On the Structure of Some Spaces of Tilings
Éric Rémila · SIAM Journal on Discrete Mathematics · 2002
We study the structure of the set of tilings of a polygon P with bars of fixed length. We obtain an undirected graph connecting two tilings if one can pass from one tile to the other one by a flip (i.e., a local replacement of tiles). Using algebraic tools (such as tiling groups and their quotients and subgroups), we give a formula to compute the distance in this graph (i.e., the minimal number of necessary flips) between two tilings. Moreover, we prove that, for each pair (T, T') of tilings, the set $\Upsilon_{T, T'}$ consisting of tilings which are in a path of minimal length from T to T' canonically has a structure of distributive lattice.