1 · The essentials
Can a single shape fill space without gaps or overlaps, yet make a pattern that repeats in every direction impossible? The manuscript constructs such a shape on the three-dimensional grid. It consists of finitely many grid points, and copies may be moved but not turned. These copies can fill the grid, but no arrangement repeats in three independent directions. Replacing each point of the shape with a unit cube gives the same outcome in ordinary three-dimensional space, even when copies may be moved by arbitrary real distances. The main result has a Lean formalization.
A picture to help (an imperfect image, not to be taken literally)
As an imperfect image, imagine filling an endless room with copies of one fixed block, sliding each copy into place without turning it. The surprising requirement is not merely that you can arrange the blocks without a repeating layout, but that no repeating layout can work.
Why it matters
This separates the ability to fill space from the ability to fill it with a pattern that repeats in all independent directions. It gives a negative resolution of the periodic tiling conjecture in dimension three, which the repository summary identifies as the smallest possible lattice dimension for such a counterexample. The continuous-space version also rules out recovering a fully repeating arrangement simply by allowing non-grid-aligned movements.
2 · Going further
The precise lattice problem concerns a finite set . A translational tiling covers every lattice point exactly once using translated copies of , without rotations. The manuscript constructs a for which such a tiling exists, but no fully periodic tiling exists: no tiling repeats under three independent translation directions. In this setting, the periodic tiling conjecture would require a tile that can tile at all to admit a fully periodic tiling. The constructed tile is a counterexample.
The abstract also states that replacing the lattice points of with unit cubes produces a tile in with the same distinction: tilings exist, but fully periodic tilings do not, even when arbitrary real translation vectors are permitted. This lattice construction followed by unit-cube thickening is the only outline supplied; the abstract does not explain the mechanism that forces the absence of full periodicity. The main manuscript is marked as Lean-formalized. No companion manuscripts or further details of the formalization's scope are supplied.
Useful notions
- The three-dimensional integer grid
- Translations without rotations
- Covering without gaps or overlaps
- Repetition in independent directions
What was known before / what is new
Before
The source identifies the periodic tiling conjecture as the question being resolved negatively in dimension three. It gives no account of earlier constructions or arguments in other dimensions, although the repository summary identifies three as the smallest possible lattice dimension for this counterexample.
New
The contribution is a finite translational tile in that admits tilings but admits no fully periodic tiling. Its unit-cube thickening has the same property in , including when arbitrary real translations are allowed. The main result has a Lean formalization.
Limits and assumptions
The lattice result concerns a finite tile in three dimensions and copies moved by translation, not rotation. The continuous-space claim concerns its particular unit-cube thickening. Absence of full periodicity does not by itself assert absence of every individual translation symmetry. The main result is Lean-formalized, but the supplied text does not specify the exact boundaries of that formalization, so it does not establish which additional claims are covered. The abstract supplies neither construction details nor an argument for the obstruction to full periodicity, and it lists no remaining open questions.
Mini glossary
- Translational tiling —
- A covering without gaps or overlaps by copies of one tile moved without rotation.
- Integer lattice —
- The set of points whose coordinates are all integers, written in three dimensions.
- Fully periodic tiling —
- In three dimensions, a tiling that repeats under translations in three independent directions.
- Unit-cube thickening —
- The conversion of a lattice tile into a solid tile by replacing each of its points with a unit cube.
- Lean formalization —
- An encoding of a mathematical result and its justification for checking by the Lean proof assistant.
Keywords : Translational tiling · Periodic tiling conjecture · Three-dimensional lattice
AI-generated explanation based on the manuscript. Unofficial. Refer to the manuscript for the authoritative version.
3 · Technical details and sources
Claim made in the manuscript (source text)
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.
Verified by Lean: the Lean software checked the part of the reasoning described below, sometimes only part of the manuscript. This does not mean independent mathematicians have validated the result.
Manuscripts in this family
A translational tile with no fully periodic tiling in dimension three
23/09/2026We construct a finite translational tile in ℤ3 that admits tilings but no fully periodic tiling. Its unit-cube thickening has the same property in ℝ3, even when arbitrary real translations are allowed. This gives a negative resolution of the periodic tiling conjecture in dimension three.