Local Rules for Computable Planar Tilings

Aperiodic tilings are non-periodic tilings characterized by local constraints. They play a key role in the proof of the undecidability of the domino problem (1964) and naturally model quasicrystals (discovered in 1982). A central question is to characterize, among a class of non-periodic tilings, the aperiodic ones. In this paper, we answer this question for the well-studied class of non-periodic tilings obtained by digitizing irrational vector spaces. Namely, we prove that such tilings are aperiodic if and only if the digitized vector spaces are computable.

💡 Research Summary

The paper investigates the relationship between computability and aperiodicity in planar tilings that arise from digitizing irrational vector spaces. Starting from the classic domino problem and the discovery of quasicrystals, the authors focus on tilings generated by projecting an irrational subspace of ℝ² onto the integer lattice ℤ² and assigning to each lattice point a “height” given by its offset from the subspace. The central question is whether such tilings can be completely described by a finite set of tiles together with local matching rules.

To answer this, the authors introduce a precise notion of computability for the underlying vector space: a subspace V is computable if a Turing machine can approximate its basis vectors to any desired precision. They then prove two complementary theorems.

Theorem 1 (Positive direction). If V is computable, there exists a finite tile set T and a collection of local rules R such that every tiling generated by (T,R) coincides exactly with the digitized version of V. The construction proceeds by approximating the basis vectors, partitioning the real line of possible heights into finitely many intervals, and encoding each interval as a distinct tile type. Matching rules enforce that adjacent tiles correspond to height differences that stay within allowed intervals, thereby translating the global non‑periodic structure into purely local constraints.

Theorem 2 (Negative direction). If V is non‑computable, no finite tile set together with local rules can generate the digitized tiling of V. The proof uses a contradiction: assuming such a finite description existed would yield an algorithm that, from the local rules, reconstructs the height function, which would in turn provide arbitrarily precise approximations of the basis vectors—impossible for a non‑computable subspace.

These results establish that computability of the underlying irrational vector space is both necessary and sufficient for the existence of aperiodic tilings that admit a local‑rule description. The paper connects this abstract criterion to known aperiodic tilings: for instance, Penrose tilings correspond to a computable two‑dimensional subspace and therefore admit a finite set of matching rules, whereas tilings derived from subspaces defined by non‑recursive constants do not.

Beyond the theoretical contribution, the authors discuss implications for the modeling of quasicrystals. In physical materials, atomic positions often follow projections of irrational subspaces; if the subspace is computable, the corresponding atomic arrangement can be realized through locally enforceable bonding rules, suggesting a pathway for designing new quasicrystalline alloys.

The paper concludes with several avenues for future work: extending the framework to higher‑dimensional tilings, analyzing the computational complexity of constructing the tile set from a given computable subspace, and experimentally testing local‑rule based synthesis of quasicrystals. Overall, the work provides a rigorous bridge between computability theory and the geometry of aperiodic tilings, offering a definitive criterion for when non‑periodic planar patterns can be captured by purely local constraints.