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Relationship between the Mandelbrot Algorithm and the Platonic Solids

Author

Listed:
  • 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)

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.

Suggested Citation

  • André Vallières & Dominic Rochon, 2022. "Relationship between the Mandelbrot Algorithm and the Platonic Solids," Mathematics, MDPI, vol. 10(3), pages 1-17, February.
    • Handle: RePEc:gam:jmathe:v:10:y:2022:i:3:p:482-:d:740708
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