Two of the most famous two-dimensional models of quasicrystals are the Penrose rhomb tiling and the Ammann–Beenker tiling. Each of these types of tiling can be used to cover the entire plane, but in an aperiodic way that has no translational symmetry.
Can aperiodic tilings be described by point sets (Delone sets)?
I think the answer is “yes”. It is true for the Penrose rhomb tiling and the Ammann—Beenker tilling. More details can be found in this paper: https://www.researchgate.net/publication/231513166_On_the_Notions_of_Symmetry_and_Aperiodicity_for_Delone_Sets