This presentation briefly explains what fractals are and how they can be created. The characteristics of fractals are illustrated with the example of the Mandelbrot set.

#### What is a Fractal ?

A fractal is a mathematical object that has a detailed structure at any scale.

When you look closely at the structure, in many cases you will see a tiny fraction in it that resembles the whole (self-similarity).

A fractal is an infinite pattern compressed into a finite space.

An example: Zooming into the Mandelbrot set

#### The Mandelbrot Set

A set discovered by Benoid Mandelbrot in 1979.

The fractal is created using the surprisingly simple formula:

zn+1 := zn2 + c

All the points in the complex plane are entered into that formula.

Each point is iterated n times using this formula, which creates a sequence of values.

#### Iteration

What does iteration mean ?

When a point is entered into the formula in the beginning, a new point is determined. This point is entered into the formula again, etc.

Thus a sequence of points is created, e.g. with the original point [0.5, 0.3]:

[.5, .3], [.66, .60], [.5756, 1.0920],
[-.3611, 1.5571], [-1.794, -.8247],
[3.0389, 3.2593], [-.8880, 20.1093],
[-403.0945, -35.4141], ...

#### Determining the Mandelbrot Set

If at least one of the values in the sequence is farther than 2 units away from the origin [0, 0], then the point that has originally been entered into the formula does not belong to the Mandelbrot set.

If all the values in the sequence are within a distance of 2 units from the origin, the original point is defined as being part of the Mandelbrot set.

#### Sequences & Orbits

Besides calculating numbers, we can also display the sequences graphically.

Let there be three points:

• p1 := [0.5, 0.3] (navy)
• p2 := [0.2, 0.5] (maroon)
• p3 := [0, 1] (green)

The plot shows so-called orbits, the paths the three points take during iteration:

You see that p1 = [0.5, 0.3] very soon leaves the radius of 2 units about the origin, thus it is not part of the Mandelbrot set.

p2 belongs to the Mandelbrot set, for its orbit does not leave the radius.

The same is for p3, but here we have a peculiar behavior: The iteration of p3 generates a periodical sequence: Its values just switch between three values: [0, 1.], [-1., 1.], and [0, -1.].

#### Creating Mandelbrot Images

After a predetermined number of iterations, it is determined whether the point belongs to the Mandelbrot set.

Thus we can draw 3-dimensional images. The higher a point is in the plot, the more iterations were performed on a point.

#### Drawing Fractals on your PC

The algorithm that creates a fractal is quite simple. An example in Maple V:

> lambdafn := proc(x, y)
>    local z, m, p1;
>    z := evalf(x+I*y);
>    for m from 0 to 50 while abs(z) < 4 do
>       z := sin(z)*1+I*0.4
>    od;
>    m
> end:

> plot3d(0, -1.5 .. 1.5, -1.1 .. 1.1, color=lambdafn, [...]);

Since in this example we have the formula zn+1 := sin(zn)*p, with p = 1 + I*0.4 (the point [1, -0.4]), whereas the Mandelbrot set is generated using the formula zn+1 := zn2 + c, you see that fractals may appear very differently.

#### Fractals that can be found in Nature

There are many fractals around us, for example:
• trees,
• ferns,
• mountains,
• clouds.

MAPLE V INTROFRACTALS #1.00 current as of May 23, 1999
Author: Alexander F. Walz, alexander.f.walz@t-online.de
Original file location: http://www.math.utsa.edu/mirrors/maple/mfrtut.htm