Theory of the structure of icosahedral quasicrystals: general principles

A unified theory of the structure of icosahedral quasicrystals is proposed, within the framework of which it is possible to describe all three types of quasi-lattices $(P, I, F)$ and both icosahedral symmetry groups. The theory is based on the combined use of three types of tilings, each of which is characterized by its own basic set of unit cells and its own substitution rules. By analogy with ordinary crystals, the problem of describing the structure of a quasicrystal splits into two stages: filling space with cells and filling cells with atoms, with the only difference that instead of one elementary cell, cells of several types are used, and to fill space with cells, instead of translations, an iterative algorithm of inflation and deflation is used.