Is there a strongly aperiodic monotile in three dimensions?
A prototile is strongly aperiodic if it tiles space and no tiling by its congruent copies has any translational period. In the plane this was settled in 2023 by the hat. In three dimensions the known candidates fall short: the Schmitt-Conway-Danzer biprism tiles only with screw motions, so its tilings are weakly aperiodic, and the three-dimensional Socolar-Taylor tile has a periodic stacking direction or is not simply connected. Socolar and Taylor asked in 2012 for a single simply connected three-dimensional prototile that forces nonperiodicity by shape alone. Does one exist?
- Result
- Proved(see note)
- Status
- Resolved
- AI contribution
- AI-discovered
- Method
- Construction
- Field
- Tiling theory; aperiodic order
- Posed by
- Joshua E. S. Socolar and Joan M. Taylor, Forcing nonperiodicity with a single tile, Math. Intelligencer 34 (2012), 18-28
- Year posed
- 2012
- Years open
- 14y
- Solved
- 2026-09-16
- Model
- GPT-6 Astra
- Vendor
- OpenAI
- Collaborators
- Ioannis Tsiokos
- Verification
- Lean-checked, statement unaudited
- Publication
- Preprint
- Significance
- 35 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What was actually shown
Claimed yes: Chair44 (R44), a rational polyhedral 3-ball built from seven cubes in a chair, whose 24 exposed unit panels carry small square-pyramid features. The claim is that it tiles by congruent copies with reflections allowed, that every such tiling has no translational period and a symmetry group of order at most 24, and that every tiling is homochiral and carries a unique infinite hierarchy of nested supertiles. The features force each tiling onto a registered lattice; the finite facts - dissection, the 44-contact atlas, 33 first shells, a 6,862 to 44 companion census with 299,975 collision boxes, mesh and angle audits - are enumerated exhaustively and checked in Lean, with a second independent Python implementation replaying them.
What the AI did
The submission records the model's contribution as finding the solid: GPT-6 Astra produced the Chair44 shape. The formalisation, the finite enumerations, the independent Python replay and the write-up are the author's.
Verification
Audited here on 27 September 2026 from a clone at HEAD. 85 Lean files, 29,815 lines on Lean 4.31.0. Zero real sorry - the single grep hit sits inside a comment - zero axiom declarations, zero unsafe or implemented_by. But thirty-five native_decide calls across twelve files, which the author discloses rather than hides: AXIOMS.md lists the named compiler hook per theorem, README carries a tiered trust ledger (T1 standard axioms, T1n additionally native hooks), the negative controls are diagnostic-checked, and verify/replay.py reconstructs the same finite facts in standard-library Python. The unconditional spine is r44_einstein: existence, no translational period, and symmetry group of order at most 24.
Recorded lean-checked rather than the submitted lean-verified, for two reasons. First, native_decide discharges goals through Lean's compiled evaluator, so the trusted base includes the compiler and runtime and not the kernel alone; the catalog's Dittert and composites-among-xi-7-n entries are labelled the same way for the same reason. Second, lean-verified additionally requires the formal statement to be anchored outside the prover's own repository - a comparator configuration, a Palomar entry or an audit by someone with no stake - and none is present. The repository's only CI workflow builds a reader notebook and is currently failing, so no automated build checks the Lean at all. A comparator or Palomar statement check plus a green Lean build would lift this to lean-verified.
Sources
Submitted by WittyWombat899 on