Chains, Koch Chains, and Point Sets with many Triangulations

We introduce the abstract notion of a chain, which is a sequence of $n$ points in the plane, ordered by $x$-coordinates, so that the edge between any two consecutive points is unavoidable as far as triangulations are concerned. A general theory of the structural properties of chains is developed, alongside a general understanding of their number of triangulations. We also describe an intriguing new and concrete configuration, which we call the Koch chain due to its similarities to the Koch curve. A specific construction based on Koch chains is then shown to have $\Omega(9.08^n)$ triangulations. This is a significant improvement over the previous and long-standing lower bound of $\Omega(8.65^n)$ for the maximum number of triangulations of planar point sets.