Newton's Fractal: Where Root-Finding Turns Chaotic

The iteration: sliding down the tangent

Newton's method turns solving f(z) = 0 into a feedback loop. From a guess zₙ, follow the tangent line of f down to where it crosses zero; that crossing is your next guess. The rule you learn for real numbers extends verbatim to complex ones:

zₙ₊₁ = zₙ − f(zₙ)/f′(zₙ)

The right-hand side defines the Newton map N(z) = z − f(z)/f′(z). Every root r of f is automatically a fixed point of N, since f(r) = 0 forces N(r) = r. What makes the method fast is the derivative there. For a simple root a short computation gives N′(z) = f(z)f″(z)/f′(z)², which vanishes at r. A fixed point with N′(r) = 0 is called superattracting, and near it the error obeys |zₙ₊₁ − r| ≈ C·|zₙ − r|²: the number of correct digits roughly doubles every step. This quadratic convergence is why Newton's method is the workhorse of numerical root-finding.

Coloring the plane: basins of attraction

Now run the loop from every point of the complex plane at once. For the cubic f(z) = z³ − 1 the three roots are the cube roots of unity — 1, ω = e^2πi/3, and ω² = e^4πi/3 — sitting at the vertices of an equilateral triangle on the unit circle. Give each root its own color. Paint every starting point z₀ with the color of the root its orbit converges to, and shade it by how many iterations that took. The result is Newton's fractal.

Concretely the map is N(z) = z − (z³ − 1)/(3z²) = (2z³ + 1)/(3z²), a rational map of the Riemann sphere of degree 3. The set of points whose orbits flow to a given root r is its basin of attraction A(r). Deep inside each basin the colors form broad, smooth regions — start near a root and you converge almost instantly. All of the structure lives at the seams between the colors, and understanding those seams is the whole game.

The fractal boundary is a Julia set

The dynamics split the plane into two opposite kinds of place. Where the iteration is tame — nearby points share the same fate — is the Fatou set, and the three basins fill it entirely. Where the iteration is wild — the map amplifies the tiniest difference — is the Julia set, and it is precisely the common boundary of the basins.

On the Julia set the behavior is genuinely chaotic: two starting points a millionth apart can land in different basins, so no finite precision can decide in advance which root you will reach. The set is a repeller, and even the point at infinity belongs to it. For a degree-d polynomial, N fixes ∞ with multiplier d/(d−1) — here 3/2 > 1, a repelling fixed point — which is why the tangled frontier reaches out at every scale instead of staying in a bounded blob. Repelling periodic points are dense in the Julia set, and its self-similar filigree is the visible fingerprint of that instability: zoom in anywhere on the boundary and you find a shrunken copy of the whole tangle.

Why the third root always appears

Look at the boundary between the red and blue basins and you will always find slivers of the green basin threaded through it, no matter how far you magnify. This is not a rendering glitch — it is forced by a theorem of Pierre Fatou: the boundary of every basin of a rational map is one and the same set, the Julia set. The frontier of the red basin is literally identical to the frontier of the blue and of the green. Consequently any point on the boundary is simultaneously on the edge of all three basins, and every neighborhood of it must contain all three colors.

Three open regions that share a single common boundary are the classic Lakes of Wada, a construction once regarded as a topological curiosity — and Newton's fractal realizes it naturally. The symmetry is exact, too. Because N(ωz) = ω·N(z) for z³ − 1 (an equivariance that follows straight from ω³ = 1), the entire picture is unchanged under a 120° rotation, so the three basins are congruent copies permuted by the same symmetry as the triangle of roots.

Cayley's problem and the birth of complex dynamics

This all started as a nineteenth-century puzzle. In 1879 Arthur Cayley posed the Newton–Fourier imaginary problem: for each starting point in the complex plane, say which root Newton's method will find. For a quadratic he solved it completely. Conjugating the Newton map of z² − 1 by the Möbius transformation w = (z − 1)/(z + 1) — which sends the roots +1 and −1 to 0 and ∞ — turns the iteration into the trivial w ↦ w². Its boundary, the unit circle |w| = 1, pulls back to a straight line: the perpendicular bisector of the two roots, i.e. the imaginary axis. Newton's method simply converges to whichever root is nearer, with no fractal anywhere.

For the cubic, Cayley admitted the problem presented no little difficulty and could not resolve it. He was right to be stuck: there is no Möbius change of variable that linearizes z³ − 1, and the honest answer is the fractal. The machinery needed to describe it — the theory of iterating rational maps — arrived only around 1918–1920 with Gaston Julia and Pierre Fatou, which is how Cayley's innocent question became a founding problem of complex dynamics.

Dimension, convergence rates, and pathologies

How rough is the boundary? Topologically it is a curve of dimension 1, yet it is nowhere smooth. Its Hausdorff dimension is strictly greater than 1, with box-counting estimates for the z³ − 1 Julia set reported near ~1.4; that fractional excess measures how densely the frontier crinkles — filling area without ever becoming genuinely two-dimensional.

Convergence is not always quadratic, either. At a root of multiplicity m the Newton map has N′(r) = 1 − 1/m ≠ 0, so a double root is only linearly attracting with rate 1/2 — one extra correct bit per step rather than a doubling of digits. Worse, Newton's method can fail to find any root at all. For f(z) = z³ − 2z + 2 the points 0 and 1 form an attracting 2-cycle, since N(0) = 1 and N(1) = 0; an entire open set of starting points spirals into that cycle forever and never reaches a solution — a whole basin belonging to no root.

• Superattracting root: N′ = 0, quadratic convergence, digits double each step.
• Multiple root: N′ = 1 − 1/m, only linear convergence.
• Spurious cycle: an attracting periodic orbit that traps orbits away from every root.

The same fractal basins surface whenever an iterative solver has several possible answers — in optimization, in computer graphics when tracing where a ray meets a surface, and in any Householder-type higher-order method. Newton's fractal is the standing reminder that an algorithm can be locally excellent and yet globally, provably unpredictable.

How the shape of the basin boundary depends on the polynomial you feed Newton's method

Polynomial	Roots	Basin boundary	Behavior
z² − 1	2	Straight line (perpendicular bisector)	Conjugate to w ↦ w²; converges to the nearer root, no fractal
z³ − 1	3	Julia-set fractal (Wada, dim > 1)	Chaotic on the boundary; all three basins meet everywhere
z³ − 2z + 2	3	Fractal plus basin of a 2-cycle	An open set never reaches any root at all
(z − 1)²	1 (double)	None — whole plane converges	Only linear convergence, rate 1/2