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Aperiodic tile sets from Sturmian lattices

This paper presents an explicit algorithm to construct aperiodic tile sets based on Sturmian words of quadratic slopes, utilizing Sturmian lattices and the bounded displacement equivalence of Delone sets to generate tiles with scaling constants that are units of real quadratic fields.

Original authors: Shigeki Akiyama, Tadahisa Hamada, Katsuki Ito

Published 2026-07-17

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CC BY 4.0

Original authors: Shigeki Akiyama, Tadahisa Hamada, Katsuki Ito

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

The Puzzle of the Infinite Floor

Imagine you are an architect tasked with designing a floor that covers an infinite plane. You have a single, unique tile shape, and you must use it to cover the entire floor without ever repeating a pattern. This is the world of aperiodic tiling, a fascinating corner of mathematics where geometry meets logic. For decades, mathematicians hunted for a "monotile"—a single shape that could tile a floor but only in a non-repeating way. Recently, a shape called the "Smith Turtle" was discovered, proving such a tile exists. But the Smith Turtle is just one specific solution; it's like finding one specific key that opens a single door. The big question remains: Can we find a whole keychain of these unique tiles, each opening a different kind of infinite, non-repeating door?

To understand the new research, we need two simple concepts. First, think of a Sturmian word as a very specific, non-repeating rhythm made of just two sounds, like "clap" and "snap." If you arrange these sounds carefully, they create a pattern that never repeats but follows a strict mathematical rule based on a "slope" (a specific ratio). Second, imagine a lattice not as a fence, but as a grid of invisible lines. In this paper, these lines are drawn in three directions, forming tiny triangles where they cross. The magic happens when the spacing of these lines follows that non-repeating "clap-snap" rhythm. The researchers are asking: Can we build physical tiles that force a floor to follow these specific, non-repeating line patterns?

The New Blueprint: From Rhythms to Tiles

In this paper, Shigeki Akiyama, Tadahisa Hamada, and Katsuki Ito provide a master blueprint for creating an infinite family of these special tiles. They don't just find one; they give an explicit algorithm to build aperiodic tile sets for any "quadratic irrational slope." In plain English, this means they can generate a unique, non-repeating tile set for a vast, infinite variety of mathematical ratios, not just the one used by the Smith Turtle.

The core of their discovery is a method to reverse-engineer the process. Instead of starting with a tile and seeing what pattern it makes, they start with the "Sturmian lattice"—that grid of lines with the non-repeating rhythm—and ask, "What tiles would force the floor to look exactly like this?" Their answer is a set of tiles that act like a strict bouncer at a club: they only let in arrangements that match the specific mathematical rhythm of the chosen slope. If you try to arrange them in a repeating pattern, the rules simply won't let you.

Note: While this paper outlines the existence of these tile sets and sketches the proof, the full classification of the lattices and complete proofs are detailed in a separate, full version of the work.

The "Nut and Bolt" Mechanism

How do they build these tiles? The authors use a clever construction involving two types of pieces: Nuts and Bolts.

Imagine the Nuts as the rule-enforcers. These are ring-shaped tiles (like a washer) with special lines drawn on them, called "Ammann bars." These bars must line up perfectly with their neighbors to form straight lines across the floor. This requirement forces the Nuts to arrange themselves into the specific "Sturmian lattice" pattern. If the lines don't match up, the tile doesn't fit.

The Bolts are the solid, disk-shaped pieces that fill the gaps. They act as the "glue" that locks the specific mathematical slope into place. The researchers discovered that the ratio of how many small, medium, and large Bolts appear in a tiling is directly tied to the mathematical slope of the pattern. By carefully choosing how many of each Bolt type to include in their tile set, they can force the entire floor to adopt a specific, non-repeating rhythm.

The paper proves that for every "quadratic irrational" slope (a specific type of number that cannot be written as a simple fraction), there exists a tile set that enforces that exact slope. The size of these tile sets is finite but can be quite large; for example, one specific slope requires a set of 29 different tile variations. The authors show that the "expansion constant" (a number that describes how the pattern grows) for these new tiles corresponds to the fundamental units of real quadratic fields, linking the physical shape of the tiles to deep number theory.

Why This Matters

This work is significant because it moves beyond the "one-off" discovery of the Smith Turtle. While the Smith Turtle is a single, beautiful solution, this paper provides a general factory. It shows that there isn't just one way to make a non-repeating floor; there are infinitely many ways, each corresponding to a different mathematical slope. The authors provide a framework to generate these sets, proving that the universe of aperiodic tiles is far richer and more structured than previously thought. They didn't just find a new tile; they found the recipe book for an entire library of them.

Technical Summary: Aperiodic Tile Sets from Sturmian Lattices

Problem Statement
The paper addresses the construction of aperiodic tile sets (sets of tiles that can tile the plane but only in non-periodic ways). While recent discoveries, such as the "Smith Turtle" monotile, have demonstrated the existence of aperiodic monotiles, the authors aim to generalize this phenomenon. Specifically, they seek an explicit algorithm to construct aperiodic tile sets based on Sturmian words of quadratic slopes. The goal is to produce tile sets where the underlying scaling constant corresponds to a unit of any real quadratic field, thereby generating infinitely many essentially different aperiodic tile sets.

Methodology
The authors propose a construction method that diverges from traditional approaches relying on hierarchical unique-composition (substitution) structures. Instead, their methodology relies on two primary components:

Sturmian Lattices: The authors utilize a grid structure generated by Sturmian words. A Sturmian lattice is defined by three families of parallel lines in directions $0, 2\pi/3,$ and $4\pi/3$ . The defining condition is that for indices $i, j, k$ satisfying $i+j+k=0$ , the lines $a=a(i), b=b(j), c=c(k)$ bound a triangle of a fixed, tiny size. The gaps between adjacent parallel lines in these lattices encode balanced sequences (Sturmian words) with a specific irrational slope $\alpha$ .
Bounded Displacement (BD) Equivalence: The core technical tool is the bounded displacement equivalence of Delone sets. The authors treat the relationship between different sets of points (specifically the centers of "Bolts" in their tiling) as a many-to-many correspondence. They partition Delone sets into subsets with specific natural densities and establish BD correspondences between them to enforce algebraic constraints.

Construction of the Tile Set
The tile set is constructed using two types of components derived from a hexagon with a small disk removed:

Nuts: Annulus-like pieces containing "Ammann bars" (segments that must align to form straight lines). These enforce the geometric structure of the Sturmian lattice.
Bolts: Disk-like pieces that carry the BD correspondence. There are three types of Bolts ( $S, M, L$ ) corresponding to the three directions of the lattice.

The construction enforces a specific slope $\alpha$ by controlling the natural densities of the Bolt centers. In a valid tiling of slope $\alpha$ , the densities of the Bolt sets $S, M, L$ must satisfy the ratio:
$\delta(S) : \delta(M) : \delta(L) = (1-\alpha)^2 : 2\alpha(1-\alpha) : \alpha^2$
To construct the tile set for a quadratic irrational $\alpha$ , the authors partition the Delone sets of the Sturmian lattice into subsets with densities proportional to integer coefficients $(x', y', z')$ and $(x'', y'', z'')$ . These coefficients define the "patch-tiles" (combinations of Bolts) that, when combined with the Nuts, enforce the required density ratios. The proof that these constraints force the specific slope relies on the elementary fact that a quadratic curve and a line intersect in at most two points.

Key Results

Theorem 3: For every quadratic irrational slope $\alpha \in [0,1]$ , there exists an aperiodic tile set $\mathcal{A}(\alpha)$ that enforces Sturmian lattices with exactly that slope (or its Galis conjugate).
Expansion Constants: The underlying expansion constant of these tile sets is a unit of the real quadratic field $\mathbb{Q}(\alpha)$ .
Aperiodicity: Since Sturmian lattices of irrational slope possess no non-trivial periods, every tiling generated by $\mathcal{A}(\alpha)$ is non-periodic.
Cardinality: The authors provide a bound on the number of tiles in such a set, $Card(\mathcal{A}_\lambda) \leq 2\lambda + O(1)$ , where $\lambda$ is the expansion constant.
Example: The paper details a construction for $\alpha = \sqrt{6}-2$ , resulting in a tile set with 29 patch-tiles and an expansion constant of $5+2\sqrt{6}$ .

Significance and Claims
The paper claims to provide a systematic, explicit algorithm for generating aperiodic tile sets from any quadratic irrational slope, extending beyond the specific case of the Smith Turtle (which corresponds to a slope in $\mathbb{Q}(\sqrt{5})$ ).

Key distinctions of this work include:

Independence from Self-Similarity: Unlike many substitution-based proofs, this construction does not require the tile sets to enforce self-similar tilings; the proof rests on density matching and geometric constraints.
Generality: The method yields infinitely many essentially different aperiodic tile sets, realizing the units of every real quadratic field as expansion constants.
Mechanism: The work highlights the role of bounded displacement equivalence as a central mechanism for matching tiles arising from Sturmian lattices, offering a new perspective on the relationship between combinatorial word properties and geometric tiling constraints.

The authors note that the full classification of Sturmian lattices, cardinality estimates, and further examples are presented in a separate full version of the work [2], while this note focuses on the existence theorem and the sketch of the proof.

The Puzzle of the Infinite Floor

The New Blueprint: From Rhythms to Tiles

The "Nut and Bolt" Mechanism

Why This Matters

Technical Summary: Aperiodic Tile Sets from Sturmian Lattices

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