5 research outputs found
Relationship between the Mandelbrot Algorithm and the Platonic Solids
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
, we show that one of them corresponds to a third Platonic
solid: the cube
Counting Involutions on Multicomplex Numbers
We show that there is a bijection between real-linear automorphisms of the
multicomplex numbers of order and signed permutations of length .
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 -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