Spaces of domino tilings
type: Item
instance of: scholarly article
title: Spaces of domino tilings
zbMATH Open document ID: 0830.05020
DOI: 10.1007/BF02570703
author: Nicolau C. Saldanha
author: M. A. jun. Casarin
author: D. Romualdo
author: Carlos Tomei
published in: Discrete \& Computational Geometry
publication date: 1996-01-28T00:00:00Z
full work available at URL: https://eudml.org/doc/131399
review text: The authors consider the set of all tilings by dominoes (\(2\times 1\) rectangles) of a surface, possibly with boundary, consisting of unit squares. Convert this set into a graph by joining two tilings by an edge if they differ by a flip, i.e., a \(90^\circ\) rotation of a pair of side by side dominoes. Tilings of a rectangle of integral sides were counted by Kasteleyn. More recently Lieb and Loss, showed how to count tilings of general regions by making use of determinants. Conway and Lagarias studied the problem of tiling subsets of \(\mathbb{R}^2\) with a given set of tiles, by group-theoretical techniques. Thurston adopted these techniques to study domino tilings, producing a necessary and sufficient condition for a simply connected region of the plane to be tiled by dominoes. Here, the authors are interested in studying \(T\), the set of all possible tilings of a fixed region. Given a tiling, they perform a ``flip'' by lifting two dominoes and placing them back in a different position. Clearly the two dominoes must form a square of side 2. Two tilings are adjacent in \(T\) if we move from one to the other by a flip. Turn \(T\) into a graph by joining adjacent tilings by edges, and define connected components of \(T\) and distance between tilings in the usual way. The authors obtain a very operational criterion for two tilings to belong to the same connected components of \(T\), giving simple formulas for distances, and a method to construct geodesics in this graph. If the region is simply connected, \(T\) is connected. By naturally adjoining to this graph higher-dimensional cells they obtain a CW-complex whose connected components are homotopically equivalent to points or circles. As a consequence for any region different from a torus or Klein bottle, all geodesics with common end points are equivalent in the following sense. Build a graph whose vertices are these geodesics, adjacent if they differ only by the order of two flips on disjoint squares. This graph …
reviewed by: Ratnakaram Nava Mohan
Mathematics Subject Classification ID: 05B45
zbMATH DE Number: 784602
zbMATH Keywords: tilings
zbMATH Keywords: dominoes
zbMATH Keywords: rectangle
zbMATH Keywords: connected region
zbMATH Keywords: flip
zbMATH Keywords: distances
zbMATH Keywords: geodesics
zbMATH Keywords: CW-complex
zbMATH Keywords: torus
zbMATH Keywords: Klein bottle
zbMATH Keywords: metric structure
zbMATH Keywords: shortest routes
zbMATH Keywords: components
MaRDI profile type: Publication
cites work: Tiling with polyominoes and combinatorial group theory
cites work: The Problem of the Calissons
cites work: On the determinant of the adjacency matrix for a planar sublattice
cites work: The statistics of dimers on a lattice. I: The number of dimer arrangements on a quadratic lattice
cites work: Fluxes, Laplacians, and Kasteleyn's theorem
cites work: Q5522742
cites work: Q5730918
cites work: Conway's Tiling Groups
cites work: Q4192669
description: scientific article; zbMATH DE number 784602