Journal
MATHEMATICS
Volume 10, Issue 3, Pages -Publisher
MDPI
DOI: 10.3390/math10030482
Keywords
generalized Mandelbrot sets; tricomplex dynamics; metatronbrot; 3D fractals; Platonic solids; Airbrot; Earthbrot; Firebrot; Stella octangula
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Funding
- Natural Sciences and Engineering Research Council of Canada (NSERC)
- FRQNT
- ISM
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This paper focuses on the dynamics of the eight tridimensional principal slices of the tricomplex Mandelbrot set and establishes a geometrical classification of these slices. It proves that the Firebrot is a regular tetrahedron and constructs the so-called Stella octangula as a tricomplex dynamical system. It also demonstrates that one of the 3D slices corresponds to a cube.
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.
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