On The Algebraic Characterization of Aperiodic Tilings Related To ADE-Root Systems

Abstract

\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.