Random tilings of compact Euclidean 3-manifolds

Abstract

Deterministic and random tilings for the ten compact Euclidean 3-manifolds are introduced. The main tools are substitution rules generating non-periodic planar patterns and a set of pre-axioms defined for each manifold. The sets of random tilings are obtained by suitable tile flips in the substitution atlas which allows us to compute their configurational entropies. The inflation rules in two dimensions together with one-dimensional substitutions in a perpendicular direction induce non-periodic three-dimensional tilings by triangular prisms which can be transformed into simplicial structures.