Triangulations are important objects of study in combinatorics, finite element simulations and quantum gravity, where its entropy is crucial for many physical properties. Due to their inherent complex topological structure even the number of possible triangulations is unknown for large systems. We present a novel algorithm for an approximate enumeration which is based on calculations of the density of states using the Wang-Landau flat histogram sampling. For triangulations on two-dimensional integer lattices we achive excellent agreement with known exact numbers of small triangulations as well as an improvement of analytical calculated asymptotics. The entropy density is C = 2.196(3) consistent with rigorous upper and lower bounds. The presented numerical scheme can easily be applied to other counting and optimization problems.Introduction. -Triangulations of spaces are relevant for a broad range of physical phenomena. They serve as discretisation of all kinds of surfaces, hypersurfaces and volumes [1], yielding applications of computational geometry in physics, material science, medical image processing or even in computer graphics and visualisation [2][3][4][5][6]. Many physical systems can be described by random surface models [7] -based on random triangulations. For instance, biological membranes and vesicles can be modelled using triangulated surfaces with curvature-dependent Hamiltonians [8][9][10][11][12][13].Triangulations are also used as a random graph model for real world networks: Random Apollonian networks [14][15][16], which are the dual graphs of classical Apollonian packed granular matter and therewith triangulations, show both small-world and scale-free behaviour, as many real world networks. The properties of triangulations of closed surfaces with arbitrary genus are of much interest, since each graph can be embedded into a closed surface with high enough genus [17,18].The (Causal) Dynamical Triangulation approach even tries to describe quantum gravity from scratch with an ensemble of random space-time triangulations as their central entity [19]. Based on a discrete version of general relativity, where spacetime is approximated by triangles or higher-dimensional analogues, the curvatures become determined purly by the topological structure of the underlying triangulation, e.g. the number of triangles. The resulting action of the theory can be used to extract a
