Decagon covering model and equivalent <i>HBS</i>-tiling model

Abstract

Abstract For decagonal Al—Co—Ni quasicrystals, coverings based on a single decagonally shaped columnar 20 Å-cluster as well as tilings of so-called hexagon-, boat- and star-tiles are used to model the atomic structure. We introduce geometrically defined local transformation rules which enable us to replace a covering of overlapping congruent decagons by an equivalent HBS -tiling at the same scale and vice versa. Starting from an obvious one-to-one correspondence in case of idealized, perfect Penrose order, we show that there is a very similar relation also for random decagon coverings with slightly relaxed structure. The proofs given here complete our statements announced in previous work.