Magic Numbers in the Discrete Tomography of Cyclotomic Model Sets

Abstract

We report recent progress in the problem of distinguishing convex subsets of cyclotomic model sets Λ by (discrete parallel) X-rays in prescribed Λ-directions. It turns out that for any of these model sets Λ there exists a 'magic number' m Λ such that any two convex subsets of Λ can be distinguished by their X-rays in any set of m Λ prescribed Λ-directions. In particular, for pentagonal, octagonal, decagonal and dodecagonal model sets, the least possible numbers are in that very order 11, 9, 11 and 13.

Discrete Tomography: Magic Numbers for N-Fold Symmetry

“…Moreover, there is a finite number m Λ such that any two convex subsets of Λ can be distinguished by their X-rays in any set of m Λ prescribed Λdirections. It was essentially shown in [6] that the least possible 'magic numbers' m Λ in the case of the practically most relevant examples of strong n-cyclotomic De-lone sets Λ with n = 5, 8 and 12 only depend on n and are (in that order) 11, 9 and 13; see also [7] for a gentle introduction. This extended a well-known result of Gardner and Gritzmann [3] on the corresponding problem for the crystallographic cases n = 3, 4 (with least possible number m Λ = 7 in both cases) to cases that are relevant in quasicrystallography.…”