  This article is about measuring and visualizing distances between domino
tilings. Given two tilings of a simply connected square tiled surface, we're
interested in the minimum number of flips between two tilings. Given a certain
shape, 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 which
result in formulas for the diameters of the flip graphs of rectangles or Aztec
diamonds.
