Wikipedia
The Mandelbrot set is the set of complex numbers c for which the function f(z) = z + c does not diverge when iterated from z = 0, i.e., for which the sequence f(0), f(f(0)), etc., remains bounded in absolute value.
The set is closely related to the idea of Julia sets, which produce similarly complex shapes. Its definition and name are due to Adrien Douady, in tribute to the mathematician Benoit Mandelbrot.
Mandelbrot set images are made by sampling complex numbers and determining for each whether the result tends towards infinity when a particular mathematical operation is iterated on it. Treating the real and imaginary parts of each number as image coordinates, pixels are colored according to how rapidly the sequence diverges, if at all.
More precisely, the Mandelbrot set is the set of values of c in the complex plane for which the orbit of 0 under iteration of the quadratic map
z = z + c
remains bounded. That is, a complex number c is part of the Mandelbrot set if, when starting with z = 0 and applying the iteration repeatedly, the absolute value of z remains bounded however large n gets. This can also be represented as
z = z + c,
c ∈ M ⇔ limsup∣z∣ ≤ 2.
For example, letting c = 1 gives the sequence 0, 1, 2, 5, 26,…, which tends to infinity. As this sequence is unbounded, 1 is not an element of the Mandelbrot set. On the other hand, c = −1 gives the sequence 0, −1, 0, −1, 0,…, which is bounded, and so −1 belongs to the Mandelbrot set.
Images of the Mandelbrot set display an elaborate boundary that reveals progressively ever-finer recursive detail at increasing magnifications. The "style" of this repeating detail depends on the region of the set being examined. The set's boundary also incorporates smaller versions of the main shape, so the fractal property of self-similarity applies to the entire set, and not just to its parts.
The Mandelbrot set has become popular outside mathematics both for its aesthetic appeal and as an example of a complex structure arising from the application of simple rules. It is one of the best-known examples of mathematical visualization.
Usage examples of "mandelbrot set".
Meanwhile Colene's mind was racing through what she remembered of the Mandelbrot set.
It's called the Mandelbrot Set (from now on, the M-Set) and you're soon going to meet it everywhere —.
Once he had recovered enough to travel, Mandelbrot set the computer in the little ship and gave him a quick course in its manual controls, in the event of emergency.
As it floated forward, it resolved into a robot drone over a meter in diameter, its surface patterned by delicate curls of the Mandelbrot set, swirls fringed by swirls fringed by swirls in an unending pattern of ever more minute lace.
Then a fan, the artist Kurt Cagle, sent me a copy, and I looked at it and saw the illustrations of the Mandelbrot Set.
The Mandelbrot Set is too complicated a matter to get into here, but those who are interested in art and mathematics should find it fascinating.
Class, today we shall take our little pencil and graph paper and define the complete Mandelbrot Set.
The Mandelbrot Set was said to be the most complicated object in mathematics.
It almost made his head spin, like a straight-line version of a Mandelbrot Set that just kept duplicating itself over and over in ever smaller dimensions.