A counterexample to periodic tiling in dimension three

The explanation

Start on the number line. Take the tile F = {0, 2}, two points. Place copies of it at the shifts 0, 1, 4, 5, 8, 9, and so on, and the matching negative ones. The copy at 0 covers 0 and 2. The copy at 1 covers 1 and 3. The copies at 4 and 5 cover 4 to 7. Every whole number is covered exactly once. A tiling is this kind of exact cover by shifted copies of one tile. This tiling repeats with period 4, so it is periodic.

In the paper's sense, a tiling is fully periodic when its set of shifts repeats under a full grid of translations. The periodic tiling conjecture asks whether every tile that tiles at all also has a fully periodic tiling. Lagarias and Wang wrote down an explicit version for ordinary spaceSource 1. On the line, every tiling by a finite tile is periodic, by an argument of NewmanSource 2. Bhattacharya proved that every finite tile of the plane grid ℤ² has a periodic tilingSource 3. Greenfeld and Tao later disproved the conjecture in some sufficiently large dimensionSource 4. That left the gap between dimension three and their very large one.

The paper builds a finite tile T in ℤ³, the grid of integer triples. T tiles ℤ³, but no tiling by T is fully periodicSource 5. The same holds for T thickened by unit cubes in ordinary space ℝ³, even with any real shiftsSource 6. Since the cases of dimensions one and two are settled positively, three is the smallest possible dimension for this on the gridSource 7.

The proof follows Greenfeld and Tao's plan. It starts from a puzzle rule, like a Sudoku on an infinite strip of columns. The rule has a solution with every column nonconstant, but no such solution repeats verticallySource 8. The puzzle rules are then turned into tiling equations on the plane grid times a finite group. The obstacle is that finite group. A general finite group needs several generators, but a cyclic group fits into one extra integer coordinateSource 9. So the paper encodes the rules while keeping the finite part cyclicSource 10. The plane grid times a finite cyclic group is a quotient of ℤ³, which lets the tile move into three dimensionsSource 11.

How sure is this? Lean, a computer proof checker, has a formal statement of the counterexample. It also covers the claim that three is the least grid dimension. So the paper proves its main result, as far as the Lean statement matches the paper. The paper gives no practical bound on the tile's sizeSource 12. No specialist review of the paper is published.

What it doesn't show. This does not give a small or connected tile; the construction is huge, and the paper does not claim the tile is in one piece. It also does not say the tilings have no repeating direction at all, only that none repeats in every direction.