The table and the chair tiling are two aperiodic tilings of the plane that are typical examples of two-dimensional quasicrystals. One way to treat such systems in dimension one, is to approximate these systems by suitable (periodic) approximations. Based on this, we raise the following questions: Is there a general method to approximate spectral properties of a given operator by the underlying geometry or dynamics? If so, can we control the approximations and which spectral properties are preserved? During the talk, we provide a short overview over such results with a special focus on dynamicallydened operator families. We will see as how to apply those results explicitly and what they tell us about the table and the chair tiling. These results are joint works with Ram Band, Jean Bellissard, Horia Cornean, Giusseppe De Nittis, Felix Pogorzelski, Alberto Takase and Lior Tenenbaum.