We explore the following percolation process. Let X be a subset of the grid Z^2, and let T either be a grid triangle with exactly one interior grid point and no grid points on the boundary other than the vertices (T is called an internal triangle), or a triangle with exactly one extra grid point on the boundary, but no interior grid point (T is called a border triangle). Whenever there is a triangle T with exactly three points in X, we add the fourth point of T to X.
We answer questions of the following flavor for both kinds of triangles.
Which sets X will eventually expand to the whole plane?
Which sets X are stable, i.e., they do not contain such triangles with exactly 3 points?
What is the maximum density of a stable set X that is a strict subset of Z^2? What are the possible densities?
This work, joint with Bryce Frederickson, Bob Krueger, Bernard Lidický, Tyrrell McAllister, Florian Pfender, Sam Spiro, and Eric Nathan Stucky, was initiated at the Graduate Research Workshop in Combinatorics (GRWC 2022).
