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In this paper, we study deployment patterns to achieve full coverage and k-connectivity (k ≤ 6) under different ratios of the sensor communication range (denoted by Rc) to the sensing range (denoted by Rs) for homogeneous wireless sensor networks (WSNs). In particular, we propose new patterns for 3- and 5-connectivity. We also discover that there exists a hexagon-based universally elemental pattern that can generate all known optimal patterns. The previously proposed Voronoi-based approach cannot be applied to prove the optimality of the new patterns due to their special features. We propose a new deployment-polygon-based methodology. We prove the optimality of deployment patterns to achieve 3-connectivity, 4-connectivity, and 5-connectivity for certain ranges of Rc/Rs, respectively, and prove the optimality of deployment patterns to achieve 6-connectivity under all ranges of Rc/Rs.