**2.2 Empires in 2D**

For the 2D case, we will consider the Penrose tiling, a non-periodic tiling, a quasicrystal configuration that can be generated using an aperiodic set of prototiles. The 2D Penrose tiling, when projected from the cubic lattice <sup>5</sup> , has eight vertex types: D, J, K, Q, S, S3, S4 and S5 [21, 22]. In **Figure 1**, we show the empires for three of the vertex types that do not present five-fold symmetry (D, K, S4) computed with the cut-and-project method along with their possibility space [16]. The S4 vertex has the densest empire, while the D vertex has no forced tiles. The density of the empire depends on the size of the empire window, individual for each vetex type. In **Figure 2**, we show the empires and the possibility space of the five-fold symmetry vertex types, S5 (the star) and S (the sun) [16]. The empires of all 8 vertex types are displayed in [16].

#### **2.3 Empires in 3D**

The cut-and-project method described above can be used also for computing the empires of a given patch in 3D, e.g., an Amman tiling defined by a projection of <sup>6</sup> to <sup>3</sup> [18]. In **Figure 3**, we show three orientations of the empires of two of the vertex types for this projection and we can see they differ in structure, as well as in density. Two other vertex types together with their empires are shown in [18].

<sup>1</sup> For a detailed description of this method and a comparison with the other methods, please refer to [16].

**Figure 1.**

*Empire calculation of the vertex types D (left), K (middle) and S4 (right). The red tiles represent the vertex patch and the green tiles represent the empire. The enlarged vertex patch is shown in the right corner for each case. The pictures for all vertex types have been published in [16], Figures 13 and 14.*

**Figure 2.**

*Empire calculation for the vertex types S5 (left) and S (right). The red tiles represent the vertex patch and the green tiles represent the empire. The enlarged vertex patch is shown in the right corner for each case. The pictures for all vertex types have been published in [16], Figures 13 and 14.*

Furthermore, the method can be used also for non-cubic lattices, but it requires some adjustment, as the proper selection of the tiles becomes more complex. It has been shown [18] that the cut-window must be sub-divided into regions, which act as acceptance domains for individual tiles and can be used to compute both the relative frequencies of the vertex's configuration and the empire of a given tile configuration. The authors have used the method to compute the frequencies and sectors for an icosahedral projection of the *D*<sup>6</sup> lattice to <sup>3</sup> , which has 36 vertex configurations [23] and have compared their findings with previous results [23–25].
