FractalGeometry.docx

1、FractalGeometryFractals: Useful Beauty(General Introduction to Fractal Geometry)Return to index BBM Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line.Benoit MandelbrotEdyta Patrzalek , Stan Ackermans Inst

2、itute, IPO, Centre for User-System Interaction, Eindhoven University of Technology AbstractFractals 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

3、. 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 defi

4、ne 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. IntroductionMany people are fascinated by the beautiful images termed

5、 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 exist

6、ing mathematical descriptions of many natural forms, such as coastlines, mountains or parts of living organisms. 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

7、, 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

8、 discovered one of the main properties of fractals. Fractals propertiesTwo 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

9、 - 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

10、 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 us

11、ing 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. Li

12、kewise, 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

13、 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.Complex number fractalsBefore describing this type of fractal, I decided to explain briefly the theory of complex numbers.A compl

14、ex 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 coordinates of the point are a (horizontal - real axis) and b (vertical - imaginary axis). The unit of imaginary numbers: .Two l

15、eading 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

16、 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 Julias work, and with the aid of computer graphics, IBM employee Man

17、delbrot was able to show the first pictures of the most beautiful fractals known today. Mandelbrot setThe 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 t

18、wo categories: points inside the Mandelbrot set, 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. It is also possible to assign a color to the points outside the Mandelbrot set. Their colors depend on ho

19、w 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

20、previous expression: using zero as the value of , we obtain C as the result. The next step consists of assigning the result to 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 represe

21、nted 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 Mande

22、lbrot 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 t

23、he plane are attracted by both: infinity and the Mandelbrot set. That makes it easy to understand why: points far from the Mandelbrot set rapidly move towards infinity, points close to the Mandelbrot set slowly escape to infinity, points inside the Mandelbrot set never escape to infinity. Julia sets

24、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

25、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

26、Julia set. How is a Julia set created? 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

27、 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 a

28、nd 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 Ma

29、ndelbrot 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 is connected; the Julia set associated with is not connected (see picture below). It

30、erated 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 pat

31、tern 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 theo

32、ry, this procedure can be repeated an infinite number of times). The most famous ISF fractals are the Sierpinski Triangle and the Koch Snowflake.Sierpinski TriangleThis 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:1) 2) 3) 4)