Inflation rules for a chiral pentagonal quasiperiodic tiling of stars and hexes

Abstract

Hexagon-boat-star (HBS) pentagonal tilings often appear in the description of decagonal quasicrystals and their periodic approximants. Being related to the Penrose tiling, they differ from the latter by a significantly higher packing density of vertices, which, in turn, depends on the relative frequency of appearance of the H, B and S tiles. Since boats (also known as “ivy leaves”) have the lowest packing density, reducing their number in the tiling leads to an increase in its packing density. The paper proposes an inflation rule for a chiral tiling, which, in principle, contains no boats and therefore has the highest possible density among HBS tilings. The relationship between the tiling and the real structures of crystal approximants of decagonal quasicrystals is discussed.