Tiling by Near Coincidence

Abstract: Moiré 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--Gähler, and the square and hexagonal Fibonacci tilings. It also uncovers new tilings not likely to arise in conventional constructions, with relative frequencies of local configurations that may take transcendental values. The near-coincidence method is algorithmically simple and already realized in an 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 moiré patterns of bilayer and trilayer graphene.