DépêcheMath - Mandelbrot sequence f₀(z) = 0; fₙ₊₁(z) = (fₙ(z))² + z

Animated development of the "Mandelbrot" sequence f₀(z) = 0; fₙ₊₁(z) = (fₙ(z))² + z with some mathematical annotations and explanations.

Music: Doug Maxwell, Cast of Pods

This animation covers the recursively defined sequence of polynomials f₀(z) = 0; fₙ₊₁(z) = (fₙ(z))² + z which is closely related to the famous Mandelbrot fractal set. Its development is depicted by coloring the points of the complex plane according to the values of fₙ(z). See the legend in the lower left corner for how to translate these colors to complex values or halt this animation at f₁(z) = z which is the identity map and study the correspondence between colors and positions in the complex plane.

Speaking in terms of dynamical systems theory, we discuss this recursion not in the dynamic plane but rather in the parameter plane and study the sequence members as functions of this parameter.

Animation imaging was created using "Complex Plane Viewer" (CPV), a program written by myself for visualizing complex functions, developed within the Lazarus IDE.

Watch this video in Full HD resolution (1080p) and full screen mode for best viewing experience.