Distances in Domino Flip Graphs

This article is about measuring and visualizing distances between domino tilings. Given two tilings of a simply-connected square-tiled surface, their distance is the minimum number of flips between them. For a given shape of surface, we're interested in computing the diameters of the flip graphs, meaning the maximal distance between any two of its tilings. Building on work of Thurston and others, we give geometric interpretations of distances that result in formulas for the diameters of the flip graphs of rectangles or Aztec diamonds.