Theory of the structure of icosahedral quasicrystals: types of packings

A unified theory of the structure of icosahedral quasicrystals is proposed. All possible variants of self-similar icosahedral packings are analyzed. These include 3 types of quasi-lattices $(P, I, F)$, which are analogues of primitive, body-centered and face-centered cubic lattices; each of them can be either centrosymmetric or non-centrosymmetric. Substitution rules for $I$ and $F$-type tetrahedral tilings are fully formalized. An example of constructing a non-centrosymmetric $I$-type packing is presented. A method is shown for generating a zonohedral packing $(P)$ from a tetrahedral packing $(I)$ by joining the neighboring tetrahedra in it. For each packing type, 3 locally isomorphic patches are possible, differing in the choice of node in its center $\mathrm{(A, B, C)}$. When the tetrahedral packings are built up, three locally isomorphic patches cyclically transform into each other after each iteration. As a consequence, the structures of the three types of characteristic clusters are not independent. An icosahedral packing of any type can be constructed based on a unified algorithm when initialized with a single tetrahedron.