Finite Heesch number at least seven

Mathematics · Geometry and tilings Epoch AI

Finite Heesch number at least seven

Construct a plane shape with finite Heesch number at least seven: congruent copies must form at least seven complete successive surrounding layers, but the shape must not tile the whole plane.

Copies may be placed by any Euclidean isometry, including reflection; restricting copies to translations or rotations alone does not establish this target. Provide an exact geometric description and a finite patch certifying the seven layers. Establish non-tiling by a rigorous argument or checkable certificate. Excluding only isohedral tilings does not exclude every tiling, so state precisely what any computational classifier proves.

General plane shapes are allowed, including polyforms of any finite cell count or bounding-box size. New obstructions or mechanisms can qualify as partial advances.

Source: Epoch AI, A shape with Heesch number 7, FrontierMath: Open Problems.

Type

closed-ended

Solve status

open

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Public