The Holey Tiling

A polyform is a figure made by gluing regular polygons edge-to-edge. Polyominoes (squares), polyhexes (hexagons), and polyiamonds (triangles) are familiar examples in the Euclidean plane. In hyperbolic tilings — regular {p,q}-tessellations with p-gons meeting q at a vertex — polyforms gain new topology.

In the Euclidean plane, a polyform with h holes needs at least a specific number of tiles, and the bound is determined by the perimeter-to-area trade-off of the tiling. In hyperbolic space, the same extremal question has a different answer because curvature changes the trade-off. The exponential growth of area with radius in hyperbolic space means that creating holes is geometrically cheaper — there’s more room for internal boundaries.

The paper establishes general lower and upper bounds on the minimum number of tiles needed to realize exactly h holes in a {p,q}-tessellation. For small values of h, they compute exact answers. There exists a structural condition — a combinatorial criterion on how the polyform’s boundary components interact — that determines when the minimum is achieved.

The problem is extremal in the classical sense: what is the smallest object with a given topological property? But the setting is non-classical. Euclidean polyform extremal problems are typically about area or perimeter, with topology as a consequence. Here, topology is the target and tile count is the cost. The curvature of the ambient space enters as a parameter that shifts the cost function.

The minimum number of tiles per hole decreases with increasing negative curvature. Hyperbolic space makes topology cheap.