§ 1.8. Quasiperiodicity compatible with tilings.

In a tiling theory for quasicrystals one wishes the tiles to carry the same arrays of atoms, independent of the position of the tile. We examine the implications of this demand for the projection method.

In H Bohr's theory, a quasiperiodic function is the restriction of lattice-periodic function in $E^{n}$ to the irrational subspace $E^{m}$ . This implies that its functional values can be specified on a single nD cell of the lattice. Even when we have constructed a quasiperiodic tiling by projection, we must analyze what subclass of quasiperiodic functions in $E^{m}$ would repeat its values on tiles, differing in the tiling by a parallel shift. For a lattice in $E^{n}$ , any periodic function by definition repeats its values on copies of cells related by a lattice translation. For quasiperiodic functions, the repetition of their values on all translates of a given tile is an additional requirement of compatibility. This compatibility problem we solved [27] with the help of the dual projection method. In this method, any tile of $E^{m}$ appears as an m-dimensional cut through an n-dimensional tile of a periodic tiling of $E^{n}$ . Since the linear cut is irrational, each intersection with a translate of the n-dimensional tile will hit it in a different perpendicular position. A finite minimal set of these n-dimensional tiles can be chosen as a fundamental domain of the lattice in $E^{n}$ . A periodic function on $E^{n}$ is then determined by its values on the points of this set. If now we restrict further the values of this periodic functions on each n-dimensional tile by choosing it independent of the perpendicular coordinate, we assure that their values on $E^{m}$ become independent of parallel shifts of the tiles on $E^{m}$ , which establishes the compatibility. Since we still restrict from periodic functions, compatibility preserves quasiperiodicity. This reasoning is illustrated in Fig. 2 for the Fibonacci tiling. If a tiling is used to model the atomic structure of a quasicrystal, compatibility implies that we need to determine only the atomic configuration on a finite minimal set of tiles. These atomic configurations will be repeated on any translated copy of a tile in the tiling.

Example 2.5: The minimal set of prototiles in the Fibonacci tiling. The minimal set of prototiles for the Fibonacci tiling Fig. 6 consists of a short and a long tile. For the Penrose tiling Fig. 7 it has each rhombus tile in 10 possible orientations. For the Triangle tiling Fig. 8 it contains the two triangles each in 10 orientations. These minimal sets of tiles generalizes the notion of a fundamental domain from crystals to quasicrystals.