Tiling by Near Coincidence

TL;DR

The paper introduces the near-coincidence method for generating quasiperiodic tilings inspired by moiré patterns in layered materials, connecting intuitive geometric ideas with rigorous formalism.

Contribution

It presents a novel, simple algorithmic approach to create quasiperiodic tilings, reproducing classical patterns and discovering new ones, with a web-based implementation.

Findings

Reproduces classical tilings like Ammann--Beenker and Fibonacci patterns.

Reveals new tilings unlikely from traditional methods.

Web application enables easy generation of tilings from parameters.

Abstract

Moir\'e patterns of twisted and scaled bilayers have recently emerged as a fertile source of quasiperiodic order in two-dimensional materials. Inspired by these systems, we introduce the \emph{near-coincidence method} for generating quasiperiodic tilings of the plane. The method is intuitive -- admitting pairs of nearly coincident points from superimposed layers -- yet rigorous, as it maps naturally to the well-established cut-and-project formalism. It reproduces classical tilings, including the Ammann--Beenker, the Niizeki--G\"ahler, and the square and hexagonal Fibonacci tilings, and also reveals new tilings that are unlikely to arise from conventional constructions. The near-coincidence method is algorithmically simple and already realized in a web-based application that generates tilings from specified layer parameters and coincidence conditions. Future extensions include trilayer systems, where preliminary results yield dodecagonal order with square layers, and very small twist angles, where the method may capture the giant moir\'e patterns of bilayer and trilayer graphene.