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

TL;DR

This paper introduces an inflation rule for a chiral pentagonal tiling that minimizes boat tiles, aiming to maximize packing density and relate to decagonal quasicrystal structures.

Contribution

It proposes a new inflation rule for a chiral HBS tiling with no boats, enhancing packing density and connecting to quasicrystal approximants.

Findings

Proposed an inflation rule for a boat-free chiral tiling.

Achieved the highest packing density among HBS tilings.

Discussed the relation to decagonal quasicrystal structures.

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.