Magic numbers in the discrete tomography of cyclotomic model sets

TL;DR

This paper investigates how to distinguish convex subsets of cyclotomic model sets using X-ray techniques, establishing specific 'magic numbers' of directions needed for unique identification in various geometric configurations.

Contribution

It introduces the concept of 'magic numbers' for cyclotomic model sets, providing explicit minimal counts of directions required for convex subset differentiation.

Findings

Existence of a 'magic number' for each model set type

Explicit minimal numbers for pentagonal, octagonal, decagonal, and dodecagonal sets

Progress in discrete tomography of cyclotomic model sets

Abstract

We report recent progress in the problem of distinguishing convex subsets of cyclotomic model sets $\varLambda$ by (discrete parallel) X-rays in prescribed $\varLambda$-directions. It turns out that for any of these model sets $\varLambda$ there exists a `magic number' $m_{\varLambda}$ such that any two convex subsets of $\varLambda$ can be distinguished by their X-rays in any set of $m_{\varLambda}$ prescribed $\varLambda$-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.