On the Algebraic Foundation of the Mandelbulb

TL;DR

This paper extends the Mandelbrot set concept using quaternion algebra and spherical coordinates, introducing new generalized sets and analyzing their properties through algebraic structures and escape time algorithms.

Contribution

It introduces a quaternionic spherical product and explores its algebraic properties, leading to new Mandelbrot set generalizations including the bulbic Mandelbrot set.

Findings

The spherical product of pure quaternions is a commutative unital magma.

A set visually identical to the Mandelbulb can be generated and shown to be bounded.

The bulbic Mandelbrot set's 2D cut exhibits Mandelbrot-like dynamics.

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.