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A counterexample to periodic tiling in dimension three
Constructs a finite translational tile in ℤ that tiles space but admits no fully periodic tiling, disproving the periodic tiling conjecture in the smallest possible lattice dimension. Its unit-cube thickening gives the same counterexample in ℝ, even with arbitrary real translation vectors.
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A translational tile with no fully periodic tiling in dimension three
Sep 23, 2026·paper.pdfLean
We construct a finite translational tile in ℤ that admits tilings but no fully periodic tiling. Its unit-cube thickening has the same property in ℝ, even when arbitrary real translations are allowed. This gives a negative resolution of the periodic tiling conjecture in dimension three.