A counterexample to periodic tiling in dimension three
What the paper claims, in its own words
Constructs a finite translational tile in ℤ3 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 ℝ3, even with arbitrary real translation vectors.
The plain-language explainer for this entry is still coming.
The original paper
- A translational tile with no fully periodic tiling in dimension three (PDF, opens in a new tab) September 23, 2026 Formal proof listed