Uniqueness in Discrete Tomography of Planar Model Sets

Abstract

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 non-zero 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 well-known examples, i.e., the cyclotomic model sets associated with the square tiling, the triangle tiling, the tiling of Ammann-Beenker, the T\"ubingen triangle tiling and the shield tiling.