Hyperdimensional generalized M–J sets in hypercomplex number space

Abstract

Doubling and truncation techniques are used to generate a hypercomplex number system of any dimension. The precondition that addition and multiplication have closure in hypercomplex number system is discussed, and the definition and constructing arithmetic of the hyperdimensional generalized Mandelbrot–Julia sets (in abbreviated form generalized M–J sets) in hypercomplex number system are listed out. By analyzing 2-D and 3-D cross sections of the hyperdimensional generalized M–J sets, the fractal feature of 2-D and 3-D cross sections is studied, and the symmetry of 2-D and 3-D cross sections has been proved. The analysis of symmetry in this paper will help to study dynamics of hypercomplex number more.