Abstract : This work is motivated by the need for software computing 3D periodic triangulations in numerous domains including astronomy, material engineering, biomedical computing, ﬂuid dynamics etc. We design an algorithmic test to check whether a partition of the 3D ﬂat torus into tetrahedra forms a triangulation (which subsumes that it is a simplicial complex). We propose an incremental algorithm that computes the Delaunay triangulation of a set of points in the 3D ﬂat torus without duplicating any point, whenever possible; our algorithmic test detects when such a duplication can be avoided, which is usually possible in practical situations. Even in cases where point duplication is necessary, our algorithm always computes a triangulation that is homeomorpic to the ﬂat torus. To the best of our knowledge, this is the ﬁrst algorithm of this kind whose output is provably correct. Proved algorithms found in the literature are in fact always computing with 27 copies of the input points in R3 , and yield a triangulation that does not have the topology of a torus. Our implementation of the algorithm has been reviewed and accepted by the Cgal Editorial Board. A video of the work was presented at SoCG'08.