To start exploring the Set on your own, simply click in the vicinity that interests you. Each time you click, the program will zoom in at the current zoom factor, using the point you clicked at as the center of the next display. To find the specifics of the new location, either use the 'Where Are We Now' button, or right-click within the applet. Of the two, right-clicking yields the more accurate number, due to the difference in precision between JavaScript (32-bit) and Java (64-bit) floating-point numbers. If you right-click, you must left-click to re-establish the display.
If you would like to build your own guided tour, and save it in your browser, click here.

Benoit B. Mandelbrot (1924- ) is a Polish-born French mathematician who developed fractal geometry as a separate field of mathematics. Born in Warsaw, Mandelbrot attended schools in France and the U.S., obtaining his doctorate in mathematics from the University of Paris in 1952. He has taught economics at Harvard University, engineering at Yale, physiology at the Albert Einstein College of Medicine, and mathematics in Paris and Geneva. Since 1958 he has worked as an IBM fellow at the Thomas B. Watson Research Center in New York.

Fractal geometry developed from Benoit Mandelbrot's study of complexity and chaos (see chaos theory). Beginning in 1961, he published a series of studies on fluctuations of the stock market, the turbulent motion of fluids, the distribution of galaxies in the universe, and on irregular shorelines on the English coast. By 1975 Mandelbrot had developed a theory of fractals that became a serious subject for mathematical study. Fractal geometry has been applied to such diverse fields as the stock market, chemical industry, meteorology, and computer graphics.

Fractal geometry is a branch of mathematics concerned with irregular patterns made of parts that are in some way similar to the whole, e.g., twigs and tree branches, a property called self-similarity or self-symmetry. Unlike conventional geometry, which is concerned with regular shapes and whole-number dimensions, such as lines (one-dimensional) and cones (three-dimensional), fractal geometry deals with shapes found in nature that have non-integer, or fractal, dimensions-linelike rivers with a fractal dimension of about 1.2 and conelike mountains with a fractal dimension between 2 and 3.

The material in this section is based closely on material extracted from the Microsoft Encarta Online Encyclopedia and from the Web Site Encyclopedia.com.

For the student of Mathematics:

The Mandelbrot set M consists of all complex c-values for which the corresponding orbits of 0 under the quadratic function X^2+c do not escape to infinity.

To actually display the set, take a square in the complex plan and overlay a grid of equally spaced point (pixels) on you monitor. Each grid point is then taken as the value of c, and the system checks whether the orbit of 0 goes to infinity (escapes) does does not go to inifnity (remains bounded). If the orbit escapes we leave the c-value (and the pixel) white; if the orbit remains bounded, we make the c-value (and the pixel) black.

As a practical matter, you can test for bounded-ness as follows: If the absolute values of both the real part of X(n) and the coefficient of the imaginary part of X(n) are both less than 2 after 200 iterations, then the orbit is bounded.

Most people who produce displays of the Mandelbrot set don't just produce black and white images. In most cases, points not in the Mandelbrot set, instead of being left white, are colored according to how fast the orbits of 0 escape to infinity. In the applet you are viewing, I used four colors based on the number of iterations before the point escapes: 0-49 iterations uses white, 50-99 iterations uses blue, 100-149 iterations uses green, and 150-199 iterations uses red.

For some really interesting additional information on the Mandelbrot Set, I would recommend the following Web Sites: