  To a given tiling a non commutative space and the corresponding C*-algebra
are constructed. This includes the definition of a topology on the groupoid
induced by translations of the tiling. The algebra is also the algebra of
observables for discrete models of one or many particle systems on the tiling
or its periodic identification. Its scaled ordered K_0-group furnishes the gap
labelling of Schroedinger operators. The group is computed for one dimensional
tilings and Cartesian products thereof. Its image under a state is investigated
for tilings which are invariant under a substitution. Part of this image is
given by an invariant measure on the hull of the tiling which is determined.
The results from the Cartesian products of one dimensional tilings point out
that the gap labelling by means of the values of the integrated density of
states is already fully determined by this measure.
