Tiling by Near Coincidence

TL;DR

The paper introduces the near-coincidence method to generate quasiperiodic plane tilings by merging nearly coincident point pairs from superimposed, rotated or scaled layers, then mapping the resulting points to a Delone set. It establishes rigorous connections to the cut-and-project formalism, reproduces classic tilings such as Ammann–Beenker, Niizeki–Gähler, and Fibonacci tilings, and uncovers new tilings (e.g., tristar) by using circular coincidence windows that yield transcendental vertex-frequency ratios. The method is algorithmically simple and adaptable, enabling efficient generation of diverse tilings and providing insight into the role of coincidence windows in shaping tile configurations and substitution rules. The work highlights potential applications to multilayer systems, including trilayer moiré patterns and graphene at small twist angles, where large-scale quasiperiodicity may emerge and inform physical properties and diffraction. Overall, the near-coincidence framework offers a physically motivated, versatile alternative to traditional tiling constructions with practical implications for moiré materials and aperiodic order.

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.

Figures (15)

• Figure 1: Moiré patterns of (a) twisted square-lattice bilayer (rotated by $45^\circ$) producing an octagonal pattern, (b) twisted triangular-lattice bilayer (rotated by $30^\circ$) producing a dodecagonal pattern, and (c) scaled honeycomb-tiling bilayer (scaled by the golden mean $(1+\sqrt{5})/2$) producing a hexagonal quasiperiodic pattern. A red dot at the center indicates the single point of perfect coincidence of the two layers.
• Figure 2: (a) Step-by-step construction of an octagonal tiling by the near-coincidence method: 1. Two layers of identical periodic square lattices---a blue lattice rotated by $45^\circ$ about the bottom-left point with respect to a red one---are superimposed; 2. Pairs of points whose separation is shorter than a prescribed threshold are identified as sites of near coincidence, and a potential tiling-vertex is placed at their midpoint, and color coded on a yellow-to-purple scale according to their degree of coincidence; 3. Edges are drawn between vertices at permitted distances. (b) A larger patch of the constructed tiling, with the section in panel (a) outlined at the bottom-left corner, after removal of the original red and blue points, showing apparent defects consisting of nearby pairs of vertices, and leading to overlapping tiles and crossing edges.
• Figure 3: (a) Apparent defect consisting of a pair of nearby vertices, obtained using an isotropic coincidence window. (b) Correcting the defect by introducing two additional edge lengths and two new prototiles---a kite and a trapezoid. (c,d) Two alternative local configurations, consisting of a square and a pair of rhombs, associated with a typical phason flip of the octagonal tiling: in (c) the edge connection is made through the lower-coincidence (yellow) vertex; and in (d) the edge connection is made through the higher-coincidence (green) vertex. One typically chooses the option in (d), discarding the lower-coincidence vertex.
• Figure 4: Octagonal tilings obtained by the near-coincidence method as described in Sec. \ref{['Sec:Generating']}. (a) The octagonal tiling obtained by admitting all the vertices, shown in Fig. \ref{['fig:steps-b']}, whose coincidence falls within the brown circular coincidence window, depicted in panel (c). The tiling requires three different edge lengths, and contains four prototiles---a square, a $45^\circ$-rhomb, a trapezoid, and a kite---displayed beneath the tiling. (b) The octagonal tiling obtained when insisting on a single edge length, thus removing the weaker-coincidence vertex in every pair of nearby vertices. This removes excess points, whose positions are indicated by empty circles, and leads to a tiling containing only the square and the $45^\circ$-rhomb prototiles, displayed beneath the tiling. As demonstrated in Sec. \ref{['Sec:Ammann']}, the vertices that are accepted into this tiling correspond to using the light-brown octagonal coincidence window, shown in panel (c), inscribed within the original circular one. This is the well-known Ammann--Beenker Beenker82Ammann92 tiling. (c) Coincidence windows selecting the point sets that are admitted into the octagonal tilings in (a) and (b).
• Figure 5: Distribution of excess points, mapped onto the centered circular coincidence window, and plotted in black together with the accepted vertices, which are color coded according to their distance from the center. The discarded points outline an octagonal boundary, revealing the octagonal coincidence window.