Characterization of the Principal 3D Slices Related to the Multicomplex Mandelbrot Set

TL;DR

This paper investigates the dynamics of 3D principal slices of multicomplex Multibrot sets, establishing their equivalence to tricomplex slices and simplifying their analysis within this framework.

Contribution

It introduces an equivalence relation among 3D slices of multicomplex Multibrot sets and shows they are all affine equivalent to tricomplex slices, simplifying their study.

Findings

Any multicomplex 3D principal slice is affine equivalent to a tricomplex slice.

Multibrot sets in 3D slices can be fully understood within the tricomplex framework.

The study reduces the need for generalization beyond tricomplex space for these slices.

Abstract

This article focuses on the dynamics of the different tridimensional principal slices of the multicomplex Multibrot sets. First, we define an equivalence relation between those slices. Then, we characterize them in order to establish similarities between their behaviors. Finally, we see that any multicomplex tridimensional principal slice is equivalent to a tricomplex slice up to an affine transformation. This implies that, in the context of tridimensional principal slices, Multibrot sets do not need to be generalized beyond the tricomplex space.