Chains, Koch Chains, and Point Sets with many Triangulations

03/15/2022

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by   Daniel Rutschmann, et al.

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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 Ω(9.08^n) triangulations. This is a significant improvement over the previous and long-standing lower bound of Ω(8.65^n) for the maximum number of triangulations of planar point sets.