FRACTALS - INTRODUCTION

The new created word "fractal" was used at the first time by Benoit Mandelbrot in the early 70ies and means an object that is self-similar on a certain length or time scale characterized by a special fractal dimension. The question raised by Mandelbrot and became famous was "How long is the coast of Great Britain". The answer is: there is no clear answer. The length of the coast always depends on the way of measurement.

To make the problem clear let us consider the so-called "Koch's curve". To generate a Koch's curve start with a straight line and split this into three equal parts. Replace the second part in the way as shown in the picture.

0. ______________________________________________________  L=1

                             /\
                            /  \
                           /    \
                          /      \
                         /        \
                        /          \
                       /            \
                      /              \
1. __________________/                \__________________ L=4/3

So you get a curve consisting of four parts. Now split every part into three equal parts and so on. When you repeat that algorithm until infinity you get the Koch's curve.

                             /\
                            /  \
                     ______/    \______
                     \                /
                      \              /
                       \            /
           /\          /            \          /\
          /  \        /              \        /  \
2. ______/    \______/                \______/    \______ L=16/9
                           __/\__
                           \    /
                     __/\__/    \__/\__
                     \                /
                     /_              _\
                       \            /
         __/\__      __/            \__     __/\__
         \    /      \                /     \    /
3. __/\__/    \__/\__/                \_/\__/    \__/\__ L=64/27


4. ... L=256/81

As you can see this curve has an infinit length (L = L₀ * ⁴/₃ * ⁴/₃ * ... ). The Koch's curve can be seen as a special coast line, so if you would measure how long is that coast line you would find out an infinit length. The Koch's Curve is a self-similar object. When you look at the first part of the curve you see that it is similar to the whole curve.

Some other well-known regular fractals are the Sierpinski's triangle and the Menger's sponge. As it's easy to realize the surface area of the Sierpinski's triangle (A = A₀ * ³/₄ * ³/₄ * ... ) goes to zero as well as the volume of the Menger's sponge (V = V₀ * ²⁰/₂₇ * ²⁰/₂₇ * ... ).

It is interesting to find out the dimension of fractal objects. First let us look at some common geometrical objects: line, square and cube. We can reduce the size of that objects by an integer factor and look how many of that new smaller objects do we need to get the original object again. The connection between that two variables is shown in the table, so we get the following equation for the dimension in depending on reduce factor and number of parts: a = ( ¹/_s )^D ==> D = ln ( a ) / ln ( ¹/_s )

Geometrical Object	Reduce Factor s	Number of Parts a	(Fractal) Dimension D
Line	¹/₃	3	1
Square	¹/₃	9	2
Cube	¹/₃	27	3
Koch's curve	¹/₃	4	1.26
Sierpinskie's triangle	¹/₂	3	1.58
Menger's sponge	¹/₃	20	2.73

Fractals are closely connected with deterministical chaos, they are two sides of one medal. To show this let us look at the so-called

Logistic Equation: ( P_n+1 - P_n ) / P_n = R * ( 1 - P_n ).

This equation can be interpreted as a model of population dynamics. P_n is the standardized quantity of population at the time n and P_n+1 at the time n+1. R is the increase factor. In a simple model the term (1-P_n) could be neglected but this would lead to an unlimited growth. In reality there is only a limited room to spread out, so the factor (1-P_n) has to be considered, that means in equilibrium there is a population quantity of P_n=1. Let us start this equation for example with P₀=0.3 and look what happens with the population after a large number of cycles. Will it be a constant, will it show periodical behaviour or will it jump chaotically?
The solution set of the Logistic equation shows that there is a clear solution for small increase factors less than 2.0. When R increases a periodical behaviour of population quantity occures. P is flopping between two states. With increasing R there is a change into period of 4 and and farther into period of 8. With an increase factor of R=2.6 there starts the chaotical behaviour. But it is a deterministical chaos that means there is only a certain range where the system can occur. It is typical for systems in the range of deterministical chaos that a small difference in starting conditions increase rapidly in time, so that it is impossible to predict the development of such a system for a long time. But it can be said for sure that the system is somewhere in the permitted sector. This sector is also called "Attractor".
It is really astonishing that there are islands of stability in the chaotical sector. For R nearby 2.85 there is a relatively broad range of the increase factor with an almost periodical behavior with a period of 3.
The Logistical equation is one of the simplest examples for the transition from order to chaos. This is connected with a transition from linear to nonlinear behavior, that means all nonlinear systems show chaotical behavior.

This is what deterministical chaos and fractals have in common: Deterministical chaos is implicated by nonlinear equations as well as fractals are attractors of nonlinear equations.
The Sierpinski's triangle shall be an example for this: At the beginning of this page it was shown that the Sierpinski's triangle is a fractal. Now we will see that this triangle can be created by a chaotical process. Imagine you have only the three edges of the triangle (1, 2 and 3) and you start for instance at edge 1. Now you create a number 1, 2 or 3 by a random generator and move from the place where you are half the way to the edge with the created number and draw a dot. After that you create a new number and go the half way to the edge with the created number and draw the next dot and so on. The result can be seen in this picture with 100,000 dots created by a Turbo Pascal program.

Disordered structures and random processes that are self-similar on certain length or time scales are very common in nature (my favourite one is cauliflower), so keep your eyes open looking for fractals!

go to the pictures