5 research outputs found

    Relationship between the Mandelbrot Algorithm and the Platonic Solids

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    This paper focuses on the dynamics of the eight tridimensional principal slices of the tricomplex Mandelbrot set for the power 2: 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\mathcal{M}_{3}, we show that one of them corresponds to a third Platonic solid: the cube

    Counting Involutions on Multicomplex Numbers

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    We show that there is a bijection between real-linear automorphisms of the multicomplex numbers of order nn and signed permutations of length 2n−12^{n-1}. This allows us to deduce a number of results on the multicomplex numbers, including a formula for the number of involutions on multicomplex spaces which generalizes a recent result on the bicomplex numbers and contrasts drastically with the quaternion case. We also generalize this formula to rr-involutions and obtain a formula for the number of involutions preserving elementary imaginary units. The proofs rely on new elementary results pertaining to multicomplex numbers that are surprisingly unknown in the literature, including a count and a representation theorem for numbers squaring to ±1\pm 1
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