
Mathematics 2007
Uniqueness in Discrete Tomography of Planar Model SetsAbstract: The problem of determining finite subsets of characteristic planar model sets (mathematical quasicrystals) $\varLambda$, called cyclotomic model sets, by parallel $X$rays is considered. Here, an $X$ray in direction $u$ of a finite subset of the plane gives the number of points in the set on each line parallel to $u$. For practical reasons, only $X$rays in $\varLambda$directions, i.e., directions parallel to nonzero elements of the difference set $\varLambda  \varLambda$, are permitted. In particular, by combining methods from algebraic number theory and convexity, it is shown that the convex subsets of a cyclotomic model set $\varLambda$, i.e., finite sets $C\subset \varLambda$ whose convex hulls contain no new points of $\varLambda$, are determined, among all convex subsets of $\varLambda$, by their $X$rays in four prescribed $\varLambda$directions, whereas any set of three $\varLambda$directions does not suffice for this purpose. We also study the interactive technique of successive determination in the case of cyclotomic model sets, in which the information from previous $X$rays is used in deciding the direction for the next $X$ray. In particular, it is shown that the finite subsets of any cyclotomic model set $\varLambda$ can be successively determined by two $\varLambda$directions. All results are illustrated by means of wellknown examples, i.e., the cyclotomic model sets associated with the square tiling, the triangle tiling, the tiling of AmmannBeenker, the T\"ubingen triangle tiling and the shield tiling.
