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A counterexample to the periodic tiling conjecture
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DOI：10.4007/annals.2024.200.1.5.png)
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Abstract
En 中文
The periodic tiling conjecture asserts that any finite subset of a lattice Zd that 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 Rd. In fact, we also obtain a counterexample in a group of the form Z2 x G0 for some finite abelian 2-group G0. 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.
Keywords:
tiling
periodicity
Journal
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5.3
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1.4K
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1.6W
