A counterexample to the periodic tiling conjecture

Abstract

The periodic tiling conjecture asserts that any finite subset of a lattice ${\mathbb{Z}}^d$ which tiles that lattice by translations, in fact tiles periodically. In this work we disprove this conjecture for sufficiently large $d$, which also implies a disproof of the corresponding conjecture for Euclidean spaces ${\mathbb{R}}^d$. In fact, we also obtain a counterexample in a group of the form ${\mathbb{Z}}^2\times G_0$ for some finite abelian $2$-group $G_0$. Our methods rely on encoding a "Sudoku puzzle" whose rows and other non-horizontal lines are constrained to lie in a certain class of "$2$-adically structured functions," in terms of certain functional equations that can be encoded in turn as a single tiling equation, and then demonstrating that solutions to this Sudoku puzzle exist, but are all non-periodic.

50 pages, 13 figures. Final version

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• A chiral aperiodic monotile
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• Undecidability of translational monotilings
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Cited by in corpus (3)

• A counterexample to the periodic tiling conjecture
• Undecidability of Translational Tiling of the 3-dimensional Space with a Set of 6 Polycubes
• Undecidability of Translational Tiling of the 4-dimensional Space with a Set of 4 Polyhypercubes