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  • Open Access

Tiling by near coincidence

Meshy Ochana and Ron Lifshitz*

  • *Contact author: ronlif@tau.ac.il
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Phys. Rev. Research 8, 023330 – Published 24 June, 2026

DOI: https://doi.org/10.1103/xjzf-xxns

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 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, 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 moiré patterns of bilayer and trilayer graphene.

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References (39)

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Phys. Rev. Research 8, 023330– Published 24 June, 2026
  • Received 5 January 2026
  • Accepted 11 May 2026
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

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