On The Algebraic Characterization of Aperiodic Tilings Related To   ADE-Root Systems
  \noindent The algebraic characterization of classes of locally isomorphic 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 invariant is considered, namely the ordered $K_0$-group of an algebra naturally assigned to the quantum space.   The result also yields an application of the $2$-dimensional abstract gap labelling theorem. 
Johannes Kellendonk