How Mandelbrot Rendering Works

From pixel to complex coordinate

Every screen pixel maps to a complex value c=x+yi. The renderer starts z at zero, iterates z←z²+c, stops when the orbit escapes or reaches the iteration limit, then maps numerical behavior to color.

• Map pixel position to the current complex viewport.
• Iterate the formula.
• Detect escape once |z|>2.
• Store iteration and optional derivative/orbit data.
• Convert that data into smooth color.
• Repeat for millions of pixels.

Why deep zoom is hard

Ordinary double precision carries only about 15–17 significant decimal digits. Deep enough, neighboring screen pixels become numerically indistinguishable. Modern deep-zoom engines use arbitrary-precision reference coordinates, perturbation, series approximation, rebasing and glitch correction to keep going.