Classification of Principle 3D Slices of Filled-in Julia Sets in Multicomplex Spaces

TL;DR

This paper extends the concept of filled-in Julia sets into multicomplex spaces, introduces visualization algorithms for 3D slices, and characterizes the number of such slices depending on polynomial degree parity.

Contribution

It generalizes filled-in Julia sets using multicomplex numbers and provides a new classification of 3D slices, differing from prior characterizations.

Findings

Nine 3D slices for odd p in polynomial $z^p + c$

Four 3D slices for even p in polynomial $z^p + c$

Different proofs from previous characterizations

Abstract

A generalization of the filled-in Julia set is presented using the multicomplex numbers and an algorithm is presented to visualize these sets in the tridimensional space. There are many ways to visualize these higher dimensional fractals sets on a computer. We therefore introduce an equivalence relation between 3D representations and show that, for the filled-in Julia sets associated to the polynomial $z^p + c$, there are nine 3D slices when $p$ is an odd integer and four when $p$ is even. These results differs from the recent characterization obtained by Brouillette and Rochon in 2019 and the proofs require different arguments in the context of the filled-in Julia sets.