We study realizations of periodic graphs in Euclidean spaces with each vertex in the center of gravity of its neighbors. As a first application, we show that every planar, 3-connected, 2-periodic graph can be drawn into the plane with convex faces such that the drawing realizes every combinatorial automorphism of the graph as an isometric symmetry. This extends results by Thomassen and by Mani-Levitska, Guigas, and Klee.