The best known fractal and one of the most
complex and beautiful mathematical objects known. It was discovered by Benoît
Mandelbrot in 1980 and named after him by
Adrien Douady and J. Hubbard in 1982.
The Mandelbrot set is produced by the incredibly simple iteration formula:
zn+1 = zn2
where z and c are complex
numbers and z0 = 0. This can be written without
complex numbers as
xn+1 = xn2
- yn2 + a, and
yn+1 = 2xnyn +
where z = (x, y) and c = (a, b).
The Mandelbrot set consists of all the points on the complex plane (see
Argand diagram) for which the function
z2 + c doesn't diverge under iteration.
A computer is essential for carrying out the necessary calculations and
for producing pictures of this remarkable structure. For the purposes
of computation, the complex plane is broken down into pixels (picture
elements), the coordinates of each of which supply the constant c
in z2 + c. For each pixel (value of c)
the function is iterated. If the function either rapidly diverges (blows
up) or rapidly converges (collapses), the pixel is left black. If the
function is more indecisive about which way it is heading, it is allowed
to iterate longer. In some cases the iterations could go on for a very
long time before it became clear that the function would ultimately diverge,
so a limit is established, known as the depth, beyond which iterations
are stopped. If the depth is reached without divergence, the corresponding
pixel is left black as though it were in the set. At locations where divergence
is indicated prior to hitting the limit, the pixel is displayed according
to a scale that represents how many iterations are needed to show divergence.
The whole Mandelbrot set lies within a circle of radius 2.5 centered at
the origin of the complex plane. Although finite in area, the Set has
a boundary that is infinitely long and has a Hausdorff
dimension of 2. The overall appearance of the Mandelbrot set is that
of a series of disks. These disks have irregular borders and decrease
in size heading out along the negative real axis; moreover, the ratio
of the diameter of one disk to the next approaches a constant. More complex
shapes branch out from the disks. One region of the Mandelbrot set containing
spiral shapes is known as Seahorse Valley because it resembles a seahorse's
tail. A computer can be used, like a microscope, to zoom in on different
parts of the set. This reveals that, although the shape is infinitely
complex, it also displays self-similarity with regions that look like
the outline of the entire set.
The Mandelbrot set also exhibits symmetry on different levels. It is identically
symmetrical about the real axis, and almost symmetrical at smaller scales.
This kind of "near-but-not-quite" symmetry is one of the most unexpected
properties to find in an object generated from such a simple formula and
process. The Mandelbrot set was created by Mandelbrot as an index to the
Julia sets. Each point in the complex
plane corresponds to a different Julia set, and those points within the
Mandelbrot set correspond precisely to the connected Julia sets.
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