Triangle Percolation on the Grid

We consider a geometric percolation process partially motivated by recent work of Hejda and Kala. Specifically, we start with an initial set $X\subseteq {\mathbb{Z}}^2$, and then iteratively check whether there exists a triangle $T\subseteq {\mathbb{R}}^2$ with its vertices in ${\mathbb{Z}}^2$ such that $T$ contains exactly four points of ${\mathbb{Z}}^2$ and exactly three points of $X$. In this case, we add the missing lattice point of $T$ to $X$, and we repeat until no such triangle exists. We study the limit sets $S$, the sets stable under this process, including determining their possible densities and some of their structure.