Quasicrystal Tilings in Three Dimensions and Their Empires

Abstract

The projection method for constructing quasiperiodic tilings from a higher dimensional lattice provides a useful context for computing a quasicrystal’s vertex configurations, frequencies, and empires (forced tiles). We review the projection method within the framework of the dual relationship between the Delaunay and Voronoi cell complexes of the lattice being projected. We describe a new method for calculating empires (forced tiles) which also borrows from the dualisation formalism and which generalizes to tilings generated projections of non-cubic lattices. These techniques were used to compute the vertex configurations, frequencies and empires of icosahedral quasicrystals obtained as a projections of the D 6 and Z 6 lattices to R 3 and we present our analyses. We discuss the implications of this new generalization.

Empires: The Nonlocal Properties of Quasicrystals

“…While the cut-and-project method and the multigrid one are mathematically equivalent in their use of generating empires, the cut-and-project method provides us with the possibility of recovering the initial mother lattice, even for defected quasicrystals [16]. The method has also been applied to projections of non-cubic lattices, making it of general use [18].…”

“…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  6 to  3 [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.…”

“…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].…”

“…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 6 lattice to  3 , which has 36 vertex configurations [23] and have compared their findings with previous results [23][24][25].…”

“…The cut-and-project method offers the most generality and has been used to calculate empires for the Penrose tiling and other quasicrystals that are projections of cubic lattices  n [16,17]. Quasicrystals that are projections of non-cubic lattices (e.g., the Elser-Sloane tiling as a projection of the E 8 lattice to  4 ) are more restrictive and some modifications to the cut-and-project method have been made in order to correctly compute the tilings and their empires [18]. The cutand-project formalism can also be altered to calculate the space of all tilings that are allowed by a given patch, wherein the set of forced tiles of the patch's empire is precisely the mutual intersection of all tilings which contain that patch [15].…”