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| Fractals | |
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 What are
fractals?
The properties of fractals What
else are fractals?
Logo fractals
Web resources Definitions
Fractal zooms Fractal Landscape generator
Moving fractal trees 4D
Magic Cube |
| A brief summary of the mathematics of fractals. | |
What are fractals?In order to understand fractals you need to understand a little bit of mathematics concerning complex numbers and the process of iteration. First you need to know about the special number known as i. You can do arithmetic with i. A complex number is a number made up of two parts added together. These parts are called the "real" part and the "imaginary" part. Examples of complex numbers are: 4 - 6i, 2.1 + 4.7i, 1-i, and so on... In fact any number of the form a + bi, where a and b are real numbers, and i is as above, is a complex number. Addition and multiplication of these numbers is achieved using the conventional rules of algebra. Complex numbers have a size. It is usually called their modulus. The idea of modulus is of central importance to fractals. It can be better illustrated by drawing pictures of complex numbers. Such pictures are called Argand diagrams. An iterative process is one by which a rule is applied in a repetitive way to produce a sequence of numbers. For example, the rule "add 2 then divide by 2", when applied to the starting value 30, produces the sequence: 30, 16, 9, 5.5, 3,75 and so on. Thus an iterative process can go on for ever. Sometimes the numbers in the sequence show convergence, sometimes divergence. It is useful to refer to the terms in sequences like these by using a special notation. The first term is called z0, the second is z1, the third is z3, and so on. Now that you have some understanding of what complex numbers are, how they can be combined by adding and multiplying, and shown on a diagram, we can introduce the idea of applying an iterative process to a complex number. It is this process that creates the fractal. Here goes... |
| Let the first number in our
sequence, z0, be 0. Now, from your argand diagram, choose a point
close to the origin, say (0.37,0.4). This is the complex number 0.37 + 0.4i.
Call this number c. We are going to decide whether this single point c should
be coloured black or white. The next number in the iterative sequence is z1. Now, z1=(z0)2 + c. Notice that (z0)2=0 so z1=0.37+0.4i The next number in our sequence is z2: z2=(z1)2 + c. You'll find that z2=0.347 + 0.696i The next number, z3=(z2)2 + c You carry this process on, over and over again and thus create a sequence of complex numbers z0, z1, z2, z3 and so on, where zn=(zn-1)2 + c After each step, find the modulus, or size, of the complex number zn obtained. Keep a record of all these moduli. If it turns out that these moduli are diverging, or to be more precise, become larger than 1, then the process is stopped. The original point representing c, (0.37, 0.4) on the argand diagram, is coloured white. If on the other hand, all the moduli keep less than 1, then the point is coloured black. Start this whole process again for another point on the argand diagram. Find out whether it should be coloured black or white. Carry on the process for every point on the argand diagram (actually only those fairly close to the origin). You will eventually obtain a picture. This picture is called a fractal. With the process described here (there are others based on different iterative rules) the picture is known as the Mandelbrot set, after Benoit Mandelbrot who worked at IBM in the seventies and eighties on iterative processes. |
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So a fractal is essentially a graph of an iterative process applied to complex numbers. What about the colour fractals? When using the above iterative process, keep a record of the number of iterations that were required before the modulus of the complex number c became bigger than 1. Make up a rule to associate this number with a colour. Then make the original point c this colour. How can you be sure that, in the above iterative process, all the moduli keep less than 1? You cannot keep checking each and every one of the terms in the infinite sequence of values of zn, so you don't! You just check the first 100 or 500 or 1000 terms and hope that gives a sufficiently accurate picture.
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The properties of fractals
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The essential and most fascinating property of
any fractal is its complexity. The rule for creating one is essentially simple
- A-Level mathematics no more. But the resulting picture has suprising depth.
If you zoom in on any part of a fractal, you find the same amount of
detail as before. It does not simplify. You find echoes of larger shapes
appearing within smaller parts of the shape. If you zoom in further, the same
thing happens. You never seem to get down to the skeleton of the picture, just
detail upon detail. Look at our fractal
zooms page for illustrations of this. There are many computer programs
available for you to do this yourself. Fractint is probably the best. It
is freely available over the Web. See Internet resources below for details.
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What else are fractals?
What else?
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DefinitionsiThe special number i is actually the square root of -1. That's right. You thought there was no such number. How can you have the square root of a negative number? Well you can, and it's known as i. |
Arithmetic with i Because i is the square root of -1, it is true that i2=-1 and i3=-i, and so on. Also i + i=2i and 5i - 8i=-3i. In fact you can add and multiply imaginary numbers like these just like you would in normal algebra. |
| Real
part The real part is an ordinary number. This could be an integer, fraction, decimal, positive or negative number. Examples are 6, -5, 10.5, 3/4 and so on. |
Imaginary
part The imaginary part is more tricky to explain. If we let the letter i stand for the square root of -1, ie., i=square root (-1), then any number like 3i, -5i, 1.77i, i/5 is an imaginary number. |
| Addition You add these numbers just like you do in algebra in general, for example 3+2i + 4-5i=7-3i and 1.7-4.2i + 1.1-1.7i=2.8 - 5.9i. Multiplication Modulus |
Argand diagrams An argand diagram is a graph. A pair of axes are drawn, the horizontal axis shows the real part of the complex number, the vertical axes shows the imaginary part of the number. The complex number itself can then be plotted as a point. For example, 5 + 2i is plotted at the point (5,2). The modulus of this complex number is the distance from the origin to the point (5,2).  |
| Convergence A sequence is said to converge when its terms are creeping closer and closer still to some fixed value, called the limit. An example is the sequence 1.1, 1.01, 1.001, 1.0001, 1.00001 ... where the terms are tending towards 1. |
Divergence A sequence is said to diverge when its terms are getting increasingly bigger and bigger until eventually they become too large to even write down. Here is a divergent sequence: 1, 2, 4, 8, 16, 32, 64, 128... |
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