Discrete tomography of icosahedral model sets

Abstract

The discrete tomography of mathematical quasicrystals with icosahedral symmetry is investigated, placing emphasis on reconstruction and uniqueness problems. The work is motivated by the requirement in materials science for the unique reconstruction of the structures of icosahedral quasicrystals from a small number of images produced by quantitative high-resolution transmission electron microscopy.

Uniqueness in Discrete Tomography of Delone Sets with Long-Range Order

“…The algorithmic reconstruction problem of discrete tomography of cyclotomic model sets has been studied in [7]. In [23], it is shown how the results for the planar case obtained in [7] and the present text can be lifted to the practically relevant case of so-called icosahedral model sets in R 3 . For a completer overview of both uniqueness and computational complexity results in the discrete tomography of Delone sets with long-range order, we refer the reader to [20].…”

“…For a completer overview of both uniqueness and computational complexity results in the discrete tomography of Delone sets with long-range order, we refer the reader to [20]. This reference also contains results on the interactive concept of successive determination of finite sets by X-rays and further extensions of settings and results that are beyond our scope here; compare also [18,21] and [23].…”

Discrete Tomography: Magic Numbers for N-Fold Symmetry

“…Note that the present text is only concerned with the uniqueness problem of determining the elements of a large collection of finite subsets of Λ by few X-rays in prescribed Λ-directions. For the algorithmic reconstruction problem in the quasicrystallographic setting, see [5].…”

Magic Numbers in the Discrete Tomography of Cyclotomic Model Sets

“…In fact, many of the model sets that describe real quasicrystallographic structures allow a slicing such that each slice is an n-cyclotomic model set, the latter being (planar) Delone sets contained in the additive subgroup of the Euclidean plane generated by the nth roots of unity; cf. [6], [7], [9] and [15] for details. It therefore suffices to study the discrete tomography of these cyclotomic model sets.…”