2. PREFACE
This books aims to provide a good background in the basic topics
of fractal geometry . it presuppose some knowledge of geometry .
many peoples 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
arts with mathematics to demonstrate that equations are more than
just a collection of numbers . what makes fractalsβ even more
interesting is that they are the best existing mathematical
descriptions of many natural forms , such as coast lines ,
mountains or parts of living organism. The first two chapters
provide a general background.
3. CONTENT
CHAPTER TITLE PAGE NO:
Chapter 1 History of Fractal geometry. 1
Chapter 2 Fractal dimension and different methods 7
used for calculating Fractal dimension.
Chapter 3 Specific fractals and some famous fractals. 18
Chapter 4 Applications of fractals. 34
5. 2
Basic Idea about Fractals
1.1 History of Fractal Geometry
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).
It took until 1872 before a function appeared whose graph would today
be considered fractal, when Karl Weierstrass gave an example of a
function with the non-intuitive property of being everywhere continuous
but nowhere differentiable.
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 snowflake. In 1915, Waclaw
Sierpinski constructed his triangle and, one year later, his carpet.
Originally these geometric fractals were described as curves rather than
the 2D shapes that they are known as in their modern constructions.
In 1918, Bertrand Russell had recognized a "supreme beauty" within the
mathematics of fractals that was then emerging. 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.
6. 3
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. However, without the aid of modern computer
graphics, they lacked the means to visualize the beauty of many of the
objects that they had discovered.
In the 1960s, Benoit Mandelbrot started investigating self-similarity in
papers such as How Long Is the Coast of Britain, 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. He illustrated this mathematical
definition with striking computer-constructed visualizations. These
images captured the popular imagination; many of them were based on
recursion, leading to the popular meaning of the term "fractal".
1.2 Fractals
A fractal is a rough or fragmented geometric shape that can be split into
parts, each of which is (atleast approximately) a reduced size copy of the
whole, a property called self similarity. The term fractal was coined by
Benoit Mandelbrot in 1975 and is derived from the Latin word Fractus
meaning βbrokenβ or βfractured. Mandelbrot is often refered to as the
father of fractal geometry.
7. 4
1.3 Features of Fractals
A fractal often has the following features
a) It has a fine structure at arbitrarily small scales.
b) It is too irregular to be described in traditional Euclidean
geometric language.
c) It is self similar
d) It has a Hausdorff dimension which is greater than its topological
dimension.
e) It has a simple and recursive definition.
1.4 Properties of Fractals
Two of the most important properties of fractals are self similarity and
non integer dimension.
1.4.1 Self similarity
Self βsimilarity means we can magnify the fractal object many times
and after every step we can see the same shape. There are three types
of self similarity found in fractals:
1) Exact self similarity
2) Quasi self similarity
3) Statistical self similarity
8. 5
1.4.1.1 Exact Self Similarity
This is the strongest type of self similarity. The fractal appears identical
at different scales. Fractals defined by iterated function systems often
display exact self similarity. For example the Sierpinski triangle and
Koch snowflake exhibit exact self similarity. Another example is the
fern leaf. Each frond of the fern is a mini copy of the whole fern and
each frond branch is similar to the whole frond and so on.
Figure: 1.1 Fern leaf
1.4.1.2 Quasi Self Similarity
This is looser form of self similarity. The fractal appears approximately
(but not exactly) identical at different scales. Quasi-self similar fractals
contain small copies of the entire fractal in distorted and degenerate
forms. Fractals defined by reccurence relations are usually quasi self
similar but not exactly self similar. The Mandelbrot set is quasi self
9. 6
similar, as the satellites are approximations of the entire set but not exact
copy.
Figure: 1.2 Mandelbrot set
1.4.1.3 Statistical Self Similarity
This is the weakest type of self similarity. The fractal has numerical or
statistical measures which are preserved across scales. Random fractals
are examples of fractals which are statistically self similar but neither
exactly nor quasi self similar. The coastline of Britain is another
example, one cannot expect to find microscopic Britains by looking at a
small section of cost with a magnifying glass.
Figure : 1.3: Root
11. 8
Fractal Dimension
Fractals themselves have their own dimension known as fractal
dimension, which is usually (but not always) a non integer dimension
that is greater than their Euclidian dimension π· πΈ
Classical geometry deals with object 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.
There are many definitions of fractal dimensions, they are the similarity
dimension π·π, the divider dimension π· π·, the Hausdorff dimension π· π» ,
the box counting dimension π· π΅, the correlation dimension π· πΆ, the
information dimension π·πΌ, the point wise dimension π· π, the average
pointwise dimension π·π΄, and the Lyapunov dimension π·πΏ.
2.1 The Similarity Dimension
Fractals with structures comprising of exact copies of themselves at all
magnifications, in other words objects possessing exact self similar are
known as regular fractals. The similarity dimension denoted by π·π, is
used to characterize the regular fractal objects.
The concept of dimension is closely associated with that of scaling.
Consider the line, surface and solid divided up respectively by self
similar sub- lengths, sub -areas, sub- volumes of side length π.
12. 9
v
length L area A volume V
Figure: 2.1
For simplicity 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 in to N smaller self similar
segments, each π in length, then π is in fact the scaling ratio
π
πΏ
= π (since L = 1)
Thus, L = Nπ = 1β¦β¦β¦β¦..(2.1.1)
That is, the unit line is composed of N self-similar parts scaled by
π = 1
π.Now consider the unit area. If we divide the area again into N
segments each π2
in area then,
A=Nπ2
=1............................ (2.1.2)
That is, the unit area is composed of N self-similar part scaled by
π = 1
(π)
1
2
13. 10
Similarly we have for the unit volume,
V= Nπ3
=1β¦β¦β¦β¦.(2.1.3)
That is, the unit solid is N self-similar parts scaled by
π = 1
(π)
1
3
(Each object consist of N elements of side length ππ is determined by
the choice of π. N for each object need not necessarily be the same).
From the equations (2.1.1), ( 2.1.2) and( 2.1.3) we see that the exponent
of π in each case is a measure of the (similarity) dimension of the object.
In general, we have
Nπ π· π = 1
Using logarithms we have
π·π =
log(π)
{log(1
π)}
Where π·π is the similarity dimension.
2.1.1 The Cantor Set
The triadic Cantor set is an example of a regular fractal object which
exhibit exact self-similarity over all scales. Cantor set was defined by
German Mathematician George Cantor (1845-1918). The Cantor set
consist of an infinite set of disappearing line segments in the unit
interval. The triadic Cantor set is generated by removing the middle
14. 11
third of the unit line segment. From the two remaining segments, each
one third in length, the middle thirds are again removed. The middle
thirds of the remaining four line segments each one β nineth in length
are then removed and so on to infinity. What is left is a collection of
infinitely many disappearing line segments lying on the unit interval
whose individual and combined lengths approach zero. This set of
points is known as Cantor set, Cantor dust or Cantor discontinum.
It is clear that the left βhand third of the set contains an identical copy of
the set. There are two such copies of the set , thus N= 2 and π = 1
3.
The similarity dimension is given by
π·π =
log(π)
log(1
π)
=
log(2)
log{1
1
3
=
log(2)
log(3) = 0.6309
For, the cantor set, the Euclidian dimension π· πΈ is obviously equal to
one, as we require one-coordinate direction to specify all the points on
the set. Also for the cantor set the topological dimension π· π is zero.
Thus for the cantor set,
π· πΈ = 1, π·π = 0.6309β¦., π· π =0
Thus ,
π· πΈ > π· π > π· π
15. 12
Thus, we can adopt, as a test for a fractal object the condition that its
fractal dimension exceeds its topological dimension β which ever
measure of fractal dimension is employed. Using this condition we can
conclude that a Cantor set is a fractal object.
2.2 Box Counting Dimension
The similarity dimension technique is employed for calculating the
fractal dimension of regular fractals that is, objects possessing exact
self- similarity. There is however, another group of fractals known as
random fractals. These fractals are not exactly self-similar. Each small
parts of a random fractal has the same statistical properties as a whole.
To examine a suspected fractal object for its box counting dimension,
we cover the object with covering elements or boxes of side length πΏ.
The number of boxes, N required to cover the object is related to πΏ
through its box counting dimension π· π΅.
Consider the following example:-
Consider a line segment of unit length. We try to cover this line segment
by squares of sides length πΏ. Then we require N = 1
πΏ1 squares to cover
this line segment. If we had used cubes of the side length πΏ, (volume
πΏ3
) or line segments (length πΏ1
),we would again have required N =
1
πΏ1 of them to cover the line ie, the number of cubes, squares or line
segments we require to cover this unit line is N = 1
πΏ.
Notice that the exponent of πΏ remains equal to one regardless of the
dimension of the probing elements and is infact the box counting
16. 13
dimension π· π΅, of the object under investigation. Now let us rename all
covering elements as hypercubes as follows :
1D hypercube = 1D element that is, a line segment
2D hypercube = 2D element that is, a square
3D hypercube = 3D element that is, a box or cube etc
And similarly rename all measure as hypervolumes, as follows :
Volume of 1D hypercube = 1D hypervolume (length)
Volume of 2D hypercube = 2D hypervolume (area)
Volume of 3D hypercube =3 D hypervolume (volume)
If we repeat the covering procedure discussed above for a plane unit
area, we would require N = 1
πΏ2 hypercubes of edge length πΏ and
Euclidian dimension greater than or equal to two. Similarly with a 3D
solid object, we would require N = 1
πΏ3 hypercubes of edge length πΏ
with Euclidian dimension greater than or equal to three to cover it. In
each case the exponent of πΏ is a measure of the dimension of the object.
In general we require N = 1
πΏ π· π΅
boxes to cover an object where the
exponent π· π΅ is the box counting dimension of the object.
Thus for objects of unit hypervolume we have
log( π) = log( 1
πΏ) π· π΅ )
17. 14
log(π)=π· π΅ log( 1
πΏ)
π· π΅ =
log(π)
log( 1
πΏ)
This method can be used only for objects of unit hypervolume. This
expression is similar to the similarity dimension π·π. However it should
be noted that the calculation of π·π requires that exactly self-similar parts
of the fractal are identified, whereas π· π΅ require the object to be covered
with self-similar boxes. Hence π· π΅ allows as greater flexibility in the
type of fractal object that may be investigated.
The general expression for the dimension of an object with a
hypervolume (that is, length, area, volume or fractal hypervolume) not
equal to unity, but rather given by πβ
is
π· π΅ =
{log π β log π β
}
log 1
πΏ
β¦β¦β¦β¦..(2.2.1)
Where N is the number of hypercubes of the side length πΏ required to
cover the object that is,
N = πβ
πΏ π· π΅
Rearranging equation (2.2.1) we get
log π = π· π΅ log 1
πΏ +log πβ
18. 15
Which is in the form of the equation of a straight line where the gradient
of the line, π· π΅ is the box counting dimension of the object. This form is
suitable for determining the box counting dimension of a wide variety of
fractal objects by plotting log π against log( 1
πΏ) for probing
elementsof various side length πΏ.r determining the fractal dimension of a
coastline using the box counting method, one may use a regular grid of
boxes and count the number of boxes N, which contain a part of the
curve for each box side length πΏ. We may obtain the box counting
dimension from the limiting gradient (as πΏ tends to zero) of a plot of
log( π) against log( 1
πΏ) that is, the derivative
Figure: 2.2
In practice box counting dimension may be estimated by selecting two
sets of (log( 1
πΏ),log( π)) coordinates at small value of πΏ (ie, large
value of log( 1
πΏ) ).
An estimate of π· π΅ is given by
19. 16
π· π΅ =
{log π2 β log π1}
{log( 1
πΏ2
) β log( 1
πΏ1
)}
2.3 Hausdorff Dimension
To find the Hausdorff dimension also, we use elements to cover the
object under inspection. But here we are trying to measure the size of
the object. To measure the size or hypervolume, we need to use the
appropriate dimension of covering hypercubes, this appropriate
dimension is the Hausdorff dimension,π· π». Suppose that, we cover a
smooth curve with line of size πΏ. Let we require N elements to cover the
curve. The measured length of the line as measured by the covering
elements is given by
πΏ π =N.πΏ1
As πΏ goes to zero, πΏ π tends to the true length of the curve,L.
That is, πΏ π β πΏ as πΏ β 0
Now if we cover with square segments of each πΏ2
in area, the measured
area π΄ π = N. πΏ2
That is, π΄ π β 0 as πΏ β0
Similarly, the measured volume,
ππ = N.πΏ3
β0 as πΏ β0.
In general if we require N hypercubes to cover an object then the
measured hypervolume is
ππ
β
= N.πΏ π· πΈ
And ππ
β
β πβ
, the actual hypervolume.
The measurement obtained is only sensible as long as the dimension of
the measuring elements and the objects are the same. ie, the measured
hypervolume tends to infinity if we use hypercubes of dimension less
20. 17
than that of the object and tends to zero for measuring elements of
dimension greater than the object. We are allowed to consider
hypercubes or test functions with hypervolume πΏ π·
, where the exponent
D is non-integer. The Hausdorff dimension π· π» of the object is defined
as the critical dimension D, for which the measured hypervolume
changes from zero to infinity. In practice, it is not possible to probe
objects with non-integer hypercubes, hence the Hausdorff dimension
estimate is not useful for determining the fractal dimension of real
objects.
22. 19
Specific Fractals
There are a lot of different types of fractals. Among these two most
important types are:
1) Complex number fractals and
2) Iterated function system fractals or IFS fractal.
3.1 Complex Number Fractals
Before describing this type of fractal, to explain brief ly 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 = a + ib,
the coordinates of the points are a (horizontal real axis) and b
(verticalimaginaryaxis).
The unit of imaginary numbers:
i = (β1)
Two leading researchers in the field of complex number fractals
are Gaston Maurice Julia and Benoit Mandelbrot.
Two most important examples of complex number fractals include
Mandelbrot set and Julia set.
23. 20
3.1.1 Mandelbrot Set
The Mandelbrot set is one of the most famous fractal in existence. It was
named after the famous mathematician Benoit Mandelbrot. The
Mandelbrot set is the set of points on a complex plane. To build the
Mandelbrot set, we have to use an algorithm based on the recursive
formula:
Zn = π πβ1
2
+ C
Separating the points of the complex plane into two categories:
1. Points inside the Mandelbrot set,
2. Points outside the Mandelbrot set.
The image below shows a portion of the complex plane. The points
of the Mandelbrot set have been colored black.
Figure: 3.1
24. 21
It is also possible to assign a color to the points outside the
Mandelbrot set. Their colors depend on how many iterations have
been required to determine that they are outside the Mandelbrot set.
Figure: 3.2
3.1.1.1 Creation of Mandelbrot set
To create the Mandelbrot set we have to pick a point (C ) on the
complex plane. The complex number corresponding with this point
has the form:
C = a + ib
After calculating the value of previous expression:
π1 = π0
2
+ C
25. 22
Using zero as the value of π0, we obtain C as the result. The next
step consists of assigning the result to π1and repeating the
calculation, now the result is the complex number πΆ2
+ πΆ. Then we
have to assign the value to π2and repeat the process again and
again.
This process can be represented as the "migration" of the initial
point C across the plane. What happens to the point when we
repeatedly iterate the function? Will it remain near to the origin or
will it go away from it, increasing its distance from the origin
without limit? In the first case, we say that C belongs to the
Mandelbrot set (it is one of the black points in the image);
otherwise, we say that it goes to infinity and we assign a color to C
depending on the speed at which the point "escapes" from the
origin.
We can take a look at the algorithm from a different point of view.
Let us imagine that all the points on the plane are attracted by both:
infinity and the Mandelbrot set. That makes it easy to understand
why:
ο· points far from the Mandelbrot set rapidly move towards
infinity,
ο· points close to the Mandelbrot set slowly escape to infinity,
ο· points inside the Mandelbrot set never escape to infinity.
26. 23
3.1.1.2 Some properties of the Mandelbrot set
1. The Mandelbrot set is connected, that is, there is always a path
from one point of the set to another point of the set so that all the
points in the path are also in the set.
2. The area of the set is finite (it fits inside a circle of radius 2, the
exact area has only been approximated) but the length of its
border is infinite.
3. If we take any part of the border of the set the length of this part
will also be infinite. This means that the border of the set has
infinite details, that is, we will never find a place where the
border is smooth with a finite length.
4. Mathematically the set is defined to have a dimension which is
larger than 2 but smaller than 3(that is, fractal dimension).
3.1.2 Julia Set
The Julia set is another very famous fractal, which happens to be very
closely related to the Mandelbrot set. It was named after Gaston Julia.
The iterative function that is used to produce them is the same as for the
Mandelbrot set. The only difference is the way this formula is used. In
order to draw a picture of the Mandelbrot set, we iterate the formula for
each point C of the complex plane, always starting with π0 = 0. If we
want to make a picture of a Julia set, C must be constant during the
whole generation process, while the value of π0 varies. The value of C
determines the shape of the Julia set; in other words, each point of the
complex plane is associated with a particular Julia set.
27. 24
3.1.2.1 Creation of Julia Set
We have to pick a point C on the complex plane. The following
algorithm determines whether or not a point on complex plane Z
belongs to the Julia set associated with C, and determines the color that
should be assigned to it. To see if Z belongs to the set, we have to
iterate the functionπ1 = π0
2
+ C using π0 = Z. What happens to the
initial point Z when the formula is iterated? Will it remain near to the
origin or will it go away from it, increasing its distance from the origin
without limit? In the first case, it belongs to the Julia set; otherwise it
goes to infinity and we assign a color to Z depending on the speed the
point "escapes" from the origin. To produce an image of the whole Julia
set associated with C, we must repeat this process for all the points Z
whose coordinates are included in this range:
-2< π < 2 ; -1.5 < π¦ < 1.5
The most important relationship between Julia sets and Mandelbrot set
is that while the Mandelbrot set is connected (it is a single piece), a Julia
set is connected only if it is associated with a point inside the
Mandelbrot set. For example: the Julia set associated with πΆ1 is
connected; the Julia set associated with πΆ 1is not connected (see picture
below).
28. 25
Figure: 3.3 Julia set
3.2 Iterated Function System Fractals
Iterated Function System (IFS) fractals are created on the basis of
simple plane transformations: scaling, dislocation and the plane axes
rotation. Creating an IFS fractal consists of following steps:
1. Defining a set of plane transformations,
2. Drawing an initial pattern on the plane (any pattern),
3. Transforming the initial pattern using the transformations defined
in first step,
4. Transforming the new picture (combination of initial and
transformed patterns) using the same set of transformations,
5. Repeating the fourth step as many times as possible (in theory,
this procedure can be repeated an infinite number of times).
29. 26
The most famous IFS fractals are the Sierpinski Triangle and the Von
Koch Curve
3.2.1 Sierpinski Triangle
This is the fractal we can get by taking the midpoints of each side of an
equilateral triangle and connecting them. The iterations should be
repeated an infinite number of times. The pictures below present four
initial steps of the construction of the Sierpinski Triangle:
Figure: 3.4
30. 27
3.2.1.1 Construction of the Sierpinski Triangle
Taking an equilateral triangle as an example:
1.Start with the equilateral triangle.
Figure: 3.5
2. Connect the midpoints of each side of the triangle to form four
separate triangles.
Figure: 3.6
3. Cut out the triangle in the center
Figure: 3.7
31. 28
4. Repeat the steps 1, 2 and 3 on the three coloured triangle left behind.
Figure: 3.8
The center triangle of each coloured triangle at the corner was cut out as
well.
5.Further repetition with adequate screen resolution will give the
following pattern
Figure: 3.9
3.2.1.2 Dimension of Sierpinski Triangle
Using this fractal as an example, we can prove that the fractal dimension
is not an integer.
We know that the similarity dimension
π·π =
log(π)
log( 1
π)
32. 29
Each pre-fractal stage in the construction is composed of three smaller
copies of the preceding stage, each copy scaled by a factor of one half.
Then the similarity dimension is
π·π =
log(3)
log( 1
1
2
)
That is, π·π =
log(3)
log( 2) = 1.585
The result of this calculation proves the non-integer fractal dimension.
3.2.2 Von Koch Curve
The curve of Von Koch is generated by a simple geometric procedure
which can be iterated an infinite number of times by dividing a straight
line segment into three equal parts and substituting the intermediate part
with two segments of the same length. The Von Koch Curve is a very
elementary example of fractal, as it follows a simple rule of
construction.
3.2.2.1 Construction of the Koch Edge
1.Start with a straight line.
Figure: 3.10
33. 30
2. The straight line is divided into 3 equal parts, and the middle part is
replaced by two linear segments at angles 60ΒΊ and 120ΒΊ.
Figure: 3.11
3. Repeat the steps 1 and 2 to the four line segments generated in two
Figure: 3.12
4. Further iterations will generate the following curves
repeat three times repeat four times
Figure: 3.13
34. 31
3.2.2.2 Dimension of Koch Edge
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, π = 1
3 and the
similarity dimension π·π is given by,
π·π=
log(π)
log( 1
π)
=
log(4)
log(3) = 1.2618
3.3 Some Famous Fractal
3.3.1 The Quadratic Koch Curve
The Koch curve is more specifically known as the triadic Koch curve.
As with the triadic cantor set, the triadic Koch curveβs name stems from
the fact that the middle thirds of the line segments are modified each
step. By changing the form of the generator a variety of Koch curves
may be produced.
Step: 1
Step: 2
Step: 3 step: 4
Figure: 3.14 creation of quadratic Koch curve
35. 32
The quadratic Koch curve, also known as Minkowski Sausage is
generated by repeatedly replacing each line segment, composed of four
quarters with the generator consisting of eight pieces, each one quarter
long. As with triadic Koch curve the Minkowski Sausage is a fractal
object. Each smaller segment of the curve is an exact replica of the
whole curve. There are eight such segments making up the curve, each
one a one-quarter reduction of the original curve.
Thus, N = 8, π = 1
4 and similarity dimension is given by
π·π =
log π
log( 1
π)
=
log(8)
log( 4) =
3log(2)
2 log( 2) = 3
2=1.5
3.3.2 The Sierpinski Carpet
A sister curve to the Sierpinski triangle is the Sierpinski carpet. Its
method of construction is similar to the that of Sierpinski triangle: here
the initiator is a square and the generator removes the middle square ,
side length one-third of the original square.
Figure: 3.15 creation of sierpinski carpet
36. 33
3.3.3 The Menger Sponge
The Menger sponge is closely related to the Sierpinski carpet. 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 prefractal composed of twenty
smaller cubes each scaled down by one third. These cubes are then
drilled out leaving 400 cubes scaled down by one-nineth from the
original cube. Repeated iteration of this construction process leads to the
Menger Sponge. The similarity dimension of the Menger Sponge is
π·π
log(π)
log( 1
π)
=
log(20)
log(3) =2.7268β¦..
Figure: 3.16 creation of menger sponge
38. 35
Applications of Fractals
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.
4.1 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.
4.2 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
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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.
4.3 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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Figure : 4.1 A fractal landscape
Figure :4.2 A fractal planet
Fractals are not just complex shapes and pretty pictures generated by
computers. Anything that appears random and irregular can be a fractal.
Fractals permeate our lives, appearing in places as tiny as the membrane
of a cell and as majestic as the solar system. Fractals are the unique,
irregular patterns left behind by the unpredictable movements of the
chaotic world at work.
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In theory, one can argue that everything existent on this world is a
fractal:
ο· the branching of tracheal tubes,
ο· the leaves in trees,
ο· the veins in a hand,
ο· water swirling and twisting out of a tap,
ο· a puffy cumulus cloud,
ο· tiny oxygen molecule, or the DNA molecule,
ο· the stock market
All of these are fractals. From people of ancient civilizations to the
makers of Star Trek II: The Wrath of Khan, scientists, mathematicians
and artists alike have been captivated by fractals and have utilized them
in their work.
Fractals have always been associated with the term chaos. One author
elegantly describes fractals as "the patterns of chaos." Fractals depict
chaotic behaviour, yet if one looks closely enough; it is always possible
to spot glimpses of self-similarity within a fractal.
To many chaologists, the study of chaos and fractals is more than just a
new field in science that unifies mathematics, theoretical physics, art,
and computer science - it is a revolution. It is the discovery of a new
geometry, one that describes the boundless universe we live in; one that
is in constant motion, not as static images in textbooks. Today, many
scientists are trying to find applications for fractal geometry, from
predicting stock market prices to making new discoveries in theoretical
physics.
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Fractals have more and more applications in science. The main reason is
that they very often describe the real world better than traditional
mathematics and physics.
4.4 Fractals in Astronomy
Fractals will maybe revolutionize the way that the universe is seen.
Cosmologists usually assume that matter is spread uniformly across
space. But observation shows that this is not true. Astronomers agree
with that assumption on "small" scales, but most of them think that the
universe is smooth at very large scales. However, a dissident group of
scientists claims that the structure of the universe is fractal at all scales.
If this new theory is proved to be correct, even the big bang models
should be adapted. Some years ago we proposed a new approach for the
analysis of galaxy and cluster correlations based on the concepts and
methods of modern Statistical Physics. This led to the surprising result
that galaxy correlations are fractal and not homogeneous up to the limits
of the available catalogues. In the meantime many more redshifts have
been measured and we have extended our methods also to the analysis
of number counts and angular catalogues.The result is that galaxy
structures are highly irregular and self-similar. The usual statistical
methods, based on the assumption of homogeneity, are therefore
inconsistent for all the length scales probed until now. A new, more
general, conceptual framework is necessary to identify the real physical
properties of these structures. But at present, cosmologists need more
data about the matter distribution in the universe to prove (or not) that
we are living in a fractal universe.
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4.5 Fractals in Nature
Take a tree, for example. Pick a particular branch and study it closely.
Choose a bundle of leaves on that branch. To chaologists, all three of the
objects described - the tree, the branch, and the leaves - are identical. To
many, the word chaos suggests randomness, unpredictability and
perhaps even messiness. Chaos is actually very organized and follows
certain patterns. The problem arises in finding these elusive and intricate
patterns. One purpose of studying chaos through fractals is to predict
patterns in dynamical systems that on the surface seem unpredictable. A
system is a set of things,- an area of study -A set of equations is a
system, as well as more tangible things such as cloud formations, the
changing weather, the movement of water currents, or animal migration
patterns. Weather is a favorite example for many people. Forecasts are
never totally accurate, and long-term forecasts, even for one week, can
be totally wrong. This is due to minor disturbances in airflow, solar
heating, etc. Each disturbance may be minor, but the change it create
will increase geometrically with time. Soon, the weather will be far
different than what was expected. With fractal geometry we can visually
model much of what we witness in nature, the most recognized being
coastlines and mountains. Fractals are used to model soil erosion and to
analyze seismic patterns as well. Seeing that so many facets of mother
nature exhibit fractal properties, maybe the whole world around us is a
fractal after all.
4.6 Fractals in Computer science
Actually, the most useful use of fractals in computer science is the
fractal image compression. This kind of compression uses the fact that
the real world is well described by fractal geometry. By this way,
images are compressed much more than by usual ways (example: JPEG
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or GIF file formats). Another advantage of fractal compression is that
when the picture is enlarged, there is no pixelisation. The picture seems
very often better when its size is increased.
4.7 Fractals in Fluid mechanics
The study of turbulence in flows is very adapted to fractals. Turbulent
flows are chaotic and very difficult to model correctly. A fractal
representation of them helps engineers and physicists to better
understand complex flows. Flames can also be simulated. Porous media
have a very complex geometry and are well represented by fractal.This
is actually used in petroleum science.
4.8 Fractals in Telecommunications
A new application is fractal-shaped antennae that reduce greatly the size
and the weight of the antennas . Fractenna is the company which sells
these antennae. The benefits depend on the fractal applied, frequency of
interest, and so on. In general the fractal parts produces 'fractal loading'
and makes the antenna smaller for a given frequency of use. Practical
shrinkage of 2-4 times are realizable for acceptable performance.
Surprisingly high performance is attained.
4.9 Fractals in Surface physics
Fractals are used to describe the roughness of surfaces. A rough surface
is characterized by a combination of two different fractals.
4.10 Fractals in Medicine
Biosensor interactions can be studied by using fractals.
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4.11 Fractals in Image Compression
Most useful applications of fractals and fractal geometry in image
compression. It is also one of the more controversial ideas. The basic
concept behind of 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.
4.12 Fractals in Film Industry
One of the more trivial applications of fractals is their visual effect. Not
only do fractals have a stunning aesthetic value, ie, they are remarkably
pleasing to the eye, but they also have a way to trick the mind. Fractals
have been used commercially in the film industry. Fractal images are
used as an alternative to costly elaborate sets to produce fantasy
landscape.
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BIBLIOGRAPHY
1. Benoit Mandelbrot βThe Fractal Geometry of Nature,
W.H Freeman and company, 1983
2. K.J Falconer βGeometry of Fractal Sets ,Cambridge
university press 1985
3. Websites :
a. Wikipedia βFractals
b. www.mathed.uta.edu
