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:- email@example.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.
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.
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..
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
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.
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
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.