A plane tiling generates a dynamical system $(X,\mathbb{R}^2)$, where $X$ is the closure (w.r.t. a certain topology) of the orbit of the tiling under the actions of $\mathbb{R}^2$. This $X$ is called the hull of the tiling. For more details, see for instance [Kel00] and references therein.


Kellendonk, J, Putnam, I
Tilings, $C^{*}$-algebras and K-theory
Directions in mathematical quasicrystals CRM Monogr. Ser., 13, Amer. Math. Soc., Providence, RI, 2000. , 177–206, MR1798993