MANDELBROT SET

Domain	Definition
Computing	Mandelbrot set (After its discoverer, Benoit Mandelbrot) The set of all complex numbers c such that \| z[N] \| < 2 for arbitrarily large values of N, where z[0] = 0 z[n+1] = z[n]^2 + c The Mandelbrot set is usually displayed as an Argand diagram, giving each point a colour which depends on the largest N for which \| z[N] \| < 2, up to some maximum N which is used for the points in the set (for which N is infinite). These points are traditionally coloured black. The Mandelbrot set is the best known example of a fractal - it includes smaller versions of itself which can be explored to arbitrary levels of detail. The Fractal Microscope (http://www.ncsa.uiuc.edu/Edu/Fractal/Fractal_Home.html/). (1995-02-08). Source: The Free On-line Dictionary of Computing.
Source: compiled by the editor from various references; see credits.

Specialty Definition: Mandelbrot set

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

z_n+1 = z_n² + c

with z₀ = 0 does not tend to infinity. If we reformulate this in terms of real numbers, replacing z_n with the point x_n + y_ni and c with the point a + bi, then we get

x_n+1 = x_n² - y_n² + a

and

y_n+1 = 2x_ny_n + b.

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

Plotting the set

Sample generated image

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.

Whilst it is of no mathematical importance, most fractal rendering programs display points outside of the Mandelbrot set in different colours depending on the number of iterations before it bailed out, creating concentric shapes, each a better approximation to the Mandelbrot set than the last.

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 not in the set, the color reflects the number of iterations it takes to reach a certain distance from the origin. 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.

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.

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 f_c(z) = z^3 + c.

For each value of c, we draw the julia set J_c of f_c(z), and determine if it is connected or not. If J_c is connected, then c is in the mandlebrot set of {f_c}, otherwise c is not in the mandlebrot 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 Mandlebrot set. Of course, only the 2-dimensional case will have an easily viewed picture.

Fractal generators

Ultra Fractal - Popular software for Microsoft Windows
Fractint (ports available for most platforms)
"Makin' Magic Fractals"
ChaosPro - for Microsoft Windows
Xaos - Realtime generator - Windows, Mac, Linux, etc.

Commercial Usage: MANDELBROT SET

Domain	Title
Books	Fractal Art: The Mandelbrot Set (reference) Fractals for the Classroom: Complex Systems and Mandelbrot Set (reference) The Mandelbrot Set, Theme and Variations (reference) (more book examples)
Source: compiled by the editor from various references; see credits.

Frequency of Internet Keywords: MANDELBROT SET

Anagrams: MANDELBROT SET

Scrabble® Enable2K-Verified Anagrams
	Words within the letters "a-b-d-e-e-l-m-n-o-r-s-t-t"
	-1 letter: demonstrable.
	-2 letters: demonstrate.
	-3 letters: banderoles, bandoleers, blastoderm, blastomere, debarments, endorsable, entodermal, lobsterman, lobstermen, resemblant, sandlotter, trematodes.
	-4 letters: abetments, adsorbent, attenders, banderole, banderols, bandoleer, batteners, betatrons, blastment, blattered, bodements, bolstered, bonemeals, debarment, demeanors, desolater, detonable, detonates, dotterels, ealdormen, elastomer, embattled, embattles, emboldens, endosteal, entoderms, lamenters, leeboards, lemonades, letterman, lobstered, mantelets, marlstone, matelotes, mestranol, moderates, modernest.
	Words containing the letters "a-b-d-e-e-l-m-n-o-r-s-t-t"
	+4 letters: demonstrabilities.
Source: compiled by the editor from various references; see credits. SCRABBLE® is a registered trademark. All intellectual property rights in and to the game are owned in the U.S.A and Canada by Hasbro Inc., and throughout the rest of the world by J.W. Spear & Sons Limited of Maidenhead, Berkshire, England, a subsidiary of Mattel Inc. Mattel and Spear are not affiliated with Hasbro.