Fractal Guide
What is a Newton Fractal?
Newton fractals emerge when Newton's method for finding equation roots is applied to the complex plane. Unlike other fractals, they arise from a different principle, demonstrating the deep connection between mathematics and computation.
Contents
1. What is Newton's Method?
Newton's method (Newton-Raphson method) is an algorithm for approximately finding roots of f(x) = 0. Starting from an initial value xโ, it repeatedly applies x(n+1) = x(n) - f(x(n))/f'(x(n)) to converge toward a root.
It's an iterative method using tangent lines, devised by Newton and Raphson in the 17th century. It remains a fundamental algorithm widely used in scientific and engineering numerical computation today.
2. Applying it to the Complex Plane
Newton's method applies to complex numbers just as it does to reals. For example, zยณ - 1 = 0 has three complex roots: 1, e^(2ฯi/3), and e^(4ฯi/3).
When Newton's method is applied with each point on the complex plane as the initial value, each point converges to one of the three roots. However, 'which root' depends on the initial value in an extremely complex way, and fractal structures emerge at the boundaries.
3. Convergence Regions and Fractals
The set of initial values that converge to each root is called a 'convergence region.' For zยณ - 1 = 0, the complex plane is divided into three convergence regions, but this division is remarkably complex.
The boundaries of convergence regions have the property that between any two regions, the third region always exists (Wigner's property). This creates infinitely complex fractal boundaries where three-colored patterns continue no matter how far you zoom in.
4. Coloring and Rendering
Newton fractal rendering assigns a hue based on which root each pixel (initial value) converges to, and varies brightness by iteration count. Points that take many iterations (slow convergence) appear dark; those that converge quickly appear bright.
Unlike other fractals, Newton fractals have no 'divergence' (Newton's method always converges to some root). The coloring is based on the target root and convergence speed rather than escape time.
Explore the Newton Fractal
Explore the beautiful convergence patterns of Newton fractals in the Fractal Gallery.
Go to Fractal Gallery