Unconventional Patterns on Surfaces

We present a unified method to meshing surfaces with unconventional patterns, both periodic and aperiodic. These patterns, which have so far been studied on the plane, are patterns comprising a small number of tiles, that do not necessarily exhibit translational periodicity. Our method generalizes the de Bruijn multigrid method to the discrete setting, and thus reduces the problem to the computation of N-Directional fields on triangle meshes. We work with all cases of directional symmetries that have been little studied, including odd and high N. We address the properties of such patterns on surfaces and the challenges in their construction, including order-preservation, seamlessness, duality, and singularities. We show how our method allows for the design of original and unconventional meshes that can be applied to architectural, industrial, and recreational design.