Abstract
The work reports the finding and the study of transformation groups with two conditional elements (binary transformations of abstract structures of the finite numerical sets with broken symmetry). The term Broken Symmetry Group (BSG) is introduced. Transformation examples of relevant structures are studied with computer visualization and application in real structure study. A special type of BSG was discovered, which describes the subsets of “evolutionary trees” with convergent and divergent properties of the oriented graph (orgraph) with structure-development direction edges and “growth spirals”.
Highlights
All the known physical effects in crystals are described based on the Curie-Neumann principle: the symmetry of the structure and the symmetry of the “impact” on the structure are summated in a way where only shared transformations remain
The first case of the broken symmetry group (BSG) finding and new properties showing up was the structure of the complex octahedral cation [Me(urea)6 ]2+,3+
The product set of two-rowed matrices of substitution numbers, which are not permutations, will be called broken symmetry groups (BSG)
Summary
All the known physical effects in crystals are described based on the Curie-Neumann principle: the symmetry of the structure and the symmetry of the “impact” on the structure are summated in a way where only shared transformations remain. Systems interaction results the transformations which break the symmetry and lead to the restructuring of each system [1] This circumstance, as with many other factors, allows highlighting of the unity of two major principles of development, growth, accumulation, evolution, and stability of systems. The substitution is the choice due to, firstly, symmetry operation record simplicity and, secondly, substitution coding regardless of space characteristics This has a particular importance when dealing with the state space or phase space, where topology and sizing may be indeterminate enough to enter a certain virtual possibilities space to visualize a calculation result. We will implement broken symmetry mathematical groups for restructuring processes description, in accordance with Curie’s principle. This was briefly reported by the article’s authors at the First Russian Crystallography Congress [11]
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