Families of convex tilings

TL;DR

This paper explores continuous families of convex polygon tilings, demonstrating how tile shapes and areas vary, and linking tiling properties to bipartite graph structures and Kasteleyn matrices for geometric and probabilistic insights.

Contribution

It introduces a framework for understanding how convex polygon tilings can be continuously deformed, with new results on shape prescription and the relation between tiling parameters and underlying graph structures.

Findings

Tile shapes can be arbitrarily prescribed in convex tilings.

Tile areas and orientations determine the tiling uniquely.

Kasteleyn matrix relates to the differential of tile area maps.

Abstract

We study tilings of polygons $R$ with arbitrary convex polygonal tiles. Such tilings come in continuous families obtained by moving tile edges parallel to themselves (keeping edge directions fixed). We study how the tile shapes and areas change in these families. In particular we show that if $R$ is convex, the tile shapes can be arbitrarily prescribed (up to homothety). We also show that the tile areas and tile ``orientations'' determine the tiling. We associate to a tiling an underlying bipartite planar graph $G$ and its corresponding Kasteleyn matrix $K$. If $G$ has quadrilateral faces, we show that $K$ is the differential of the map from edge intercepts to tile areas, and extract some geometric and probabilistic consequences.