§1 Quasicrystals

§ 1.9. Coverings in quasicrystals.

Quasiperiodic coverings can be built from overlapping clusters. We show that coverings and tilings are unified by canonical projection.

If one relaxes the condition in tiling theory that the tiles should not overlap, but keeps the condition that any point of space must be assigned to at least one tile, one arrives at the concept of a covering. Coverings of periodic sets are studied in great detail by Conway and Sloane in [8]. For quasiperiodic sets and for quasicrystals, coverings were first constructed in 1996 by Gummelt [12]. A conceptual advantage of the covering approach is that a single cluster may suffice to give a covering. A drawback is the fact that the Fourier coefficients of a covered tiling in general cannot be expressed in terms of integrals over the atomic density on the cluster(s). The covering of quasiperiodic sets is studied in [34]. In [34] pp. 97-164 I studied the relation of covering to the dual projection method and showed that covering clusters are projections of cells from the geometry of the lattice \Lambda. A distinction must be made between the covering of sets of points on one hand, and the stronger concept of covering of tiles in a tiling on the other [34].

Example 2.6: Covering of the Fibonacci tiling: In Fig. 2 we indicate a covering of the Fibonacci tiling by parallel projections of Delone cells. In Fig. 7 we show a covering of the Penrose rhombus tiling by decagons, and in Fig. 8 a covering of the Tübingen triangle tiling by two types of pentagons. In all cases the covering clusters turn out to be projections of Voronoi or Delone cells from the lattice in E^{n}.
There is a connection between the concept of covering and of a fundamental domain for canonical quasiperiodic tilings: If one looks at the covering clusters for the Fibonacci tiling Fig. 6 and the Triangle tiling Fig. 8, one finds that any set of covering clusters comprises exactly a minimal a nd hence fundamental set of prototiles: a short and a long tile in the Fibonacci tiling, 10 rhombus tiles in all orientations in the Penrose tiling, 10 triangles in all orientations respectively in the triangle tiling.

These and other aspects of coverings for icosahedral tilings are discussed in [34].