Abstract. 2014 We study the undulation instability in a smectic A in the presence of shear flow parallel to the layers. We look for the dilation at the threshold where an undulation appears with a wave vector inclined at an angle 03B8 to the direction of flow. We find an integral relation showing that the dilation is necessarily positive. We develop a perturbation method to calculate the dilation explicitly as a function of the angle 03B8 and the shear rate. We show that an undulation with wave vector perpendicular to the shear direction is not affected by the flow and has a minimum threshold dilation equal to the static value. We conclude that a rectangular texture develops at higher dilations in agreement with the rectangular focal domain pattern observed experimentally.

A likely mechanism for the formation of quasicrystals is by maximally covering space with overlapping, stable atomic clusters. This notion is here applied to the experimentally determined layered structure of octagonal MnSiAl quasicrystals, which can be described in terms of a decoration of the octagonal Ammann-Beenker tiling. This decoration is abstractly represented by a two-color version of the tiling. The covering cluster of the quasicrystal corresponds to an octagonal covering patch of the colored tiling. This covering patch appears in two variants with complementary colors. The three-dimensional quasicrystal has a centered octagonal translation module, and its space group is I8 4 /mcm. ͓S0163-1829͑99͒15525-5͔

Defective vertex configurations are important for the whole range of models for quasicrystalline structures from quasiperiodic tilings through random tilings to polyhedral glasses. The combinatorially possible vertex configurations are enumerated for the 1D Fibonacci chain, for the 2D Penrose pattern with its generalizations, as well as for the Beenker pattern and the triangle pattern, and for the 3D simple icosahedral tiling. The methods for quantifying the deviation of vertex configurations from perfection are reviewed. The simple method of partial dual overlap provides a means to estimate the abundancy of vertex configurations within random tilings. More sophisticated is the method of the defectivity functional; it is particularly suitable to deal with nearly perfect tilings. Local configurations are formally classified by characteristic integers: degree, rank and order. Some possible applications are hinted at.