Newton Fractal for z³ − 1

Esta fórmula convierte un algoritmo conocido de búsqueda de raíces en un mapa del plano complejo. A diferencia del conjunto de Mandelbrot, no pregunta si una órbita escapa. Pregunta a cuál raíz de z³ − 1 llega el método de Newton desde un punto de partida dado. Las tres raíces cúbicas de la unidad (1, e^{2πi/3}, e^{4πi/3}) reclaman cada una su cuenca de atracción: todos los valores iniciales cuya órbita converge a esa raíz.

Las tres cuencas son fáciles de distinguir. Su frontera compartida no. Un punto exactamente sobre ella no llega a ninguna raíz; sigue un camino caótico que no se repite. Esta frontera es el conjunto de Julia del mapa de Newton —la función racional N(z) = z − (z³ − 1)/(3z²). Acércate y sigue apareciendo más estructura.

Es un fractal de convergencia clásico: un algoritmo numérico cotidiano con una frontera inquieta en su interior. Arthur Cayley planteó el problema de clasificación de las cuencas de Newton en 1879. Resolvió el caso cuadrático, pero el cúbico «parece presentar una dificultad considerable». La dinámica compleja posterior de Fatou y Julia hizo legible el problema; las primeras visualizaciones por computadora llegaron a principios de los años ochenta.

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.