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Fractals: Useful Beauty (General Introduction to Fractal Geometry) |
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Benoit Mandelbrot |
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Edyta Patrzalek , Stan Ackermans
Institute, |
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Fractals is a new branch of mathematics and art. Perhaps this is the reason why most people recognize fractals only as pretty pictures useful as backgrounds on the computer screen or original postcard patterns. But what are they really? Most physical systems of nature and many human artifacts are not regular geometric shapes of the standard geometry derived from Euclid. Fractal geometry offers almost unlimited waysof describing, measuring and predicting these natural phenomena. But is it possible to define the whole world using mathematical equations? This article describes how the four most famous fractals were created and explains the most important fractal properties, which make fractals useful for different domain of science.
Introduction Many people are fascinated by
the beautiful images termed fractals. Extending beyond the typical perception of
mathematics as a body of complicated, boring formulas, fractal geometry mixes art
with mathematics to demonstrate that equations are more than just a collection of
numbers. What makes fractals even more interesting is that they are the best existing
mathematical descriptions Although fractal geometry is closely connected with computer techniques, some people had worked on fractals long before the invention of computers. Those people were British cartographers, who encountered the problem in measuring the length of Britain coast. The coastline measured on a large scale map was approximately half the length of coastline measured on a detailed map. The closer they looked, the more detailed and longer the coastline became. They did not realize that they had discovered one of the main properties of fractals.
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.The non-integer dimension is more difficult to explain. Classical geometry deals with objects of integer dimensions: zero dimensional points, one dimensional lines and curves, two dimensional plane figures such as squares and circles, and three dimensional solids such as cubes and spheres. However, many natural phenomena are better described using a dimension between two whole numbers. So while a straight line has a dimension of one, a fractal curve will have a dimension between one and two, depending on how much space it takes up as it twists and curves. The more the flat fractal fills a plane, the closer it approaches two dimensions. Likewise, a "hilly fractal scene" will reach a dimension somewhere between two and three. So a fractal landscape made up of a large hill covered with tiny mounds would be close to the second dimension, while a rough surface composed of many medium-sized hills would be close to the third dimension. There are a lot of different types of fractals. In this paper I will present two of the most popular types: complex number fractals and Iterated Function System (IFS) fractals.
Before describing this type of fractal, I decided to explain briefly the theory of complex numbers. A complex number consists of a real number added to an imaginary number. It is common to refer to a complex number
as a "point" on the complex plane. If the complex number is
The unit of imaginary numbers:  .
Two leading researchers in the field of complex number fractals are Gaston Maurice Julia and Benoit Mandelbrot. Gaston Maurice Julia was born at the end of 19th century in Algeria. He spent his life studying the iteration of polynomials and rational functions. Around the 1920s, after publishing his paper on the iteration of a rational function, Julia became famous. However, after his death, he was forgotten. In the 1970s, the work of Gaston Maurice Julia was revived and popularized by the Polish-born Benoit Mandelbrot. Inspired by Julia’s work, and with the aid of computer graphics, IBM employee Mandelbrot was able to show the first pictures of the most beautiful fractals known today. Mandelbrot set The Mandelbrot set is the set of points on a complex plain. To build the Mandelbrot set, we have to use an algorithm based on the recursive formula:
separating the points of the complex plane into two categories:
The image below shows a portion of the complex plane. The points of the Mandelbrot set have been colored black.
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.
How is the Mandelbrot set created? 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:
After calculating the value of previous expression:
using zero as the value of  and repeating the calculation:
now the result is the complex number  . Then we
have to assign the value to  and 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:
Julia sets are strictly connected with the Mandelbrot set. 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 . If
we want to make a picture of a Julia set, C
must be constant during the whole generation
process, while the value of  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.
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 function
using  . 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:
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
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:
The most famous ISF fractals are the Sierpinski Triangle and the Koch Snowflake. 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:
Using this fractal as an example, we can prove that the fractal dimension is not an integer. First of all we have to find out how the "size" of an object behaves when its linear dimension increases. In one dimension we can consider a line segment. If the linear dimension of the line segment is doubled, then the length (characteristic size) of the line has doubled also. In two dimensions, if the linear dimensions of a square for example is doubled then the characteristic size, the area, increases by a factor of 4. In three dimensions, if the linear dimension of a box is doubled then the volume increases by a factor of 8.This relationship between dimension D , linear scaling L and the result of size increasing S can be generalized and written as:
Rearranging of this formula gives an expression for dimension depending on how the size changes as a function of linear scaling:
In the examples above the value of D is an integer - 1, 2, or 3 - depending on the dimension of the geometry. This relationship holds for all Euclidean shapes. How about fractals?Looking at the picture of the first step in building the Sierpinski Triangle, we can notice that if the linear dimension of the basis triangle ( L) is doubled, then the area of whole fractal (blue triangles) increases by a factor of three ( S).
Using the pattern given above, we can calculate a dimension for the Sierpinski Triangle:
The result of this calculation proves the non-integer fractal dimension. To construct the Koch Snowflake, we have to begin with an equilateral triangle with sides of length, for example, 1. In the middle of each side, we will add a new triangle one-third the size; and repeat this process for an infinite number of iterations. The length of the boundary is -infinity. However, the area remains less than the area of a circle drawn around the original triangle. That means that an infinitely long line surrounds a finite area. The end construction of a Koch Snowflake resembles the coastline of a shore.
Four steps of Koch Snowflake construction:
Another IFS 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. 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. Fractals in the Biological Sciences Biologists have traditionally modeled nature using Euclidean representations of natural objects or series. They represented heartbeats as sine waves, conifer trees as cones, animal habitats as simple areas, and cell membranes as curves or simple surfaces. However, scientists have come to recognize that many natural constructs are better characterized using fractal geometry. Biological systems and processes are typically characterized by many levels of substructure, with the same general pattern repeated in an ever-decreasing cascade. Scientists discovered that the basic architecture of a chromosome is tree-like; every chromosome consists of many 'mini-chromosomes', and therefore can be treated as fractal. For a human chromosome, for example, a fractal dimension D equals 2,34 (between the plane and the space dimension).Self-similarity has been found also in DNA sequences. In the opinion of some biologists fractal properties of DNA can be used to resolve evolutionary relationships in animals. Perhaps in the future biologists will use the fractal geometry to create comprehensive models of the patterns and processes observed in nature. 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.
Many scientists have found that fractal geometry is a powerful tool for uncovering secrets from a wide variety of systems and solving important problems in applied science. The list of known physical fractal systems is long and growing rapidly. Fractals improved our precision in describing and classifying "random" or organic objects, but maybe they are not perfect. Maybe they are just closer to our natural world, not the same as it. Some scientists still believe that true randomness does exist, and no mathematical equation will ever describe it perfectly. So far, there is no way to say who is right and who is wrong. Perhaps for many people fractals will never represent anything more than beautiful pictures.
Bibliography Lewis R. Fractals In Your Future. Chapter 1. Ontario 2000. Mandelbrot, B.B. The Fractal Geometry Of Nature. San Francisco 1982. Turner, M.J. Modeling Nature With Fractals. Leicester 2000 (Click here) |
, the coordinates of the point are
,
, we obtain 
; 
is connected; the Julia set
associated with 
is not connected (see picture
below). 
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