For each \(c = Re(c) + i\, Im(c)\) in the complex plane, let the sequence \((z_{n})\) be defined by recurrence \(z_{n+1}=z_{n}^{2}+c\) and \(z_{0}=0\).

The sequence is computed until the soonest of (i) \(Abs(z_{n})>EscapeRadius=2\), or (ii) \(n>MaxIterations\).

If \(n>MaxIterations\) then the sequence is deemed to be bounded and the point whose coordinates are \(\{Re(c),\, Im(c)\}\) is colored in black.

Else let \(p\) be the first \(n\) for which \(Abs(z_{n})>EscapeRadius\), then the point representing \(c\) is colored according to a gradient of colors defined as follows RGB(f(RedFrequency*p), f(GreenFrequency*p), f(BlueFrequency*p)), with f() being the function that transforms the sequence of positive natural integers {0, 1, 2, 3, etc} into the following sequence {0, 1, 2,.., 254, 255, 254,.., 1, 0, 1, etc}.

EscapeRadius is set to 2 because it can be proven that if there is a \(n\) such that \(Abs(z_{n})>EscapeRadius\) then the sequence will escape to infinity.

The Number of Workers is the number of parallel processes uses to calculate the image.

Click on a point to zoom in.

Shift+Click on a point to zoom out.

ZoomFactor determines the amplitude of the zooming in/out.

Press respectively J/K/F/R to move Left/Right/Down/Up.

Enter the coordinates of the center in Re(Image Center)/Im(Image Center) and the level of Zoom in Zoom.

(The initial Zoom is set to 1 and corresponds to Image Height = 3 centered around zero).