The Mandelbrot set reference article from the English Wikipedia on 24-Jul-2004 (provided by Fixed Reference: snapshots of Wikipedia from wikipedia.org)

Mandelbrot set

In mathematics, the Mandelbrot set is a fractal that is defined as the set of points c in the complex number plane for which the iteratively defined sequence

with z₀ = 0 does not tend to infinity.

The sequence is thus expanded mathematically as follows for each point c in the complex plane:

and so on.

If we reformulate this in terms of the real and imaginary parts (x and y co-ordinates of the complex plane), looking at each iteration, n replacing z_n with the point x_n + y_ni and c with the point a + bi, then we get

and

The Mandelbrot set can be divided into an infinite set of black figures: the largest figure in the center is a cardioid. The rest of the figures are all circles which branch out from this central cardioid. There is a (countable) infinity of circles which are in direct (tangential) contact with the cardioid, but they vary in size, tending asymptotically to zero diameter. Then each of these circles has in turn its own (countable) infinite set of smaller circles which branch out from it, and this set of surrounding circles also tends asymptotically in size to zero. The branching out process can be repeated indefinitely, producing a fractal.

The total number of figures is uncountably infinite: the cardioid is the first figure, which branches out in the first step to a countable infinity of circles, each of which branches out on the second step to another countable infinity of circles, and so on. At each successive step, the number of figures is multiplied by (aleph-null), so the total number of figures is which is uncountable.

Table of contents

1 History of the Mandelbrot Set 2 Relationship with Julia sets 3 Plotting the set 3.1 Adding color 3.1.1 Optimization 3.2 Art and the Mandelbrot set 4 Other Mandelbrot sets 5 Mandelbrot Set in Song 6 External links

History of the Mandelbrot Set

The Mandelbrot set was first defined in 1905 by Pierre Fatou, a French mathematician working in the field of complex analytic dynamics. Fatou studied recursive processes like

Starting with some point on the complex plane, successive points may be generated by repeatedly applying this formula. The sequence of points thus obtained is called the orbit of under the transformation

Fatou realised that the orbit of under this transformation would provide some insight into the behaviour of such systems. There are an infinite number of such functions - one for each value of c. Fatou did not have access to a computer capable of plotting the orbits of all these functions, but attempted to do so by hand. He proved that once a point moved to a distance greater than 2 from the origin, then the orbit would escape to infinity.

Fatou never saw the image of what we now call the Mandelbrot set as we do as the number of calculations required to generate this is far more than could be calculated by hand. Professor Benoît Mandelbrot was the first person to use a computer to plot the set.

Fractals were popularised by Mandelbrot in 1975 in his book . In this book, Mandelbrot used the term fractal to describe a number of mathematical phenomena that seemed to exhibit chaotic or surprising behaviour. All of these phenomena involved the definition of some curve or set through the use of some recursive functions or algorithms. The Mandelbrot set is one such phenomena that is named after its discoverer.

Relationship with Julia sets

The Mandelbrot set was created by Benoît Mandelbrot as an index to the Julia sets: each point in the complex plane corresponds to a different Julia set. The points within the Mandelbrot set correspond precisely to the connected Julia sets, and the points outside correspond to disconnected ones.

Intuitively, the "interesting" Julia sets correspond to points near the boundary of the Mandelbrot set; those far inside tend to be simple geometric shapes, while those well outside look like dust surrounded by blobs of color. Some programs, such as Fractint, let the user choose a point, and jump to the corresponding Julia set, making it easy to find Julia sets that the user is likely to enjoy.

Plotting the set

It can be shown that once the modulus of z_n is larger than 2 (in cartesian form, when x_n² + y_n² > 2²) the sequence will tend to infinity, and c is therefore outside the Mandelbrot set. This value, known as the bail-out value, allows the calculation to be terminated for points outside the Mandelbrot set. For points inside the Mandelbrot set, i.e. values of c for which z_n doesn't tend to infinity, the calculation never comes to such an end, so it must be terminated after some number of iterations determined by the program. This results in the displayed image being only an approximation to the true set.

Adding color

Mathematically speaking, the pictures of the Mandelbrot set and Julia sets are "black and white". Either a point is in the set or it is not. Most computer-generated graphs are drawn in color. For the points that diverge to infinity, and are therefore not in the set, the color reflects the number of iterations it takes to reach a certain distance from the origin. This creates concentric shapes, each a better approximation to the Mandelbrot set than the last. One possible scheme is that points that diverge quickly are drawn in black; then you have brighter colors for the middle; then you have white for the points in the set, and near-white for the points that diverge very slowly.

In order to determine if the point, Z₀, is going to be in the mandelbrot set (traditionally coloured black) or outside the set (coloured according to the escape speed), the distance of the Z_i must be calculated at each iteration in the sequence:

If , then . Note that one of the many possible optimisations for calculating mandelbrots can be applied here. Rather than testing to see if , we can simply test to see if — thus saving the square root operation.

So, if then colour the point black, otherwise colour it according to the value of i. Using the number of iterations required to determine that the point has escaped is the easiest and most common way of recording the "speed" of escape. This number can be directly mapped to a colour via a lookup table or palette, or by use of some suitable algorithm.

So far, we have worked out how to tell if the point is outside the set — it moves more than a distance of 2 away from the origin. How do we tell it is inside the set? There are lots of strategies for this — this is not so simple. If it were, there would not be so many different algorithms out there for calculating mandelbrots. This is the essence of the problem. Clearly most points will not actually get to zero in a reasonable number of iterations. The simplest algorithm is to simply limit the number of iterations and if you have not gone outside the boundary by the time you have done your last iteration, then you assume the point is in the set.

Indeed, points nearer the set take longer to escape. This is why plotting slows down near the dark region — because more terms in the expansion are having to be calculated to determine the escape speed.

Optimization

One way to improve calculations is to find out beforehand whether the given point lies within the cardioid. The cardioid's polar equation is

but with the center of polar coordinates located at (1/4,0) — the cardioid's cusp — instead of at the origin. Given point (x,y), calculate the following

If then point (x,y) is in the Mandelbrot set (color it black), and the usual calculations can be skipped.

Art and the Mandelbrot set

Some people have a hobby of searching the Mandelbrot set for interesting pictures. They have a collection of pictures, along with the coordinates for generating that picture. For example, the image to the right is a closeup centred at coordinates (0.282, -0.01).

Many people have discovered a set of fractal-generating programs by Stephen Ferguson such as Sterling Fractal and the Tierazon series of programs. These programs are notable in that they are free programs, available without fee to allow exploration into fractal art.

Other Mandelbrot sets

When people speak of the Mandelbrot set, they usually are referring to the set described above. Any function that maps to and from the complex number plane has a Mandelbrot set, which characterizes whether or not the Julia set corresponding to that function is connected.

Example:

Let .

For each value of c, we draw the Julia set of , and determine if it is connected or not. If is connected, then c is in the Mandelbrot set of {}, otherwise c is not in the Mandelbrot set.

This can also be generalized to Julia sets parameterized by more than two real numbers. For example, a collection of Julia sets parametrized by three real numbers will have a three dimensional Mandelbrot set. Of course, only the 2-dimensional case will have an easily viewed picture.

Mandelbrot Set in Song

The Australian band GangGajang has a song Time (and the Mandelbrot set) where the term Mandelbrot set is used liberally in the lyrics.

External links

• Julia and Mandelbrot Set Explorer

• The Fractal Microscope provides a nice Java interface (figure to right). Unlike some other Java-based Mandelbrot set interfaces on the internet, this one works for Macintosh browsers (tested on these OS X browsers: Safari, Mozilla, Camino, Explorer).

• Detailed mandelbrot set, see bottom of page

• 3D Mandelbrot Set Explorer Applet

• A gallery of Fractal images and copies of Fractal-exploring programs.

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