Chapter 33 - Julia sets in the quaternions

Abstract

Several studies have appeared in recent years demonstrating the visual effects obtainable from applying 2Dcomputer graphics to complex polynomials. Fascinating as such pictures are, they are only slim fragments compared to the 3D or 4D physical reality. Quadratics reside in higher dimensions, and this chapter presents one such extension––that is, the usage of 4D quaternion algebra to define structures possessing complex patterns of infinitely repeating geometric structure. Recent mathematical work on the dynamics of complex analytic functions has given rise to a new subject matter for computer graphics. The combination of mathematical theory and computer graphics has resulted in new insight into the nature of some of the simplest of mathematical objects: second-degree polynomials. Most of that work has focused on the possibilities within the 2D complex plane. This chapter shows the extension of these investigations to higher dimensions, thereby resulting in fractals that naturally reside in the 4D quaternions. Particular attention is paid to the formula ax2 + b. A method is given for obtaining various interconnection patterns for the Julia sets in 4-space, and the results are displayed in 3D computer graphics.