Relationship between the Mandelbrot Algorithm and the Platonic Solids

TL;DR

This paper explores the geometric structures of 3D slices of the tricomplex Mandelbrot set, revealing connections to Platonic solids and constructing a dynamical system involving these shapes.

Contribution

It provides a geometric classification of the principal 3D slices of the tricomplex Mandelbrot set and links them to Platonic solids, including the tetrahedron and cube.

Findings

Firebrot is a regular tetrahedron.

Constructed the Stella octangula from Firebrot and its dual.

Identified a 3D slice corresponding to a cube.

Abstract

This paper focuses on the dynamics of the eight tridimensional principal slices of the tricomplex Mandelbrot set: the Tetrabrot, the Arrowheadbrot, the Mousebrot, the Turtlebrot, the Hourglassbrot, the Metabrot, the Airbrot (octahedron) and the Firebrot (tetrahedron). In particular, we establish a geometrical classification of these 3D slices using the properties of some specific sets that correspond to projections of the bicomplex Mandelbrot set on various two-dimensional vector subspaces, and we prove that the Firebrot is a regular tetrahedron. Finally, we construct the so-called "Stella octangula" as a tricomplex dynamical system composed of the union of the Firebrot and its dual, and after defining the idempotent 3D slices of $\mathcal{M}_{3}$, we show that one of them corresponds to a third Platonic solid: the cube.