Socolar, JES, *Weak matching rules for quasicrystals*,
Communications in Mathematical Physics, vol. 129 no. 3
(1990),
pp. 599-619 [doi] .
**Abstract:**

*Weak matching rules for a quasicrystalline tiling are local rules that ensure that fluctuations in "perp-space" are uniformly bounded. It is shown here that weak matching rules exist for N-fold symmetric tilings, where N is any integer not divisible by four. The result suggests that, contrary to previous indications, quasicrystalline ground states are not confined to those symmetries for which the incommensurate ratios of wavevectors are quadratic irrationals. An explicit method of constructing weak matching rules for N-fold symmetric tilings in two dimensions is presented. It is shown that the generalization of the construction yields weak matching rules in the case of icosahedral symmetry as well. © 1990 Springer-Verlag.*