Geometric Shapes and Forms Associated With Three Kinds of Genetic Code Equivalences

Abstract

It is well known that a genetics system in biology delivers the biological organism’s self-reproduction in their generations. Similarly, the “golden ratio” in mathematics keeps its self-reproduction property in their iterations. The biological system divides the genetic four-letter alphabet (A, C, G, T/U) into various three pairs of letters. There are three kinds of genetic equivalences among these four-letter alphabets (A=C, G=U; A=G, C=U; A=U, C=G). In this paper, we investigate the geometric shapes and forms associated with these three kinds of genetic code equivalences. We show that each equivalence has its own geometric shape and form. These geometric properties include attracting fixed point, repelling fixed point, basin of attractions, Julia sets and corresponding Mandelbrot sets. We further study the golden ratio, ring ratio and unity ratio matrices associated with three kinds of genetic code equivalences.