In this paper, we propose an efficient algorithm to compute the centroidal Voronoi tessellation in 2D periodic space. We first present a simple algorithm for constructing the periodic Voronoi diagram (PVD) from a Euclidean Voronoi diagram. The presented PVD algorithm considers only a small set of periodic copies of the input sites, which is more efficient than previous approaches requiring full copies of the sites (9 in 2D and 27 in 3D). The presented PVD algorithm is applied in a fast Newton-based framework for computing the centroidal Voronoi tessellation (CVT). We observe that full-hexagonal patterns can be obtained via periodic CVT optimization attributed to the convergence of the Newton-based CVT computation.