Making fractals

Date: Sat, 8 Jul 1995 15:26:28 -0400
From: Anonymous
Subject: How do you make a fractal ?

Question: How do you make a fractal?

-- Gustavo Moreira

Date: Mon, 10 Jul 1995 09:34:05 -0400 (EDT)
From: Heather Mateyak
Subject: Re: How do you make a fractal ?

Hi Gustavo,

Fractals are part of the mathematical field called dynamical systems, or,
sometimes, chaos theory.  Any object that does not have an integer
dimension (1D, 2D, etc.) is called a fractal.  Some fractals are
very geometrical in shape, and these are the easiest to understand without
much difficult math to go along with them.  

In one of these types of figures, the von Koch curve, you start
with a triangle, and then add a triangle onto each side with sidelength 1/3
as long as the original size of the triangle's size. The length of each
side of the triangle increases by 4/3 with each new addition of a
triangle onto each side of the previous form.  If you are familiar with
exponential growth, you will see that if something increases by 4/3 each
time, 4/3 x 4/3 X 4/3 ...., or (4/3) to the nth power, this number goes off
to infinity as n gets infinitely large.  To find the area enclosed by this
curve, you can look at the area of the shape at each step.  Take the length
of a triangle side to be 1.  Figure the area of the triangle (you need to
use some trigonometry to find the height, and then use 1/2 x base x
height).  Now add the triangle to each side of the previous form, and 
calculate the area.  If you keep doing this for a while, you may notice a
pattern.  If you want to try this, you will need to know something about
geometric series.  It's a really interesting problem, though.

There are many other kinds of fractals, simpler than the von Koch curve.
Take a line segment of length 1.  In the next step, remove the middle third
of the line.  In the next step, remove the middle thirds of the remaining
segments.  If you do this forever, you have a set that is called the
Cantor Ternary Set.  This is another example of a fractal.  This has
dimension between 0 and 1.  One way that mathematicians measure the
dimension of highly geometric sets (you will see what I mean by this; those
that are formed by removing pieces, but pieces that look exactly like the
original whole) is a measure called the Capacity Dimension.  The precise
definition is a little more complicated, but I will give you a definition
that is easy to handle (that is, if you are familiar with logarithms).

Dc = (ln m)/(ln 1/r)

Now, the m stands for the number of copies of the original figure that exist
in a shrunken size at the next step.  For example, with the Cantor Ternary
Set, when we go from the first step to the second step, we end up with 2
copies of the original line.  At step 3, we end up with 2 copies of the
picture that is in step 2.  So, m=2.  Now, r stands for the amount of
'shrinkage' of the sidelength from one step to the next.  For the Cantor
Ternary Set, we see that the length of each part in step 2 is 1/3 as long as
the original line.  So, r=1/3.  So, the capacity dimension of the Cantor
Ternary Set is Dc = (ln 2)/(ln 3), or I think it's about .63 or somewhere
around there.  I highly suggest drawing the pictures as you go along,
looking at the set from one step to the next, seeing how the set develops.

There are several other sets of which you can measure the capacity
dimension.  For example, there is a set called the Sierpinski Gasket.  To
form this set, start with an equilateral triangle (filled in).  For step 2,
remove the middle triangle (imagine the triangle being split into four
sections).  For the next step, remove the middle triangles or the three
triangles that remain.  If you repeat this process infinitely, you will have
the Sierpinski Gasket.  Can you calculate the capacity dimension of this set?

There is another figure called the Sierpinski Carpet.  If you start with a
filled-in square, split up the square into 9 squares.  For the first step,
remove the middle square.  Then, for the next step, remove the middle square
of the remaining squares.  If you keep doing this, you'll have the
Sierpinski Carpet.  An example of a fractal that is between 2 and 3
dimensions is the Menger Sponge.  Start with a Cube that is split into 27
equal smaller cubes.  For the first step, remove the middle cube, and the
cube that is in the middle of each face of the cube.  Repeat this action on
the remaining cubes for the next step.  

One thing to notice about all of the highly geometric fractals is that if
you keep focusing in on a part of the set, say of the Menger Sponge, and
then blow up the size of that small part, it will look exactly the same as
the set as a whole.  Some objects in nature exhibit a fractal form, like a
fern, for example.  Each part of the fern resembles the fern as a whole.  Or
a tree, to a certain extent.  Each branch has a part with no branches (like
the trunk), and then it has more branches with leaves, like the tree as a
whole.  There are many examples of fractals in nature.  Just look around you.

I have mentioned only these 'highly geometric' fractals.  There are other
sets that we encounter in dynamical systems that have non-integer dimension,
though there are several that cannot be easily measured (or accurately
measured) using the capacity dimension.  There is another measure of
dimension called the Lyapunov Dimension, which is used to measure the
dimension of sets that are less regularly shaped than the ones I have
mentioned.  The definition of the Lyapunov Dimension is quite complex, so if
you want to study in detail, you will have to look it up in a textbook.  The
textbook I used for my class was 'Encounters With Chaos' by Denny Gulick.
Anyway, I'll throw out a name of a set that one uses the Lyapunov Dimension
to measure the dimension of -- the Henon Attractor.  Knowledge of linear
algebra is necessary for an understanding of this object, so if you want to
look it up, be my guest.

The most famous fractals are the Julia Sets and the Mandelbrot Set.  There
are lots of places on-line that have pictures of these sets.  If you've ever
seen the pictures of these fractals, you know they are quite beautiful.
However, as Ken mentioned, to understand these fractals, you need a thorough
understanding of complex numbers and in addition, a thorough understanding
of dynamical systems.  The Julia Sets and the Mandelbrot Set exist in the
complex plane.  Mathematicians can clearly define these sets, and many of
the properties they exhibit, but there may be more to discover. These things
have been studied only since the latter part of this century.  So whoever
says math is dead, having been established by stodgy old-timers from a long
time ago, is just plain wrong!  

Anyway, if you want me to further explain anything of what I said, feel free
to write back.  If you want to learn even more, I could try to tell you
more, or at least tell you where to start. One good place to look is:

   http://mathforum.org/library/topics/fractals/   

Hope you like this,

Heather Mateyak
(occasionally moonlighting as Dr.Math)