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a) The Cartesian coordinate system spans the three-dimensional Euclidean space in eight cubical segments. What is the constitution of a local part (?) in any of them? b) Regardless of size it retains the Cartesian representation. c) Hence, the smallest composite space portion is a Lie neighbour-hood of eight indivisible ground unit CuBits.
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In the immediate intuitive as well as penetrating logical contemplation, the universe stands forth as an ever-present entity; consisting solely of itself, and fulfilling solely itself. A truism and a truth, and also a conclusion by proxy because its shape expressions are covered by curvilinear mathematical functions which completely exhaust their s...
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... The stage is set for stepwise lattice transformations projecting symmetry-and volume-preserving ellipsoids and spheroids from the unit sphere identity. This has been described and illustrated in previous reports [10][11][12][13][14][15][16][17][18][19][20][21] so is here only exemplified by the basic u,d,s supermultiplet baryons (Figures 2 e, 3) and their major semiaxis steps in the lattice with the resulting Figure 3. Lateral, frontal and horizontal cross-section projection of major semiaxis of the nucleon, 0 , +-, Δ ++,+,0,-, Ξ 0,-, (1385 )+,0,and (1405) 0 from folding out in successive unit SU(3) root vector steps along the existing charged or neutral channels from end-and focal points of preceding states shape alterations shown in Figure 4 and the masses in Table 1 as calculated according to the quark pressure formula from the minor semiaxis contraction in inverse proportionality to the radius of the unit Proton. All of this has been thoroughly described in earlier papers for all observed u,d,s baryons in extensive and exact correspondence with recorded states and their masses, charge levels and angular momenta [Ib.] so will not be repeated here. ...
... The sphere-ellipsoid/spheroid transformation scheme in effect holds a number of interconnecting subgroups, e.g. spatial channel/flavor orientations alluded to in Figure 4, and moreover offers commutating separate degrees of freedom for the charm and bottom quark classes as well [16,17], and this will be described more in detail to further clarify the model. ...
... There are also unflavored mesons, so a more realistic classification is offered by the Lie product group SU(2) × U(1) of their geometrical constitution by which their mass can be approximated as the symmetric area times the antisymmetric tightness between the interchanging particles in the transformation lattice. This has also been reported in further detail before [10][11][12][13][14][15][16][17][18][19][20][21] so is here only summarized for some basic u,d,s mesons in Figure 6 and Table 2. Table 2. Computed and observed ground meson masses. ...
As reported in a series of previous PIRT conferences, a direct SU(3) structural realization of the Standard Model has been developed based upon Marius Sophus Lie’s original Norwegian Ph.D. thesis Over en Classe Geometriske Transformationer from 1871 (and thus due for a most deserved 150-year anniversary). It elucidates how “the theory of main tangential curves can be brought back to that of rounded curves”, anticipating a coherent linear representation of the elementary particles instead of the rotational chosen since they were considered point-like and amorphous when they many years later entered the stage. Under these premises the Standard Model has built a magnificent, undoubtedly true but congested multi-particle system whereas the Lie continuous transformation element, the partial derivative ’straight line of length equal to zero’ outlines an isotropic vector matrix lattice of crystallographic Killing root space diagram A 3 form which from the Nucleon and inwards can backtrack the Standard Model geometrically, as well as continue outward iterating to a space-filling solid state R ³ ×SO(3) wave-packet complex tessellating the whole periodic table with electron shells and subshells, isotope spectrum, neutron captures, radiative channels, oxidation states, molecular binding sites etc. in successive layers also including the Lanthanides in the sixth period and the Actinides in the seventh, in which now the concluding Oganesson has been reached in perfectly well-built saturated noble gas shape and condition.
... In previous PIRT proceedings [1][2][3][4] and elsewhere [5][6][7][8][9][10][11][12][13][14][15][16], a direct SU(3) structural realization of the elementary particle spectroscopy has been reported (Figure 1), which can shortest be described as a faithful real space implementation of the original "nye (new) Classe Geometriske Transformationer" disclosed by Marius Sophus Lie in 1871 [17,18] according to their full differential equation scope of iterating their universal partial derivative "geodetic curves of length equalling zero" to "transform two spaces into each other", and more specifically in the SU(3) case how the "Plücker line geometry can be transferred into a sphere geometry" by trigonometrically "transforming space r's straight lines into space R's. . . spheres' rectilinear generatrices" ( = infinitesimal generators), where "x, y, z are perceived as parameters and. . . ...
... consequently, there cannot exist a space separate from body, since all spatial extension simply is body" [26]. Lie, who in the foreword of his thesis declared that it "stands in an intimate dependence on philosophical reflections upon the nature of Cartesian geometry" [Ib.], identified "not the point" but the infinitesimal "straight line whose length equals zero" [17,18] as its differential "space-element" both as a mechanistic matter of fact and in a deeper exegesis as the categorical contrast to nothing at all [11][12][13][14][15][16][17][18]. ...
The geometric Lie algebra SU(3) isotropic vector matrix realization of the periodic table reported at PIRT 2017 has now been broken up to a disjoint-set modular R ³ ×SO(3) building kit, exactly sufficing to stage the Big Bang and ensuing nucleosynthesis events: First the ultrashort radiation/plasma inflation of the Big Bang phase transition moment with release of photons, neutrinos and module precursors, which in next seconds recombine to the Protium proton/electron compound and its neutron conversion to continue separately or in fusion of the two get on to Deuterium and from there to Tritium, Helium in isotope and α form, and traces of Lithium and possibly Beryllium. That is, literally the whole primordial start-gas delivered within a few minutes to billion-year wait for sufficiently energetic perturbations with itself for astrophysical/cosmogenic/experimental nucleosynthesis of the full periodic table and likewise replicable by systematic space-filling assembly/disassembly of the here disclosed neutrino and photon lattice vector, β particle, α wave-packet and neutron building bricks, providing clues also on isotope/neutron excess, shell/subshell, spectroscopy, and chemical bond structural make-up and disposition. Furthermore, the absolute trigonometric sharpness of the nucleosynthesis phase transition burst and expansion is reciprocal to the absolute speed of light and hence a specific test and verification of the relativity theory.
... At the infinitesimal level this stepwise composition outlines a reciprocal trigonometric function, namely, in the present case, the SU(3) and subgroups Lie algebra [7,8] ; and more particularly their real geometric automorphisms in which the R 3 scaffold naturally appears as a unit cube, or 'CuBit'. Physically projecting the analytical neighborhood of the respective Lie algebra it may further modulate itself into isomorphic parallelepiped and/or relativistically or otherwise skewed layout, setting a virtual R 3 ×SO(3) wavepacket incubation and mechanics stage which actually blueprints the entire elementary particle spectroscopy [9][10][11][12][13][14][15][16][17][18][19][20]. And also in a static block embodiment it has interesting, in especial mathematical uses as a whole-number counting frame bead with straightforward abacus bearings and solutions to Fermat's last theorem by cumulatively tessellating the infinite Diophantine Equation Universe both according to classical Sumerian building principles as adopted and adapted by Euclid [9,16,17], and in a likewise infinitely ascending Pascal binomial expansion order [18]. ...
... The SO(3) moiety of the system has a double identity as its in ground state spherical eigenelement as well as algorithm of its own volume-preserving spheroidal, that is, in QCD sense, gauge-and symmetry-invariant transactions and transformations, which in effect enable an exact and exhaustive reproduction of the entire elementary particle spectroscopy including charge, channels, states, angular momenta and masses [9][10][11][12][13][14][15][16][17][18][19][20]. The present report focuses on the joint R 3 ×SO(3) geodesics according to the full differential equation scope of the original Lie algebras, which generally iterate their universal partial derivative "geodetic curves of length equaling zero" to "transform two spaces into each other", and more specifically in the SU(3) case how the "Plücker line geometry can be transferred into a sphere geometry" by trigonometrically "transforming space r´s straight lines into space R's…spheres' rectilinear generatrices" (= infinitesimal generators in today's terminology), where "x, y, z are perceived as parameters and…dx, dy, dz as direction cosines" ( The so absolutely sharp transition of the respective geometries' infinitesimal generators can thus be uniquely effectuated by the corresponding root space diagrams of their commutation relations. ...
... It is accomplished by a real geometry coset decomposition of SU (3], that interlinks to the infinitesimal generator root vector space of the sphere directly spanned by the duplicated A2 diagram, whose neutral isospin vectors coincide with the Cartesian x and z axes but whose charged t isospin axes set up a hexagonal lattice with unit side continuing in the surrounding as space-filling regular tetra-and octahedrons. It is a template for both inward nucleon [19,20], surface muon [9][10][11][12][13][14][15][16][17][18][19][20] and external baryon and meson [Ib.] transformations, whereas the electron (and positron) geodesics are generated by a coherent continuation of the charged t isospin vectors as will be focused on in the following. Figure 2 gives an overview of the game plan and its pieces. ...
... At the infinitesimal level this stepwise composition outlines a reciprocal trigonometric function, namely, in the present case, the SU(3) and subgroups Lie algebra [7,8] ; and more particularly their real geometric automorphisms in which the R 3 scaffold naturally appears as a unit cube, or 'CuBit'. Physically projecting the analytical neighborhood of the respective Lie algebra it may further modulate itself into isomorphic parallelepiped and/or relativistically or otherwise skewed layout, setting a virtual R 3 ×SO(3) wavepacket incubation and mechanics stage which actually blueprints the entire elementary particle spectroscopy [9][10][11][12][13][14][15][16][17][18][19][20]. And also in a static block embodiment it has interesting, in especial mathematical uses as a whole-number counting frame bead with straightforward abacus bearings and solutions to Fermat's last theorem by cumulatively tessellating the infinite Diophantine Equation Universe both according to classical Sumerian building principles as adopted and adapted by Euclid [9,16,17], and in a likewise infinitely ascending Pascal binomial expansion order [18]. ...
... The SO(3) moiety of the system has a double identity as its in ground state spherical eigenelement as well as algorithm of its own volume-preserving spheroidal, that is, in QCD sense, gauge-and symmetry-invariant transactions and transformations, which in effect enable an exact and exhaustive reproduction of the entire elementary particle spectroscopy including charge, channels, states, angular momenta and masses [9][10][11][12][13][14][15][16][17][18][19][20]. The present report focuses on the joint R 3 ×SO(3) geodesics according to the full differential equation scope of the original Lie algebras, which generally iterate their universal partial derivative "geodetic curves of length equaling zero" to "transform two spaces into each other", and more specifically in the SU(3) case how the "Plücker line geometry can be transferred into a sphere geometry" by trigonometrically "transforming space r´s straight lines into space R's…spheres' rectilinear generatrices" (= infinitesimal generators in today's terminology), where "x, y, z are perceived as parameters and…dx, dy, dz as direction cosines" ( The so absolutely sharp transition of the respective geometries' infinitesimal generators can thus be uniquely effectuated by the corresponding root space diagrams of their commutation relations. ...
... It is accomplished by a real geometry coset decomposition of SU (3], that interlinks to the infinitesimal generator root vector space of the sphere directly spanned by the duplicated A2 diagram, whose neutral isospin vectors coincide with the Cartesian x and z axes but whose charged t isospin axes set up a hexagonal lattice with unit side continuing in the surrounding as space-filling regular tetra-and octahedrons. It is a template for both inward nucleon [19,20], surface muon [9][10][11][12][13][14][15][16][17][18][19][20] and external baryon and meson [Ib.] transformations, whereas the electron (and positron) geodesics are generated by a coherent continuation of the charged t isospin vectors as will be focused on in the following. Figure 2 gives an overview of the game plan and its pieces. ...
Seeking rapport with the Copenhagen interpretation of quantum mechanics, Erwin Schrödinger in 1926 devised the Wellenpaket encapsulation of his wave equation, which eventually even Werner Heisenberg accepted: but only “in multi-dimensional configuration space, not in ordinary space.” However, in recent development including string and loop quantum gravity theories operating in real space, there are now a “growing number of suggestive connections between classical stochastic models and an assortment of quantum phenomena”. We have made a direct electron wave-packet modeling of the elements and parameters of the solid state configuration space R3×SO(3) by a structural realization of the SU(3) Lie algebra into a cyclically expanding isotropic vector matrix unfolding of its infinitesimal root vectors, faithful to his prescriptions how these partial derivative “geodetic curves of length equalling zero” interact so that the “Plücker line geometry can be transferred into a sphere geometry” by trigonometrically projecting its “straight lines into space R’s spheres’ rectilinear generatrices” and vice versa, where “x, y, z are perceived as parameters and dx, dy, dz as direction cosines”. This outlines a unit lattice cell of the two interconnecting root vector sets, spanning an interior spacefilling 12-step one-octahedron/two-tetrahedrons SO(3) electron helical loop walled in and coordinated by the likewise 12-edged and by itself three-dimensionally close-packing parallelepiped infinitesimal ground eigenelement of R3. By a hierarchically accumulating coherent brick-laying of this wave-packet hybrid in a recurring Bohr orbital sequence order, the primordial hydrogen ion plasma phase of the universe and its self-inflation into the full atomic and periodic table organization with all its static and dynamic, single as well as collective characteristics is replicated exactly and exhaustively as in reality.
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 R ³ ×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.
