Maxwell's equations describe the classical electromagnetic properties of all systems, including metamaterials, which are
periodic and highly inhomogeneous. In studies of metamaterials, however, one typically further assumes that their low-frequency
properties are described by Maxwell's equations in an equivalent homogenous medium. Hence, tremendous
recent efforts have focused on discovering structures with unusual properties in electrical permittivity and magnetic
permeability tensors. Here we offer an alternative viewpoint, by designing three-dimensional metamaterials, which
may be best described by effective uniform media that is non-Maxwellian. In the low-frequency limit, these
metamaterials support multi-component effective fields, with the numbers of field-components designable by geometry.
Our work indicates that the physics of metamaterials is far richer than previously anticipated. In particular, new effective
low-energy theory with high symmetry can emerge from topological complexity alone.