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CHAPTER-2

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CHAPTER-2

INTRODUCTION

Fractal is a new branch of Mathematics and Art. Perhaps this is the reason why most

people recognize fractals only as pretty pictures useful as backgrounds on the computer

screen or original postcard patterns. But what are they really? Most physical systems of

nature and many human artifacts are not regular geometric shapes of the standard

geometry derived from Euclid. Fractal geometry offers almost unlimited ways of

describing, measuring and predicting these natural phenomena. But is it possible to

define the whole world using mathematical equations. This article describes how the

four most famous fractals were created and explains the most important fractal

properties, which make fractals useful for different domain of science.

From the beginning of our education, formal and informal, we have been given

simplified categories for organizing the world. The world is a sphere. The mountain is

triangular. Throw a baseball in the air, and its trajectory is a parabola. All of these

statements have a strong element of truth, but none of them turns out to be accurate

when we look closely. We have known since the Apollo days that the earth is really

pear-shaped. After allowing for air resistance, the pear-shape of the earth, and even the

gravitational field of the moon, the path of a baseball is not exactly a parabola. Then the

question arises, what should be the path of a baseball, or what exactly is the earth‘s

shape. Similarly, how can we model the shape of an animal or the shape of a tree? The

solutions of above questions are in the form of Fractal Geometry, a relatively new

branch of Mathematics.

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Many people are fascinated by the beautiful images termed fractals. Extending beyond

the typical perception of Mathematics as a body of complicated, boring formulas,

fractal geometry mixes art with Mathematics to demonstrate that equations are more

than just a collection of numbers. What makes fractals even more interesting is that

they are the best existing mathematical descriptions of many natural forms, such as

coastlines, mountains or parts of living organisms.

Although fractal geometry is closely connected with computer techniques, some people

had worked on fractals long before the invention of computers. Those people were

British cartographers, who encountered the problem in measuring the length of Britain

coast. The coastline measured on a large scale map was approximately half the length

of coastline measured on a detailed map. The closer they looked, the more detailed and

longer the coastline became. They did not realize that they had discovered one of the

main properties of fractals.

2.1 Fractal Geometry

The name ‗fractal came from a Latin word ‗Fractus‘, which means ‗to break‘ .Indeed

the word fractal given by B.B.Mandelbrot in 1975, which is also called the father of

Fractal Geometry. (For details [3], [14], [15] [27], [43], [52], [56], [60], [83] [84], [90],

and Web [125], Web [126]).The basic concept of fractal is that they contain a large

degree of self similarity. This means they usually contain little copies of themselves

buried deep within the original and also have infinite detail. The repetition of form over

a variety of scales is called self-similarity. A fractal looks similar to itself on a variety

of scales. A little piece of mountain looks a lot like a bigger piece of a mountain, and

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vice versa. The bigger eddies in turbulent flow look much the same as the smaller ones,

and vice versa.

In other words of Mandelbrot [16]:

Fractal is a rough fragmented geometric shape that can be subdivided in parts each of

which is (at least approximately) a reduced copy of the whole.

In words of F. Kenton:

Fractal is a complex object, the complexity of which arises from the repetition of a

given shape at a variety of scales.

Almost all geometric forms used for building man made objects belong to Euclidean

geometry; they are comprised of lines, planes, rectangular volumes, arcs, cylinders,

spheres, etc. These elements can be classified as belonging to an integer dimension, 1,

2, or 3. This concept of dimension can be described both intuitively and

mathematically.

Intuitively we say that a line is one dimensional because it only takes 1 number to

uniquely define any point on it. That one number could be the distance from the start of

the line. This applies equally well to the circumference of a circle, a curve, or the

boundary of any object.

Figure: 2.1.1 one dimensional Curve

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A plane is two dimensional since in order to uniquely define any point on its surface we

require two numbers. There are many ways to arrange the definition of these two

numbers but we normally create an orthogonal coordinate system. Other examples of

two dimensional objects are the surface of a sphere or an arbitrary twisted plane.

Figure: 2.1.2 Two Dimensional Surfaces

The volume of some solid object is 3 dimensional on the same basis as above; it takes

three numbers to uniquely define any point within the object.

Figure: 2.1.3 Three Dimensional Surfaces

A more mathematical description of dimension is based on how the "size" of an object

behaves as the linear dimension increases. In one dimension consider a line segment. If

the linear dimension of the line segment is doubled then obviously the length

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(characteristic size) of the line has doubled. In two dimensions, if the linear dimensions

of a rectangle for example is doubled then the characteristic size, the area, increases by

a factor of 4. In three dimensions if the linear dimension of a box are doubled then the

volume increases by a factor of 8. This relationship between dimension D, linear

scaling L and the resulting increase in size S can be generalized and written as

S= DL

If we scale a two dimensional object for example then the area increases by the square

of the scaling. If we scale a three dimensional object the volume increases by the cube

of the scale factor. Rearranging the above gives an expression for dimension depending

on how the size changes as a function of linear scaling, namely

D=)log(

)log(

L

S

In the examples above the value of D is an integer, either1, 2, or 3, depending on the

dimension of the geometry. This relationship holds for all Euclidean shapes. There are

however many shapes which do not conform to the integer based idea of dimension

given above in both the intuitive and mathematical descriptions. That is, there are

objects which appear to be curves for example but which a point on the curve cannot be

uniquely described with just one number. If the earlier scaling formulation for

dimension is applied the formula does not yield an integer. There are shapes that lie in a

plane but if they are linearly scaled by a factor L, the area does not increase by L

squared but by some non integer amount. These geometries are called fractals. One of

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the simpler fractal shapes is the von Koch snowflake [91]. The method of creating this

shape is to repeatedly replace each line segment with the following 4 line segments.

Figure: 2.1.4 Single line segment for Koch Snowflake

The process starts with a single line segment and continues for ever. The first few

iterations of this procedure are shown below.

Figure: 2.1.5 Koch Snowflake

This demonstrates how a very simple generation rule for this shape can generate some

unusual (fractal) properties. Unlike Euclidean shapes this object has detail at all levels.

If one magnifies a Euclidean shape such as the circumference of a circle it becomes a

different shape, namely a straight line. If we magnify this fractal more and more detail

is uncovered, the detail is self similar or rather it is exactly self similar. Put another

way, any magnified portion is identical to any other magnified portion.

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2.2 Classification of Fractals

From the preceding discussion, several properties of fractals provide the basis for a

classification system, including dimensionality, self-similarity, and generation

technique. One property which cleanly distinguishes two distinct classes of fractals is

the role of chance in their generation and resulting structure. Following figure

summarizes this classification scheme. Classification scheme for fractals based on the

role of chance.

Figure: 2.2.1 Block Diagram of Fractal Classification

2.2.1 Deterministic Fractals

Deterministic fractals have structures which are fixed uniquely by the algorithm

employed in their creation. That is, for a given set of parameters, a deterministic fractal

generator will produce identical structures each time it is run. Chance plays no role in

the final structure of the deterministic fractal. Interestingly, identical fractals may be

produced by distinct algorithms. The Sierpinski gasket, for instance, can be generated

by the "Chaos Game" algorithm or by an iterated function system algorithm. Some of

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these algorithms do employ random number generators within the deterministic

program. The final structure of the fractal, however, shows no indication of random

processes and is a function of the control parameters only.

2.2.2 Stochastic Fractals

The distinctive characteristic of stochastic fractals is that random processes play a

central role in determining the structure of the fractal object. Phenomena such as

turbulence, seacoast, and mountain formation are deterministic at a physical level but

are extremely sensitive to the initial conditions, a property common to all chaotic

systems. The complex interaction of the system with the initial conditions and

subsequent environment results in apparently random turbulent behavior and fractal

mountain landscapes. It is virtually impossible to accurately represent the initial

condition and successfully simulate the subsequent system development of such natural

systems. A far more tractable approach has been to simulate the fractal geometry of

such objects by introducing random processes in creating the scenes.

The approach has proven successful in simulating Brownian motion, percolation

behavior, and mountain geography.

2.3 Fractals’ Properties

Two of the most important properties of fractals are self-similarity and non-integer

dimension.

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2.3.1 Self-Similarity

If you look carefully at a fern leaf, you will notice that every little leaf - part of the

bigger one - has the same shape as the whole fern leaf. You can say that the fern leaf is

self-similar. The same is with fractals: you can magnify them many times and after

every step you will see the same shape, which is characteristic of that particular fractal.

The non-integer dimension is more difficult to explain. Classical geometry deals with

objects of integer dimensions: zero dimensional points, one dimensional lines and

curves, two dimensional plane figures such as squares and circles, and three

dimensional solids such as cubes and spheres. However, many natural phenomena are

better described using a dimension between two whole numbers. So while a straight

line has a dimension of one, a fractal curve will have a dimension between one and two,

depending on how much space it takes up as it twists and curves. The more the flat

fractal fills a plane, the closer it approaches two dimensions. Likewise, a "hilly fractal

scene" will reach a dimension somewhere between two and three. So a fractal

landscape made up of a large hill covered with tiny mounds would be close to the

second dimension, while a rough surface composed of many medium-sized hills would

be close to the third dimension.

There are a lot of different types of fractals. Here I will present two of the most popular

types: complex number fractals and Iterated Function System (IFS) fractals.

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2.3.1.1 Complex Number Fractals

Before describing this type of fractal, I decided to explain briefly the theory of complex

numbers.

A complex number consists of a real number added to an imaginary number. It is

common to refer to a complex number as a "point" on the complex plane. If the

complex number is, Z= iba . the coordinates of the point are a (horizontal - real axis)

and b (vertical imaginary axis). The unit of imaginary numbers: Ii

Two leading researchers in the field of complex number fractals are Gaston Maurice

Julia and Benoit Mandelbrot.

Gaston Maurice Julia spent his life studying the iteration of polynomials and rational

functions. Around the 1920s, after publishing his paper on the iteration of a rational

function, Julia became famous. However, after his death, he was forgotten.

In the 1970s, the work of Gaston Maurice Julia was revived and popularized by the

Polish-born Benoit Mandelbrot. Inspired by Julia‘s work, and with the aid of Computer

Graphics, Mandelbrot was able to show the first pictures of the most beautiful fractals

known today.

2.3.2 Dimension

Dimension is a term used to measure the size of a set. Usually, this set will be an

image, but not always. It helps to think of images or objects when thinking about

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dimension. We are all familiar with objects that are one-dimensional (a line segment),

two-dimensional (a square), and three-dimensional (a cube). We can also say that these

objects have dimension 1, 2, and 3, respectively.

Fractals have dimension, also. But the value of their dimension does not necessarily

need to be an integer. This fact is what gives fractals many of their unique properties;

for example, the Sierpinski triangle has an area of 0.

There are two main types of dimension that we measure: box dimension and

topological dimension. We now explore the two.

2.3.2.1 Box Dimension

Suppose we have a number of boxes, all with the same side length. We denote this

length by r. Now suppose we have an object in nR and we want to cover this object

with these boxes. Let rN denote the number of such boxes it takes to cover this

object. These boxes have area nr , and this they were scaled by a factor of

n

r

1 .

Now, if we took a simple square of length s and covered it with boxes of area , we

could determine as follows:

nn

n

rsrN

r

srN

rrNs

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2

2

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Since 2s is a constant, we can denote it by C, thus giving us:

nr

Cr 1

Solving for n yields:

n is the dimension of our object. Since C is a constant, we can ignore it for our

purposes. If we take the limit of this formula as r approaches 0, we get the formula for

box dimension:

Using the Sierpinski triangle as an example, we have N(r) = 3 (three smaller triangles

created from one large one) and 2

1r (each triangle is scaled by a factor of

2

1). Putting

these values in the above formula, we get

This is the box dimension for the Sierpinski triangle.

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2.3.2.2 Topological Dimension

Topological dimension is more difficult to explain and compute. A few common

definitions are used, all of which are analogous to one another. The definition we will

use deals with line segments.

If a set S cannot be restated as the union of two or more sets, and this set does not

contain any line segments, then the topological dimension of S is 0.

If a set S is a union of two or more sets of topological dimension k, where k is a

nonnegative integer, then the topological dimension of S is k + 1.

The set of rational numbers and the set of irrational numbers do not contain any line

segments, and thus both have topological dimension 0. The union of these two sets is

the set of real numbers, which thus has topological dimension 1.

Often, the topological dimension cannot be computed exactly. For simple objects, like

lines, polygons, and polyhedrons, the topological dimension is equal to the box

dimension.

A fractal is a set for which SDSD Bt .this is formal mathematical definition.

2.4 L-System

L-systems are a mathematical formalism proposed by the biologist Aristid

Lindenmayer in 1968 as a foundation for an axiomatic theory of biological

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development. More recently, L-systems have found several applications in Computer

Graphics[74][76]. Two principal areas include generation of fractals and realistic

modeling of plants. Central to L-systems, is the notion of rewriting, where the basic

idea is to define complex objects by successively replacing parts of a simple object

using a set of rewriting rules or productions. The rewriting can be carried out

recursively.

The most extensively studied and the best understood rewriting systems operate on

character strings. Chomsky's work on formal grammars [65] spawned a wide interest in

rewriting systems. Subsequently, a period of fascination with syntax, grammars and

their application in computer science began, giving birth to the field of formal

languages.

Aristid Lindenmayer's work introduced a new type of string rewriting mechanism,

subsequently termed L-systems. The essential difference between Chomsky grammars

and L-systems lies in the method of applying productions. In Chomsky grammars

productions are applied sequentially, whereas in L-systems they are applied in parallel,

replacing simultaneously all letters in a given word. This difference reflects the

biological motivation of L-systems. Productions are intended to capture cell divisions

in multicellular organisms, where many divisions may occur at the same time.

An L-system or Lindenmayer system is a parallel rewriting system, namely a variant of

a formal grammar, most famously used to model the growth processes of plant

development, but also able to model the morphology of a variety of organisms. It is a

particular type of symbolic dynamical system with the added feature of a geometrical

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interpretation of the evolution of the system. L-systems were introduced and developed

in 1968 by the Hungarian theoretical biologist and botanist from the University of

Utrecht, Aristid Lindenmayer (1925–1989).

2.4.1 Components of L-System

The components of an L-system are as follows:

Alphabet:

The alphabet is a finite set V of formal symbols, usually taken to be letters a, b, c, etc.,

or possibly some other characters.

Axiom:

The axiom (also called the initiator) is a string w of symbols from V. The set of strings

(also called words) from V is denoted V*. Given V= {a, b, c}, some examples of words

are aabca, caab, b, bbc, etc. The length |w| of a word w is the number of symbols in the

word.

Productions:

A production (or rewriting rule) is a mapping of a symbol a Є V to a word w Є V*.

This will be labeled and written with notation:

P: a w

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We allow as possible productions mappings of a to the empty word, denoted ø, or to a

itself. If a symbol a Є V does not have an explicitly given production, we assume it is

mapped to itself by default. In that case, a is a constant of the L-system [6]. L-System

is a kind of form language, if we want to get L-System and graphics to be together, we

should take on every letter in L-System to a specific graphics meaning, that is, explain

L-System by the graphics that using turtle.

2.4.2 Variants and Notations of L-System

A 0L system is an ordered triple Π = (V, w0, P) where V is the set of alphabet, w0 a

non-empty word over V which is called the axiom or initial word and P is finite set of

rules of the form a w, a Є V and w Є V*. Furthermore, for each a Є V, there is at

least one rule with a on the left-hand side (This is called the completeness condition).

A 0L system Π = (V, w0, P) is deterministic if for every a Є V, there is exactly one rule

in P with a on the left hand side. It is propagating (ε – free), if ε is not on the right-hand

side of any production. Notations D0LS, P0LS, and DP0LS are used for these systems.

The languages generated by these systems are called D0L, P0L, and DP0L languages,

respectively.

Other possible variants are T0L, DT0L, PT0L, DPT0L, E0L, ET0L etc. and the

corresponding languages [7].

Example: Consider the following DP0L system:

Π1 = ( {a, b}, ab, {aaa, bbb} )

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The derivation steps are:

ab aabb aaaabbbb …….

L (Π1) = { a2n

b2n

| n≥0}

2.5 Traditional Fractal Generation Methods

Four common techniques for generating fractals are:

2.5.1 Escape-Time Fractals

It can also be known as "orbits" fractals. These are defined by a formula or recurrence

relation at each point in a space (such as the complex plane). Examples of this type are

the Mandelbrot set, Julia set, the Burning Ship fractal, the Nova fractal and the

Lyapunov fractal. The 2d vector fields that are generated by one or two iterations of

escape-time formulae also give rise to a fractal form when points (or pixel data) are

passed through this field repeatedly.

2.5.2 Iterated Function Systems

These have a fixed geometric replacement rule. Cantor set, Sierpinski carpet, Sierpinski

gasket, Peano curve, Koch snowflake, Harter-Highway dragon curve, T-Square,

Menger sponge, are some examples of such fractals.

2.5.3 Random Fractals

In this method fractals are generated by stochastic rather than deterministic processes,

for example, trajectories of the Brownian motion, fractal landscapes and the Brownian

tree. The latter yields so-called mass fractals, for example, diffusion-limited

aggregation or reaction-limited aggregation clusters.

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2.5.4 Strange Attractors

The fractals are generated by iteration of a map or the solution of a system of initial-

value differential equations that exhibit chaos.

2.6 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. You have probably

noticed that 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. Some of the fractal plants

he created became classic examples. Here are some of them in addition to several other

ones:

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Figure: 2.6.1 Fractal Plant 1

Another way of creating fractal plants is using fractal canopies or Pythagoras trees.

Fractal canopies are formed by splitting lines, which is very similar to branching.

Pythagoras trees, such as the one below do the same more realistically by using squares

and triangles instead of lines:

Figure: 2.6.2 Fractal plant 2

One of the properties of fractal canopies is the endpoints being interconnected. This is

especially interesting in its similarity to broccoli, where the branches‘ endpoints form

an interconnected surface:

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Figure: 2.6.3 Fractal Plant 3

The final way of creating plant models is by using IFS fractals such as the Barnsley

Fern below, which resemble plant shapes:

Figure: 2.6.4 Fern

2.7 Fractals Applications

Fractal geometry has permeated many area of science, such as astrophysics, biological

sciences, and has become one of the most important techniques in Computer Graphics.

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2.7.1 Fractals in Astrophysics

Nobody really knows how many stars actually glitter in our skies, but have you ever

wondered how they were formed and ultimately found their home in the Universe?

Astrophysicists believe that the key to this problem is the fractal nature of interstellar

gas. Fractal distributions are hierarchical, like smoke trails or billowy clouds in the sky.

Turbulence shapes both the clouds in the sky and the clouds in space, giving them an

irregular but repetitive pattern that would be impossible to describe without the help of

fractal geometry.

2.7.2 Galaxies

Looking at the structure of our universe, you can find it to be very self-similar. It is

composed of gigantic super clusters, which are in turn composed of clusters. Every

cluster is composed of galaxies, which are in turn composed of star systems such as the

solar system, which are further composed of planets with moons revolving around

them[Web 127]. Truly, every detail of the universe shows the same clustering patterns.

The cluster fractals, such as the Cantor Square below are indeed useful in modeling the

universe:

Figure: 2.7.2.1 Cantor Square

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Cluster fractals are formed by repeatedly cutting out pieces of a polygon. The fractal

above is obviously not a good model, and making it more random helps a lot. The

fractal dimension of such fractals can be found very easily using the similarity method.

In the cantor square, for example there are 4 smaller squares, the sides of each of which

are 1/3 of the entire picture. The fractal dimension will thus be log 4 / log 3 = 1.26.

This is remarkably close to the fractal dimension of the universe according to one of the

experiments, where it was found to be about 1.23. The fact that this is a fraction is yet

another proof of the universe‘s fractal geometry.

2.7.3 Fractals in the Biological Sciences

Biologists have traditionally modeled nature using Euclidean representations of natural

objects or series. They represented heartbeats as sine waves, conifer trees as cones,

animal habitats as simple areas, and cell membranes as curves or simple surfaces.

However, scientists have come to recognize that many natural constructs are better

characterized using fractal geometry. Biological systems and processes are typically

characterized by many levels of substructure, with the same general pattern repeated in

an ever-decreasing cascade. Scientists discovered that the basic architecture of a

chromosome is tree-like; every chromosome consists of many 'mini-chromosomes', and

therefore can be treated as fractal. For a human chromosome, for example, a fractal

dimension D equals 2,34 (between the plane and the space dimension). Self-similarity

has been found also in DNA sequences. In the opinion of some biologists fractal

properties of DNA can be used to resolve evolutionary relationships in animals.

Perhaps in the future biologists will use the fractal geometry to create comprehensive

models of the patterns and processes observed in nature.

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2.7.4 The Lungs

The first place where this is found is rather obvious to anyone who knows fractals in

the pulmonary system, which you 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 canopie, which is formed by splitting lines [Web

115].

Figure: 2.7.4.1 Lungs (fractal Canopies)

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.

2.7.5 The Alveoli

Supporting evidence that your lungs are fractal comes from measurements of the

alveolar area, which was found to be 80 m2 with light microscopy and 140 m

2 at higher

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magnification with electron microscopy. From the geometric method we know that the

increase in size with magnification is one of the properties of fractals!

2.7.6 The Blood Vessels

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 [Web 115].

2.7.7 The 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. Geometrically, the increase in

folding means the increase in. 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, [Web 115].

2.7.8 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

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increasing the sensing surface. However, in humans this membrane is less fractal than

in other animals, which makes them less sensitive to smells.

The fractal dimensions of some anatomical structures are given below. Note that all

dimensions are greater than you would expect them to be, and most are fractions, which

automatically implies that the structures are fractal.

Anatomical Structure Fractal Dimension

Bronchial Tubes very close to 3

Arteries 2.7

Brain 2.73 – 2.79

Alveolar Membrane 2.17

Mitochondrial Membrane (outer) 2.09

Mitochondrial Membrane (inner) 2.53

Endoplasmic Reticulum 1.72

Table: 2.7.8.1 Fractal Dimension

Fractals, in addition to the anatomical structures above can be found in the body on

smaller scales in various molecules.

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2.7.9 Rings of Saturn

Saturn is perhaps most famous for the ring that it has around itself. Originally, it was

believed that the ring is a single one. After some time, a break in the middle was

discovered, and scientists considered it to have 2 rings. However, when Voyager I

approached Saturn, it discovered that the two rings were also broken in the middle and

the 4 smaller rings were broken as well. Eventually, it identified a very large number of

breaks, which continuously broke even small rings into smaller pieces. The overall

structure is amazingly similar to...Cantor Set, which is formed by continuously cutting

out middles of the segments:

Figure: 2.7.9.1 Cantor set line Segment

If you put circles through the points in the last picture above, you will get a simple

model of the rings of Saturn:

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Figure: 2.7.9.2 Rings of Saturn

2.7.10 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 the population, our formula will become

new P = (1+r) · P

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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

If we now do some algebra

new P = [1 + r – r(P)] · P

new P = P + r(P) – r(P)

new P = (1 + r) · P – r(P)

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. So may be this will give us a fractal pattern as

well! Let‘s make a graph in which we plot the values of the population for all values of

r from 1.9 to 3:

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Figure: 2.7.10.1 Population Graph

2.7.11 Fractals in Computer Graphics

The biggest use of fractals in everyday live is in computer science. Many image

compression schemes use fractal algorithms to compress Computer Graphics files to

less than a quarter of their original size.

Computer graphic artists use many fractal forms to create textured landscapes and other

intricate models.It is possible to create all sorts of realistic "fractal forgeries" images of

natural scenes, such as lunar landscapes, mountain ranges and coastlines. We can see

them in many special effects in Hollywood movies and also in television

advertisements. The "Genesis effect" in the film "Star Trek II - The Wrath of Khan"

was created using fractal landscape algorithms, and in "Return of the Jedi" fractals were

used to create the geography of a moon, and to draw the outline of the dreaded "Death

Star". But fractal signals can also be used to model natural objects, allowing us to

define mathematically our environment with a higher accuracy than ever before.

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A fractal landscape

Figure: 2.7.11.1 Fractal Landscape

A fractal planet

Figure: 2.7.11.2 Fractal Planet

2.7.12 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

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it, the probability of various changes of the market can be shown using the standard bell

curve:

Figure: 2.7.12.1 Economy 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. You will get a broken line with some rises and falls. Now, if you take one of the

months and graph it in a more detailed way with every week shown, you will get a very

similar line with some rises and falls. If you make it more and more detailed by

showing every day, every hour, and even every minute or second... you will still get the

same, only smaller, rises and falls. There is your Brownian self-similarity! Mandelbrot

came up with a method of creating fractals that fit the above description. He based in

on simple generator iteration and created base-motif fractals that could model the

market. In the February 1999 issue of Scientific American, he published some of his

fractal "forgeries" next to real market lines, showing how remarkably similar they were.

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In his method, you start with a shape, called the generator. The generator must be

composed of three line segments, in order to contain both a rise and a fall in price:

Figure: 2.7.12.2 Generator 1

You then take this exact picture and substitute every line segment with it:

Figure: 2.7.12.3 Generator 2

Continuing to substitute you get something looking like this:

Figure: 2.7.12.4 Generator 3

Now, compare it to a Portfolio Theory model:

Figure: 2.7.12.5 Generator 4

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2.8 Conclusion

After completing the introduction we have seen the fractal have been used in many

sciences as well as in other technology. We have observe from fundamental of fractals

that all basic concepts of fractal are evaluated from the Mathematics and gradually with

the help of computer science it has been converted into fractal graphics. The basic

growth of plant science is easily and deeply understood with the advent of fractal

technology specially the generation of trunks, branches and leaves with self similar

property of fractal. Further chapter will built of this theory and more focus on fractal

uses in plants and its growth.