Tricomplex dynamical systems generated by polynomials of odd degree

TL;DR

This paper characterizes the geometric structure of Multibrot sets generated by odd-degree polynomials in complex, hyperbolic, and tricomplex numbers, revealing their interval bounds, square shapes, and octahedral forms.

Contribution

It provides exact interval bounds for complex Multibrot sets, shows hyperbolic Multibrots are squares, and introduces a 3D octahedral generalization in tricomplex dynamics.

Findings

Exact interval of the cross section of Multibrot sets in complex plane

Hyperbolic Multibrots are always squares

Tricomplex Multibrot generalization forms an octahedron

Abstract

In this article, we give the exact interval of the cross section of the Multibrot sets generated by the polynomial $z^p+c$ where $z$ and $c$ are complex numbers and $p > 2$ is an odd integer. Furthermore, we show that the same Multibrots defined on the hyperbolic numbers are always squares. Moreover, we give a generalized 3D version of the hyperbolic Multibrot set and prove that our generalization is an octahedron for a specific 3D slice of the dynamical system generated by the tricomplex polynomial $\eta^p+c$ where $p > 2$ is an odd integer.