Quadratic Dynamics in Matrix Rings: Tales of Ternary Number Systems

Abstract

We describe the quadratic dynamics in certain three-component number systems, which like the complex numbers, can be expressed as rings of real matrices. This description is accomplished using the properties of the real quadratic family and its various first-and second-order phase and parameter derivatives. We demonstrate that the fundamental dichotomy of defining the Mandelbrot set either in terms of filled Julia sets or in terms of the orbit of the origin extends to these ternary number systems.

The method of the ternary complex based on three-variable rational interpolation function

“…And Horton. MD described the quadratic dynamics in certain three-component number systems, which like the complex numbers, can be expressed as rings of real matrices [9].…”

Relationship between the Mandelbrot Algorithm and the Platonic Solids

“…Quadratic polynomials iterated on hypercomplex algebras have been used to generate multidimensional Mandelbrot sets for several years [1][2][3][4][5][6][7][8][9][10]. Although this approach is widespread in the literature, other attempts at generalizing the classic fractal to higher dimensions have been made [11][12][13]. While Bedding and Briggs [1] established that possibly no interesting dynamics occur in the case of the quaternionic Mandelbrot set, the generalization given in [7], which uses the four-dimensional commutative algebra of bicomplex numbers, possesses an interesting fractal aspect reminiscent of the classical Mandelbrot set.…”

Relationship between the Mandelbrot Algorithm and the Platonic Solids

“…Quadratic polynomials iterated on hypercomplex algebras have been used to generate multidimensional Mandelbrot sets for several years [3,9,11,13,17,23,28,34]. While Bedding and Briggs [1] established that possibly no interesting dynamics occur in the case of the quaternionic Mandelbrot set, the generalization given in [23], which uses the four-dimensional algebra of bicomplex numbers, possesses an interesting fractal aspect reminiscent of the classical Mandelbrot set.…”