Newton's Method Fractal: Basins of Attraction in the Complex Plane

Newton's method, run on every pixel

Newton's method finds a root of a function f by repeatedly following the tangent line down to where it crosses zero: starting from a guess z, the next guess is z - f(z)/f'(z). For a real function this converges fast near a root and can misbehave far from one. Run the same iteration with complex starting guesses z = x + iy, colour each starting pixel by which root its iteration converges to, and you get a Newton fractal - a picture of the basins of attraction of every root, with all the ordinary convergence guarantees breaking down in spectacular fashion along the basin boundaries.

The iteration and a concrete polynomial

For the simplest interesting case, f(z) = zⁿ - 1, whose n roots are the n-th roots of unity evenly spaced on the unit circle, Newton's update simplifies to:

z_(k+1) = z_k - (z_kⁿ - 1) / (n·z_kⁿ⁻¹)
        = ((n-1)·z_kⁿ + 1) / (n·z_kⁿ⁻¹)

Every pixel of the plane is an independent starting value z0. Iterate a fixed number of times (typically 30-60, enough for convergence almost everywhere except right on a boundary), check which of the n roots the final value is closest to, and paint the pixel that root's colour. Shading by the iteration count to convergence, rather than a flat colour, turns the flat basins into banded contour rings that reveal the fractal structure hiding inside what looks like a solid-coloured region.

Why the boundaries are fractal

Near a root, Newton's method converges quadratically - the number of correct digits roughly doubles every step - so deep inside a basin the picture is smooth and boring. The interesting structure lives entirely on the boundary between basins. Because the iteration function z - f(z)/f'(z) is a rational map of the Riemann sphere, its Julia set (the boundary where nearby starting points diverge to different attractors) has the self-similar, infinitely detailed structure typical of complex dynamics: zoom into any boundary point and you find smaller copies of the same three-or-more-basin tangle, forever.

This is a special case of a much older surprising fact from complex dynamics: the boundary between basins of attraction for any rational map with three or more distinct attracting fixed points is either connected or has infinitely many components, and in the Newton's-method case for a generic cubic, the three basins famously all share the same boundary - every boundary point is arbitrarily close to all three basins at once, not just two of them, a genuinely three-dimensional-feeling pathology living in a 2D picture.

Where Newton's method fails to converge at all

Not every starting point settles down. If the derivative f'(z) is ever exactly zero during the iteration the update blows up; more commonly a starting point can fall into a periodic cycle that never reaches any root - for z³ - 2z + 2, the real starting value z0 = 0 famously cycles forever between 1 and 0 without ever converging, a textbook counterexample to naive convergence claims. The fractal image marks such points (or points that fail to converge within the iteration budget) with a distinct 'no convergence' colour, usually black, and these points are exactly where the basin boundaries are thickest and most tangled.

Relaxed Newton and switching polynomials

A damping (relaxation) factor λ changes the update to z_(k+1) = z_k - λ·f(z)/f'(z). At λ=1 you get standard Newton; smaller λ slows convergence but can shrink or reshape the fractal boundary, and complex λ rotates the basin colouring entirely, since it changes the direction each step is allowed to move relative to the tangent line. Switching the polynomial from zⁿ-1 to something asymmetric like z³-2z+2 removes the rotational symmetry of the roots-of-unity case and produces a visibly different, less symmetric tangle of basins - the demo lets you flip between both and re-render live.

Frequently asked questions

Try it live

Everything above runs in your browser — open Newton's Method Fractal and change the parameters while it is running. Nothing is installed, nothing is uploaded, the whole model lives in one tab.