Ultimate Symmetry

Mathematically, the nested levels of time can be represented by imaginary or complex numbers where space is treated as a plane or spherical wave, and time is the orthogonal imaginary axis. However, in addition to the normal complex number plane: , that can describe Euclidean space, split-complex, or hyperbolic, numbers: , are required to express the relation between space-like and time-like dimensions, which are the inner and outer levels of time, respectively. Normal complex numbers can describe homogeneous or isomorphic space, without (the outer) time, where each number  defines a circle, or sphere, because its modulus is given by , while in split-complex numbers the modulus is given by , so , which defines a hyperbola. This negative sign in calculating the modulus of complex time  reflects the essential fact that the perpetual re-creation of space and matter particles in the inner levels of time  is interrupted and re-newed every instance of the outward time , which produces kinetic motions on the physical level, as dynamic local deformations of the otherwise flat and homogeneous Euclidean space.

Therefore the non-Euclidean Minkowski space-time coordinates  are an approximation of the complex space-time coordinates , or complex time-time coordinates:  where  or  represent the innerreal part of time: , and  represents the outerimaginary time, so the totalcomplex timeis , where  is the split-imaginary unit which defines hyperbolic numbers, thus: , and its modulus is . These hyperbolic numbers have been also called: tessarines, motors, bireal, perplex, semi-complex, or split-complex, but in this article, unless otherwise stated,imaginaryandcomplexrefer to these hyperbolic numbers .

In this abstract complex frame, space and time are absolute, or mathematical, just as they had been originally treated in the classical Newtonian Mechanics, but now empty space isvoid, which is a pure mathematical space, because it does not have any material reality, to differentiate it from the physicalvacuum, which is the dynamic aether, or the ground state of matter. So, for void both the real and imaginary parts are null: , while for vacuum only the imaginary part is null: , which indicates infinite and inert space that is the ground state of matter particles, that are then described by , or: , which means that they are internally being re-created at the speed of , that is the real part, and moving outwardly at the apparent velocity , that is the imaginary part; given by equation 3.1. We shall see in sections 2.1.5 and 3.3.3 that the complex-time space provides simple alternative mass generation mechanism that could alter or replace the Higgs.