Newton's Method: From Calculus to Fractal Geometry
Isaac Newton described his root-finding method in De Analysi (1669), and Joseph Raphson published an independent version in 1690. For a real function f(x), the method is geometrically intuitive: at each step, draw a tangent to the curve at the current guess and use its x-intercept as the next guess. Formally:
Setting y = 0: xn+1 = xn − f(xn) / f′(xn)
For well-behaved functions with a simple root, the method converges quadratically — the number of correct decimal digits roughly doubles with each iteration. Starting within the "basin of attraction" of a root guarantees convergence to it.
Extending to the Complex Plane
The same formula applies verbatim in the complex plane ℂ, where zn ∈ ℂ and f′(z) is the complex derivative. Every polynomial of degree d has exactly d roots in ℂ (Fundamental Theorem of Algebra), so every starting point z0 belongs to the basin of one of these roots — unless it lies on the fractal boundary between basins, where the iteration never settles.
Arthur Cayley observed in 1879 that for f(z) = z² − 1 (roots +1 and −1), the two basins are simply the right and left half-planes — no fractal structure. But for degree 3 and higher, the basins are unmistakably fractal. The boundaries form Julia sets, studied rigorously by Gaston Julia (1918) and Pierre Fatou (1919) long before the invention of computer graphics.
Roots of Unity and Symmetric Cases
For f(z) = zd − 1, Newton's method simplifies to zn+1 = ((d−1)zn + zn−(d−1)) / d. The d roots of unity are evenly spaced at angles 2πk/d on the unit circle, giving the fractal perfect d-fold rotational symmetry. The case d = 3 produces the classic symmetric Newton fractal with three interleaved basins.
Convergence Speed and Brightness
The iteration count n — how many Newton steps until |zn − root| < ε — encodes how "decisively" the starting point belongs to its basin. Points deep inside a basin converge in 2–5 steps (bright colour); points on the basin boundary may oscillate for 50+ steps before slowly settling (dark bands, showing fine structure when zoomed in).
Self-Similarity at All Scales
The boundary between basins is a fractal: zooming in at any boundary region reveals detail that never resolves into smooth curves. This is a consequence of the Fatou-Julia theorem: every neighbourhood of a boundary point contains points that converge to all d different roots. The fractal dimension of the boundary exceeds 1 (it is "more than a curve"), and for the z³−1 Newton fractal it has been computed to be approximately 1.85.
Frequently Asked Questions
What is a Newton fractal?
A Newton fractal colours each point of the complex plane according to which root Newton's method converges to from that starting point. The intricate boundary between differently coloured regions is a fractal (Julia set) with self-similar structure at every zoom level.
Why is the boundary fractal rather than smooth?
Near the boundary between two basins, the iteration is sensitive to the exact starting value — an exponentially small perturbation can redirect the orbit to a different root. The Fatou–Julia theorem states that every neighbourhood of a boundary point contains points from all basins, making the boundary infinitely complicated and non-differentiable everywhere.
How does Newton's method relate to the Mandelbrot set?
Both Newton fractals and the Mandelbrot/Julia sets are examples of complex dynamical systems — iteration of a complex function. The Mandelbrot set studies escape vs. non-escape for f_c(z)=z²+c; Newton fractals study convergence to roots of a polynomial. Both generate infinite fractal complexity from simple arithmetic rules applied repeatedly in the complex plane.
Is Newton's method the fastest root-finder?
Within its basin of attraction, Newton's method has quadratic (order-2) convergence — each step squares the error. Halley's method achieves cubic convergence at the cost of computing the second derivative. For polynomials, the Durand-Kerner method finds all roots simultaneously. For global root-finding outside a known basin, more robust methods (bisection, Brent's method) are preferred, as Newton's method can cycle or diverge.
Related Simulations & Articles
Explore 40+ interactive simulations
Physics, biology, mathematics, and computer science — all free to start.
Browse All Simulations →