Arbitrary pattern formation on infinite regular tessellation graphs

Given a set R of robots, each one located at different vertex an infinite regular tessellation graph, we aim to explore the Arbitrary Pattern Formation (APF) problem. multiset F grid vertices such that |R|=|F|, APF asks for distributed algorithm moves robots so as reach configuration similar F. Similarity means must be disposed regardless translations, rotations, reflections. So far, possible graph discretizing Euclidean plane only standard square has been considered in context classical OBLOT model. However, it is natural consider also other graphs, are triangular and hexagonal grids. In particular, former can most general terms symmetries trajectories. We provide resolution when initial asymmetric topology any graph. The its correctness based on rigorous methodology.