Tiling by incongruent equilateral triangles without requiring local finiteness. II
A family \(\mathcal{T}=\{T_i \mid i\in I\}\) of subsets of \(\mathbb{R}^2\) is called a 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 \(T_i\) are disjoint. A tiling will be called perfect if, for any different \(T_i\) and \(T_j\), they are similar but incongruent. In a previous work by the author [Elem. Math. 67, No. 4, 157--163 (2012; Zbl 1264.51009)], it was proved that every open subset of \(\mathbb{R}^2\) admits a perfect tiling. In this work, it is proved that the following statements are equivalent for any subset \(S\subseteq \mathbb{R}^2\): {\parindent=6mm\begin{itemize}\item[--] \(S\) admits a perfect tiling by equilateral triangles. \item[--] \(S\) admits a tiling by equilateral triangles. \item[--] There exists a family of non-overlapping equilateral triangles \(E_i\subseteq S\) that covers \(S\setminus \text{int}(S)\). \end{itemize}} Some applications are given in order to characterize which polygons and closed convex sets satisfy the previous equivalent conditions.
scientific article; zbMATH DE number 6881800