he Mandelbrot Set ... the Canonical Fractal

(and Julia Sets)

Under Construction!

If you see nothing but text above this line, your browser probably doesn't support Java.

The Mandelbrot set is the most familiar of fractals, having achieved a kind of emblematic status for the whole field of Chaos. The Applet above is still being constructed, but it is currently possible to drill down into the details of the Set by dragging the mouse to select a rectangle. It is also possible to do the same for the Julia sets, a related class of object. Each point on the complex plane around and within the Mandelbrot set has its own Julia set. To see the Julia set for any point click that point with the mouse without dragging. You can then drill down into the Julia set by dragging the mouse again.

These sets are fractal in nature, a term derived from the concept of fractional dimension. This implies self-similarity at all scales - no matter how much you magnify the details of the sets' border, there is always detail, on a similar degree of complexity and often of a similar form. The Mandelbrot set, in particular, has a fractal dust of small copies of itself around its periphery.

The algorithm used to plot these sets is deceptively simple. The 2-dimensional plane of which they are plotted is a representation of the complex plane. The horizontal coordinate of a point on this plane corresponds to a real number. The vertical position corresponds to an imaginary number. Any point on the plane is completely defined by a complex number, which is the sum of a real number and an imaginary number. Imaginary numbers are traditionally notated as a multiple of i, which is the square root of -1. Thus, a complex number might be represented as 3 + 2i.

Complex numbers can be operated on in a similar way to real numbers, and this is crucial to understanding the Mandelbrot set. To be continued...