In the Euclidean plane, making a hole in a polyform — a connected figure assembled from regular polygons — costs tiles. You need to build a boundary around the emptiness. The minimum number of tiles required to enclose h holes grows predictably because flat space forces you to invest material in every boundary.
In hyperbolic space, holes come cheaper.
Eldridge, Prabha, Roger, Roger, Roldan, and Toala-Enriquez study polyforms assembled from regular polygons in hyperbolic tessellations and ask the optimization question: what is g_{p,q}(h), the minimum number of tiles needed to realize exactly h holes? The answer depends on the tessellation parameters, but the qualitative finding is that negative curvature subsidizes topological complexity. The exponential growth of area with radius in hyperbolic space means that enclosing a hole requires fewer tiles relative to the surrounding structure. The ambient geometry is doing work that, in flat space, the tiles have to do themselves.
This is not a metaphor. It is a direct consequence of the Gauss-Bonnet theorem. The total curvature integral over a surface with boundary relates the number of boundary components, the vertex angles, and the Gaussian curvature of the ambient space. In hyperbolic geometry, the intrinsic negative curvature contributes a negative term that relaxes the topological constraints on the boundary. Each hole that would require extensive tile investment in flat space can be realized with fewer tiles because the curvature is already doing some of the boundary's geometric work.
The result inverts the usual relationship between complexity and cost. In Euclidean polyforms, topology is expensive — you pay for every handle, every hole, every departure from simple connectivity. In hyperbolic polyforms, the geometry makes topology affordable. The environment provides structural subsidy for complexity.
This distinction matters beyond recreational mathematics. Any system where structure is assembled from discrete units in a curved ambient space — molecular assemblies on curved membranes, network topologies on non-flat manifolds — inherits the same cost structure. The geometry of the embedding space doesn't just contain the topology. It sets its price.