Imaginary scators quadratic mapping in 1+2D dynamic space

The elliptic scator algebra quadratic iteration is evaluated in 1+2 dimensions in dynamic space. There exists a non divergent K set in the scator three dimensional space with a highly complex boundary J = 8K. Two and three dimensional renderings of the sets in dynamic space are published for the first time. The sets exhibit a rich fractal boundary in all three directions. Some of the salient features of the sets can be described in terms of square nilpotent iterations. The Julia and filled in Julia sets are identically reproduced at two perpendicular planes where only one non-vanishing hypercomplex director component is present. The fixed points of K in 81+2 with real constant c, can be obtained from the roots of a scator quadratic polynomial equation. In 81+2 there can be, in addition to the usual complex roots, hypercomplex roots that give rise to four additional fixed points. The inverse orbits of the hypercomplex roots reveal a rich complex structure, that allow for the evaluation of an infinite set of points in 8K. The 8K set of the origin is equal to the unit magnitude scator surface, named a cusphere. The J set exhibits self similar structures in 3D at different scales, typical of fractal phenomena. The ix set, is the three dimensional equivalent of the M-set in three dimensions. It is conjectured that the ix-set with some restrictions, is the set of parameters where the J set is connected.& COPY; 2023 Elsevier B.V. All rights reserved.