3D RENDERING OF THE QUATERNION MANDELBROT SET WITH MEMORY

Abstract

In this paper, we explore the quaternion equivalent of the Mandelbrot set equipped with memory and apply various visualization techniques to the resulting $4$-dimensional geometry. Three memory functions have been considered, two that apply a weighted sum to only the previous two terms and one that performs a weighted sum of all previous terms of the series. The visualization includes one or two cutting planes for dimensional reduction to either $3$ or $2$ dimensions, respectively, as well as employing an intersection with a half space to trim the $3$D solids in order to reveal the interiors. Using various metrics, we quantify the effect of each memory function on the structure of the quaternion Mandelbrot set.