  Penrose hyperbolic tilings are tilings of the hyperbolic plane which admit,
up to affine transformations a finite number of prototiles. In this paper, we
give a complete description of the C*-algebras and of the K-theory for such
tilings. Since the continuous hull of these tilings have no transversally
invariant measure, these C*-algebras are traceless. Nevertheless, harmonic
currents give rise to 3-cyclic cocycles and we discuss in this setting a
higher-order version of the gap-labelling.
