[Math Lair] Mandelbrot Set

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The Mandelbrot set, perhaps the most famous fractal, is perhaps the emblem of fractal geometry. With the advent of computer graphics in the 1980s, images of this set, often coloured in various stunning shades, became iconic.

This fractal is named after Benoit Mandelbrot, who produced images of this set in the 1970s. The actual set consists of the points plotted in black in the image below. The colour values represent how quickly a point goes to infinity when iterated; black points never reach infinity. If you were to look at this set under greater and greater magnification, you would see more detail that can't be seen in this picture.

[34K picture of Mandelbrot set]

One might think that, with this degree of complexity, that the Mandelbrot set is impossible to comprehend. The Mandelbrot set is the set of values c for which the Julia set is disconnected. See the Julia set page for a discussion of the complex plane, a definition of the Julia set, and what "connected" means.

The Mandelbrot set is a very complicated form. It is cardioid-shaped, with a series of prominent circular "buds" attached. Each of these buds is surrounded by other, smaller buds, which in turn are surrounded by smaller ones, and so on. In addition, there are very fine, branching "hairs" that grow outwards from the buds. These hairs contain miniature copies of the Mandelbrot set along their length.

The Mandelbrot set is a connected set.

Sources used (see bibliography page for titles corresponding to numbers): 17.