Phase transition creates the geometry of the continuum from discrete space

Models of discrete space and space-time that exhibit continuum-like behavior at large lengths could have profound implications for physics. They may help tame the infinities arising from quantizing gravity, and remove the need for the machinery of the real numbers; a construct with no direct observational support. However, despite many attempts to build discrete space, researchers have failed to produce even the simplest geometries. Here we investigate graphs as the most elementary discrete models of two-dimensional space. We show that if space is discrete, it must be disordered, by proving that all planar lattice graphs exhibit a taxicab metric similar to square grids. We then give an explicit recipe for growing disordered discrete space by sampling a Boltzmann distribution of graphs at low temperature. Finally, we propose three conditions which any discrete model of Euclid's plane must meet: have a Hausdorff dimension of 2, support unique straight lines, and obey Pythagoras' theorem. Our model satisfies all three, resulting in a discrete model in which continuum-like behavior emerges at large lengths.