The Hyperbolic Surround

The Heesch number of a tile is the maximum number of complete layers of copies that can surround it without tiling the entire plane. A tile with Heesch number zero cannot even complete a first corona. A tile with Heesch number k admits k layers of surrounding copies before the process jams. If the Heesch number is infinite, the tile can tile. The question, open since Heesch posed it: can the Heesch number be arbitrarily large? Can a convex tile surround itself with a hundred layers, a thousand, without ever being able to tile?

In the Euclidean plane, the largest known Heesch number for a convex tile is 1 — and finding tiles with higher values has resisted decades of effort. The problem has the flavor of an impossibility that nobody can prove.

The authors of arXiv:2603.27827 (March 2026) resolve the problem in the hyperbolic plane: the Heesch number for convex monotiles in hyperbolic geometry is unbounded. For any positive integer k, there exists a convex tile in the hyperbolic plane that admits exactly k surrounding layers but cannot tile. As a corollary, the duals of homogeneous tilings produce the first known weakly aperiodic convex monotiles — single convex tiles that can fill the hyperbolic plane but only non-periodically.

The hyperbolic plane separates cleanly from the Euclidean case. In flat geometry, the rigidity of Euclidean isometries severely constrains how tiles can fit together, and the constraint appears to prevent high Heesch numbers for convex tiles. In hyperbolic geometry, the exponential growth of area with distance provides enough room for more layers to form before the combinatorial obstruction to tiling takes effect. The curvature of space is the structural variable that unlocks unbounded surrounding.

The structural observation: a question that appears intractable in flat space has a definitive answer in curved space. The Euclidean and hyperbolic Heesch problems are the same question asked in different geometries, and the geometry changes the answer qualitatively — from "probably bounded" to "provably unbounded." The obstacle was not in the combinatorics of tiling but in the geometry of the space.