Newton's method iterative fractal - Basin of attraction visualization on complex plane
Newton's method (also called Newton-Raphson method) was developed by Isaac Newton in 1669 and later refined by Joseph Raphson in 1690. It is a powerful iterative technique for finding successively better approximations to the roots (or zeroes) of a real-valued function. The extension to complex numbers and the study of its fractal behavior came much later, with the visualization of Newton fractals becoming possible with modern computing in the late 20th century.
For a complex polynomial f(z), Newton's method iterates using the formula: z_{n+1} = z_n - f(z_n)/f'(z_n). Starting from each point z_0 in the complex plane, the iteration typically converges to one of the roots of f(z). The 'basin of attraction' for each root consists of all starting points that converge to that root. The boundaries between these basins form infinitely intricate fractal patterns - this is the Newton fractal.
The fractal boundaries occur due to the sensitive dependence on initial conditions. Near the boundary between two basins, tiny changes in the starting point can lead to convergence to different roots. This sensitivity creates infinitely complex boundary patterns at all scales - a hallmark of fractal geometry. The boundary has a fractal dimension greater than 1 (the dimension of a smooth curve), meaning it's more 'space-filling' than a simple line.