Symmetry : De Rerum Structura

De Rerum Structura means “About the Structure of Things”. Symmetry is a property of structures such as those associated with the cosmos and with the roots of non linear equations.

The book describes how cubic and quadratic roots are organized as symmetrical structures by means of a methodology that is both novel and generic.

The roots are organized as algebraic and geometric structures that can be either partially symmetrical (PSS's) or totally symmetrical (TSS's). PSS's are defined as structures with parts some of which are organized as symmetrical structures and others that are not. TSS's are defined as structures all the parts of which are symmetrically organized.

The salient features of all symmetrical structures are as follows:

1. Their parts are symmetrically located around one or more centers.

2. The sum of all the parts with respect to each center is equal to zero.

The centers behave as the center of mass of a distribution of mass in space. They cannot be considered as black holes because the parts are not actually located over them.

As the first step, the cubic roots are organized as an algebraic PSS and then the non symmetrical parts are systematically replaced by PSS's until a final structure is obtained that is a TSS.

Up to this point, structures and formulas are expressed as functions of the roots. They are converted to functions of the coefficients by means of the Elementary Symmetric Functions.

Carlo Faustini was born and raised in Italy, where he enjoyed studying mathematics and Latin. He went on to gain a BSEE from Rutgers University and an MSEE from the University of Pennsylvania, and then he built a career doing classified work relating to national defense.

His ambition to do original applied research led him to seek an independent project he could conduct on a part-time basis. It took twelve years from that decision to synthesize a partially symmetrical structure with cubic roots as parts and then to find a first formula for the cubic.

It took another fifty years of upgrades before selecting as the problem's best solution a pair of formulas that together generate three totally symmetrical but different structures.