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
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|
Fractals are closely connected with deterministical chaos, they are
two sides of one medal. To show this let us look at the so-called
( Pn+1 - Pn ) / Pn = R * ( 1 - Pn ).
This equation can be interpreted as a model of population dynamics. Pn is the
standardized quantity of population at the time n and Pn+1 at the
time n+1. R is the increase factor. In a simple model the term (1-Pn)
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-Pn) has to be considered, that means in
equilibrium there is a population quantity of Pn=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
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
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!
(c) Lutz Tautenhahn 1994, 1/99
go to the pictures