Homogenization of complex lattices for metamaterials: open problems and conjectures

Based on concrete examples dealing with one-level periodic and quasi-periodic, and two- level periodic lattices, we show that the homogenization of complex lattices generically requires a multi-field continuum approach. We do not present results for specific cases, rather we discuss some related open problems and formulate pertinent conjectures. In this way, we suggest in essence a possible research program addressed towards the design of metamaterials.