Irregular Tiling Methods

Updated 8 July 2026

Irregular tiling method is a collection of strategies that impose local constraints, seed rules, and substitution systems to prevent repetitive patterns in tiling designs.
These techniques are applied across fields—from computer graphics for texture realism to engineering for modular array design—by managing local and global tiling regularity.
Methods include dappled tiling repairs, graph-based region completions, atlas rule reductions, variable-geometry substitutions, and near-coincidence constructions to achieve desired tiling properties.

An irregular tiling method is a procedure for generating, transforming, or selecting tilings whose organization departs from simple periodic repetition. In the cited literature, the phrase covers several technically distinct constructions: local anti-repetition filters on rectangular grids, graph-based completion of arbitrary square-grid regions with holes, seed-dependent or atlas-based forcing of nonperiodicity, substitution systems with decorated local rules, and bilayer coincidence schemes that recover quasiperiodic order from superposed periodic layers (Kaji et al., 2016, Derouet-Jourdan et al., 2016, Klaassen, 2021, Ochana et al., 1 Jan 2026). The common theme is not a single formal definition of “irregularity,” but a family resemblance: each method suppresses, avoids, or reorganizes regular repetition by imposing constraints on runs, local neighborhoods, growth order, substitution ancestry, or coincidence windows.

1. Meanings and scope

In this literature, “irregular” has several non-equivalent meanings. In computer graphics, it often denotes suppression of visually objectionable repetition on a square grid, especially long horizontal or vertical strips of the same tile. In Wang tiling and related completion problems, it denotes tiling arbitrary finite regions, including non-rectangular regions and regions with holes, under boundary constraints. In aperiodic-order research, it usually denotes nonperiodic or quasiperiodic organization forced by seeds, matching rules, substitutions, or bilayer constructions. In engineering and computational mechanics, the same phrase may refer to irregular cluster layouts or irregular simplex partitions used for synthesis or homogenization rather than to classical prototile theory (Kaji et al., 2016, Derouet-Jourdan et al., 2016, Klaassen, 2021, Anselmi et al., 13 Aug 2025).

Family	Representative paper	Core mechanism
Local anti-repetition	"Dappled tiling" (Kaji et al., 2016)	Bounded same-tile run lengths on a rectangular grid
Irregular-region completion	"A linear algorithm for Brick Wang tiling" (Derouet-Jourdan et al., 2016)	Cycle/tree graph decomposition with boundary constraints
Seed or atlas forcing	"Forcing nonperiodic tilings with one tile using a seed" (Klaassen, 2021); "Aperiodic tilings with one prototile and low complexity atlas matching rules" (Fletcher, 2010)	Growth rules, seed patches, or 1-corona atlases
Substitution and variable geometry	"An aperiodic tiling of variable geometry made of two tiles, a triangle and a rhombus of any angle" (Dongen, 2021)	Decorated substitution with a continuous rhombus-angle parameter
Bilayer/quasiperiodic generation	"Tiling by Near Coincidence" (Ochana et al., 1 Jan 2026)	Midpoints of accepted near-coincident pairs from transformed layers

A recurrent distinction is between local irregularity and strong aperiodicity. Some methods only suppress specific local artifacts, while others force global nonperiodicity. This distinction is explicit in the literature and governs both theorems and applications.

2. Local anti-repetition on rectangular grids

The most explicit grid-based irregular tiling method in this set is dappled tiling, introduced for rectangular planar grids and motivated by texturing (Kaji et al., 2016). The underlying region is

$G_{m,n}=\{(i,j)\in \mathbb{Z}\times \mathbb{Z}\mid 0\le i<m,\ 0\le j<n\},$

and a tiling is a function

$f:G_{m,n}\to T$

for a finite tile set $T$ with $\#T\ge 2$ . The method does not attempt strong nonperiodicity. Instead, it forbids long monochromatic runs along the coordinate axes. For $p>1$ and tile $t\in T$ , the condition $H_t^p$ requires that there be no horizontal strip with more than $p$ consecutive $t$ 's, and $V_t^q$ is the analogous vertical condition. A tiling satisfying all conditions in a set $f:G_{m,n}\to T$ 0 is called $f:G_{m,n}\to T$ 1-dappled.

This model is irregular only in a precise local sense. It suppresses the stripe artifacts that make grid-based textures look artificial, but it does not optimize global randomness, does not enforce aperiodicity in the Penrose-like sense, and does not control diagonal repetition except indirectly. The paper states this explicitly by contrasting dappled tilings with “draughtboard” tilings, which are always valid but “look very artificial and are not suitable for texturing” (Kaji et al., 2016).

The central algorithm is formulated for $f:G_{m,n}\to T$ 2. The grid is processed in increasing order of the weight

$f:G_{m,n}\to T$ 3

Thus the sweep proceeds along anti-diagonals. If $f:G_{m,n}\to T$ 4 is the minimum-weight cell where a violation occurs, the repair is local. Step (I) tries to flip the tile,

$f:G_{m,n}\to T$ 5

provided this does not create a violation at $f:G_{m,n}\to T$ 6. Step (II), used when the flip fails, performs a $f:G_{m,n}\to T$ 7-local surgery:

$f:G_{m,n}\to T$ 8

The key invariant is that the violation at $f:G_{m,n}\to T$ 9 is removed without introducing any new violation at cells of weight $T$ 0. The violation is therefore “pushed” forward. Since each repair either increases the minimum weight of violating cells or reduces the number of violations at the current minimum weight, termination follows on the finite grid. The main correctness statement is that Algorithm 1 takes any tiling $T$ 1 and outputs an $T$ 2-dappled tiling, and if $T$ 3 is already $T$ 4-dappled, it outputs $T$ 5. The paper describes this as a retraction from the set of all tilings onto the set of $T$ 6-dappled tilings (Kaji et al., 2016).

The method admits two notable extensions. First, the paper allows non-uniform conditions

$T$ 7

so the allowed run lengths may vary by position. Second, it gives a cyclic variant for seamless periodic textures, where constraints wrap around boundaries. That cyclic algorithm is more limited: for $T$ 8 it requires $T$ 9, and unlike the basic algorithm it may alter an already valid cyclic tiling because it imposes stronger intermediate conditions. The basic method also has an explicit edge-case limitation: it “does not always work when $\#T\ge 2$ 0 or $\#T\ge 2$ 1 for some $\#T\ge 2$ 2,” and the paper gives a counterexample $\#T\ge 2$ 3 (Kaji et al., 2016).

Its main applications in the paper are Brick Wang tiles and flow tiles. For Brick Wang tiles, the set

$\#T\ge 2$ 4

is split as $\#T\ge 2$ 5, and an $\#T\ge 2$ 6-dappled tiling with $\#T\ge 2$ 7 suppresses long horizontal runs of one brick orientation and long vertical runs of the other. The method is therefore best understood as a local anti-repetition filter on grid tilings rather than as a global randomness optimizer or a strong aperiodicity construction (Kaji et al., 2016).

3. Boundary-constrained completion on irregular regions

A distinct irregular tiling method appears in the graph-based treatment of Brick Wang tiles on arbitrary planar square-grid regions, including irregular shapes and holes (Derouet-Jourdan et al., 2016). Here the issue is not suppression of repetition but exact completion under boundary colors.

The region is modeled as a finite undirected graph

$\#T\ge 2$ 8

where $\#T\ge 2$ 9 are the actual cells and $p>1$ 0 is a distinguished vertex called the constrainer. Missing neighbors, outer boundaries, and hole boundaries are all represented uniformly by edges incident to $p>1$ 1. For square-grid tiling, each cell has four ordered legs $p>1$ 2, and a boundary coloring is a map $p>1$ 3. This graph formalism is the mechanism by which irregular regions and holes are handled.

The paper introduces sequentially permissive tilesets. A set $p>1$ 4 is $p>1$ 5-sequentially permissive if, whenever colors on any $p>1$ 6 sides are fixed, there exists at least one legal color on the remaining side. The informal interpretation is that a cell can always be tiled as long as it has at least one free leg. Brick Wang tiles satisfy

$p>1$ 7

with $p>1$ 8, and this set is $p>1$ 9-sequentially permissive (Derouet-Jourdan et al., 2016).

The main theorem reduces irregular geometry to graph topology: a square-grid board is always solvable for arbitrary boundary constraints iff the full subgraph on the cells contains a cycle. If the cell graph is a tree, one can choose boundary colors that force inconsistency. If it contains a cycle, the cycle provides enough flexibility to absorb arbitrary boundary conditions. This yields the algorithmic decomposition: detect a cycle by DFS; if none exists, solve the tree by propagating local conditions; if a cycle exists, use it as a solvable core, propagate conditions from attached trees toward the cycle, tile the cycle, and then propagate colors back outward. The total complexity is linear in the number of cells (Derouet-Jourdan et al., 2016).

This method differs structurally from dappled tiling. Dappled tiling starts from an arbitrary initial pattern and locally repairs anti-repetition violations. The Brick Wang algorithm instead decides and constructs a globally valid completion under prescribed boundary constraints. The former targets visual stripe suppression; the latter targets exact solvability on arbitrary planar regions with holes.

4. Seed rules, atlas rules, and hierarchical local-rule systems

A second major branch of irregular tiling methods uses seeds, growth rules, atlas rules, or richer local constraints to force nonperiodic organization. Bernhard Klaassen’s seed-based construction is exemplary (Klaassen, 2021). It uses a decorated nonconvex tile $t\in T$ 0 carrying a marked arc and the growth rule

$t\in T$ 1

Starting from a single seed tile, tiles are added sequentially. The union of all marked arcs is then a continuous and self-avoiding curve, and the paper proves that any monohedral tiling carrying such a curve and reaching every tile cannot have translational symmetry. The result is a seed-dependent method for forcing nonperiodicity with one prototile plus a growth rule. The same paper also presents an undecorated variant: a specific three-tile seed patch forces a nonperiodic tiling without any decoration or further matching rule. The method is explicitly weaker than a true undecorated einstein, and the paper states that the nonperiodicity obtained is weaker than that of canonical aperiodic sets such as Penrose tiles (Klaassen, 2021).

Fletcher’s atlas construction addresses a related but different problem: reducing the number of prototiles by replacing facet matching rules with a finite atlas of permitted patches (Fletcher, 2010). The main theorem states that a $t\in T$ 2-tiling is MLD to a $t\in T$ 3-tiling for some $t\in T$ 4-corona atlas rule $t\in T$ 5 and a prototile set $t\in T$ 6 with $t\in T$ 7. The shift is conceptual. Information usually stored on tile boundaries is moved into local neighborhood type. The paper’s examples compress a $t\in T$ 8-tile Wang system to a pair of square prototiles in $t\in T$ 9, and Kari’s $H_t^p$ 0 Wang cubes to a single cubic prototile in $H_t^p$ 1, but only under atlas matching rules rather than pure facet rules (Fletcher, 2010).

A recent square-tile example, tileset As, also belongs to this family but emphasizes hierarchical emergence from richer local constraints (Dongen, 3 Mar 2025). The system uses three square tiles—blue, red, and green—with edge colors, edge thickness values, and corner values. The local rules require adjacent border colors to match, adjacent border thicknesses to sum to $H_t^p$ 2 or $H_t^p$ 3, and the sum of the four corner values around every interior lattice vertex to be $H_t^p$ 4, $H_t^p$ 5, or $H_t^p$ 6. A brute-force classification shows exactly 9 supertiles, and at that macro-scale border color constraints alone suffice. Each supertile has a fixed-point replacement rule, and the induced line sequence obeys

$H_t^p$ 7

The paper identifies the resulting sequence with OEIS A159684 and uses that as the non-periodicity witness. It also states a limitation: non-periodic full-plane tilings are demonstrated, but a full proof that As is an aperiodic tileset is still pending (Dongen, 3 Mar 2025).

Across these constructions, a common misconception is addressed explicitly: low tile count, one-tile formulations, or local rules do not automatically imply a true undecorated monotile theorem. Seed rules may be process-dependent rather than static local checks, atlas rules may replace boundary information rather than eliminate it, and some systems demonstrate non-periodic tilings without yet proving aperiodicity of the entire tileset.

5. Substitution, grid superposition, and near-coincidence generation

Another major family uses substitution, continuously variable geometry, or superposition of periodic grids to produce aperiodic or quasiperiodic order. In the variable-geometry construction built from an equilateral triangle and a rhombus of angle $H_t^p$ 8, the square-triangle tiling is reinterpreted as a rhombus-triangle system with a continuous parameter $H_t^p$ 9 (Dongen, 2021). The paper describes three rule-count levels for the $p$ 0 rhombus case— $p$ 1, $p$ 2, and $p$ 3 substitution rules—and claims that the same substitution structure can be adapted for any $p$ 4. It treats $p$ 5 as the square-triangle case and $p$ 6 as periodic degenerate endpoints. At the same time, it explicitly does not provide a symbolic substitution operator, a substitution matrix $p$ 7, an inflation factor $p$ 8, or a formal proof of aperiodicity for all $p$ 9. The construction is therefore best read as a parameterized decorated substitution system with local matching constraints rather than as a completed formal classification (Dongen, 2021).

A more geometric real-space method appears in the grid construction from superposed periodic tilings by lattice fundamental domains (Sadoc et al., 2023). Two grids are built from the same lattice, one rotated relative to the other. Their overlap domains are assigned reference points, typically the midpoint of the two tile centers, and the quasiperiodic tiling is extracted from the Delaunay triangulation of these points after keeping only edges of a specific common length. For the Stampfli-type example with two hexagonal grids rotated by $t$ 0, the relevant angles are $t$ 1, $t$ 2, and $t$ 3, producing regular triangles, squares, and rhombuses. For the square-grid example rotated by $t$ 4, the paper identifies the result with the Ammann–Beenker tiling. The method is constructive and planar, but the paper does not provide a full general theorem for all possible lattice/domain/rotation choices (Sadoc et al., 2023).

The most explicit bridge between moiré intuition and cut-and-project formalism is the near-coincidence method (Ochana et al., 1 Jan 2026). Its vertex set is

$t$ 5

where $t$ 6 are points from two transformed layers and $t$ 7 is a coincidence window. In the isotropic case, the condition becomes $t$ 8. The midpoint is the physical-space point, the difference vector is the internal-space selector, and the coincidence window plays the role of the cut-and-project acceptance domain. The paper states that the method reproduces classical tilings including the Ammann–Beenker, the Niizeki–Gähler, and the square and hexagonal Fibonacci tilings, while also yielding new tilings “not likely to arise in conventional constructions,” with relative frequencies of local configurations that may take transcendental values (Ochana et al., 1 Jan 2026).

This branch makes explicit another important distinction. Some constructions are canonical because their windows are polygonal and aligned with substitution structure. Others become irregular precisely because they keep physically natural but noncanonical windows, such as circles. In that sense, irregularity can mean deviation from the standard acceptance domain rather than loss of rigor.

6. Applications, limitations, and recurring distinctions

The application range of irregular tiling methods is unusually broad. In computer graphics, dappled tiling is motivated by texturing, Brick Wang wall patterns, and flow tiles (Kaji et al., 2016). In antenna engineering, irregular domino or multi-square tilings are used to organize modular phased arrays. The divide-and-conquer tiling method for large aperiodic phased arrays partitions an aperture into sub-areas, tiles them locally while preserving full coverability of the untiled remainder, and evaluates each candidate through a global mask-violation functional; it is presented as a scalable alternative to exhaustive or global optimization methods for large LEO satcom arrays (Anselmi et al., 13 Aug 2025). A related isophoric design uses two square tile sizes, graph-encoded admissible layouts, and an integer-coded genetic algorithm to induce an effective tile-size taper while preserving full aperture coverage (Rocca et al., 2021). In self-assembly, a three-patch DNA-particle model with intermediate geometry and sufficient bound-state rotational flexibility self-assembles into a periodic false tiling composed of irregular polygons and three vertex types, showing that complex periodic structure can emerge from controlled mismatch plus flexibility (Preisler et al., 2017). In computational mechanics, the quasicontinuum method is extended to irregular particle systems by using triangulations or tetrahedralizations as interpolation and homogenization cells; there the “tiling” is a computational partition rather than a prototile theory (Mikeš et al., 2017).

Several recurring limitations are explicit. Dappled tiling does not optimize global randomness and does not in general produce the closest valid tiling in Hamming distance (Kaji et al., 2016). Seed-based one-tile methods may require decorations or a seed patch and need not give strong aperiodicity (Klaassen, 2021). Atlas-rule reductions replace facet information with neighborhood information rather than eliminating constraints outright (Fletcher, 2010). Variable-geometry substitution systems may state aperiodicity claims without providing a fully rigorous proof for every parameter value (Dongen, 2021). Tileset As demonstrates non-periodic full-plane tilings, but its full aperiodicity proof remains open (Dongen, 3 Mar 2025). In applied engineering papers, “irregular tiling” may describe an optimization architecture or multiscale partition rather than a mathematical tiling theory in the classical sense (Anselmi et al., 13 Aug 2025, Mikeš et al., 2017).

A plausible implication is that “irregular tiling method” functions less as the name of a single technique than as a cross-disciplinary category. In one regime, it means controlled local irregularity on a grid; in another, it means exact completion of irregular regions; in another, it means nonperiodic order forced by seeds, atlases, substitutions, or near coincidences. What unifies these methods is not a common proof schema, but a shared strategy: regular repetition is not removed by noise alone, but by a formal mechanism that constrains how local structure can extend to global tilings.