Newton Fractal for z³ − 1

This formula turns a familiar root-finding algorithm into a map of the complex plane. Unlike the Mandelbrot set, it does not ask whether an orbit escapes. It asks which root of z³ − 1 Newton's method reaches from a given start. The three cube roots of unity (1, e^{2πi/3}, e^{4πi/3}) each claim a basin of attraction: all starting values whose orbit converges to that root.

The three basins are easy to pick out. Their shared boundary is not. A point exactly on it reaches no root; it follows a chaotic, non-repeating path instead. This boundary is the Julia set of the Newton map—the rational function N(z) = z − (z³ − 1)/(3z²). Zoom in, and more structure keeps arriving.

It is a classic convergence fractal: an everyday numerical algorithm with a restless border inside it. Arthur Cayley framed the classification problem for Newton's basins in 1879. He solved the quadratic case, but the cubic case “appears to present considerable difficulty.” Fatou and Julia’s later complex dynamics made the problem legible; the first computer visualizations arrived in the early 1980s.

At the heart of this image is Newton's method, discovered by Isaac Newton in 1669 and independently refined by Joseph Raphson in 1690. For real functions it is a practical root‑finding tool; for complex polynomials it reveals fractal basins no one could have foreseen at the time.

The mathematical study of those basins began in 1879 with Arthur Cayley's paper "The Newton–Fourier imaginary problem" (American Journal of Mathematics, vol. 2, p. 97). For a complex quadratic polynomial, Cayley showed that the basins of attraction are simply the two half-planes separated by the perpendicular bisector of the line segment joining the two roots. At cubic polynomials, though, he found the problem "considerable difficult." His inability to extend the simple quadratic result to the cubic case became known as Cayley's problem.

The resolution came through complex dynamics, developed by Pierre Fatou and Gaston Julia in the years around 1918. Their theory of iterated rational functions supplied the language for the basins and their infinitely convoluted boundaries. But the first pictures of the Newton fractal for z³ − 1 had to wait for computer graphics in the early 1980s.

In 1986, Heinz-Otto Peitgen and Peter H. Richter included a chapter titled "Newton's Method for Complex Polynomials: Cayley's Problem" (Chapter 6, pp. 93–106) in *The Beauty of Fractals: Images of Complex Dynamical Systems*. It presents some of the earliest published computer renderings of these basins and helped introduce a wide audience to the fractal structure hidden inside Newton's method.

Three large basins, one for each cube root of unity, occupy the image. They meet in wheel-spoke lines from the centre, making the three-part symmetry plain. Inside each basin, convergence is quick and the colour stays smooth.

Between any two basins—and especially at the central three-way meeting—spirals, miniature three-basin copies, and dendritic threads crowd in. Follow a spiral arm inward: the three-basin arrangement returns at smaller scales, bent by local dynamics. Broad calm regions; then a restless frontier.