In this paper, we propose deployment patterns to achieve full coverage and three-connectivity and full coverage and five-connectivity under different ratios of sensor communication range (denoted by R) over sensing range (denoted by R) for wireless sensor networks (WSNs). We also discover that there exists a hexagon-based universally elemental pattern which can generate all known optimal patterns. The previously proposed Voronoi-based approach can not be applied to prove the optimality of the new patterns due to their special features. We propose a new deployment-polygon based methodology, and prove their optimality among regular patterns when R/R ≥ 1. We conjecture that our patterns are globally optimal to achieve full coverage and three-connectivity, and full coverage and five-connectivity, under all ranges of R/R. With these new results, the set of optimal patterns to achieve full coverage and κ-connectivity (κ ≤ 6) is complete, for the first time. Copyright 2008 ACM.