A bottom-up ‘Game of Lie’ cellular automaton evolution in SO(3) root space both nucleating, crystallizing and space-filling the complete atomic realm

Abstract

Abstract Re-examining Marius Sophus Lie’s Ph.D. thesis Over en Classe geometriske Transformationer from 1871 I found that its bearings on the “nature of Cartesian geometry” are that it “translates any geometric theorem into an algebraic one and thus…of the geometry of space a representation of the algebra of three variable quantities” with both the algebraic and “geometrical transformation…consisting of a transition from a point to a straight line as element…through a particularly remarkable transformation” in which “the Plücker line geometry can be transformed into a sphere geometry”, by its “straight lines of length equal to zero” partial derivative elements “turning…into the sphere’s rectilinear generatrices” both as “partial differential equations of the first order” and physical “geodetic curves” under the “general equation system f(x, y, z, dx, dy, dz) = 0” and its “spatial reciprocity” that “relative to the given line complex” in x, y, z “corresponds a certain curve-net” in the simultaneous “line element (dx, dy, dz)”. This is a revolutionary geometric algebra discovery of the constitution of our Universe as a stru ctural phase transition between Straight and Round, transcending Hermann Grassmann’s Ausdehnungslehre (1844/1862) and William Kingdom Clifford’s Space-theory of Matter (1876) by the infinitesimal transformations opening up spherical geometry in its real form identity state later assigned as SO(3). The continuous groups, whose vast field occupied the rest of his all too short life were not mentioned in the thesis but implied both in the term Classe (class) geometriske transformationer and the coverage of their principles in an “unlimited manifold of possible systems”; e.g. “turning rounded curves into rounded curves” and “straight lines into straight lines”, where “one can choose any space-curve which depends upon three parameters as the element of the geometry of the space”, and it “is possible to create a representation of an algebra that embraces an arbitrary number of variables.” Important examples were given, except of course of the still dormant elementary particles. Now, as the prime of matter’s appearance they step forth as the most natural to examine by the true Lie algebras in their original form. When over a long series of years I have scientifically carried out this, the result as here so far concluded is a concrete cellular automaton building kit of the Standard Model and all its features, and their mechanisms and dealings in a structural R3×SO(3) wave-packet organization, both inwards from the elementary particles and outwards via the periodic table of the atoms over the further hierarchical growth of this in molecular and crystal stages to an isotropic space-filling of the whole classical Euclidean Universe in harmonic exchange with its relativistic spherical moiety, and the dark mass and energy collectively worked out in the differential interstice between them.