Delone Sets and Tilings: Local Approach

We present new results in the local theory of Delone sets, regular systems, and isogonal tilings. In particular, we prove a local criterion for isogonal tilings of the Euclidean space. This criterion is then applied to the study of $2R$-isometric Delone sets, where $R$ is the covering radius for these sets. For regular systems in the plane we establish the exact value $\widehat {\rho }_2=4R$ of the regularity radius. We prove that in any cell of the Delone tiling in an arbitrary Delone set in the plane, there is a vertex at which the local group is crystallographic. Hence, the subset of points with local crystallographic groups in a Delone set in the plane is itself a Delone set with covering radius at most $2R$.