Fractals are Fun!

What is a Fractal ? 
A fractal is a rough or fragmented geometric shape that can be subdivided
in parts, each of which is (at least approximately) a reducedsize copy
of the whole.
The core ideas behind it are of feedback and iteration. The creation
of most fractals involves applying some simple rule to a set of geometric
shapes or numbers and then repeating the process on the result. This feedback
loop can result in very unexpected results, given the simplicity of the
rules followed for each iteration.

Lets make a Fractal 
Consider a triangle: 

Rule: add a smaller triangle to each edge : 

This is the first iteration, now repeat the rule on each of the new
edges: 

If we continue this loops we get a beautiful fractal shape: 

Finite Area, but Infinite
Perimeter! 
This process of iteration (looping) can continue for ever. Lets see
what is happening to the perimeter of the shape as we interate. If we define
the initial length of one side of the triangle as 1 unit then at the start
the shapes perimeter was 3 units. After the first iteration with the addition
of three small triangles the perimeter increased to 4 units. Then it went
up to 5.33 and our last image has a perimeter of 7.11 units.
We can make the observation that if we continue with this indefinitely
we will end up with a figure that has a finite area, but an infinite perimeter.
Think about that for a moment! This is a general property of fractals.
Another property of fractals is selfsimilarity. If we look at the perimeter
it looks similar  not the same  at all levels of magnification.

More examples of Fractals: 
Dragon : 

Snowflake : 

Sierpinsky Carpet : 

Its interesting to note that the total perimeter of all the holes (in
the original square) continues to increase as the remaining area (dark)
decreases. Eventually we will have something of no area but infinite perimeter
of all the holes?

Sierpinsky Carpet : 3D Version 

Sierpinsky Triangle : 



Hilbert Curve : 

In the end, this onedimensional line completely fills a two dimensional
square and never once crosses itself in the process! 
Torn Square : 

Fractals in Nature 
Fractal geometry models natural objects more closely than does other
geometries.
Examples include :

Clouds

Mountains

Turbulence

Coastlines

Roots

Branches of trees

Blood vesels and lungs of animals.

Mandelbrot Set 
The most famous fractal is the Mandelbrot Set.
The set is defined as those values of C which when iterated according
to the complex number function Zn+1 < Zn^2 + C with inital value
Z= 0 + 0i result in a sequence that does not go off to infinity.
Plotted, the set looks like:


The black region is the mandlebrot set, like other fractals its perimeter
is infinitely convoluted, these regions can look beautiful when plotted.
The colour regions represent numbers which when iterated "escaped" to infinity.
How may iterations it took a number to escape is indicated by the different
colours. Many of the images in our "Eternal Signs in Fractals" CDROM were generated from areas of the Mandelbrot
and other similar Sets. 
Fractal Sound 
Its been describes as "music composed by mother nature herself", but
what is it?
Fractal Sound is sound composed using the same types of feedback processes
used to create fractal images.During fractal sound composition the numerical
outputs from the feedbackprocess are mapped to sound parameters to produce
melodies, harmonies, rhythms, textures, etc. Since there are numerous,
if not an infinite number of, different ways to map the numerical outputs
to these parameters the choice of a specific mapping has a substantial
impact on the compositional results.
Some people are amazed that this kind of mathematically generated music
sounds much better than they expected. One reason is that selfsimilarity,
a characteristic of fractals, translates well into music giving it that
balance between predictability and randomness, between repetition and surprise,
between the familiar and the strange.
The sound track of our "Eternal Signs in Fractals" CDROM was generated solely from this process  you
are listening to parts of the Madelbrot set! 