On the algebraic characterization of aperiodic tilings related to ADE-root systems

The algebraic characterization of sets of locally equivalent aperiodic tilings, being examples of quantum spaces, is conducted for a certain type of tilings in a manner proposed by A. Connes. These 2-dimensional tilings are obtained by application of the strip method to the root lattice of an ADE-Coxeter group. The plane along which the strip is constructed is determined by the canonical Coxeter element leading to the result that a 2- dimensional tiling decomposes into a cartesian product of two 1- dimensional tilings. The properties of the tilings are investigated, including selfsimilarity, and the determination of the relevant algebraic is considered, namely the ordered K0-group of an algebra naturaly assigned to the quantum space. The result also yields an application of the 2-dimensional abstract gap labelling theorem. (orig.)