Symmetry of quasicrystals

The definition of an aperiodic crystal (quasicrystal) as a solid that is characterized by the forbidden symmetry suggests the existence of an unsolved problem, because, in a mutually exclusive manner, it appeals to the fundamental theorem of classical crystallography. Using the Penrose tiling as an example, we have investigated the symmetry properties of aperiodic tilings for the purpose to establish the allowed symmetry groups of quasicrystals. The filling of the Euclidean space according to an aperiodic law is considered as the action of an infinite number of group elements on a fundamental domain in the non-Euclidean space. It is concluded that all locally equivalent tilings have a common “parent” structure and, consequently, the same symmetry group. An idealized object, namely, an infinitely refined tiling, is introduced. It is shown that the symmetry operations of this object are operations of the similarity (rotational homothety). A positive answer is given to the question about a possible composition of operations of the similarity with different singular points. It is demonstrated that the transformations of orientation-preserving aperiodic crystals are isomorphic to a discrete subgroup of the Möbius group $PSL(2,\mathbb{C})$; i.e., they can be realized as discrete subgroups of the full group of motions in the Lobachevsky space. The problem of classification of the allowed types of aperiodic tilings is reduced to the procedure of enumeration of the aforementioned discrete subgroups.