#### 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 [11]. 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 [32]. In [32] 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 Λ. 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
[32].

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 and 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
[32].