Distances in domino flip graphs

TL;DR

This paper explores the measurement and visualization of distances between domino tilings, providing formulas for the diameters of flip graphs for specific shapes like rectangles and Aztec diamonds.

Contribution

It offers geometric interpretations and formulas for the diameters of flip graphs in domino tilings, extending prior work by Thurston and others.

Findings

Formulas for diameters of flip graphs of rectangles and Aztec diamonds

Geometric interpretations of domino tiling distances

Enhanced understanding of tiling flip graph structures

Abstract

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.