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Mathematics  2007

# Uniqueness in Discrete Tomography of Planar Model Sets

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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.

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