On the Number of Lattice Triangulations

For n a positive integer, we consider triangulations of the n × n lattice set straight line embedded geometric graphs on this point set—thus with (n + 1) 2 vertices, 3n 2 + 2n edges and 2n 2 triangular faces. Figure 1: A triangulation of the 20 × 20 lattice. Extending a previous argument by Emile Anclin [1], we show that the number of triangulations of the n × n lattice is at most O(6.86 n 2), improving on the previous bounds of O(64 n 2) and O(8 n 2) in [4] and [1], respectively. It compares to a lower bound of Ω(4.15 n) given in [2].