Point processes for decagonal quasiperiodic tilings

TL;DR

This paper introduces a new method for generating decagonal quasiperiodic tilings using inflation rules and point decoration processes, resulting in diverse tilings with fractal boundaries and chirality.

Contribution

It proposes a general construction principle for inflation rules of decagonal tilings with polygonal prototiles, including two new tiling families and detailed analysis of ternary tilings.

Findings

Generated broad range of decagonal tilings, many chiral and with fractal boundaries.

Presented two new families of decagonal tilings: quarternary and ternary.

Analyzed properties of ternary tilings with various prototiles.

Abstract

A general construction principle of inflation rules for decagonal quasiperiodic tilings is proposed. The prototiles are confined to be polygons with unit edges. An inflation rule for a tiling is the combination of an expansion and a division of the tiles, where the expanded tiles can be divided arbitrarily as far as the set of prototiles is maintained. A certain kind of point decoration processes turns out to be useful for the identification of possible division rules. The method is capable of generating a broad range of decagonal tilings, many of which are chiral and have atomic surfaces with fractal boundaries. Two new families of decagonal tilings are presented; one is quarternary and the other ternary. Properties of the ternary tilings with rhombic, pentagonal, and hexagonal prototiles are investigated in detail.