For a detailed description of the Mandelbrot Set, see the Mandelbrot Set Wikipedia Page.
In brief, the Mandelbrot Set is the set of complex numbers C for which the following iterative formula does not diverge to infinity:
Zₙ₊₁ = Zₙ² + C, where Z₀ = 0
In other words:
- Take any complex number C:
- Multiply it by itself, and add C again.
- Take that result, multiply it by itself again, and add C again.
- Repeat the previous step over and over.
If the result above never increases toward infinity, the number C is part of the Mandelbrot Set. For example, if C = 1 the result obviously increases forever, so 1 is not in the Mandelbrot Set. If C = -1, though, the result alternates between 0 and -1, so -1 IS in the Mandelbrot Set.
Such a simple formula creates an infinitely detailed image:
C is a complex number, meaning it has a "real" part X and an "imaginary" part Y. A complex number represents a point in the X-Y plane, where X is a distance on the horizontal axis and Y is a distance on the vertical axis.
In the image of the Mandelbrot Set, points in the set are black. Points not in the set are colored by how fast the iterations diverge to infinity. Blue points are complex numbers that diverge quickly, and are "far away" from the set. Red points must be iterated many times before they diverge; they are "almost" in the set.
We can't iterate every point forever to determine for certain whether the point is in the set. Instead we iterate for a limited number of times, between 200 and 5000. If the result doesn't diverge by the time the limit is reached, we assume the point is in the set. Higher limits produce more accurate results, but take longer to generate an image. When zooming in to small scales, reduce the iteration limit for faster drawing, then increase the limit for more detail.
The Mandelbrot Set is thought to be connected, which means it is one single shape. It has infinitely many tendrils that are too thin to appear black in the image, but they're revealed by the points that are "close to" the set.
As you zoom into the Mandelbrot, you will see features that repeat infinitely at smaller and smaller scales. They never repeat exactly, though. Each one is slightly different than the others.
Even though the edge of the Mandelbrot Set is infinitely long, the set has a finite area of about 1.5 square units. The exact area is unknown.
This web page runs best on a PC with a multi-core processor and a mouse for dragging and zooming. On some tablets zooming and resizing appear jerky because it takes significant time to generate a new image.
To return to the Mandelbrot Set image, select the web browser's "Mandelbrot Set" tab. This information page displays in a new tab so that your current location within the image is not lost.