The general quaternionic M–J sets on the mapping z←zα+c(α∈N)

Fractal nature exists not only in complex plane but also in higher dimensional space. It is a focus to find new modals to construct new 3-D fractals. In this paper, the general Mandelbrot sets and Julia sets on the mapping f:z←zα+c(α∈N) are discussed. The 3-D projections of M–J sets are constructed using escape time algorithm and ray-tracing method. And their properties are theoretically analysed. The connectness of the general quaternionic M sets is proved and the boundary of the stability region of the fixed point is calculated. It is found that if the parameter c of Julia sets is chosen from the M sets, they share the same cycle number and stable points. It can be concluded that the quaternionic M sets contain sufficient information of quaternionic Julia sets.