SlideShare a Scribd company logo
1 of 43
HIDDEN DIMENSIONS OF NATURE(AN INTRODUCTION TO FRACTAL GEOMETRY AND ITS APPLICATIONS)ByMilan A. Joshi & Dr S.M.PadhyeDEPT OF MATHEMATICS SHRI RLT COLLEGE OF SCIENCE AKOLA.Email:- mlnjsh@gmail.com
ABSTRACT:- In this presentation we introduce the very basics of fractal geometry discovered by French/American mathematician Dr Benoit Mandelbrot and its applications. The most amazing thing about fractals is the variety of their applications. Besides theory, they were used to compress data in the Encarta Encyclopedia and to create realistic landscapes in several movies like Star Trek. The places where you can find fractals include almost every part of the universe, from bacteria cultures to galaxies to your body. In this paper, we have picked out the most important applications, trying to include them from as many areas of science and everyday life as possible. Here we list the area where fractals are applied. Astronomy:  Galaxies, Rings of Saturn Bio / Chem.  Bacteria cultures, Chemical Reactions, Human Anatomy, Molecules, Plants, Population Growth  Other:Clouds, Coastlines and Borderlines ,Data Compression, Diffusion, Economy, Fractal ArtFractal Music, Landscapes, Newton's Method, Special Effects (Star Trek),Weather.  
INTRODUCTION:- We are used to Euclidian geometry, where every thing is extremely  regular for example straight lines , circles, triangles, spheres, cones , cylinders, and our regular calculus. We are always scared to study the patterns calling them monsters (Weirstrass nowhere differential function), pathological curve(Koch curve) and rejecting them all the time. But  clouds are not spheres, mountains are not cones coast lines are straight lines ,barks are not regular ,but these patterns are in nature. Then Mandelbrot came up and say “Hey Guys” you can describe these patterns by mathematical formulas  only it requires different kind of formulas. And he gave us a beautiful Mandelbrot set and fractal geometry.
FRACTALS ARE EVERYWHERE
History and Motivation The story begins with the young French  mathematician Gaston Julia(1893 – 1978)  who introduce the problem of  iterated function (IFS)during world war I,  which is just like a regular function except that it performs  over and over again with each out put used as next input. Then he describes Julia sets.  Few Julia sets are
Dr Benoit Mandelbrot born(1924) at Warsaw (Poland) French /American Mathematician student of Gaston Julia has studied Julia's concept of iterated function and has done something that Julia could never do. He took a function f(z) = z2 + c ,for complex variable zand a complex parameter c and started to seethe Patterns emerging out in computer at IBM, (In 1980’s )and what he found is infinitely complex structure which he called as Mandelbrot set. The Mandelbrot set is visual representation of an iterated function on the complex plane
MATHEMATICAL DEFINITION (MANDELBROT SET) Mathematically Mandelbrot set is a set of all complex numbers c for which the orbit of 0 under iteration of the function z      z2 + c, remains unbounded.
When computed and graphed on the complex plane the Mandelbrot Set is seen to have an elaborate boundary which does not simplify at any given magnification. This qualifies the boundary as a fractal.
ZOOMING M-SET
HOW TO GRAPH MANDELBROT SET
PROPERTIS OF MANDELBROT SET It is a compact set. It is contained in the closed disk of radius 2. It is connected. The area of Mandelbrot set is 1.50659177 ± 0.00000008 (Approximately) It is conjectured that the Mandelbrot set is locally connected. It is a fractal. It is a set of all points whose Julia sets are connected.
WHAT IS FRACTAL The word fractal is derived  from a Latin word fractus means broken. It is defined to be geometric figure that repeats itself under several levels of magnification, a shape that appears irregular at all scales of length, e.g. a fern
Fractals’ properties Two of the most important properties of fractals are  self-similarity and non-integer dimension.  What does self-similarity mean?  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.   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. 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..
BROCCLI
FRACTAL IN VEINS
ROUGHNESS- FRACTAL DIMENSION
Types of Fractals    There is an infinite variety of fractals and tons of ways to create them, from formulas to folding paper. We used the ways they are created to split them into several basic categories. Fractals not fitting into any categories were grouped together into nonstandard fractals. Please realize that you might not understand some concepts used in the descriptions if you haven't read the tutorial. However, concepts not discussed in the tutorial were not used in this section as well.  Base-Motif FractalsDusts and ClustersFractal CanopiesIFS FractalsJulia SetsMandelbrot SetsNonstandard FractalsPaper-Folding Fractals Peano CurvesPlasma FractalsPythagoras TreesQuaternionsStar FractalsStrange AttractorsSweeps
FRACTAL DIMENSION Let’s calculate the fractal dimension. To explain the concept of fractal dimension, it is necessary to understand what we mean by dimension in the first place. Obviously, a line has dimension 1, a plane dimension 2, and a cube dimension 3. But why is this? You can think about it. We often say that a line has dimension 1 because there is only 1 way to move on a line. Similarly, the plane has dimension 2 because there are 2 directions in which to move. And space has dimension 3.
With these three examples, you should see a clear pattern. If you take the magnification and raise it to the power of dimension... you will always get the number of shapes! Writing it as a formula, you get:                                 N = rd Since we are trying to calculate the dimension, we should solve this equation for d. If you are familiar with logs, you should easily find that                       d = log N / log r .
DimensionTHE SIMPLEST METHOD One way to calculate fractal dimension is by taking advantage of self-similarity. For example, suppose you have a 1-dimensional line segment. If you look at it with the magnification of 2, you will see 2 identical line segments. Let’s use a variable D for dimension, r for magnification, and N for the number of identical shapes.
DIMENSION Dimension =        log(No of self-similar pieces)                                       log(Scaling factor)
SIERPINSKI TRIANGLE D = log(N)/log(r) = log(3)/log(2) = 1.585. We get a value between 1 and 2.
VON-KOCH CURVE Koch constructed his curve in 1904 as an example of a non-differentiable curve, that is, a continuous curve that does not have a tangent at any of its points. Karl Weierstrass had first demonstrated the existence of such a curve in 1872. The article by Sime  Ungar  provides a simple geometric proof. The length of the intermediate curve at the nth iteration of the construction is (4/3)^n, where n = 0 denotes the original straight line segment. Therefore the length of the Koch curve is infinite. Moreover, the length of the curve between any two points on the curve is also infinite since there is a copy of the Koch curve between any two points
D = log(N)/log(r) D = log(4)/log(3) = 1.26.
KOCH SNOW FLAKE AND SURFACE
APPLICATIONS:-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. Although the first algorithms and ideas are owed to the discoverer of fractals himself, the artistic field of using fractals was started by Richard Voss, who generated the landscapes for Mandelbrot’s book. This sparked the imagination of many artists and producers of science fiction movies. A little later, Loren Carpenter generated a computer movie of a flight over a fractal landscape. He was immediately hired by Pixar,. Fractals were used in the movie Star Trek II: The Wrath of Khan, to generate the landscape of the Genesis planet and also in Return of the Jedi to create the geography of the moons of Endor and the Death Star outline. The success of fractal special effects in these movies lead to making fractals very popular. Today, numerous software allows anyone who only knows some information about computer graphics and fractals to create such art. For example, we ourselves were able to generate all landscapes throughout this website.
LANDESCAPES
MAKING MOUNTAIN SURFACE
STAR TREK
STAR TREK FRACTAL SWIRL
APPLICATION:-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. Using a different term, you can say that the weather behaves very chaotically. This should automatically tell you what we are getting too. 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
Lorenz Attractor
HEART BEAT Heart beat is not constant over time. It fluctuates and fluctuates lot . One of the most powerful application of fractal is in rhythms of heart something that Boston cardiologist Ary Goldberger has been studying in his entire professional life. Initially Galileo stated that normal heart beats like a metronome. But Ary and his colleagues has proved that this theory was wrong .Healthy heart beat has fractal architecture. It has a distinctive fractal pattern. HEART BEAT TIME SERIES
It definitely helps us to understand  lot of things about pattern of heart and one day many cardiologist spot heat problem sooner
Application :- FRACTAL ANTEENA A fractal antenna is an antenna that uses a fractal, self-similar design to 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. Cohen use this concept of fractal antenna. And it is theoretically it is proved that fractal design is the only design which receives multiple signals.
FRACTAL ANTEENA
APPLICATION TO GLOBAL WARMING
REFRENCES:- Fractal geometry of nature: Dr Mandelbrot Color of infinity              : Aurther s documentry www.pbs.org
THANK YOU

More Related Content

What's hot (14)

2. fractal presentation
2. fractal presentation2. fractal presentation
2. fractal presentation
 
Mandelbrot
MandelbrotMandelbrot
Mandelbrot
 
Fractals
Fractals Fractals
Fractals
 
Fractals and symmetry by group 3
Fractals and symmetry by group 3Fractals and symmetry by group 3
Fractals and symmetry by group 3
 
Fractals
FractalsFractals
Fractals
 
Fractals
FractalsFractals
Fractals
 
Paper
PaperPaper
Paper
 
Does God play dice ?
Does God play dice ?Does God play dice ?
Does God play dice ?
 
Fractals And Chaos Theory
Fractals And Chaos TheoryFractals And Chaos Theory
Fractals And Chaos Theory
 
Master's Thesis: Closed formulae for distance functions involving ellipses.
Master's Thesis: Closed formulae for distance functions involving ellipses.Master's Thesis: Closed formulae for distance functions involving ellipses.
Master's Thesis: Closed formulae for distance functions involving ellipses.
 
104
104104
104
 
Lectures on Analytic Geometry
Lectures on Analytic GeometryLectures on Analytic Geometry
Lectures on Analytic Geometry
 
Secrets of fractals dfs-yuc
Secrets of fractals dfs-yucSecrets of fractals dfs-yuc
Secrets of fractals dfs-yuc
 
Ch3
Ch3Ch3
Ch3
 

Similar to HIDDEN DIMENSIONS IN NATURE

Fractals -A fractal is a natural phenomenon or a mathematical set .pdf
Fractals -A fractal is a natural phenomenon or a mathematical set .pdfFractals -A fractal is a natural phenomenon or a mathematical set .pdf
Fractals -A fractal is a natural phenomenon or a mathematical set .pdf
LAMJM
 
Gupte - first year paper_approved (1)
Gupte - first year paper_approved (1)Gupte - first year paper_approved (1)
Gupte - first year paper_approved (1)
Shweta Gupte
 

Similar to HIDDEN DIMENSIONS IN NATURE (20)

Fractal geometry
Fractal geometryFractal geometry
Fractal geometry
 
Fractals a research
Fractals a researchFractals a research
Fractals a research
 
Fractals, Geometry of Nature and Logistic Model
Fractals, Geometry of Nature and Logistic ModelFractals, Geometry of Nature and Logistic Model
Fractals, Geometry of Nature and Logistic Model
 
An Exploration of Fractal Geometry
An Exploration of Fractal GeometryAn Exploration of Fractal Geometry
An Exploration of Fractal Geometry
 
Presentation
PresentationPresentation
Presentation
 
Fractals in physics
Fractals in  physicsFractals in  physics
Fractals in physics
 
Fractals in physics
Fractals in physicsFractals in physics
Fractals in physics
 
Fractal Theory
Fractal TheoryFractal Theory
Fractal Theory
 
Correlation between averages times of random walks on an irregularly shaped o...
Correlation between averages times of random walks on an irregularly shaped o...Correlation between averages times of random walks on an irregularly shaped o...
Correlation between averages times of random walks on an irregularly shaped o...
 
Conformal matching
Conformal matchingConformal matching
Conformal matching
 
Fractal Geometry Course
Fractal Geometry CourseFractal Geometry Course
Fractal Geometry Course
 
05 from flatland to spaceland
05 from flatland to spaceland05 from flatland to spaceland
05 from flatland to spaceland
 
Fractal Analyzer
Fractal AnalyzerFractal Analyzer
Fractal Analyzer
 
the minkowski curve
the minkowski curvethe minkowski curve
the minkowski curve
 
A MATLAB Computational Investigation of the Jordan Canonical Form of a Class ...
A MATLAB Computational Investigation of the Jordan Canonical Form of a Class ...A MATLAB Computational Investigation of the Jordan Canonical Form of a Class ...
A MATLAB Computational Investigation of the Jordan Canonical Form of a Class ...
 
Fractales bartolo luque - curso de introduccion sistemas complejos
Fractales   bartolo luque - curso de introduccion sistemas complejosFractales   bartolo luque - curso de introduccion sistemas complejos
Fractales bartolo luque - curso de introduccion sistemas complejos
 
Fractals -A fractal is a natural phenomenon or a mathematical set .pdf
Fractals -A fractal is a natural phenomenon or a mathematical set .pdfFractals -A fractal is a natural phenomenon or a mathematical set .pdf
Fractals -A fractal is a natural phenomenon or a mathematical set .pdf
 
Order, Chaos and the End of Reductionism
Order, Chaos and the End of ReductionismOrder, Chaos and the End of Reductionism
Order, Chaos and the End of Reductionism
 
Gupte - first year paper_approved (1)
Gupte - first year paper_approved (1)Gupte - first year paper_approved (1)
Gupte - first year paper_approved (1)
 
BSc dissertation np
BSc dissertation npBSc dissertation np
BSc dissertation np
 

More from Milan Joshi (6)

Yolo9000.pdf UFHUFHFUHFUDDFHVDSFHDHNDSHN
Yolo9000.pdf UFHUFHFUHFUDDFHVDSFHDHNDSHNYolo9000.pdf UFHUFHFUHFUDDFHVDSFHDHNDSHN
Yolo9000.pdf UFHUFHFUHFUDDFHVDSFHDHNDSHN
 
Unit 1
Unit 1Unit 1
Unit 1
 
Milan Joshi _Edvancer_R Programer
Milan Joshi _Edvancer_R ProgramerMilan Joshi _Edvancer_R Programer
Milan Joshi _Edvancer_R Programer
 
Graph theory and life
Graph theory and lifeGraph theory and life
Graph theory and life
 
hidden dimension in nature
hidden dimension in naturehidden dimension in nature
hidden dimension in nature
 
Hidden dimensions in nature
Hidden dimensions in natureHidden dimensions in nature
Hidden dimensions in nature
 

Recently uploaded

Spellings Wk 4 and Wk 5 for Grade 4 at CAPS
Spellings Wk 4 and Wk 5 for Grade 4 at CAPSSpellings Wk 4 and Wk 5 for Grade 4 at CAPS
Spellings Wk 4 and Wk 5 for Grade 4 at CAPS
AnaAcapella
 
SPLICE Working Group: Reusable Code Examples
SPLICE Working Group:Reusable Code ExamplesSPLICE Working Group:Reusable Code Examples
SPLICE Working Group: Reusable Code Examples
Peter Brusilovsky
 
SURVEY I created for uni project research
SURVEY I created for uni project researchSURVEY I created for uni project research
SURVEY I created for uni project research
CaitlinCummins3
 

Recently uploaded (20)

ĐỀ THAM KHẢO KÌ THI TUYỂN SINH VÀO LỚP 10 MÔN TIẾNG ANH FORM 50 CÂU TRẮC NGHI...
ĐỀ THAM KHẢO KÌ THI TUYỂN SINH VÀO LỚP 10 MÔN TIẾNG ANH FORM 50 CÂU TRẮC NGHI...ĐỀ THAM KHẢO KÌ THI TUYỂN SINH VÀO LỚP 10 MÔN TIẾNG ANH FORM 50 CÂU TRẮC NGHI...
ĐỀ THAM KHẢO KÌ THI TUYỂN SINH VÀO LỚP 10 MÔN TIẾNG ANH FORM 50 CÂU TRẮC NGHI...
 
AIM of Education-Teachers Training-2024.ppt
AIM of Education-Teachers Training-2024.pptAIM of Education-Teachers Training-2024.ppt
AIM of Education-Teachers Training-2024.ppt
 
diagnosting testing bsc 2nd sem.pptx....
diagnosting testing bsc 2nd sem.pptx....diagnosting testing bsc 2nd sem.pptx....
diagnosting testing bsc 2nd sem.pptx....
 
Spellings Wk 4 and Wk 5 for Grade 4 at CAPS
Spellings Wk 4 and Wk 5 for Grade 4 at CAPSSpellings Wk 4 and Wk 5 for Grade 4 at CAPS
Spellings Wk 4 and Wk 5 for Grade 4 at CAPS
 
How To Create Editable Tree View in Odoo 17
How To Create Editable Tree View in Odoo 17How To Create Editable Tree View in Odoo 17
How To Create Editable Tree View in Odoo 17
 
SPLICE Working Group: Reusable Code Examples
SPLICE Working Group:Reusable Code ExamplesSPLICE Working Group:Reusable Code Examples
SPLICE Working Group: Reusable Code Examples
 
Book Review of Run For Your Life Powerpoint
Book Review of Run For Your Life PowerpointBook Review of Run For Your Life Powerpoint
Book Review of Run For Your Life Powerpoint
 
Major project report on Tata Motors and its marketing strategies
Major project report on Tata Motors and its marketing strategiesMajor project report on Tata Motors and its marketing strategies
Major project report on Tata Motors and its marketing strategies
 
24 ĐỀ THAM KHẢO KÌ THI TUYỂN SINH VÀO LỚP 10 MÔN TIẾNG ANH SỞ GIÁO DỤC HẢI DƯ...
24 ĐỀ THAM KHẢO KÌ THI TUYỂN SINH VÀO LỚP 10 MÔN TIẾNG ANH SỞ GIÁO DỤC HẢI DƯ...24 ĐỀ THAM KHẢO KÌ THI TUYỂN SINH VÀO LỚP 10 MÔN TIẾNG ANH SỞ GIÁO DỤC HẢI DƯ...
24 ĐỀ THAM KHẢO KÌ THI TUYỂN SINH VÀO LỚP 10 MÔN TIẾNG ANH SỞ GIÁO DỤC HẢI DƯ...
 
Graduate Outcomes Presentation Slides - English (v3).pptx
Graduate Outcomes Presentation Slides - English (v3).pptxGraduate Outcomes Presentation Slides - English (v3).pptx
Graduate Outcomes Presentation Slides - English (v3).pptx
 
When Quality Assurance Meets Innovation in Higher Education - Report launch w...
When Quality Assurance Meets Innovation in Higher Education - Report launch w...When Quality Assurance Meets Innovation in Higher Education - Report launch w...
When Quality Assurance Meets Innovation in Higher Education - Report launch w...
 
Stl Algorithms in C++ jjjjjjjjjjjjjjjjjj
Stl Algorithms in C++ jjjjjjjjjjjjjjjjjjStl Algorithms in C++ jjjjjjjjjjjjjjjjjj
Stl Algorithms in C++ jjjjjjjjjjjjjjjjjj
 
male presentation...pdf.................
male presentation...pdf.................male presentation...pdf.................
male presentation...pdf.................
 
Including Mental Health Support in Project Delivery, 14 May.pdf
Including Mental Health Support in Project Delivery, 14 May.pdfIncluding Mental Health Support in Project Delivery, 14 May.pdf
Including Mental Health Support in Project Delivery, 14 May.pdf
 
Mattingly "AI and Prompt Design: LLMs with NER"
Mattingly "AI and Prompt Design: LLMs with NER"Mattingly "AI and Prompt Design: LLMs with NER"
Mattingly "AI and Prompt Design: LLMs with NER"
 
Observing-Correct-Grammar-in-Making-Definitions.pptx
Observing-Correct-Grammar-in-Making-Definitions.pptxObserving-Correct-Grammar-in-Making-Definitions.pptx
Observing-Correct-Grammar-in-Making-Definitions.pptx
 
SURVEY I created for uni project research
SURVEY I created for uni project researchSURVEY I created for uni project research
SURVEY I created for uni project research
 
An overview of the various scriptures in Hinduism
An overview of the various scriptures in HinduismAn overview of the various scriptures in Hinduism
An overview of the various scriptures in Hinduism
 
TỔNG HỢP HƠN 100 ĐỀ THI THỬ TỐT NGHIỆP THPT TOÁN 2024 - TỪ CÁC TRƯỜNG, TRƯỜNG...
TỔNG HỢP HƠN 100 ĐỀ THI THỬ TỐT NGHIỆP THPT TOÁN 2024 - TỪ CÁC TRƯỜNG, TRƯỜNG...TỔNG HỢP HƠN 100 ĐỀ THI THỬ TỐT NGHIỆP THPT TOÁN 2024 - TỪ CÁC TRƯỜNG, TRƯỜNG...
TỔNG HỢP HƠN 100 ĐỀ THI THỬ TỐT NGHIỆP THPT TOÁN 2024 - TỪ CÁC TRƯỜNG, TRƯỜNG...
 
VAMOS CUIDAR DO NOSSO PLANETA! .
VAMOS CUIDAR DO NOSSO PLANETA!                    .VAMOS CUIDAR DO NOSSO PLANETA!                    .
VAMOS CUIDAR DO NOSSO PLANETA! .
 

HIDDEN DIMENSIONS IN NATURE

  • 1. HIDDEN DIMENSIONS OF NATURE(AN INTRODUCTION TO FRACTAL GEOMETRY AND ITS APPLICATIONS)ByMilan A. Joshi & Dr S.M.PadhyeDEPT OF MATHEMATICS SHRI RLT COLLEGE OF SCIENCE AKOLA.Email:- mlnjsh@gmail.com
  • 2. ABSTRACT:- In this presentation we introduce the very basics of fractal geometry discovered by French/American mathematician Dr Benoit Mandelbrot and its applications. The most amazing thing about fractals is the variety of their applications. Besides theory, they were used to compress data in the Encarta Encyclopedia and to create realistic landscapes in several movies like Star Trek. The places where you can find fractals include almost every part of the universe, from bacteria cultures to galaxies to your body. In this paper, we have picked out the most important applications, trying to include them from as many areas of science and everyday life as possible. Here we list the area where fractals are applied. Astronomy: Galaxies, Rings of Saturn Bio / Chem. Bacteria cultures, Chemical Reactions, Human Anatomy, Molecules, Plants, Population Growth Other:Clouds, Coastlines and Borderlines ,Data Compression, Diffusion, Economy, Fractal ArtFractal Music, Landscapes, Newton's Method, Special Effects (Star Trek),Weather.  
  • 3. INTRODUCTION:- We are used to Euclidian geometry, where every thing is extremely regular for example straight lines , circles, triangles, spheres, cones , cylinders, and our regular calculus. We are always scared to study the patterns calling them monsters (Weirstrass nowhere differential function), pathological curve(Koch curve) and rejecting them all the time. But clouds are not spheres, mountains are not cones coast lines are straight lines ,barks are not regular ,but these patterns are in nature. Then Mandelbrot came up and say “Hey Guys” you can describe these patterns by mathematical formulas only it requires different kind of formulas. And he gave us a beautiful Mandelbrot set and fractal geometry.
  • 5. History and Motivation The story begins with the young French mathematician Gaston Julia(1893 – 1978) who introduce the problem of iterated function (IFS)during world war I, which is just like a regular function except that it performs over and over again with each out put used as next input. Then he describes Julia sets. Few Julia sets are
  • 6.
  • 7.
  • 8. Dr Benoit Mandelbrot born(1924) at Warsaw (Poland) French /American Mathematician student of Gaston Julia has studied Julia's concept of iterated function and has done something that Julia could never do. He took a function f(z) = z2 + c ,for complex variable zand a complex parameter c and started to seethe Patterns emerging out in computer at IBM, (In 1980’s )and what he found is infinitely complex structure which he called as Mandelbrot set. The Mandelbrot set is visual representation of an iterated function on the complex plane
  • 9. MATHEMATICAL DEFINITION (MANDELBROT SET) Mathematically Mandelbrot set is a set of all complex numbers c for which the orbit of 0 under iteration of the function z z2 + c, remains unbounded.
  • 10. When computed and graphed on the complex plane the Mandelbrot Set is seen to have an elaborate boundary which does not simplify at any given magnification. This qualifies the boundary as a fractal.
  • 12. HOW TO GRAPH MANDELBROT SET
  • 13. PROPERTIS OF MANDELBROT SET It is a compact set. It is contained in the closed disk of radius 2. It is connected. The area of Mandelbrot set is 1.50659177 ± 0.00000008 (Approximately) It is conjectured that the Mandelbrot set is locally connected. It is a fractal. It is a set of all points whose Julia sets are connected.
  • 14. WHAT IS FRACTAL The word fractal is derived from a Latin word fractus means broken. It is defined to be geometric figure that repeats itself under several levels of magnification, a shape that appears irregular at all scales of length, e.g. a fern
  • 15. Fractals’ properties Two of the most important properties of fractals are self-similarity and non-integer dimension. What does self-similarity mean? 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. 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. 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..
  • 19. Types of Fractals    There is an infinite variety of fractals and tons of ways to create them, from formulas to folding paper. We used the ways they are created to split them into several basic categories. Fractals not fitting into any categories were grouped together into nonstandard fractals. Please realize that you might not understand some concepts used in the descriptions if you haven't read the tutorial. However, concepts not discussed in the tutorial were not used in this section as well. Base-Motif FractalsDusts and ClustersFractal CanopiesIFS FractalsJulia SetsMandelbrot SetsNonstandard FractalsPaper-Folding Fractals Peano CurvesPlasma FractalsPythagoras TreesQuaternionsStar FractalsStrange AttractorsSweeps
  • 20. FRACTAL DIMENSION Let’s calculate the fractal dimension. To explain the concept of fractal dimension, it is necessary to understand what we mean by dimension in the first place. Obviously, a line has dimension 1, a plane dimension 2, and a cube dimension 3. But why is this? You can think about it. We often say that a line has dimension 1 because there is only 1 way to move on a line. Similarly, the plane has dimension 2 because there are 2 directions in which to move. And space has dimension 3.
  • 21.
  • 22. With these three examples, you should see a clear pattern. If you take the magnification and raise it to the power of dimension... you will always get the number of shapes! Writing it as a formula, you get: N = rd Since we are trying to calculate the dimension, we should solve this equation for d. If you are familiar with logs, you should easily find that d = log N / log r .
  • 23. DimensionTHE SIMPLEST METHOD One way to calculate fractal dimension is by taking advantage of self-similarity. For example, suppose you have a 1-dimensional line segment. If you look at it with the magnification of 2, you will see 2 identical line segments. Let’s use a variable D for dimension, r for magnification, and N for the number of identical shapes.
  • 24. DIMENSION Dimension = log(No of self-similar pieces) log(Scaling factor)
  • 25. SIERPINSKI TRIANGLE D = log(N)/log(r) = log(3)/log(2) = 1.585. We get a value between 1 and 2.
  • 26. VON-KOCH CURVE Koch constructed his curve in 1904 as an example of a non-differentiable curve, that is, a continuous curve that does not have a tangent at any of its points. Karl Weierstrass had first demonstrated the existence of such a curve in 1872. The article by Sime Ungar provides a simple geometric proof. The length of the intermediate curve at the nth iteration of the construction is (4/3)^n, where n = 0 denotes the original straight line segment. Therefore the length of the Koch curve is infinite. Moreover, the length of the curve between any two points on the curve is also infinite since there is a copy of the Koch curve between any two points
  • 27. D = log(N)/log(r) D = log(4)/log(3) = 1.26.
  • 28. KOCH SNOW FLAKE AND SURFACE
  • 29. APPLICATIONS:-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. Although the first algorithms and ideas are owed to the discoverer of fractals himself, the artistic field of using fractals was started by Richard Voss, who generated the landscapes for Mandelbrot’s book. This sparked the imagination of many artists and producers of science fiction movies. A little later, Loren Carpenter generated a computer movie of a flight over a fractal landscape. He was immediately hired by Pixar,. Fractals were used in the movie Star Trek II: The Wrath of Khan, to generate the landscape of the Genesis planet and also in Return of the Jedi to create the geography of the moons of Endor and the Death Star outline. The success of fractal special effects in these movies lead to making fractals very popular. Today, numerous software allows anyone who only knows some information about computer graphics and fractals to create such art. For example, we ourselves were able to generate all landscapes throughout this website.
  • 34. APPLICATION:-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. Using a different term, you can say that the weather behaves very chaotically. This should automatically tell you what we are getting too. 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
  • 36. HEART BEAT Heart beat is not constant over time. It fluctuates and fluctuates lot . One of the most powerful application of fractal is in rhythms of heart something that Boston cardiologist Ary Goldberger has been studying in his entire professional life. Initially Galileo stated that normal heart beats like a metronome. But Ary and his colleagues has proved that this theory was wrong .Healthy heart beat has fractal architecture. It has a distinctive fractal pattern. HEART BEAT TIME SERIES
  • 37. It definitely helps us to understand lot of things about pattern of heart and one day many cardiologist spot heat problem sooner
  • 38. Application :- FRACTAL ANTEENA A fractal antenna is an antenna that uses a fractal, self-similar design to 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. Cohen use this concept of fractal antenna. And it is theoretically it is proved that fractal design is the only design which receives multiple signals.
  • 41.
  • 42. REFRENCES:- Fractal geometry of nature: Dr Mandelbrot Color of infinity : Aurther s documentry www.pbs.org