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Geometrochemistry vs Soft Computing of Mendeleev's Brain
Gottvald, Aleš
The role of projective geometry in nature remains somewhat enigmatic for centuries. It is very strange indeed, as the projective geometry is the mother of all geometries with more restrictive symmetry groups, as clearly recognized yet by seminal insights of Felix Klein, Arthur Cayley, Paul Dirac and other eminent scientists. We usually imagine that Euclidean geometry is primary for the geometrization of our (nonrelativistic) spaces, and the Euclidean-Pythagorean metric is natural for measuring the distances in such a space. However, how to measure distances in spaces associated with statistical thermodynamics or quantum mechanics? We show that projective geometry and associated "geometrochemistry" is manifest in nature. In particular, it offers a novel soft-computing rationale for recovering basic structure of Mendeleev's periodic table of chemical elements, and elucidates some mysteries of brain information processing, including a new understanding of Artificial Neural Networks.
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Projective Geometry and the Law of Mass Action
Gottvald, Aleš
A new law of nature asserts that chemical equilibria and chemical kinetics are governed by fundamental principles of projective geometry. The equilibrium constans of chemical reactions are the invariants of projective geometry in disguise. Chemical reactions may geometrically be represented by incidence structures, which are preserved under projective transformations. Theorems of Ceva, Menelaus, and Carnot for triangles and n-gons represent the chemical equilibria, while Routh's theorem represents non-equilibria. Intrinsically projective Riccati's differential equation, being also generic to many other equations of mathematical physics, governs parametric dependence of the equilibrium constants. The theory offers tangible geometrizations and generalizations to the Law of Mass Action, including a new projective-geometric approach to soft computing of very complex problems.
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