Multiple-scale structures: from Faraday waves to soft-matter quasicrystals

Savitz, S.; Babadi, M.; Lifshitz, R.

doi:10.1107/S2052252518001161

Figure 20
Feasible $[({\tilde \rho}_1, {\tilde \rho}_q)]$ sets for decagonal and dodecagonal quasicrystals in the BDL model: these are the convex regions over which the Fourier coefficients can vary without violating the vacuum constraint. There is a mirror line given by $[{\tilde \rho}_1]$ = $[{\tilde \rho}_q]$ . Solid dark lines in the first quadrant show the path of the minimizing values of $[({\tilde \rho}_1, {\tilde \rho}_q)]$ as the temperature is changed. While the polygonal vertices are exact, we believe that there is no simple expression for the two smooth curves in the dodecagonal feasibility boundary. Note that the peakedness of the decagonal structure jutting far into the first quadrant essentially explains its surprising stability in the BDL model. Similar shapes exist for the additional phases considered in Fig. 18