 The Mandelbrot Set and Julia Sets
Mandelbrot Set
To generate an image of the Mandelbrot Set or a Julia Set, a relatively
simple algorithm is used that requires the squaring of a complex number followed
by the addition of a constant.
In the region of the complex plane being investigated, each pixel position
is iterated through that simple equation, and the color of the pixel determined
by the behavior observed. In most cases, the color relates to the number of
iterations required before the value does one of two things: 1) it reaches
some maximum number of iterations, or 2) it becomes too large. A typical
method for deciding when an iteration is too large is to calculate the
distance from the origin to that point. If the distance exceeds 2 units,
successive iterations will get extremely large, approaching infinty.
if a value is too large, typically a test of the distance from the
The same basic algorithm used for both the Mandelbrot and Julia Set
programs. Bothe fractals use Complex Numbers, and the only difference
relates to the use of the constant that is added on each time. Here are the
equations:

Z_{n+1} = Z_{n}^{2} + C,
where Z_{n} = a_{n} + b_{n}i
and
C = a_{0} + b_{0}i
The value i of
course stands for the square root of
1.
In order to graph the pixels in the coordinate plane, the above equation
is separated into its Real and Imaginary components for each arbitrary
point (a_{n}, b_{n}).
The Real part is
graphed along the horizontal axis and the Imaginary part graphed along the
vertical axis. The equations to generate the next real or imaginary
portion is as follows:

Real Part: a_{n+1} = a_{n}^{2} 
b_{n}^{2} + a_{0}
Imaginary Part: b_{n+1} = 2a_{n}b_{n}
+ b_{0}
It is also necessary to check the distance of that point from the origin
after each iteration, so the distance formula becomes:
d = sqrt [ ( a_{n+1}  0 )^{2} + (
b_{n+1}  0 )^{2} ]
Since the square root process causes the computer to do extra computation,
it is often faster to eliminate that step and just check for when
d^{2} is greater than 4.
Since it is wise to avoid unnecessary computation, it is a good idea to
calculate the numerical position of each pixel. For instance, if a tiny
Mandelbrot 10 pixels wide were to be drawn in the range 2 < x <
+2, the first pixel would be drawn at 2 and the last pixel would
be drawn at +2, but where would the other 8 pixels be located? To
calculate this, the following rationale might help. With n pixels
there will be n1 intervals between them. Since the total distance
between the starting and stopping point must be divided into n1
equal intervals, this example becomes:

Total Distance = x_{2}  x_{1} = 2  (2) = 2 + 2 =
4
Number of Intervals = n  1 = 10  1 =
9
Distance Between Intervals = Total Distance / Number of Intervals = 4 / 9 =
0.444444444...
A simple loop can be used to cycle between the first and last pixels by
starting with the first pixel location ( 2 ) and adding the step
between adjacent pixels ( 0.444444444 ) each time through the loop. There might be
some slight rounding inaccuracies, but this approach will give each pixel a unique
numerical value in the proper range.
An Interesting Aside
There was an interesting hoax about a Benedictine Monk ( circa 12001270 AD)
who was reprted to be the first one to actually discover the
existence of the Mandelbrot Set. Students should be aware that web
resources are not reviewed for content nor can they be verified as accurate.
However, the following link does make an amusing April Fool's Joke:

http://www.raygirvan.co.uk/apoth/udo.htm 
The Mandelbrot Monk
by Ray Girvan
In any case, we who study science and mathematics are often just discovering
principles that already exist. Surely there are many more discoveries that
will be made in years to come, and those who are experimenting with concepts
today may be laying the groundwork for future enlightenment centuries from now.
We should applaud all those who contribute to and perpetuate knowledge
in our society.
Julia Sets
For the Mandelbrot Set, the value for C is the initial pixel position at the
start of each iteration sequence but for a Julia Set, the value for C is the
same for every iteration, regardless of the starting pixel
position. Most of the
constants that produce interesting Julia Sets are near the boundaries of
the Mandelbrot Set. Here are some
constants that produce interesting Julia Sets:
 C = 0.12375 + 0.56508i (Basin with one attractor)
 C = 0.12 + .74 i (Three Attractors)
 C = 0.11 + 0.6557i (Three arttractors but with slight spiral
 C = 0.194 + 0.6557i (Threeattractor Cantor Set)
 C = 0 + i (Dendrite filament near 3attractor basin)
 C = 0.125 + 0i (Parabolic case: boundary between 2 and 4 attractors)
 C = 0.11031  0.67037i (Fatou dust near 7attractor basin)
 C = 0.27334 + 0.00742i (Parabolic case near 20attractor basin)
For either the Mandelbrot Set or a Julia Set, the region in the Complex
Plane that is graphed is typically 2 < a
< 2 and 2 < b < 2 . To zoom into a region, the
values for the glOrtho( ) can be easily modified to produce a
closeup view.
For more information on the Mandelbrot Set or Julia Sets, check out the
web page developed by
Sean Dobbs' Fractals: Mandelbrot
Set, Julia Sets, Orbits of Points
Interesting things to try:
 Use mouse routines to generate a zoom into any region of the Mandelbrot
Set.
 Try plotting the path of selected points as they are iterated in the
Mandelbrot set or specific Julia sets.
 Using mouse routines, interactively select points from the Mandelbrot
Set to use for Constants in creating Julia Sets.
 Try higher order Mandelbrot type calculations: Z^{3} + C,
Z^{4} + C, etc. What do the Julia sets look like?
 Create an animation to show what happens through fractional powers
going from Z^{2} + C to higher powers of Z.
 Newton's Method in the Complex Plane
Newton's method is a Calculusbased process used to arrive at roots of
high order polynomials. This iterative approach relates to a line tangent
to a polynomial at a specific point on the curve which can be used to
give a better approximation of the root with another iteration.
The process for general equations in the Complex Plane can be defined as:

Z_{n+1} = Z_{n}  [ f(Z_{n}) /
f'(Z_{n}) ]
where Z moves from some initial guess to an approximation of the root
through some maximum number of iterations. The iteration process can be halted when consecutive values of Z are within some arbitrarily small distance,
epsilon, of one another.
To generate an interesting fractal, all points in some region of the Complex
plane can be colored in some way depending upon how these points converge
to the various roots or "Basins of Attraction". For example:

Given the function f(Z) = Z^{3}  1
The derivative would be f'(Z) = 3Z^{2}
The equation to calculate the next iteration will be
Z_{n+1} = Z_{n}  [(Z_{n}^{3}  1) /
3Z^{2}]
This function has three roots:
the real number 1 and two imaginary roots, one near
(0.5 + 0.866i)
and the other near (0.5  0.866i).
An arbitrary point a + bi in the complex plane should eventually
converge to one of
these three roots, assuming the iteration process is long enough and the
approximation technique does not divide by zero some where along the way.
Newton's Method calculations can be very computationally intensive though,
especially since it will be necessary to eliminate values of i
in the denominator of any fractions by using the complex conjugate.
After that, isolate the real and imaginary portions of each calculation
carefully since even a misplaced sign will significantly distort the
picture. Look for ways to save on computation, too. The following
page might be helpful for understanding more about
Complex Numbers and DeMoivre's Theorem. It will show you some
tricks for working with fractional powers.
For more examples of Newton's Method in the Complex Plane, check out:

Newton's
Method in the Complex Plane A Supercomp Page by former student Adrian Porter. He experimented with fractional powers in order to generate some intersting animations. With fractional powers, it is necessary to go to polar coordinates instead and apply DeMoivre's Theorem.
 Newton Basins A discussion of generating fractals using the Basins of Attraction for
Newton's Method by Dr. David E. Joyce at Clark University
 Newton's
Method Fractals A collection of graphic images generated by people in the
Math Department at Iowa State University
 LSystems
Some interesting graphics can be produced using a process developed by
Dr. Aristid Lindenmayer, a colleague of Dr. Mandelbrot. The technique has been
called "LSystems" after Dr. Lindenmeyer, and involves recursive string
replacement using a set of simple rules.
LSystems can be used to generate rather complex graphic images from some
simple rules that define such things as lines, polygons, angles, and such.
For each Lsystem, a set of string replacement rules
are defined, and the objects defined in the string are graphed after a
certain number of iterations have been applied. In a book by Prusinkiewicz
and others titled The Algorithmic Beauty of Plants, a number of
incredibly realistic biological images such as flowers, trees,
and ferns are defined by these relatively simple rules.
For more examples on Lsystems, check out:
 LSystems  A Supercomp Page by former student Tom Dixon
 An Introduction to Lindenmayer Systems by Gabriela Ochoa at the School of
Cognitive and Computing Sciences at the University of Sussex
 Modelling of Plants and Fractals with Lindenmayer Systems A collection of Web references by Christian Jacob in Germany