The Mandelbrot Set and Julia Sets

Julia Sets - Coloring Schemes

If the iterates of z₀ fail to satisfy the escape criterion, we assume z₀ belongs to the filled-in Julia set K_c and paint black the pixel containing z₀.

If the iterates of z₀ do satisfy the escape criterion, we usually paint the pixel containing z₀ with a color indicating how many iterations were needed to first get farther than 2 from the origin.

For example,

paint the pixel containing z₀ white if z₁ is farther than 2 from the origin.

Paint the pixel light to dark pink if a few more iterates are needed to get farther than 2 from the origin.

Paint the pixel orange to yellow if a few more iterates are needed to get farther than 2 from the origin.

Paint the pixel green to blue if a few more iterates are needed to get farther than 2 from the origin.

and so on. The successive magnifications - left to right, top to bottom - illustrate this coloring.

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