A translational tile with no fully periodic tiling in dimension three
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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Updates: Sharing AI progress on mathematics (amazing!); and my lecture plansCombinatorics and moreOct 7, 2026
that admits no fully periodic tiling, reaching the smallest possible dimension for the lattice problem. Covering density of convex bodies. I described the breakthrough of Ordentlich, Regev, and Weiss on efficient lattice coverings in this post . OpenAI announces a universal lattice-covering bound…