A Julia toolkit for dual-space phasing in arbitrary dimensions is presented. It offers a flexible platform for developing and testing charge-flipping algorithms under symmetry constraints.

Natural quasicrystals from the Khatyrka meteorite reveal that hypervelocity impacts can generate and preserve quasiperiodic phases under extreme non-equilibrium conditions. Their occurrence highlights the role of quasicrystals as robust markers of shock processes, offering new insights into the collisional and chemical evolution of planetary bodies.

The internal space expansion with variable system size is investigated for three different oxide quasicrystal systems. Patches of 7800, 4800 and 3600 vertices are examined in Ba–Ti–O/Pt(111), Eu–Ti–O/Pd(111) and Sr–Ti–O/Pd(111), respectively. This internal space inspection provides unique structural information for quasicrystalline systems, which goes far beyond a tiling analysis in the physical space.

The decoration of cluster centers, while not modifying the surface plane selection rule established for Tsai-type quasicrystals and approximants, affects the superstructure type stabilized at the surface.

The study introduces new primitive icosahedral quasicrystals in Zn–Mg–Li–(Dy, Ho, Er, Tm) systems. Quasicrystals are studied with an X-ray diffraction technique resolving the local atomic structure.

Employing density functional theory calculations, we identify the complex adsorption energy landscape of C₆₀ on Al₁₃Fe₄(010). We show that the most favorable adsorption sites are consistent with molecule–surface symmetry matching. We highlight the role of surface Al atoms in molecule–surface interactions.

We present new families of octagonal substitution tilings with three prototiles, motivated by the observation of structural phases exhibiting octagonal symmetry. Some of their properties are discussed.

The study characterizes the symmetries of a symbolic dynamical system arising from k-free integers in the biquadratic field . It proves that the extended symmetry group is a semi-direct product , with the stabilizer explicitly linked to the field's unit and Galois groups. The analysis demonstrates how these number-theoretic shifts mirror the complex symmetry structures typically seen in aperiodic order.

In this work, renormalization methods for quantities related to the diffraction of inflation systems are surveyed.

The 2D faces of the Voronoi cell of the lattice A₄* are of two types: regular hexagons and squares in 4D, but project into two types of hexagons and two types of rhombuses with edges of two lengths in proportion to the golden ratio.

The paper presents a parsimonious structure model for the icosahedral Cd_5.7Yb quasicrystal. It employs only spherical and elliptical occupation domains, which significantly decreases the number of parameters and the required calculation effort. Still, the model describes the characteristic structural features with considerable accuracy.