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 M3, we show that one of them corresponds to a third Platonic solid: the cube.
Highlights
Quadratic polynomials iterated on hypercomplex algebras have been used to generate multidimensional Mandelbrot sets for several years [1–10]
This suggests another approach to generating regular polyhedra within tricomplex dynamics: defining and visualizing 3D slices in a basis that is directly linked to the simple dynamics of the real line
We first established new results in the algebra of tricomplex numbers related to idempotent elements and invertibility, paving the way for interesting extensions valid in the multicomplex setting M(n), n ≥ 3
Summary
Quadratic polynomials iterated on hypercomplex algebras have been used to generate multidimensional Mandelbrot sets for several years [1–10]. This approach is widespread in the literature, other attempts at generalizing the classic fractal to higher dimensions have been made [11–13]. The same authors introduced an equivalence relation between the fifty-six principal 3D slices of the tricomplex Mandelbrot set M3 in order to establish which slices have the same dynamics and appearance in. The authors of [15,16] determined the exact intervals corresponding to M p ∩ R, depending on whether the integer p is odd or even, and showed that the tricomplex Mandelbrot set M33 generated by the cubic polynomial z3 + c has only four principal 3D slices [17]. We talk about the possibility to generate, in the same 3D subspace as that of the Firebrot and its geometric dual, a regular compound called the stellated octahedron ( named the Stella octangula) as a tricomplex dynamical system
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