# FRACTALS A RESEARCH by iaemedu

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FRACTALS: A RESEARCH

Dr. MAMTA RANI1, SALONI 2
1
Department of Mathematics, Statistics & Computer Science, Central University of Rajasthan, Patni
College, Kishangarh, Ajmer, Rajasthan, India
2
Department of MCA, Krishna Engg. College, 95, Loni Road Mohan Nagar, Ghaziabad, India,

ABSTRACT

Fractal research is a fairly new field of interest and has wide range of applications in science
and engineering. Inspired by the study of Mandelbrot, I have reviewed the key research work that has
undertaken in the field of Fractals. I have surveyed various aspects of fractals, its characteristics,
Fractal Geometry and how the fractal dimension is calculated. Different techniques for generating
the fractals are also explored. Fractals are self similar objects and they are classified according to
their self similarity. So I have also reviewed various types of fractals. New approaches in fractal
graphics are also studied along with its scope, importance and applications in real life.

Keywords: fractals, dimension, classification, generation, Superior Iteration Method

1. INTRODUCTION

Large number of people believes that the geometry of nature is centred on simple figures
such as a lines, circles, conic sections, polygons, sphere, and quadratic surfaces and so on. For
example, tires of the vehicle are circular, Solar system moves around the sun in elliptical orbit. Poles
are cylindrical, etc. Have we ever thought, what is the shape of a mountain? Can we describe the
structure of animals and plants? How can the networks of veins that supply blood be described by
classical geometry? Many objects in nature, which are so complicated and irregular that it is
hopeless to use classical geometry to model them. To analyze many of these questions fractals and
mathematical chaos are appropriate tools.
The mathematics behind fractals began to take shape in the 17th century when mathematician
and philosopher Leibniz considered recursive self-similarity (although he made the mistake of
thinking that only the straight line was self-similar in this sense).
In 1872 Karl Weierstrass gave an example of a function whose graph would today be
considered fractal, with property of being everywhere continuous but nowhere differentiable.

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In 1904, Helge von Koch, dissatisfied with Weierstrass's very abstract and analytic definition,
gave a more geometric definition of a similar function, which is now called the Koch curve. Waclaw
Sierpinski constructed his triangle in 1915 and one year later his carpet.
In 1918, Bertrand Russell recognized a "supreme beauty" within the emerging mathematics
of fractals. The idea of self-similar curves was taken further by Paul Pierre Levy, who, in his 1938
paper Plane or Space Curves and Surfaces Consisting of Parts Similar to the Whole described a new
fractal curve, the Levy C curve. Georg Cantor also gave examples of subsets of the real line with
unusual properties - these Cantor sets are also now recognized as fractals.
Iterated functions in the complex plane were investigated in the late 19th and early 20th
centuries by Henri Poincare, Felix Klein, Pierre Fatou and Gaston Julia. In the 1960s, Benoit
Mandelbrot started investigating self-similarity in papers such as How Long Is the Coast of Britain?
[8]. Statistical Self-Similarity and Fractional Dimension, which built on earlier work by Lewis Fry
Richardson. Finally, in 1975 Mandelbrot coined the word ‘fractal’ to denote an object whose
Hausdorff - Besicovitch dimension is greater than its topological dimension.

2. FRACTALS

Fractal is a set, which is self-similar under magnification. A fractal is "a rough or fragmented
geometric shape that can be split into parts, each of which is (at least approximately) a reduced-size
copy of the whole” [9].
The term "fractal" was coined by Beno Art Mandelbrot in 1975 and was derived from the
Latin fractus meaning "broken" or "fractured." “A mathematical fractal is based on an equation that
undergoes iteration, a form of feedback based on recursion”[3]. Because fractals appear similar at all
levels of magnification, fractals are often considered to be infinitely complex (in informal terms).
Natural objects that approximate fractals to a degree include clouds, mountain ranges, lightning
bolts, coastlines, snowflakes, various vegetables (cauliflower and broccoli), and animal coloration
patterns. However, not all self-similar objects are fractals for example, the real line (a straight line) is
formally self-similar but fails to have other fractal characteristics; for instance, it is regular enough to
be described in Euclidean terms[1].
Classical Euclidean geometry works with objects which exist in integer dimensions: zero-
dimensional points, one-dimensional lines and curves, two-dimensional surfaces like planes, and
three-dimensional solid objects such as balls and blocks (think spheres and cubes). However, many
things in nature are better described as having a dimension which is not a whole number, because of
a property called self-similarity: if you magnify some part of the object, you will find that part is
identical to the whole object, on a smaller scale. A common example is a fern branch, where each
leaf resembles the entire branch in miniature (Fig.1).

Fig.1: A fractal fern leaf

The fern leaf, when pressed flat, is part of a two-dimensional plane, and appears to take up
two-dimensional space, even though any individual point on the fern appears to be simply part of a
one-dimensional curve that helps make up the leaf. One may argue that the fern's dimension is then
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somewhere between one and two: it doesn't really take up two-dimensional space in the same way
the interior of a square does, but it does take up more than just the one dimension of a simple curve.
Such an object is said to have fractal dimension, by virtue of its self-similarity, and such objects are
said to be fractal. While a straight line has a dimension of exactly one, a fractal curve will have a
dimension between one and two, depending on how much space it takes up as it curves and twists.
The more a fractal curve fills up a plane, the closer it approaches two dimensions. In the same
manner of thinking, a fractal surface (like, say, the surface of a cloud) will cover a dimension
somewhere between two and three. Hence, a fractal landscape which consists of a hill covered with
tiny bumps would be closer to two dimensions, while a landscape composed of a rough surface with
many average sized hills would be much closer to the third dimension. Self-similarity, in its strictest
interpretation (also called scale invariance), implies that a fractal object has an infinite level of
detail, as no matter how much you magnify the fractal, you will see the same amount of detail[4].

2.1 The Difference between Non-Fractal and Fractal Objects

Non-Fractal: As a non-fractal object is magnified, no new features are revealed (Fig.2).

Fig. 2: Non-Fractal Object

Fractal: As a fractal object is magnified, even finer new features are revealed. The shapes of the
smaller features are kind-of-like the shapes of the larger features(Fig.3).

Fig. 3: Fractal Object

2.2 Characteristics of a Fractal

i. The construction of a Fractal is based on an iterative (or recursive) process(Fig.4).

Fig. 4: Construction of Sierpinski’s triangle

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ii. A Fractal is infinitely complex. It can be magnified infinitely (Fig.5).

Fig. 5: Magnification of Mendelbrot Set

iii. Upon magnification of a Fractal, we can find subsets of it that look like the whole figure.
This feature is called self similarity. An object is self-similar only if you can break the object
down into an arbitrary number of small pieces, and each of those pieces is a replica of the entire
structure. Some examples of self-similarity follow. The red outlining indicates a few of the self-
similarities of the object. (Fig.6).

Fig. 6. Self-similarities of the object

iv. The dimension of a Fractal is typically a non-integer and hence has non-integer complexity.

3. FRACTAL GEOMETRY

While the classical Euclidean geometry works with objects which exist in integer dimensions,
fractal geometry deals with objects in non-integer dimensions. Euclidean geometry is a description
lines, ellipses, circles, etc. Fractal geometry, however, is described in algorithms- a set of instructions
on how to create a fractal.
The world as we know it is made up of objects which exist in integer dimensions, single
dimensional points, one dimensional lines and curves, two dimension plane figures like circles and
squares and three dimensional solid objects such as spheres and cubes. However, many things in
nature are described better with dimension being part of the way between two whole numbers. While
a straight line has a dimension of exactly one, a fractal curve will have a dimension between one and
two, depending on how much space it takes up as it curves and twists. The more a fractal fills up a
plane, the closer it approaches two dimensions. In the same manner of thinking, a wavy fractal scene
will cover a dimension somewhere between two and three. Hence, a fractal landscape which consists
of a hill covered with tiny bumps would be closer to two dimensions, while a landscape composed of
a rough surface with many average sized hills would be much closer to the third dimension [6].

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3.1 Fractal Dimension
There are many definitions of dimension which give a non-integer or fractal dimension.
These dimensions are particularly useful in characterizing fractal objects. Here we will concentrate
on the similarity dimension, denoted by Ds, to characterize the construction of regular fractal objects.
The concept of dimension [16] is closely associated with that of scaling. Consider the line,
surface and solid divided up respectively by self-similar sub-lengths, sub-areas and sub-volumes of
side length ε. In the following derivation assume that the length L, area A, and volume V, are all
equal to unity.
Consider first the line. If the line is divided into N smaller self-similar segments, each ε in
length, then ε is in fact the scaling ratio, i.e. ε /L= ε since L=1.
Thus

L=Nε =1

i.e. the unit line is composed of N self-similar parts scaled by ε =1/N.

Now consider the unit area. If we divide the area again into N segments each ε2 in area, then

A=N ε2 =1

i.e. the unit surface is composed of N self-similar parts scaled by ε= 1/N1/2 .

Applying similar logic, we obtaining for a unit volume

V=N ε3 =1

i.e. the unit solid is N self-similar parts scaled by ε=1/N1/3

Examining expressions we see that the exponent of ε in each case is a measure of the (similarity)
dimension of the object, and we have in general

N ε Ds =1
Using logarithms leads to the expression,

log(N ε Ds ) =log 1=0

log N+ Ds log ε =0

log N = - Ds log

or log N= Ds log(1/ ε)

Ds =       ε
(1)

Here the subscript‘s’ denotes the similarity dimension.

The above equation may also be used to produce dimension estimates of fractal objects where Ds is
non-integer.

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For Example:

i. Cantor Set:

Fig.7: Construction of the Cantor Set

From Fig.7 we see that the left-hand third of the set contains an identical copy of the set.
There are two such identical copies of the set contained within the set, thus N=2 and ε =        .
According to equation 1 the similarity dimension is then

Ds =                 =       =0.6309......

Thus, for the cantor set, Ds is less than one greater than zero. In fact it has a non-integer
similarity dimension of 0.6309.... due to the fractal structure of the object.
Instead of considering each sub-interval of the cantor set scaled down by one-third we could
have looked at each subinterval scaled down by one-ninth. As in the Fig.7 there are four such
segments, each an identical copy of the set. In this case N=4 and ε =1/9 and again this leads to a
similarity dimension of

Ds =                     =    =          = 0.6309....

Ds =                 =           =           = 0.6309....

Ds =             =           =         = 0.6309....

Where the scaling constant, c depends on the scale used to identify the self-similarity of the
object.

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ii. The Koch curve
The Koch curve is simply constructed using an iterative procedure beginning with the
initiator of the set as the unit line segment (step k=0 in the Fig. 8). The unit line segment is divided
into thirds and the middle third removed. The middle third is then replaced with two equal segments,
both one-third in length, which form an equilateral triangle (step k=1); this step is the generator of
the curve. At the next step (k=2), the middle third is removed from each of the four segments and
each is replaced with two equal segments as before. This process is repeated an infinite number of
times to produce the Koch curve.
The Koch curve is a fractal object possessing a fractal dimension. Each smaller segment of
the Koch curve is an exact replica of the whole curve. At each scale there are four sub-segments
making up the curve, each one a one third reduction of the original curve. Thus, N=4, ε= and the
similarity dimension based on expression

Ds =      ε
= 1.2618…..

That is, the Koch curve has a dimension greater than that of the unit line and less than that of the unit
area.

Fig. 8: Construction of the Koch Curve

iii. The Menger sponge
So far we have looked at construction on the line (cantor set) and in the plane (Koch curve).
The Menger sponge is constructed in 3D space. The initiator in the construction is a cube.
The first iteration towards the final fractal object, the generator, is formed by ‘drilling through’ the
middle segment of each face. This leaves a pre fractal composed of twenty smaller cubes each scaled
down by one-third. These cubes are then drilled out leaving 400 cubes scaled down by one-ninth
from the original cube. Repeated iteration of this construction process leads to the Menger sponge.
The similarity dimension of the Menger sponge is
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Ds =       ε
=           = 2.7268

Fig.9: Construction of Menger Sponge

4. GENERATION OF FRACTALS

Common techniques for generating fractals are:
i. Escape-time fractals
ii. Iterated function systems
iii. Random fractals

i. Escape Time fractals

Escape-time fractals are defined by a recurrence relation at each point in a space. It use a formula or
recurrence relation at each point in a space (such as the complex plane); usually quasi-self-similar;
also known as "orbit" fractals.

The Mandelbrot set M is defined by a family of complex quadratic polynomials

fc : C     C                        (1)

given by
fc = z2 + c                         (2)

where c is a complex parameter. For each c, one considers the behaviour of the sequence

(0, fc(0), fc (fc (0)), fc (fc (fc (0))), . . .)   (3)

Thus, we map all the points in which the sequence

(0, c, c2 + c, (c2 + c)2 + c, . . .), c      C      (4)

does not escape to infinity.

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Examples of escape time fractals

,      fractal                                        ).
Burning Ship fractal(fig. 10a), Nova fractal(fig. 10b) and Lyapunov fractal(fig. 10c)

(a)                   (b)             (c)

Fig. 10: Examples of escape time fractals

ii. Iterated Function System
replacement rule. One of them is the Menger sponge
These fractals have a fixed geometric repl                                        sponge.

Examples of Iterated function System

carpet        ,
Koch snowflake, Cantor set, Haferman carpet, Sierpinski carpet(Fig. 11), Sierpinski gasket, Peano
Heighway               T-Square, Menger sponge
curve(Fig. 11), Harter-Heighway dragon curve, T

Sierpinski carpet                             Peano curve construction

Fig. 11: Examples of Iterated Function System

iii. Random Fractals
rando         ike
These kinds of fractals are generated in a more random way, like the trajectories of Brownian
motion. These are generated by stochastic rather than deterministic processes.
Examples of these types are trajectories of the Brownian motion, Lacvy flight, fractal
landscapes and the Brownian tree. Random fractal is shown by the following figure:

Fig. 12: Fractal Terrain

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4. CLASSIFICATION OF FRACTALS

Fractals can be classified according to their self-similarity. There are three types of self-
similarity found in fractals:
i. Exact self-similarity
ii. Quasi-self-similarity
iii. Statistical self-similarity

i. Exact self-similarity
This is the strongest type of self-similarity where fractal appears identical at different scales.
Fractals defined by iterated function systems often display exact self similarity. E.g. Koch Snowflake
which is a fractal that begins with an equilateral triangle and then replaces the middle third of every
line segment with a pair of line segments that form an equilateral(Fig. 12).

Fig. 12: Koch Snowflake

ii. Quasi self-similarity
Fractal appears approximately (but not exactly) identical at different scales. This is a loose
form of self-similarity. Quasi-self-similar fractals contain small copies of the entire fractal in
distorted and degenerate forms. Fractals defined by recurrence relations are usually quasi-self similar
but not exactly self-similar. e.g., the Mandelbrot set's satellites are approximations of the entire set,
but not exact copies, as shown in Fig.13.

Fig.13: Basic Mandelbrot set, Six times magnification of the Mandelbrot set, Hundred times
magnification, Two Thousand times magnification

iii. Statistical self-similarity
These type of fractals repeat a pattern stochastically so numerical or statistical measures are
preserved across scales. This is the weakest type of self-similarity. Random fractals are examples of
fractals which are statistically self-similar, but neither exactly nor quasi-self similar. e.g., the well-
known example of the coastline of Britain.(Fig.14)

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Fig. 14: coastline of Britain

5. NEW APPROACH IN FRACTAL GRAPHICS

In 2002, Rani introduced superior iterations in the study of fractals [14]. This was a new
iterative approach to study fractal models, which was found superior to the conventional Peano-
Picard iterative approach. Rani called the new gallery of fractals superior fractals. Subsequently, In
2002 Rani and other workers used the superior approach and generated escape-time fractals [14,15]
fractals via iterated function systems[4] and strange attractors[15].

SUPERIOR ITERATION METHOD
Iteration methods are one way to achieve the self-similarity exhibited by fractals. Basically,
this is done by using two types of feedback machines, one-step machines and two-step machines
[12]. Both types of machines can be characterized by iterative procedures. One-step feedback
machines are characterized by Peano-Picard iterations (generally called Picard or function iterations)
represented by the formula +1 = f( ), where f can be any function of x.

Definition (Picard Orbit)

Let X be a non-empty set of numbers and f: X X. For a point           in X, the Picard orbit (generally
called orbit) of f is the set of all iterates of the point , that is:
O(f, ) = { :             = f(      ), n = 1, 2, . . .}.
The orbit of f at the initial point , O(f, ), is the sequence { ( )}.

For a long time, fractal theory was based on one-step feedback machines; all the fractal
models were studied in the Picard orbit until the two-step feedback machine was introduced by Rani
in 2002. In two-step feedback machines, the output is computed by the formula         =g( ,          ),
which requires two numbers as input and returns a new number. For example, the Fibonacci numbers
are generated using g ( ,        ) =     +       . (Rani, 2002) characterized the two-step feedback
machine by superior iterates in fractal graphics and generated superior fractals. It was a new
approach in computation, visualization and analysis of fractal models. The following is the definition
of superior iterates.

Definition (Superior iterates)
Let X be a non-empty set of real numbers and f : X     X. For an         X, construct a sequence
{ } in the following manner:

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f    + 1−

f    + 1−

f        + 1−

Where 0 <        1 and {    } is convergent away from 0.

The sequence {     } constructed this way is called a superior sequence of iterates, denoted by
SO f,   ,    . At     1, SO f, ,     reduces to O f,    .

This procedure was essentially given by Mann [10] and Krasnosel’skii was the first to study it for
1
in 1955. Since the results obtained in fractal modelling via Mann iterates are the super set of
2
their corresponding fractal models in the Picard orbit, Rani called the same superior iterates[14].
Researchers have since developed superior fractal models for           , n = 1, 2. . . for various values
of .

6. REAL-LIFE RELEVANCE, IMPORTANCE & APPLICATIONS OF FRACTALS

Fractals have and are being used in many different ways. Both artist and scientist are
intrigued by the many values of fractals. Fractals are being used in applications ranging from image
compression to finance. We are still only beginning to realize the full importance and usefulness of
fractal geometry.
One of the largest relationships with real-life is the similarity between fractals and objects in
nature. The resemblance many fractals and their natural counter-parts is so large that it cannot be
overlooked. Mathematical formulas are used to model self-similiar natural forms. The pattern is
repeated at a large scale and patterns evolve to mimic large scale real world objects.
One of the most useful applications of fractals and fractal geometry is in image compression.
It is also one of the more controversial ideas. The basic concept behind fractal image compression is
to take an image and express it as an iterated system of functions. The image can be quickly
displayed, and at any magnification with infinite levels of fractal detail. The largest problem behind
this idea is deriving the system of functions which describe an image.
One of the more trivial applications of fractals is their visual effect. Fractals have been used
commercially in the film industry, in films such as Star Wars and Star Trek. Fractal images are used
as an alternative to costly elaborate sets to produce fantasy landscapes.
Another seemingly unrelated application of fractals and chaos is in music. Some music,
including that of Back and Mozart, can be stripped down so that is contains as little as 1/64th of its
notes and still retain the essence of the composer. Many new software applications are and have been
developed which contain chaotic filters, similar to those which change the speed, or the pitch of
music.
Fractal geometry also has an application to biological analysis. Fractal and chaos phenomena
specific to non-linear systems are widely observed in biological systems. A study has established an
analytical method based on fractals and chaos theory for two patterns: the dendrite pattern of cells

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during development in the cerebellum and the firing pattern of intercellular potential. Variation in the
development of the dendrite stage was evaluated with a fractal dimension [7].

6.1 Applications in various fields

DATA COMPRESSION
In December 1992, Microsoft released a compact disk entitled the Encarta Encyclopaedia. It
contains thousands of articles, 7000 photographs, 100 animations, and 800 colour maps. All of this is
in less than 600 megabytes of data. It was possible only by the fractals and the answer lies in the
mathematics of fractal data compression.

DIFFUSION
Fractals can be used to describe the spreading of substances, such as gases diffusing or oil
spilling in water. In order to make a diffusion fractal, we need to start from the centre and spread the
points outwards. The way this is done is by randomly moving points around the screen, similarly to
real-life molecules “Brownian motion”. When a point hits the centre point, we make it stay there
permanently. When some other point hits either the centre point, or the new point, we make it stay as
well. Similarly, this can be used to model not only diffusion, but the spilling of oil in water as well.

ECONOMY
In economy, perhaps the most important thing is to be able to predict more or less accurately
what will happen to the market after some time. Until very recently, the dominant theory that was
used for this was the so called Portfolio Theory. According to it, the probability of various changes
of the market can be shown using the standard Bell curve.

Here we can show the relation between change in price and the probability of the changes.
Now with the concept of this curve we can predict the forthcoming fluctuations in the market. The
standard Bell curve is shown by the following Fig.15.

Fig.15: Standard Bell curve

Assuming this theory is accurate; we can conclude that very small changes happen most
often, while very big changes happen extremely rarely. However, this is not true in practice. While
on the bell curve, one can observe the probability of rapid changes to approach zero, they can, be
seen almost every month at the real market. Recently, in about 20 years after discovering fractals,
Benoit Mandelbrot introduced a new fractal theory that can be used much more efficiently than the
Portfolio Theory to analyze the market. Consider taking a year of market activity and graphing the
price for every month. We will get a broken line with some rises and falls. Now, if we take one of the
month and graph in a more detailed way with every week shown, we will get a very similar line with
some rises and falls. If we make it more and more detail by showing every day, every hour and even
every minute or second, We will still get the same, only smaller, rises and falls. There is the
Brownian self-similarity, Mandelbrot came up with a method of creating fractals that fit the above
description.

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FRACTAL ANTENNA
A fractal antenna (Fig. 16) is an antenna that uses a fractal, self-similar design maximize the
length, or increase the perimeter (on inside sections or the outer structure), of material that can
receive or transmit electromagnetic radiation within a given total surface area or volume. Such
fractal antennas are also referred to as multilevel and space filling curves, but the key aspect lies in
their repetition of a motif over two or more scale sizes, or 'iterations'. For this reason, fractal
antennas are very compact, are multiband or wideband, and have useful applications in cellular
telephone and microwave communications.

Fig. 16: An example of a fractal antenna: a space-filling curve called a Minkowski Island

NEWTON’S METHOD
Besides exotic applications in nature, fractals have important mathematical applications as
well. One of them is in the analysis of the Newton’s method, which is used for approximating roots
of an equation. For example, it can be used to solve an equation xn = 1, where n is some positive
integer. One of the roots of this equation is always 1, and if n is even -1 is also one of the roots. The
rest of the roots are complex numbers. If we graph these numbers on a complex plane, we will see
that they are equally spaced on the unit circle. The Newton’s method is based on making a guess of
the root and then iterating a certain formula to change that number (it is the method used by
computers). The number we will end up with is the root that is the closest to the guess, unless we are
unlucky enough to pick a number that is equidistant from two different roots. In such case the
number will jump around chaotically and create fractal shapes if we graph it. The fractal will have a
different number of "chaos lines" depending on the number of roots (the value of n determines the
number of roots).

CHEMICAL REACTIONS:
We are probably familiar with the concept of forward and backward reactions. Most reactions
are accompanied by a backward reaction, in which the products turn back into the reactants. At
equilibrium, the rates of these reactions become equal and the overall composition of the system does
not change. However, the fact that is usually missed here is that talking about the rates of reactions
we are talking about average rates, since the rates depend on the movement of particles, which
involves a lot of chance. Sometimes, however, the rates become different for a short interval of time
and the composition of the system changes. We might guess, these changes would be very chaotic.
Maybe if we view every three consecutive concentrations of a substance as coordinates of a point in
space. We can get something that is fractal in shape. Such fractal would be a strange attractor

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because we know that this is the type of fractals based on changing numbers. Indeed, fractal shapes
were found after graphing many different systems, even such common ones as hydrogen and oxygen
reacting to make water. One of the scientists who tried to study this mathematically was Otto
Rossler. He came up with three formulas that could model chemical reactions. When these three
formulas are used to create a strange attractor, they create the famous 3-dimensional Rossler
Attractor.

Fig. 17: Rossler Attractor

HUMAN BODY
Fractals, being a math topic, are very important in real life also. We will find out that our
organs are made of fractals too.

Lungs
The first place where this is found is in the pulmonary system, which we use to breathe. The
pulmonary system is composed of tubes, through which the air passes into microscopic sacks called
alveoli. The main tube of the system is trachea, which splits into two smaller tubes that lead to
different lungs, called the bronchi. The bronchi are in turn split into smaller tubes, which are even
further split. This splitting continues further and further until the smallest tubes, called the
bronchioles which lead into the alveoli. This description is similar to that of a typical fractal,
especially a fractal canopy, which is formed by splitting lines. The endpoints of the pulmonary tubes,
the alveoli, are extremely close to each other. The property of endpoints being interconnected is
another property of fractal canopies.

Blood Vessels
Similarly to bronchial tubes, splitting can also be found in blood vessels. Arteries, for
example start with the aorta, which splits into smaller blood vessels. The smaller ones split as well,
and the splitting continues until the capillaries, which, just like alveoli, are extremely close to each
other. Because of this, blood vessels can also be described by fractal canopies.

Brain
The surface of the brain, where the highest level of thinking takes place contains a large
number of folds. Because of this, a human, who is the most intellectually advanced animal, has the
most folded surface of the brain as well. Instead of 2, which is the dimension of a smooth surface, the
surface of a brain has a dimension greater than 2. In humans, it is obviously the highest, being as
large as between 2.73 - 2.79. Here’s another topic for science fiction: super-intelligent beings with a
fractal brain of dimension up to 3.

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Membranes
The surface folding similar to that of a brain was found in many other surfaces, such as the
ones inside the cell on mitochondria, which is used for obtaining energy and the endoplasmic
reticulum, which is used for transporting materials. The same kind of folding was found in the nasal
membrane, which allows sensing smells better by increasing the sensing surface. However, in
humans this membrane is less fractal than in other animals, which makes them less sensitive to
smells.

POPULATION GROWTH
We hear about the rapid growth of population in developing countries the all the time. With
the problems that it is constantly causing, it is rather obvious how important it is to analyze the
population growth. Last century, Thomas Malthus came with a theory in which he said that with
every generation, the population increases a certain amount of times depending on the growth rate.
Mathematically, if we make r the percent growth rate, and P be the old population, then our formula
will become

New (P) = (1+ r) · P                       (1)

For example, if r = 1/2 the population will increase 50%, or become 1.5 times larger. However,
something about this theory seems not right.

According to this theory, the population will increase infinitely. However, the population is
really limited by natural resources, such as space and food. Let’s pretend the maximum possible
population the environment can hold is 1, so P is a number from 0 to 1. As the population gets closer
to 1, the growth rate is going to decrease and get close to 0. We can achieve this by multiplying the
growth rate by (1– P). This way, as P is getting closer to 1, the growth rate will be multiplied by a
number that is getting close to 0. We now determined that the growth rate should really be r (1 – P).
If we use it in the above formula, we get

New (P) = [1 + r (1 – P)] · P        (2)

If we now do some algebra

New (P) = [1 + r – rP] · P
or
New (P) = P + rP – rP2
or
New (P) = (1 + r). P – rP2

We will now use this formula. Knowing this formula, it is easy to determine what the
population becomes after a long period of time. For example, when r is between 0 and 2, the
population becomes 1 and stays there, no matter what it was at the beginning. When it is 2.25, it will
always end up jumping between 1.17 and 0.72. When r is 2.5, it ends up jumping between 1.22, 0.54,
1.16 and 0.70. When it is 2.5, it ends up jumping between 8 values, and when r gets higher, it jumps
between 16 values. As we increase r, the number of these values doubles. We call this bifurcation.
So this will give us a fractal pattern as well.

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Weather
Weather behaves very unpredictably. Sometimes, it changes very smoothly from day to day.
Other times, however, it changes very rapidly. Although weather forecasts are often accurate, there is
never an absolute chance of them being right. Indeed, weather can create fractal patterns. This was
discovered by Edward Lorenz, who was mathematically studying the weather patterns. Lorenz came
up with three formulas that could model the changes of the weather. When these formulas are used to
create a 3D strange attractor, they form the famous Lorenz Attractor:
Landscapes
Fractal landscapes are a very classic application of fractals. If we look at a mountain, we will
not find its shape being a cone, but instead we will find a more complicated shape with some smaller
hills and valleys. Looking at every hill, we will find it to be composed of even smaller hills and
valleys. Even if we pick up a small rock from the mountain, we will find it to be similar to the entire
mountain — a property of fractals which we defined as self-similarity. To form a 3D landscape out
of a fractal, all we have to do is assign a height to every point depending on the color and draw a
picture with those heights. Usually plasma fractals are used for the landscapes because they give the
most realistic pictures.
Fractal Art
Pictures like this are one of the applications of fractals. It is an example of abstract art, which
can be used for various decorative purposes when the colors are chosen appropriately. The natural
intricate designs of fractals allow the artist not to worry about shapes, but only take care of other
things such as colors, shading and 3D effects. A simple 2-dimensional strange attractor, for
example, can be turned into something very different like this:
Plants: Most plants show some form of branching. This happens when the main stem (of trunk) splits
into a number of branches. Each of those branches splits into smaller branches, and this kind of
splitting continues until the smallest branches. A tree branch looks similar to the entire tree and a
fern leaf looks almost identical to the entire fern. This property, called self-similarity is one of the
most important properties of fractals. Because of numerous ways branching can be achieved
geometrically, there are several ways of creating models of plants as well.
One classic way of creating fractal plants is by means of l-systems. Lindenmayer, who is the
founder of l-systems, introduced them in a book called The Algorithmic Beauty of Plants, where he
first used them to create models of plants. Another way of creating fractal plants is using
fractal canopies or Pythagoras trees.
Benoit Mandelbrot, the founder of fractals has first noticed the properties of fractals on the
coast of Britain. He realized that no matter how small a piece of the coast is, it will still have its own
bays, harbors, and capes. Basing himself on Richardson’s data, he was able to prove that many coasts
as well as borderlines are fractal.
Richardson searched many encyclopedias to find data about the lengths of certain
borderlines. He found enormous differences in data from different countries. For example, Portugal
claimed its border with Spain to be 1214 km, while Spain claimed it to be 987 km. Portugal, as a
smaller country would definitely measure its border more accurately. Thus, we know that the
increase of accuracy increased the measurement... which is one of the properties of fractals! This
happens because fractals are figures with an infinite amount of detail, and measuring more accurately
adds more of these details, which adds to the overall size. Mandelbrot claimed that the difference in
the two measurements were due to the fact that Spain used a "yardstick" that was bigger than
Portugal’s. If, for example, Spain measured the border with a 2 kilometer yardstick, its measurement
would be less exact than Portugal, which used a 1 kilometer yardstick. If we graph log(total length)
against log(length of yardstick), we get lines with negative slopes since the total length decreases
with the increase of the size of the yardstick:
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Fig.18: Graph to find fractal dimension of the coasts and borders

Using this graph, we can find fractal dimensions of the coasts and borders. Simple models
of coasts can be made with base-motif fractals that use polygons for the bases. Such fractals are also
called Koch Islands.

Special Effects
Computer graphics has been one of the earliest applications of fractals. Indeed, fractals can
achieve realism, beauty and require very small storage space because of easy compression. Very
beautiful fractal landscapes were published as far back as in Mandelbrot’s Fractal Geometry of
Nature.

7. CONCLUSION

In this survey paper we have attempted to review the concept of fractals. Fractals are irregular
geometric objects made of parts that are in some way similar to the whole. These figures and the
study of them, Fractal geometry, allow the connection between math and nature. In this paper fractal
Geometry is explained by calculating fractal dimension of various fractal objects. Various fractal
generation techniques are also discussed along with its classification which depends upon self-
similarity. New approaches in fractal graphics are also studied giving the brief review on superior
fractals which can be used to analyze and generate new fractals. There are hundreds of applications
of fractals from different aspects. In this paper we have also described some of the practical
applications of fractals.

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