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 [25] 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.