Tilings whose members have finitely many neighbors
The article studies plane tilings in which each tile meets at most finitely many other tiles. Let S be the set of singular points in a tiling and let W denote the set of boundary points which belong to ''no nondegenerate'' edge of the tiling. It is shown that exactly one of the following must occur: Either W is uncountable and so is S; or both are empty. Furthermore it is proven that the closure of W is exactly S. The inclusion \(S\subseteq cl W\) also holds but in general \(S\varsubsetneq W\), though \(S\supseteq W\) is always true.
tiling by closed topological discs in the Euclidean plane
scientific article; zbMATH DE number 3937955