In the 70's and 80's, a French mathematician by the name of Benoit
Mandelbrot was studying the concept of fractional dimensions. He
defined what it meant for a set of points to have 'dimensions' between
the normal integer dimensions (e.g. 1, 2, or 3, etc.). He presented
as an example a set that arises from a very simple equation and yet has
a very complex structure.

Consider a complex number *c = c _{r} + c_{
i} I* that has the property that

With a description so simple, it is quite suprising to see the shape
of the set that results. In the xy plane we say that each point will
represent a complex number by the convention that the point *(x,
y)* will represent *x + y I*.
We cannot graph every point in the complex plane, just to the resolution
of our picture. For every point in our picture we assign a colour,
black if the point is in the Mandelbrot set and some other colour depending
on the number of steps before *|z _{i}| > 2*.
The graph of the region

The Mandelbrot
set are the points in black, the other colours in this picture are
based on the number of iterations before *|z _{i}|*
is greater than

For this project, create a function that produces a picture of the Mandelbrot set and allows you to zoom in and display parts of the boundary. The better the picture you create as a final product, the better your grade will be. The image created above was a crudely displayed with a very short Maple procedure. You will probably have to use procedures other than the built in 'plot' functions to create something more detailed. This image has a resolution of 160x160 pixels.

It will not take you long to produce a picture similar to the one above, there are probably functions on the web that will draw something similar in Maple. With enough computer time, you might even be able to produce a very detailed image. An 'A' project demonstrates that you have used some feature of the Mandelbrot set or computer programming which helps speed up drawing this picture. A project that simply produces a lot of pretty pictures is (probably) worth a 'C.'

An alternative project:

Another interesting fractal arises in the study of Newton's method for
finding the roots of an equation. Say that we want to find the roots
to equation *f(z) = z ^{3} -1*.
Pick a point in the complex plane and iterate using Newton's method.
Eventually the process will settle towards one of the three roots of this
equation

Recall that Newton's method starts with a value *z _{
1}* and then we put

As an alternate project, create functions that allow you to view the
above image in more detail. You should also color each point with
a darker/lighter color depending on how many iterations it takes to converge
to a given root (i.e. points that take 30 iterations before they converge
should be colored with light green/blue/red, while points that converge
after 3 iterations should be colored with near black). For this example
you may also experiment with other equations such as *x ^{4}-1*
where you should see four different roots. Do you see the same self-similar
patterns in the image that arises?

Here is a gallery of pictures by produced
by previous classes.

zabrocki@math.yorku.ca