Heesch’s problem: boundedness and spectrum of Heesch numbers

Determine whether the set of finite Heesch numbers realized by shapes (topological disks) in the Euclidean plane is bounded, and characterize precisely which positive integers occur as Heesch numbers.

Background

Heesch numbers measure how many concentric layers of copies of a shape can surround a central copy before failure, unless the shape tiles the plane (in which case the Heesch number is infinite). Known examples achieve finite values up to six, but there is no general theoretical bound or complete characterization.

The author explicitly identifies Heesch’s problem as unsolved, underscoring the lack of understanding about whether arbitrarily large finite Heesch numbers exist or which integers can occur.