Relationship between the Mandelbrot Algorithm and the Platonic Solids

André Vallières and Dominic Rochon
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André Vallières: Département de Mathématiques et D’informatique, Université du Québec à Trois-Rivières, C.P. 500, Trois-Rivières, QC G9A 5H7, Canada
Dominic Rochon: Département de Mathématiques et D’informatique, Université du Québec à Trois-Rivières, C.P. 500, Trois-Rivières, QC G9A 5H7, Canada

Mathematics, 2022, vol. 10, issue 3, 1-17

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 M 3 , we show that one of them corresponds to a third Platonic solid: the cube.

Keywords: generalized Mandelbrot sets; tricomplex dynamics; metatronbrot; 3D fractals; Platonic solids; Airbrot; Earthbrot; Firebrot; Stella octangula (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2022
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