This ppt. is about Fractals-where we find them in Nature and life- Its Applications in real-life.

1. FRACTALS
In Nature, Arts and
Science
Dr. Farhana Shaheen
Assistant Professor
YUC- Women Campus

2. FRACTALS???
For centuries, mathematicians rejected
complex figures, leaving them under a
single description: “formless”. For
centuries, geometry was unable to
describe trees, landscapes, clouds, and
coastlines. However, in the late 1970’s a
revolution of our perception of the world
was brought by the work of Benoit
Mandelbrot who introduced FRACTALS.

3. Mandelbrot; inventor of fractals

4. “Fractua” means
Irregular
Fractals are geometric figures like
circles, squares, triangles etc., but
having special properties. They are
usually associated with irregular
geometric objects, that look the same
no matter at what scale they are
viewed at.
A fractal is an object in which the
individual parts are similar to the
whole.

5. Fractals exhibit
self-similarity
Fractals have the property of self-
similarity, generated by iterations,
which means that various copies of
an object can be found in the original
object at smaller size scales.
The detail continues for many
magnifications -- like an endless
nesting of Russian dolls within dolls.

6. Russian dolls within dolls

7. What is a Fractal?
A fractal is a rough or fragmented geometric
shape that can be subdivided into parts, each of
which is (at least approximately) a reduced-size
copy of the whole. The core ideas behind it are of
feedback and iteration. The creation of most
fractals involves applying some simple rule to a
set of geometric shapes or numbers and then
repeating the process on the result. This feedback
loop can result in very unexpected results, given
the simplicity of the rules followed for each
iteration.
Fractals have finite area but infinite perimeter.

8. Examples of Fractals
A cauliflower is a perfect example of a
fractal where each element is a
perfect recreation of the whole.

9. A naturally occurring Cauliflower
Fractal
Take a close look at a cauliflower:
Take a closer look at a single floret
(break one off near the base of your
cauliflower). It is a mini cauliflower with
its own little florets all arranged in spirals
around a centre.

10. Computer-generated
Fractal patterns
These days computer-generated
fractal patterns are everywhere. From
squiggly designs on computer art
posters to illustrations in the most
serious of physics journals, interest
continues to grow among scientists
and, rather surprisingly, artists and
designers.

11. Examples of Fractals

13. Air Balloon

14. Bat Fractal

15. Coastlines Fractal

19. How to make a Koch Snowflake
Fractal

20. Let’s make a Snowflake Fractal
Fractals are Fun!

21. Pentagram Fractal

22. Animated fractal mountain

23. The Koch Curve Forming of a
Fractal

24. Fractal Canopies

25. Dragon Curve

26. Jurassic Park Fractal

27. The Sierpinski Triangle
Let's make a famous fractal called the
Sierpinski Triangle.
Step One Draw an equilateral triangle with
sides of 2 triangle lengths each.
Connect the midpoints of each side.
How many equilateral triangles do you now
have?

28. Cut out the triangle in the
center.
Step Two
Draw another equilateral triangle with sides
of 4 triangle lengths each. Connect the
midpoints of the sides and cut out the
triangle in the center as before.

29. The Sierpinski Triangle

30. The Sierpinski Triangle
Unlike the Koch Snowflake, which is
generated with infinite additions, the
Sierpinski triangle is created by infinite
removals. Each triangle is divided into four
smaller, upside down triangles. The center
of the four triangles is removed. As this
process is iterated an infinite number of
times, the total area of the set tends to
infinity as the size of each new triangle
goes to zero.

31. Sierpinski Carpet

32. Sierpinski Carpet Cube

33. Theory of Fractals
Mandelbrot introduced and
developed the theory of fractals --
figures that were truly able to describe
these shapes. The theory was
continued to be used in a variety of
applications. Fractals’ importance is in
areas ranging from special TV effects
to economy and biology.

34. The term fractal was coined by
Benoit Mandelbrot in 1975 in his book
Fractals: Form, Chance, and
Dimension. In 1979, while studying
the Julia set, Mandelbrot discovered
what is now called the Mandelbrot
set and inspired a generation of
mathematicians and computer
programmers in the study of fractals
and fractal geometry.
Mandelbrot’s
discovery

35. The Mandelbrot Set
Named after Benoit Mandelbrot, The
Mandelbrot set is one of the most
famous fractals in existence. It was
born when Mandelbrot was playing
with the simple quadratic equation
z=z2+c. In this equation, both z and c
are complex numbers. In other words,
the Mandelbrot set is the set of all
complex c such that iteration z=z2+c
does not diverge.

36. FRACTALS- Mandelbrot Set
The Mandelbrot Set

37. Mandelbrot Zoom

39. The Julia set

40. The 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, who
studied the iteration of polynomials
and rational functions during the early
twentieth century, making the Julia
set much older than the Mandelbrot
set.

41. Difference between the Julia set
and the Mandelbrot set
The main difference between the Julia set
and the Mandelbrot set is the way in which
the function is iterated. The Mandelbrot set
iterates z=z2+c with z always starting at 0
and varying the c value. The Julia set
iterates z=z2+c for a fixed c value and
varying z values. In other words, the
Mandelbrot set is in the parameter space,
or the c-plane, while the Julia set is in the
dynamical space, or the z-plane.

42. The Lorenz Model

43. Lorenz Model
The Lorenz Model, named after
E. N. Lorenz in 1963, is a model for
the convection of thermal energy.
This model was the very first example
of another important point in chaos
and fractals, dissipative dynamical
systems, otherwise know as strange
attractors.

44. A magnification of the Phoenix Set

45. Hidden Mandarin Fractal
Sterling

46. Fractals in Jewelry

47. Microwaved-DVD

48. FRACTALS IN NATURE

49. Objects in Nature
Many objects in nature aren’t formed of
squares or triangles, but of more
complicated geometric figures. e.g. trees,
ferns, clouds, mountains etc. are shaped
like fractals. Other examples include snow
flakes, crystals, lightning, river networks,
cauliflower or broccoli, and systems of
blood vessels and pulmonary vessels.
Coastlines may also be considered as
fractals in nature.

50. Similarity between fractals and
objects in nature.
One of the largest relationships with real-
life is the similarity between fractals and
objects in nature. The resemblance of
many fractals and their natural counter-
parts is so large that it cannot be
overlooked. Mathematical formulas are
used to model self similar natural forms.
The pattern is repeated at a large scale
and patterns evolve to mimic large scale
real world objects.

51. Fractals in Nature
As fractals are patterns that reveal greater
complexity as it is enlarged, they portray
the notion of worlds within worlds.
Trees and ferns are fractals in nature and
can be modeled on a computer by using a
recursive algorithm. This recursive nature
is obvious in these examples—a branch
from a tree or a frond from a fern is a
miniature replica of the whole: not identical,
but similar in nature. The connection
between fractals and leaves are currently
being used to determine how much carbon
is contained in trees.

52. Fern leave

53. Fractal Trees

54. Tree Decomposition

55. Fractal Julius Tree
The rainbow fractal Julius tree below
was created with the help of the
computer program Fractal Imaginator.

56. Fractals in Architecture

57. Fractals in Lightning

58. Clouds as fractals

59. Rocky mountains as Fractals

61. Waterfall Fractals

62. RIVERS as Fractals

63. Tsunami

64. Sea shell fractal

65. Natural fractal pattern - air
displacing a vacuum formed by
pulling two glue-covered acrylic
sheets apart.

66. Fractal Geometry
Fractal geometry is a new language
used to describe, model and analyze
complex forms found in nature. Chaos
science uses this fractal geometry.
Fractal geometry and chaos theory
are providing us with a new way to
describe the world.

67. 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 for lines,
ellipses, circles, etc. Fractal
geometry, however, is described in
algorithms -- a set of instructions on
how to create a fractal.

68. Applications of fractals in science
Fractals have a variety of applications in science
because its property of self similarity exists
everywhere. They can be used to model plants,
blood vessels, nerves, explosions, clouds,
mountains, turbulence, etc. Fractal geometry
models natural objects more closely than does
other geometries.
Engineers have begun designing and constructing
fractals in order to solve practical engineering
problems. Fractals are also used in computer
graphics and even in composing music.

69. Fractals in music

70. 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.

71. Fractals in Music

72. Special features of fractals
A fractal often has the following features:
It has a fine structure at arbitrarily small scales.
It is too irregular to be easily described in
traditional Euclidean geometric language.
It is self-similar (at least approximately or
stochastically).
It has a Hausdorff dimension which is greater than
its topological dimension (although this
requirement is not met by space-filling curves such
as the Hilbert curve).
It has a simple and recursive definition.

73. Application to biological analysis
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 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. The order in
many ion channels generating the firing pattern
was also evaluated with a fractal dimension,
enabling the high order seen there to be
quantized.

75. Fractal Geometry and The
Golden Mean

76. Real-Life Relevance And
Importance of Fractals and Fractal
Geometry
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.

77. Fractals in Finance and Risk

78. Fractals in Finance
Finance played a crucial role in the
development of fractal theory.
Fractals are used in finance to make
predictions as to the risk involved for
particular stocks.

80. Why does it matter?
How is the stock market associated with a
fractal? Easily, if one looks at the market
price action taking place on the monthly,
weekly, daily and intra day charts where
you will see the structure has a similar
appearance. Followers of this approach
have determined that market prices are
highly random but with a trend. They claim
that stock market success will happen only
by following the trend.

81. Applications of fractals
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.

82. 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,
that is, 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, in films
such as Star Wars and Star Trek. Fractal
images are used as an alternative to costly
elaborate sets to produce fantasy
landscapes.

83. Other Applications of Fractals
As described above, random fractals can be used
to describe many highly irregular real-world
objects. Other applications of fractals include:
Classification of histopathology slides in medicine
Fractal landscape or Coastline complexity
Enzyme/enzymology (Michaelis-Menten kinetics)
Generation of new music
Signal and image compression
Creation of digital photographic enlargements
Seismology
Fractal in soil mechanics

84. Computer and video game design, especially
computer graphics for organic environments and
as part of procedural generation
Fractography and fracture mechanics
Fractal antennas – Small size antennas using
fractal shapes
Small angle scattering theory of fractally rough
systems
T-shirts and other fashion
Generation of patterns for camouflage, such as
MARPAT
Digital sundial
Technical analysis of price series (see Elliott wave
principle)

85. Applications of Fractals in C.Sc.
fractal techniques for data analysis
fractals and databases, data mining
visualization and physical models
automatic object classification
fractal and multi-fractal texture
characterization
shape generation, rendering techniques
and image synthesis
2D, 3D fractal interpolation
image denoising and restoration
image indexing, thumbnail images

86. fractal still image and video compression,
wavelet and fractal transforms,
benchmarking, hardware
watermarking, comparison with other
techniques
biomedical applications
engineering (mechanical & materials,
automotive)
fractal and compilers, VLSI design
internet traffic characterization and
modeling
non classical applications

87. THANK YOU