Evolution of Symmetry in Elemental Crystals

In a recent communication in the scientific journal symmetry [1] we outlined the harmonic evolution of structural symmetry on the basis of three principles: atomic attraction, repulsion, and the limitations in 3D space. We described a simple bottom-up atomistic crystal growth model derived for equivalent atoms and a pair potential entailing a maximum number of equilibrium distances in the atoms local vicinity to obtain high symmetric structural motifs, among them the Platonic Solids, as well as a phase diagram summarizing stable regions of close-packed fcc and hcp, next to bcc symmetry. In dependence on the pair potentials short- and long-range characteristics the, by symmetry, rigid lattices relax isotropically within the potential well. The first few coordination shells with lattice-specific fixed distances not necessarily determine, which equilibrium symmetry prevails.
To highlight the applicability of the model to scientific cases of solid state physics, the present chapter recaptures the evolution of structural symmetry in crystals accompanied with the basic principles outlined above and underlying weights. This guides us to a holistic view of similarities to structural characteristics in other fields, such as music and art. An impulse is given by Goldschmidts
work on fundamental rational series in sciences reflecting the potential of mathematical series as fundamental tool for further quantitative insights.