Generation and graphical analysis of Mandelbrot and Julia sets in more than four dimensions

Abstract

A method to define and generate Mandelbrot and Julia sets in more than four dimensions is presented. A doubling process is used to create from the set of real numbers a hypercomplex number system of arbitrary dimension. Since the new number system is closed under addition and multiplication, it can be used to generate Mandelbrot and Julia sets of corresponding dimension. Generation of these sets in more than four dimensions is discussed. A graphical analysis manifests the sets are fractal in these higher dimensions. Symmetrical properties of Mandelbrot and Julia sets are observed and reported.