Bobbin lace is a 500-year-old fiber art form created by braiding together fine threads. In its design, lacemakers employ doubly periodic textures to create contrast and interest in a predominantly monochromatic fabric. In the past we have created a model for these doubly periodic patterns which employs graph drawings to describe the flow of threads. In this paper, we demonstrate that these graph drawings, which we call `tesselace patterns', exist for each of the 17 planar periodic symmetry groups. We provide an algorithm for exhaustively generating patterns with a particular symmetry on a grid of fixed size. We also explore the symmetry of the interlaced fabric resulting from these patterns.