Tiling by incongruent equilateral triangles without requiring local finiteness
A family \(\mathcal{T}=\{T_i\;|\;i\in I\}\) of subsets of \(\mathbb{R}^2\) is called a \textit{tiling} of a certain set \(S\subseteq\mathbb{R}^2\) provided \(S=\bigcup_{i\in I}T_i\) and the interiors of any two different elements of \(\mathcal{T}\) are disjoint. A tiling will be called \textit{perfect} if, for any different \(T_i\) and \(T_j\), they are similar but incongruent. This paper deals with the construction of perfect tilings for certain sets \(S\subseteq\mathbb{R}^2\), where the elements of the family \(\mathcal{T}\) are equilateral triangles. In particular, it is proved that every open subset of \(\mathbb{R}^2\) admits one such tiling.
scientific article; zbMATH DE number 6144255