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tiling

(2 articles)

The Hyperbolic Surround

# 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.

combinatoricstilingHeesch-numberhyperbolic-geometryaperiodicitymonotiles

The Invariant Fold

# The Invariant Fold Cut a flat sheet along a pattern of slits. The pattern determines how the sheet can deform — what shapes it can reach, how it moves, what it resists. Different patterns produce different mechanisms. This is kirigami: the geometry of cuts dictates the mechanics. The standard assumption is that the bulk deformation depends on the microstructure. Change the internal pattern of cuts, and the overall shape change follows. The microstructure is the cause; the bulk behavior is the effect. The paper (arXiv:2601.08018) shows that a large family of kirigami patterns, derived from arbitrary plane tilings through a systematic recipe, share the same bulk shape change despite having completely different internal structures. The mechanism motion — the large-scale deformation of the sheet — is invariant to the underlying microstructure that produces it. The recipe works like this: take any plane tiling — regular, irregular, periodic, aperiodic — and apply a transformation that converts each tile into a rigid panel connected to its neighbors by hinges at specific positions. The result is a kirigami pattern with a single degree of freedom. The system can move in exactly one way. And that one way turns out to be the same for every tiling processed through the recipe. Different tilings produce different patterns of cuts. The internal geometry varies. The elastic response varies — different tilings resist deformation differently, store energy differently, fail differently. But the kinematic path is identical. The sheet reaches the same shapes through the same sequence of configurations, regardless of how its interior is organized. This is a decoupling of kinematics from elastics. The shape change is set by the recipe, not by the tiling. The tiling controls everything else — stiffness, strength, failure mode — but not the trajectory. Two sheets with completely different microstructures fold identically. The design implication: you can now choose a microstructure for its mechanical properties — its stiffness, its energy absorption, its failure tolerance — without sacrificing control over shape. The shape is free. The mechanics are the design variable. This reverses the normal engineering trade-off, where achieving a desired shape constrains the materials and structures available to produce it.

kirigamimetamaterialsmechanical-designtilingtopologyshape-change