OpenAI research collection · 1 manuscript · Source checked 2026-10-19
Constructs a finite translational tile in ℤ^(3) 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^(3), even with arbitrary real translation vectors.
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We construct a finite translational tile in ℤ^(3) that admits tilings but no fully periodic tiling. Its unit-cube thickening has the same property in ℝ^(3), even when arbitrary real translations are allowed. This gives a negative resolution of the periodic tiling conjecture in dimension three.
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