This picture is a Newton pattern given by the dynamical system with plotted on a small rectangular neighborhood of the point Advanced Method for Newton Patterns: If we can find the roots of a given polynomial prior to the fractal plotting, we can choose a color for each initial value z0 (which is a pixel in the canvas for a Julia pattern) depending upon which root the corresponding sequence zn converges to. In plotting the picture above, the computer is programmed to find all five roots of g through the method called Müller's algorithm and then color an initial value red, for instance, if the sequence converges to the first root. This causes the canvas to be separated into the five regions that are painted red, yellow, sky, blue, and green, and that are called the basins of attraction of the roots of g. Point of Mathematical Interest: The picture shows that the complex plane is divided into five basins of attraction of roots colored red, blue, sky, green, and amber. The basins are intricately interlaced and we can easily imagine from their fundamental property that no two of them can be in direct contact. So, there must be a nonempty set of points not belonging to any of the basins. The set is called the basin boundary. Can we visualize the basin boundary? If we zoom in on any curved segment that appears to separate distinct basins of attraction, we find a fractal pattern painted by all five colors, hence it cannot be a subset of the basin boundary. Thus, the basin boundary is more like dust. In fact, it happens to be a Julia set and the dusty Julia set is often classified as Cantor dust in topology. The fractal structure of the Julia set is quite complex and often escapes our intuition. It also illustrates the chaotic nature of Newton's algorithm characterized by the fact that a very slight change in the initial value z0 near the Julia set has an unpredictable consequence. This is the shortcoming of the otherwise extremely useful algorithm. Return to 
Main Gallery
Fractal Home
Gallery Annex
| |||
