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For soft matter to form quasicrystals an important ingredient is to have twocharacteristic lengthscales in the interparticle interactions. To be moreprecise, for stable quasicrystals, periodic modulations of the local densitydistribution with two particular wavenumbers should be favored, and the ratioof these wavenumbers should be close to certain special values. So, for simplemodels, the answer to the title question is that only these two ingredients areneeded. However, for more realistic models, where in principle all wavenumberscan be involved, other wavenumbers are also important, specifically those ofthe second and higher reciprocal lattice vectors. We identify features in theparticle pair interaction potentials which can suppress or encourage densitymodes with wavenumbers associated with one of the regular crystalline orderingsthat compete with quasicrystals, enabling either the enhancement or suppressionof quasicrystals in a generic class of systems.

Which wavenumbers determine the thermodynamic stability of soft matter quasicrystals?