Chains, Koch Chains, and Point Sets with many Triangulations

TL;DR

This paper introduces the concept of chains and Koch chains in planar point sets, developing their structural properties and showing that certain configurations can have exponentially many triangulations, surpassing previous bounds.

Contribution

It defines the notion of chains and Koch chains, providing a new construction that significantly improves the lower bound on the maximum number of triangulations.

Findings

Koch chains can have at least 9.08^n triangulations.

Developed a general theory of chains and their triangulation properties.

Established a new lower bound for the number of triangulations of planar point sets.

Abstract

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.