The Mandelbrot Set(left) is a fractal coloration of the complex plane. A fractal is essentially a geometric shape that shows similarity to its overall shape on all levels of magnification, so called "self-similarity." Fractals are also characterized by having non-integer dimensions(according to a suitable definition of dimension that overlaps with the usual one). On an ideal picture of the set(an infinitely complex object), each black point would represent a complex number, call it c, that remains "bounded" under repeated applications of the mapping c -> c^2 + c. Each white point would represent a complex number that does not stay bounded.This means that if we mapped a given complex number, c, to c^2 + c, then to (c^2+c)^2 + c, then to ((c^2+c)^2 + c)^2 + c and so on... and if we could find a fixed circle, centered on the origin of the complex plane, so that no matter how many times we repeated the mapping, |(the latest value of the mapping)<(the radius of the fixed circle), then the point corresponding to c would be colored black.

Instead of requiring that the the complex number c NEVER goes outside of a certain fixed circle, which we could not test by brute force(the testing would never stop if the sequence for a given c is actually bounded), Mandelbrot Explorer tests whether c stays bounded under only a finite number of iterations of the mapping. The accuracy of this approximation is increased by the fact that if |the latest value of the iteration on c|>2, then the iteration on c does not remain bounded: it is said to "escape," and is promptly colored white. The accuracy of the picture is increased incrementally by testing with higher and higher numbers of iterations. After 10 seconds or so, deeply zoomed pictures of the set should appear to be carved out, and refined. This shows points escaping, points that had not escaped under less iterations.

Each point on Mandelbrot plane corresponds to another fractal called a Julia set. The Mandelbrot set can, in fact, be said to be a "map" or "catalog" of Julia sets, where the color of a given point on the Mandelbrot set is the same color as the origin of the Julia set corresponding to that point. The coloration a point on the Julia set image represents: REMAINDER[(the number of iterations on that point before it escapes)/10].