References & Citations
Mathematics > Dynamical Systems
Title: Relationship between the Mandelbrot Algorithm and the Platonic Solids
(Submitted on 6 Jul 2021 (v1), last revised 2 Feb 2022 (this version, v5))
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.
Submission history
From: Dominic Rochon [view email][v1] Tue, 6 Jul 2021 17:50:53 GMT (2470kb,D)
[v2] Fri, 9 Jul 2021 13:20:56 GMT (2470kb,D)
[v3] Wed, 8 Sep 2021 15:54:42 GMT (2469kb,D)
[v4] Thu, 18 Nov 2021 15:11:24 GMT (8686kb,D)
[v5] Wed, 2 Feb 2022 14:03:40 GMT (8686kb,D)
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