A family of ternary decagonal tilings

TL;DR

This paper introduces a new family of decagonal quasiperiodic tilings constructed via generalized point substitution processes, featuring fractal windows and multiple symmetry classes, advancing understanding of complex aperiodic patterns.

Contribution

The paper develops a novel substitution formalism for decagonal tilings, deriving fractal window boundaries and classifying an infinite variety of local isomorphism classes.

Findings

Tilings are composed of three prototiles: rhombus, pentagon, and hexagon.

Windows in perpendicular space have fractal boundaries, analytically derived.

Multiple symmetry classes and transformations via simpleton flips are identified.

Abstract

A new family of decagonal quasiperiodic tilings are constructed by the use of generalized point substitution processes, which is a new substitution formalism developed by the author [N. Fujita, Acta Cryst. A 65, 342 (2009)]. These tilings are composed of three prototiles: an acute rhombus, a regular pentagon and a barrel shaped hexagon. In the perpendicular space, these tilings have windows with fractal boundaries, and the windows are analytically derived as the fixed sets of the conjugate maps associated with the relevant substitution rules. It is shown that the family contains an infinite number of local isomorphism classes which can be grouped into several symmetry classes (e.g., $C_{10}$, $D_5$, etc.). The member tilings are transformed into one another through collective simpleton flips, which are associated with the reorganization in the window boundaries.