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An aperiodic set of Wang tiles for every quadratic irrational

This paper establishes a geometric sufficient condition for the non-periodicity of striped Wang tiles and demonstrates that for any pair of irrational numbers within the same quadratic field, there exists a corresponding finite aperiodic set of such tiles with prescribed stripe densities.

Original authors: Jarkko Kari, Sébastien Labbé, Pieter Mostert

Published 2026-06-24

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Jarkko Kari, Sébastien Labbé, Pieter Mostert

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you have a giant, infinite floor and a box of square tiles. Each tile has colored edges. The rule for tiling the floor is simple: whenever two tiles touch, their touching edges must have the same color.

Usually, if you have a set of tiles that can cover the floor, you can arrange them in a pattern that repeats over and over again, like a wallpaper design. This is called a periodic tiling.

However, some special sets of tiles are aperiodic. This means they can cover the entire infinite floor, but they never form a repeating pattern. No matter how far you zoom out, the design never looks exactly the same twice. Proving that a set of tiles is aperiodic is notoriously difficult, like trying to prove a maze has no exit without walking the whole thing.

This paper by Kari, Labbé, and Mostert introduces a new, elegant way to prove that certain sets of tiles are aperiodic. They do this by turning the problem into a geometry puzzle involving stripes and parabolas.

The "Striped" Tiles

The authors focus on a specific type of tile set where the tiles naturally form "stripes."

Imagine some tiles have a horizontal stripe painted on them, and others have a vertical stripe.
The rules of the tiles force these stripes to line up, creating long, continuous horizontal or vertical lines across the floor.
The authors ask: "If we look at a huge section of the floor, what percentage of the floor is covered by horizontal stripes, and what percentage by vertical stripes?" Let's call these percentages $\alpha$ (vertical) and $\beta$ (horizontal).

The Geometric Trick: The Quadrilateral and the Parabola

The paper's main discovery is a mathematical relationship between these stripe percentages and a shape called a quadrilateral (a four-sided polygon).

The Quadrilateral: For any set of these striped tiles, you can calculate four specific points in space (let's call them A, B, C, and D) based on the colors of the tile edges. These points form a four-sided shape.
The Equation: The authors prove that the stripe percentages ( $\alpha$ and $\beta$ ) must satisfy a specific equation. Geometrically, this equation describes a line that passes through the center of the shape and touches a parabola (a U-shaped curve) that is perfectly wrapped around the four sides of the quadrilateral.
The "Irrational" Key: Here is the magic part. If the shape of the quadrilateral is "just right," the only possible values for $\alpha$ and $\beta$ that satisfy the equation are irrational numbers (numbers like $\sqrt{5}$ or the Golden Ratio that cannot be written as a simple fraction).

Why does this prove the tiles are aperiodic?
If a tiling repeats itself (is periodic), the pattern must fit into a neat, repeating box. This forces the percentages of stripes to be simple fractions (rational numbers).

The Logic: If the math proves that the stripe percentages must be irrational, but a repeating pattern requires them to be rational, then a repeating pattern is impossible.
The Conclusion: Therefore, the tiles can only form non-repeating, aperiodic patterns.

What They Did with This New Tool

The authors didn't just invent the tool; they used it to solve old puzzles and build new ones:

New Proofs for Old Tiles: They applied their method to famous tile sets, like the Ammann tiles (16 tiles) and an encoding of Penrose tiles (24 tiles). Instead of using long, complex arguments about self-similarity (which were the old way), they simply showed that the "quadrilateral" for these tiles forces irrational stripe densities. This provides a much shorter, cleaner proof that these tiles are aperiodic.
Building New Tiles: They went in the opposite direction. They asked: "Can we build a set of tiles for any pair of irrational numbers we choose?"
- They proved that for almost any pair of irrational numbers (specifically those from "quadratic number fields," which include numbers involving square roots), you can construct a finite set of Wang tiles that will force the floor to have exactly those stripe densities.
- This means they can create a custom aperiodic tile set for any specific irrational ratio you want.

Summary

Think of the paper as introducing a new "metal detector" for tile sets.

Old way: You had to walk through the entire maze of the tile set to prove it had no repeating pattern.
New way: You just look at the "shape" of the tile rules (the quadrilateral). If the shape forces the "stripe density" to be an irrational number, you instantly know the tiles cannot form a repeating pattern.

The authors used this detector to quickly verify known aperiodic sets and to build entirely new families of aperiodic tiles, showing that the connection between geometry (parabolas and quadrilaterals) and number theory (irrational numbers) is a powerful key to unlocking the secrets of these infinite floor puzzles.

Technical Summary: An Aperiodic Set of Wang Tiles for Every Quadratic Irrational

Problem Statement
The paper addresses the problem of proving the non-periodicity (aperiodicity) of sets of Wang tiles. While the existence of aperiodic tile sets is well-established, proving that a specific set admits no periodic tiling is often difficult, typically requiring complex combinatorial arguments based on hierarchical structures, unique composition properties, or self-similarity. The authors aim to provide a more direct, arithmetic-based method for proving non-periodicity, specifically for sets of Wang tiles possessing a "stripe" structure. Furthermore, the paper seeks to answer the inverse problem: for any pair of irrational numbers $(\alpha, \beta)$ within the same quadratic number field, can one construct a finite aperiodic set of Wang tiles with stripes that admits a valid tiling with exactly those stripe densities?

Methodology
The core methodology relies on a geometric-arithmetic correspondence between Wang tiles with stripes and planar geometry.

Stripe Partitioning: The authors consider sets of Wang tiles $T$ that can be partitioned into four subsets based on the presence or absence of horizontal and vertical stripes: $T = T_\square \cup T_\urcorner \cup T_\sqsubset \cup T_\boxplus$ .
Vector Encoding: Labels on the edges of the tiles are mapped to vectors in $\mathbb{R}^2$ via functions $v$ (vertical labels) and $h$ (horizontal labels).
Quadrilateral Determination: A set of tiles $T$ is said to "determine" a quadrilateral $ABDC$ if the vector sum $v(a) + h(b) - v(c) - h(d)$ equals a specific vertex ( $A, B, C,$ or $D$ ) depending on which subset of the partition the tile belongs to.
Density Equation: For any doubly-periodic valid configuration, the densities of vertical stripes ( $\alpha$ ) and horizontal stripes ( $\beta$ ) must satisfy a specific linear equation derived from the quadrilateral vertices:
$\mathbf{0} = (1-\alpha)((1-\beta)A + \beta B) + \alpha((1-\beta)C + \beta D)$
This equation is interpreted geometrically: solutions $(\alpha, \beta)$ correspond to lines passing through the origin that are tangent to a unique parabola circumscribed to the quadrilateral $ABDC$.
Non-Periodicity Criterion: If the solutions to this equation are irrational (specifically, if the discriminant of the resulting quadratic equation is positive but not a perfect square), then no periodic tiling can exist, as periodic tilings require rational densities.

Key Contributions and Results

Theorem A (Sufficient Condition for Non-Periodicity): The authors establish a sufficient condition for a set of Wang tiles with stripes to be aperiodic. If the tiles determine a quadrilateral $ABDC$ with integer vertices satisfying specific determinant conditions (ensuring the quadrilateral is non-degenerate in a specific way) and the discriminant of the associated quadratic equation is not a perfect square, the set admits no periodic tiling.
- Application: This theorem provides new, short proofs of non-periodicity for known sets, including the Ammann 16-tile set, a 24-tile encoding of Penrose tilings, and the family of metallic mean Wang tiles. These proofs bypass traditional hierarchical decomposition arguments.
Theorem B (Construction for Quadratic Irrationals): The paper proves that for every pair $(\alpha, \beta)$ of elements in the same quadratic number field (where both are rational or both are irrational), there exists a finite set of Wang tiles with stripes that:
1. Determines a quadrilateral $ABDC$ such that $(\alpha, \beta)$ is a solution to the density equation.
2. Admits a valid tiling where the vertical and horizontal stripe densities are exactly $\alpha$ and $\beta$ .
3. Is aperiodic if $\alpha$ and $\beta$ are irrational.
- Construction: The construction uses a "balanced representation" of real numbers (similar to Sturmian sequences) to define tile colors, ensuring the stripe densities match the target values. The authors provide explicit bounds on the number of tiles required.
Algorithmic Approach (Section 5): The paper presents a linear algebra algorithm (based on pullbacks of linear transformations) to compute the vectors $v, h$ and the quadrilateral $ABDC$ for a given set of Wang tiles. This allows for the automated verification of the sufficient condition in Theorem A for various known tile sets.

Significance and Claims
The authors claim that their work offers a "short proof" of non-periodicity for several significant tile sets, replacing lengthy combinatorial arguments with a direct arithmetic argument based on the irrationality of stripe densities. This approach is analogous to Kari's earlier work using multiplicative arithmetic but utilizes additive arithmetic and geometric properties of parabolas.

The paper also claims a constructive solution to the problem of generating aperiodic tiles for any pair of quadratic irrationals, filling a gap in the literature where examples were previously sparse or relied on complex substitution rules (like Mozes' construction). The authors explicitly state that their construction is "as simple as possible" in the context of algebraic numbers of degree 2.

Limitations and Open Questions
The paper maintains modesty regarding the scope of its results:

The sufficient condition in Theorem A applies specifically to sets of Wang tiles with stripes.
While the construction in Theorem B guarantees the existence of a tiling with specific densities, the authors pose Question 9.1: Is it true that every valid configuration in the constructed tile set has these specific densities, or could other configurations with different densities (solutions to the same equation) exist?
Question 9.2 asks whether the Wang shifts generated by these new tile sets are minimal dynamical systems.
Question 9.3 inquires whether these shifts are self-similar (admitting an expansive 2D substitution), a property known for metallic mean tilings but not yet proven for the general construction.

The authors do not claim to have solved the general domino problem or to have found the minimal number of tiles for aperiodicity in all cases, but rather to have provided a new, efficient tool for proving non-periodicity and a systematic method for constructing aperiodic sets with prescribed quadratic irrational densities.

The "Striped" Tiles

The Geometric Trick: The Quadrilateral and the Parabola

What They Did with This New Tool

Summary

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