  For many years, quasicrystals were observed only as solid-state metallic
alloys, yet current research is now actively exploring their formation in a
variety of soft materials, including systems of macromolecules, nanoparticles
and colloids. Much effort is being invested in understanding the thermodynamic
properties of these soft-matter quasicrystals in order to predict and possibly
control the structures that form, and hopefully to shed light on the broader
yet unresolved general questions of quasicrystal formation and stability.
Moreover, the ability to control the self-assembly of soft quasicrystals may
contribute to the development of novel photonic or other applications based on
self-assembled metamaterials. Here a path is followed, leading to quantitative
stability predictions, that starts with a model developed two decades ago to
treat the formation of multiple-scale quasiperiodic Faraday waves (standing
wave patterns in vibrated fluid surfaces) and which was later mapped onto
systems of soft particles, interacting via multiple-scale pair potentials. The
article reviews, and substantially expands, the quantitative predictions of
these models, while correcting a few discrepancies in earlier calculations, and
presents new analytical methods for treating the models. In so doing, a number
of new stable quasicrystalline structures with octagonal, octadecagonal and
higher-order symmetries, some of which may, it is hoped, be observed in future
experiments.
