The Mandelbrot Set

The following Java applet displays the Mandelbrot set. You can click on the image to zoom in by a factor of two, and again after that to zoom in again by another factor of two, and so on. The zoom factor is displayed just below the back button. You can press the back button to zoom back out once you’ve zoomed in, or you can press the left mouse button while holding down the control key to zoom out at that point.

You can share an interesting or artistic view of the Mandelbrot Set with your friends. Just copy the URL text from the applet below and paste it into an email or blog post. Click here for details.

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The Mandelbrot set is a function of the complex plane. When the set is first drawn, the real number line extends left-to-right from -3.0 to 3.0, and the imaginary line extends bottom-to-top from -3.0i to 3.0i, and the origin of the complex plane is at the center of the drawing area. Each pixel in the drawing area is translated to the associated coordinate in the complex plane, and then that complex number is tested to see if it belongs to the Mandelbrot set or not. A complex number z belongs to the Mandelbrot set if |z| < 2, and |z + z²| < 2, and |z + (z + z²)²| < 2, and so on. |z| represents the modulus of the complex number z, where z = a + bi, and |z| = (a² + b²)½.

Obviously, it takes an infinite expansion of this series to determine if a number actually belongs to the set, so the best you can do is determine if a number does not belong to the set. This applet makes up to 400 iterations of squaring and adding each complex number before it gives up and colors the associated pixel black. Other bands of color indicate numbers that were determined to not be in the set after fewer iterations.