culled from literature with reference tag.



52

Function ( iw ) IFS-1

A b c d e F

1 0.5000 0.0000 0.0000 0.5000 0.0000 0.0000

2 0.5000 0.0000 0.0000 0.5000 0.5000 0.0000

3 0.5000 0.0000 0.0000 0.5000 0.2500 0.5000

IFS-2

1 0.3333 0.0000 0.0000 0.3333 0.0000 0.0000

2 0.1667 -0.2887 0.2887 0.1667 0.3333 0.0000

3 0.1667 0.2887 -0.2887 0.1667 0.5000 0.2887

4 0.3333 0.0000 0.0000 0.3333 0.6667 0.0000

IFS-3

1 0.3333 0.0000 0.0000 0.3333 0.0000 0.0000

2 0.3333 0.0000 0.0000 0.3333 0.3333 0.0000

3 0.3333 0.0000 0.0000 0.3333 0.6667 0.0000

4 0.3333 0.0000 0.0000 0.3333 0.0000 0.3333

5 0.3333 0.0000 0.0000 0.3333 0.6667 0.3333

6 0.3333 0.0000 0.0000 0.3333 0.0000 0.6667

7 0.3333 0.0000 0.0000 0.3333 0.3333 0.6667

8 0.3333 0.0000 0.0000 0.3333 0.6667 0.6667

IFS-4

1 0.3333 0.0000 0.0000 0.3333 0.0000 0.0000

2 0.3333 0.0000 0.0000 0.3333 0.6667 0.0000

3 0.3333 0.0000 0.0000 0.3333 0.0000 0.6667

4 0.3333 0.0000 0.0000 0.3333 0.6667 0.6667

5 0.3333 0.0000 0.0000 0.3333 0.3333 0.3333

IFS-5

1 0.1950 -0.4880 0.3440 0.4430 0.4431 0.2452

2 0.4620 0.4140 -0.2520 0.3610 0.2511 0.5692

3 -0.0580 -0.0700 0.4530 -0.1110 0.5976 0.0969

4 -0.0350 0.0700 -0.4690 -0.0220 0.4884 0.5069

5 -0.6370 0.0000 0.0000 0.5010 0.8562 0.2513



53

Table 1 contd.: IFS-Parameters of Selected fractal images culled from literature with reference tag.

Function ( iw ) IFS-6

A b c D e F

1 0.0000 0.0000 0.0000 0.1600 0.0000 0.0000

2 0.8500 0.0400 -0.0400 0.8500 0.0000 1.6000

3 0.2000 -0.2600 -0.2300 0.2200 0.0000 1.6000

4 -0.1500 0.2800 0.2600 0.2400 0.0000 0.4400

IFS-7

1 0.5000 0.0000 0.0000 0.5000 0.0000 0.0000

2 0.5000 0.0000 0.0000 0.5000 0.5000 0.5000

3 0.2500 0.0000 0.0000 0.2500 0.0000 0.7500

4 0.2500 0.0000 0.0000 0.2500 0.2500 0.5000

5 0.2500 0.0000 0.0000 0.2500 0.5000 0.2500

IFS-8

1 0.5000 0.0000 0.0000 -0.5000 0.0000 0.5000

2 -0.5000 0.0000 0.0000 -0.5000 1.0000 1.0000

3 0.3333 0.0000 0.0000 0.3333 0.6667 0.0000

4 0.2500 0.0000 0.0000 0.2500 0.2500 0.5000

IFS-9

1 0.2500 0.0000 0.0000 0.2500 0.0000 0.7500

2 0.2500 0.0000 0.0000 0.2500 0.2500 0.5000

3 0.2500 0.0000 0.0000 0.2500 0.5000 0.7500

4 0.2500 0.0000 0.0000 0.2500 0.7500 0.5000

5 0.7500 0.0000 0.0000 0.5000 0.0000 0.0000

IFS-10

1 0.3333 -0.3333 0.3333 0.3333 0.3333 0.0000

2 0.6667 0.0000 0.0000 0.6667 0.3333 0.3333

3 0.3333 0.0000 0.0000 0.3333 0.6667 0.0000

2.2 Simulation and Fractal Characterization

Parameters

The simulation codes in FORTRAN90 was driven

with arbitrarily picked random number generating

seed value of 9876 starting from arbitrary co-

ordinate’s point (1, 0.5) and iterated continuously

over 500 and 5000 transient and steady solution

points respectively. The choices of the positive non-

zero n-order power were made from 0.5 10n at

fixed step of 0.5. However, the disk count was

performed in space of five arbitrary starts coordinates

points and over each of ten scales of observation

systematically tied to the corresponding attractor

characteristic length. The characteristic length was

the distance between farthest coordinate pair on the

attractor. Thereafter, the optimum disk counted

(minimum) determine from the range of five

observed possibilities per observation scale. The

estimated fractal disk dimension for either the

untransformed (standard) or the transformed fractal

is obtained from the slope of line of best fit of the

logarithms of observation scale and minimum disk

counted using least square regression analysis.

Relevant graphs were made using Microsoft Excel-

2003.

2.3 Special cases

Both IFS-1 and IFS-6 were selected for further

investigation within the limits of the positive non-

zero n-order power ( 0.1 20,0.1n ) with respect

to multiplication and division transformations while

keeping other simulation parameters the same. It is



54

interesting to recall that Euler’s theorem [5] gave the

connection among five of the most important

numbers of mathematics (0, 1, i, e, and π) as

1 0ie utilizing the most important

transformation operations (multiplication,

exponentiation, negation, and addition). This is the

basis that informed the quest for visual investigation

of the transformation of IFS-1 at the positive non-

zero n-order powers: the golden ratio ( 1.618 ),

exponential ( 2.718e ) and phi ( 3.142 ).

3. RESULTS AND DISCUSSIONS

Table 2: Comparison of current estimated disk dimension (EDD) with literature value

IFS-1 IFS-2 IFS-3 IFS-4 IFS-5 IFS-6 IFS-7 IFS-8 IFS-9 IFS-10

L-V 1.585 1.262 1.893 1.465 NA NA 1.585 1.528 NA NA

C-V 1.497 1.168 1.662 1.421 1.400 1.561 1.517 1.444 1.616 1.493

E-R 5.6 7.5 12.2 3.0 CD CD 4.3 5.5 CD CD

Note: L-V=literature value, C-V=current value, E-R=absolute relative percentage error, NA=not available and

CD=cannot be determined.

Table 2 refers. The reliability of the current method

of estimating the disk dimension based on fractals

with the corresponding literature value fall within the

range of 3.0 and 12.2 absolute relative percentage

error. Thus, the current method can be adjudged to be

acceptable and hence the corresponding current disk

dimension serves as reference standard for

subsequent relevant comparison.

Table 3: Variation of estimated disk dimension (EDD) of transformed fractal images with increasing transformation

power (TP= 0.5 10;0.5n )

TP IFS-1 IFS-2 IFS-3 IFS-4 IFS-5 IFS-6 IFS-7 IFS-8 IFS-9 IFS-10

0.5 1.494 1.099 1.683 1.466 1.360 1.261 1.465 1.606 1.518 1.597

1.0 1.485 1.119 1.635 1.460 1.403 1.163 1.533 1.464 1.602 1.437

1.5 1.506 1.129 1.660 1.410 1.529 1.353 1.596 1.524 1.393 1.477

2.0 1.471 1.152 1.665 1.413 1.519 1.537 1.507 1.550 1.369 1.522

2.5 1.457 1.165 1.748 1.484 1.528 1.532 1.455 1.441 1.476 1.495

3.0 1.427 1.154 1.770 1.460 1.574 1.349 1.437 1.482 1.336 1.481

3.5 1.510 1.140 1.731 1.459 1.572 1.330 1.441 1.467 1.366 1.500

4.0 1.563 1.146 1.740 1.476 1.594 1.533 1.458 1.519 1.386 1.507

4.5 1.599 1.159 1.718 1.529 1.603 1.500 1.442 1.535 1.383 1.511

5.0 1.580 1.140 1.707 1.465 1.470 1.499 1.456 1.576 1.479 1.524

5.5 1.636 1.162 1.700 1.446 1.475 1.474 1.454 1.558 1.473 1.551

6.0 1.603 1.108 1.713 1.463 1.595 1.523 1.489 1.421 1.450 1.539

6.5 1.580 1.105 1.706 1.448 1.577 1.535 1.453 1.581 1.418 1.554

7.0 1.622 1.146 1.694 1.458 1.537 1.569 1.462 1.568 1.409 1.544

7.5 1.575 1.100 1.703 1.430 1.592 1.511 1.438 1.445 1.464 1.549

8.0 1.488 1.074 1.691 1.426 1.599 1.493 1.418 1.426 1.451 1.535

8.5 1.458 1.105 1.668 1.428 1.546 1.522 1.409 1.438 1.463 1.542

9.0 1.484 1.103 1.674 1.424 1.546 1.496 1.424 1.423 1.424 1.508



55

9.5 1.535 1.101 1.632 1.401 1.424 1.541 1.411 1.417 1.448 1.530

10.0 1.490 1.061 1.519 1.448 1.546 1.503 1.445 1.425 1.463 1.526

Table 4: Absolute relative percentage Difference in the estimated disk dimension (EDD) of transformed fractal

images with increasing transformation power (TP= 0.5 10;0.5n )

TP IFS-1 IFS-2 IFS-3 IFS-4 IFS-5 IFS-6 IFS-7 IFS-8 IFS-9 IFS-10

0.5 0.2 5.9 1.3 3.2 2.8 19.2 3.4 11.2 6.1 7.0

1.0 0.8 4.2 1.6 2.8 0.2 25.5 1.1 1.4 0.9 3.7

1.5 0.6 3.3 0.1 0.8 9.2 13.3 5.2 5.6 13.8 1.0

2.0 1.7 1.3 0.2 0.6 8.5 1.5 0.6 7.3 15.3 2.0

2.5 2.7 0.3 5.2 4.4 9.1 1.8 4.1 0.2 8.7 0.2

3.0 4.7 1.2 6.5 2.8 12.5 13.6 5.2 2.7 17.4 0.8

3.5 0.9 2.4 4.2 2.7 12.3 14.8 5.0 1.6 15.5 0.5

4.0 4.4 1.8 4.7 3.9 13.8 1.8 3.9 5.2 14.2 1.0

4.5 6.8 0.8 3.4 7.6 14.6 3.9 4.9 6.3 14.4 1.2

5.0 5.5 2.4 2.8 3.1 5.1 3.9 4.0 9.1 8.5 2.1

5.5 9.3 0.5 2.3 1.8 5.4 5.5 4.1 7.9 8.9 3.9

6.0 7.1 5.1 3.1 2.9 14.0 2.4 1.8 1.6 10.3 3.1

6.5 5.5 5.4 2.7 1.9 12.7 1.7 4.2 9.5 12.3 4.1

7.0 8.4 1.8 1.9 2.6 9.8 0.5 3.6 8.6 12.8 3.5

7.5 5.2 5.8 2.5 0.7 13.8 3.2 5.2 0.1 9.4 3.8

8.0 0.6 8.1 1.8 0.4 14.2 4.3 6.5 1.3 10.2 2.9

8.5 2.6 5.4 0.4 0.5 10.4 2.5 7.1 0.4 9.5 3.3

9.0 0.9 5.6 0.7 0.2 10.5 4.2 6.1 1.5 11.9 1.1

9.5 2.6 5.8 1.8 1.4 1.7 1.2 7.0 1.8 10.4 2.5

10.0 0.4 9.1 8.6 1.9 10.4 3.7 4.7 1.3 9.5 2.2

MND 0.2 0.3 0.1 0.2 0.2 0.5 0.6 0.1 0.9 0.2

MAD 9.3 9.1 8.6 7.6 14.6 25.5 7.1 11.2 17.4 7.0

Note: MND and MAD are respectively the minimum and maximum difference expressed in absolute percentage.

Tables 3 and 4 refer. The measured absolute

percentage difference range (0.1-25.5) in the

estimated disk dimension for all fractals and different

transformations power studied is wide. This is to

suggest that there are transformations powers that can

drastically change the structural details of the IFS-

based fractals accepting that dimension as indicator

of structural details. The bolded entries in table 4

shows the dependence of transformation power that

can change the estimated dimension either minimally

or maximally on the fractals studied. The maximum

change in estimated dimension are noted for fractals

(IFS-1 to IFS-10) at respective power 5.5, 10.0, 10.0,

4.5, 4.5, 1.0, 8.5, 0.5, 3.0 and 0.5.



56

Figure 1: Variation of estimated disk dimension (EDD) of transformed fractal (IFS-1) relative to its standard

(STD) with increasing transformation power (TP= 0.1 20;0.1n )

Figure 2: Variation of estimated disk dimension (EDD) of transformed fractal (IFS-6) relative to its standard

(STD) with increasing transformation power (TP= 0.1 20;0.1n )

Figures 1 and 2 refer. The estimated disk dimension

(EDD) fluctuate erratically reference to the

corresponding standard disk dimension (STD).

However, there is observed distinction between the

fluctuation pattern for IFS-1 and that of its

counterpart IFS-6. In IFS-1, the fluctuation stay

1.00

1.10

1.20

1.30

1.40

1.50

1.60

1.70

0.0 5.0 10.0 15.0 20.0

TP

ED

D IFS-1

STD

1.00

1.10

1.20

1.30

1.40

1.50

1.60

1.70

0.0 5.0 10.0 15.0 20.0

TP

ED

D IFS-6

STD



57

averagely around the STD line whereas in the IFS-6, it remains dominantly below the STD line.

(a) IFS-1 (b) IFS-6

Figure 3: Scatter plots of the Untransformed IFS-Based fractals (a) IFS-1 and (b) IFS-6

Figure 4: Scatter plots of the Transformed fractal (IFS-1 Basis) using the golden ratio as transformation

power (i.e. 1.618n )

U nt t ransf o rmed IFS- 1

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

I FS - 6

0.0

2.0

4.0

6.0

8.0

10.0

12.0

-2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0

Transfromed IFS-1 (Golden ratio)

0.0

0.2

0.4

0.6

0.8

1.0

1.2

-0.3 -0.1 0.1 0.3 0.5 0.7 0.9



58

Figure 5: Scatter plots of the Transformed fractal (IFS-1 Basis) using the exponential value as

transformation power (i.e. 2.718n e ).

Figure 6: Scatter plots of the Transformed fractal (IFS-1 Basis) using the phi value as transformation power

(i.e. 3.142n ).

Transfromed IFS-1 (Exponential)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

-1.3 -0.8 -0.3 0.2 0.7

Transfromed IFS-1 (Phi)

-0.6

-0.4

-0.2

0.1

0.3

0.5

0.7

-1.4 -0.9 -0.4 0.2 0.7



59

Figures 3 to 6 refer. Visual evaluation of each of the

fractal for both the untransformed and the

transformed showed that they are aesthetically

appealing. The overall structural details of figures 4

to 6 showed that they originated from the

untransformed IFS-1. In addition , figures 4 to 6

shows that as non-zero positive transformation power

(TP) increases the resulting transformed fractal

displayed discernable evident of turn around. In view

of this fact , a trial and error exercise using increasing

value of TP affirmed that the transformed fractal

turned round a full circle when the TP is 5.678 (see

figure 7).

Another interesting visual result obtained was that of

the IFS-5 fractal transformed at TP=2.0 shown in

figure 8. The untransformed fractal resembled a

natural standing forest tree while its transformed

counterpart resembled a falling tree. The falling

action seems to be provoked by storm weather or

engineered by bulldozer.

Figure 7: Scatter plots of the Transformed fractal (IFS-1 Basis) using the transformation power (i.e.

5.678n TP ).

Transformed IFS-1 (TP=5.678)

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

-0.6 -0.1 0.4 0.9 1.4 1.9



60

(a) Untransformed IFS-5 (b) Transformed IFS-5

Figure 8: Scatter plots of the Untransformed and Transformed fractal (IFS-5 Basis) using the transformation

power (i.e. 2n TP ).

4. CONCLUSIONS

This study has utilized complex mathematical

operations (multiplication, exponentiation and

division) to effect the transformation of selected IFS-

based fractals over range of positive non-zero powers

including the popular special mathematical numbers:

the golden ratio ( 1.618 ),the exponent (

2.718e ) and the phi ( 3.142 ). The study

has correspondingly shown the existence of both

quantitative and aesthetically appealing qualitative

differences between the untransformed (standard) and

the transformed fractals using respectively the fractal

disk dimension and visual evaluations. The observed

differences are noted for their dependence on both

the fractal and transformation power. The results of

this study can found applications in beautification,

exhibition and in engineering such as in the robots

driven industries.

REFERENCES

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Transf o rmed IFS- 5

-0.1

0.1

0.3

0.5

0.7

0.9

1.1

1.3

-0.7 -0.5 -0.3 -0.1 0.1 0.3 0.5 0.7

Unt r a nsf or me d I FS - 5

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

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