On the Notions of Symmetry and Aperiodicity for Delone Sets

Abstract

Non-periodic systems have become more important in recent years, both theoretically and practically. Their description via Delone sets requires the extension of many standard concepts of crystallography. Here, we summarise some useful notions of symmetry and aperiodicity, with special focus on the concept of the hull of a Delone set. Our aim is to contribute to a more systematic and consistent use of the different notions.

Aperiodicity, rotational tiling spaces and topological space groups

“…In general, sufficiently regular (for example, repetitive) aperiodic patterns have an analogous 'point group' which likewise captures richer structure about the pattern than is capable with the purely translationally defined tiling space alone; in the aperiodic case this group can even be infinite, for example it is the group O(2) in the case of the Pinwheel tiling [36]. The notion of point group is already documented in the literature, see for example [3], but we discuss it in detail for general patterns in Section 3, defining precisely the class of aperiodic tilings considered in Section 2. This includes all repetitive tilings (Definition 2.11), which are shown to always have well-defined point groups in Proposition 3.8.…”

Aperiodicity, rotational tiling spaces and topological space groups

“…In general, aperiodic patterns have an analogous 'point group' which likewise captures richer structure about the pattern than is capable with the purely translationally defined tiling space alone; in the aperiodic case this group can even be infinite, for example it is the group O(2) in the case of the Pinwheel tiling [25]. The notion of point group is already documented in the literature, see for example [2], but we discuss it in detail for general patterns in Section 3, defining precisely the class of aperiodic tilings considered in Section 2.…”

Hexagonal Inflation Tilings and Planar Monotiles

“…Indeed, the non-periodicity here is a result of a screw axis with an incommensurate rotation, wherefore the repetitive cases are aperiodic, but not strongly aperiodic; see [4] for a discussion. Also, unlike the situation above, the local rules for the SCD tile have to explicitly exclude the use of a reflected version, which is perhaps not fully satisfactory either.…”

“…Moreover, it is a face to face stone inflation (in the sense of Danzer), which means that each inflated tile is precisely dissected into copies of the prototile so that the final tiling is face to face. This rule defines an aperiodic tiling of the plane, but it does not originate from an aperiodic prototile set (for the terminology, we refer to [4] and references therein). In principle, the procedure of [5] can be applied to add local information to the prototile and to the inflation rule (via suitable markers and colours), until one arrives at a version with an aperiodic prototile set.…”