How Physical Systems Evolve Invariants of Symmetries, Their Geometries And Measures
Physical systems are treated in this investigation as abstract objects, described in a mathematical setting. The meaning of undefined words can be taken from their physical content. The purpose of the investigation was to find a symmetry based extension of the standard model of physics which includes gravity. For this a theory describing states of a deuteron atomic kernel was developed.
In a series of articles the author presents the MINT-Wigris theory of her book  in connection with the 8 models of her new MINT-Wigris Tool Bag. They are here useful for describing how energy sytems and interactions evolve.
In the first section the tool Horn torus (figure 1) is taken in a decay process for a big bang situation.
The figure is for the geometrical location of a Higgs bosons energy as dark matter. In chemistry such simplifications are used by drawing for electrons main energy distribution in an atoms shell a Bohr sphere, a ahndle ot a torus. The chosen shapes are in this article taken as subsets in an octonian space, not necessarily the unverses spacetime. The focus is, also in the other sections, on coordinate constructions, on different sets of invariants in the sense of the Noether theorem, on geometrical shapes and symmetries for or connected with them. Equations, also in differential form, are rarely used. Measures and geometrical shapes arise in many forms. Different mathematical oriented presentations are useful for the understanding of the new theory for physical systems. The basic vector space are octonians. In other forms the 8 dimensions are for the 8 Gell-Mann matrices generating the symmetry of SU(3). In a third projectve geometrical treatment
a real 8-dimensional space space is used, complex coordinates are from a 4-dimensional complex space. The difference between octonians and SU(3) generators is in their multiplication tables which allows the seven Gleason frame GF spin like octonian coordinate triples as new measures in the sense of the Copenhagen interpretation of physics. From SU(3) is taken its geometry as trivial fiber bundle product of a 3- with a 5-dimensional sphere. For the field quantums of gravity is assumed that they map the sphere S³ with its Gell-Mann projection 3x3-matrices λ1,2,3 down to the Pauli matrices σ1,2,3 which carry as symmetry the xyz-coordinates of space. The 5-dimensional sphere is a fiber bundle, has a 1-dimensional circle as fiber and as base a complex 2-dimensional space CP² with a bounding 2-dimensional sphere S². This is taken as location for a deuteron atomic kernel and its inner dynamics is set such that gravity can be added to the standard model of phyics. Important as new S² symmetry are its Moebius transformations which give rise to the six cross rations representing six energies. Projective geometry has many applications, for instance of getting quadrics for orbits of systesm in spacetime or for shapes. Complex numbers in 4 dimensions are the standard way to describe the global vector space in whch all physicsl systems live. How these spaces, symmetries, energy forces, interactions and local energy systems can evolve from the Horn torus is mathematical described in this article.