Cover illustration: Understanding the properties of primitive substitution tilings is of increasing relevance not only to various fields of mathematics, but also to the study of crystals and quasicrystals for which tilings are used as structure models. One property is the frequency module of the hull of a primitive substitution tiling, which contains the frequency of each patch of the tiling. Because the frequency module of the hull of a primitive substitution tiling cannot always be derived from its corresponding substitution, a technique used by Say-awen et al. [Acta Cryst. (2022). A78, 36–55] is to introduce a new substitution, which arises from a partition of the tiling satisfying certain conditions. The cover image shows a portion of a partition of a particular primitive substitution tiling.
A new algorithm for lattice reduction based on a series of directional and hyperplanar shears and driven by the decrease of the basis rhombicity is proposed. It can be used to reduce unit cells in dimension 3 and higher.
Two-dimensional point patterns for cyclic twins of six-, eight-, ten- and 12-fold symmetry are generated based on a general parametrization in combination with specific integer sequences following from a common construction scheme.