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Table of contents
However there are many other methods for creating this shape — and here are just a few. The Sierpinski pattern will continue, producing bigger and bigger triangles.
In the Chaos Game, we start with an empty triangle and select a random point in the middle. We then choose one of the three vertices of the triangle at random, and mark the point at the centre of the line from the random point to the vertex. Then we repeat the process, starting with that new point…. The cellular automaton is a grid consisting of black and white squares.
We start with one black square in the first row; the squares in all following rows are coloured automatically depending on the three squares immediately above. The eight rules at the top determine what a square red will look like, depending on the three squares above. Modify the rules by clicking them, and try to find the set of rules that produces something like the Sierpinski Gasket. Here is how you can create it:. We repeat this process for every point in the coordinate system.
The collection of all the black points is the Mandelbrot set. Move the blue pin below to explore what happens at various points:. This sequence will always increase and tends to infinity, so 1 is not part of the Mandelbrot set. The point is coloured white. This sequence is always bounded, so 1 is part of the Mandelbrot set.
The point is coloured black. A computer can do these computations very quickly for millions of numbers c — like all pixels on a screen. The code required is simple, but the resulting fractal is unbelievable complex. But using mathematics he was able to predict its complexity. The first computer generated image of the Mandelbrot set was produced by an IBM supercomputers in ; today everybody can do the same calculations on a normal laptop. Black points in the image below are part of the Mandelbrot set. Coloured areas are not in the Mandelbrot set, and the colour indicates the speed with which the respective sequence of complex numbers diverges tends to infinity.
However, there are many shapes in nature which are very similar to fractals:. These shapes appear to be completely random, but — as with fractals — there is an underlying pattern that determines how the shapes are formed and what they will look like.
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 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 s, after publishing his paper on the iteration of a rational function, Julia became famous. However, after his death, he was forgotten. 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:.
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. 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:. 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 represented as the "migration" of the initial point C across the plane.
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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.
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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.
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.
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 have to iterate the function using. What happens to the initial point Z when the formula is iterated? 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 is connected; the Julia set associated with is not connected see picture below. 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:. Sierpinski Triangle. In he invented the word fractal to describe his discoveries. Extending fractals into the plane of complex numbers followed in But the breakthrough that made them famous was the ability of computers to plot them in a way that is easy on the eye.
Introduction to Fractal Geometry
Thus were launched the posters, the cards and the T-shirts. Before all this Dr Mandelbrot worked in the obscurity that modern mathematicians have resigned themselves to. He had followed, albeit belatedly, a path familiar to Jewish intellectuals driven from eastern Europe by the rise of the Nazis.
Once there, he worked for IBM. Among other things, he modelled electrical noise. Which, it turns out, is fractal. That it was the transmogrification of his formula by computers which brought him fame is thus appropriate. For a time, fractals seemed the answer to everything: the shape of clouds, the growth of organisms, even why the night sky is dark. Then the world lost interest.
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Perhaps it should not have. If Dr Mandelbrot's belief was correct, trading models based on Gauss's distribution are wrong. That markets are not Gaussian has now been accepted. Dr Mandelbrot's interpretation, however, has not. Even if it had been, the bankers might not have noticed. They preferred algorithms to geometry.