Newton fractal

Newton's method is one of the oldest and most powerful algorithms in mathematics: given a function f, start from a guess z₀ and iterate zₙ₊₁ = zₙ − f(zₙ)/f'(zₙ). In the real numbers, this converges rapidly to a root when you start close enough. But in the complex plane, the question "which root do you converge to?" has a fractal answer. The basins of attraction — the sets of starting points that converge to each root — are separated by infinitely intricate boundaries.

The boundary between basins is a Julia set: a fractal curve where three or more regions meet at every point. Near the boundary, an infinitesimal change in starting position can send Newton's method to a completely different root. The coloring here assigns each root a distinct hue, and maps convergence speed to brightness — points that converge quickly are bright, while points near the fractal boundary linger in indecision and darken.

The damping slider controls a generalization called "relaxed Newton's method," where the step size is multiplied by a factor α. When α = 1 you get the standard method; other values reshape the basins of attraction entirely, producing new fractal geometries from the same polynomial.