GummeltversusLück decagon covering and beyond. Implications for decagonal quasicrystals

Abstract

Specific structural repeat units can be used as quasi-unit cells of decagonal quasicrystals. So far, the most famous and almost exclusively employed one has been the Gummelt decagon. However, in an increasing number of cases Lück decagons have been found to be more appropriate without going into depth. The diversities and commonalities of these two basic decagonal clusters and of some more general ones are discussed. The importance of the type of underlying tiling for the correct classification of a quasi-unit cell is demonstrated.

A General Approach to Ammann Bars for Aperiodic Tilings

“…She came up with a decagon covering model, equivalent to the HBS tiling model, as did Lück earlier [Lü90]: their decorated decagons superimpose according to specific assembly rules, and any valid covering with those can be interchanged with an HBS tiling. Their models were recently compared by Steurer [Ste21], following a more recent short review on quasicrystals [Ste18].…”