Erik Trell

Abstract:

When the English translation of Marius Sophus Lie’s Norwegian Doctoral Dissertation Over en Classe geometriske Transformationer from 1871 now becomes internet open accessible it will be widely recognized that it concretely describes how “through a particularly remarkable transformation…main tangential curves can be brought back to…rounded curves.” This “fundamental relation between the Plucker line geometry and a spatial geometry whose element is the sphere” is a “limit transition” by a “polygon…projection” of the “space’s straight lines into the sphere’s rectilinear generatrices” through which the “Plucker line geometry can be transformed into a sphere geometry.” <br />A faithful modern nanotechnology version of this simultaneous straight-to-round phase transition in which the infinitesimal generators of the flat Plucker/Euclidean space are turned into the infinitesimal generators of the sphere is a thereby directly hybridized “curve-net” lattice where a hexagonally symmetric isotropic vector matrix (IVM) of octahedrons and tetrahedrons is juxtaposed to the orthogonal spatial framework in a trigonometrically determined absolute 60 degrees inclination. The utility is the same as in Lie’s time when “we find between the corresponding transformations of R:…all movements (translation movement, rotation-movement and the helicoidal movement), semblability-transformation, transformation by reciprocal radii, parallel transformation…etc”. Today, the elementary particles have added to the field and it is a significant and lasting fact that the lattice system verbatim blueprinted from Lie’s thesis generates exact replications of them and their spectroscopy, too.<br />Here the IVM is of particular sustainable mathematical as well as material interest. It is outlined by the electron which expands in self-similar cycles to an instant inflation of the mainly Hydrogen ion primordial universe. Inevitable ripples in the isotropy then accumulate to increasing perturbations and ultimately to the nucleogenetic era, when a backtracking in the IVM mesh leads to retrograde crystallization of a full centrally symmetric periodic system of the elements.