On the Algebraic Foundation of the Mandelbulb

Abstract

In this paper, we generalize the Mandelbrot set using quaternions and spherical coordinates. In particular, we use pure quaternions to define a spherical product. This product, which is inspired by the product of complex numbers, add the angles and multiply the radii of the spherical coordinates. We show that the algebraic structure of pure quaternions with the spherical product is a commutative unital magma. Then, we present several generalizations of the Mandelbrot set. Among them, we present a set that is visually identical to the so-called Mandelbulb. We show that this set is bounded and that it can be generated by an escape time algorithm. We also define another generalization, the bulbic Mandelbrot set. We show that one of its 2D cuts has the same dynamics as the Mandelbrot set and that we can generate this set only with a quaternionic product, without using the spherical product.