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4 as exceptions among the members of AЈAЉBX 4 family because their thermotropic phase transitions do not follow the commonly observed sequence and because their modulation wave vector, unusually, lies on the pseudohexagonal plane. Until now, the modulated phases of these compounds have been interpreted using hexagonal symmetry. The present work indicates, however, that a model consisting of threefold twinned orthorhombic modulated domains better describes the satellites observed in the temperature range of 620-740 K. The Ccmm space group is proposed for the average structure of the whole family K 2 Mo x W 1−x O 4 , whereas the Ccmm͑␣00͒000 with the wave vector q = ͑0.5+ ␦͒a ‫ء‬ is the appropriate superspace group for the description of the modulated structures.The coexistence of three orthorhombic domains, each of which is rotated by 120°from each other, indeed reproduces a pseudohexagonal basis, which is a common feature in the AЈAЉBX 4 family. The usage of the twinning model allows a much simpler interpretation of the diffraction pattern and provides a more consistent way to describe the phase-transition sequence shown by the mixed K 2 Mo x W 1−x O 4 crystals.

The covering cluster approach offers a plausible mechanism for the formation of aperiodic (as well as periodic) crystals. Thus, decagonal and octagonal tilings can be constructed by covering with a single cluster (or patch). The approach fails in the dodecagonal case where two clusters are needed to guarantee a quasiperiodic structure. Considering the cut-and-project method one may guess that this is due to the extra freedom when projecting from 6D to 2D. To explore this line of thought, we projected a 6D simple cubic lattice into 3D so that a further projection to 2D yields a dodecagonal structure. A 6D cube projects into a 3D triacontahedron different from the Keplerian; its symmetry is-3m. The final result is a dodecagonal layer structure quasiperiodic in the basal plane and periodic in the perpendicular direction. We also explore this approach in the octagonal case where it yields a structure quasiperiodic in one direction and periodic in the two perpendiculars.