  The localization of waves in two-dimensional clusters of scatterers arranged
in relatively twisted lattices is studied by multiple scattering theory. It is
found that, for a given frequency, it is always possible to find localized
modes for a discrete set of rotation angles, analogous to the so-called "magic
angles" recently found in two-dimensional materials like graphene. Similarly,
for small rotations of the lattices, a large number of resonant frequencies is
found, whose position strongly depends on the rotation angle. Moreover, for
angles close to those that make the two lattices commensurable a single mode
appears that can be easily tuned by the rotation angle. Unlike other twisted
materials, where the properties of the bilayers are mainly explained in terms
of the dispersion relation of the individual lattices, the special angles in
these clusters happen because of the formation of dipolar scatterers due to the
relative rotation between the two lattices, enhancing therefore their
interaction. While the presented results are valid for any type of wave, the
specific case of flexural waves in thin elastic plates is numerically studied,
and the different modes found are comprehensively explained in terms of the
interaction between pairs of scatterers. The analysis presented here shows that
these structures are promising candidates for the inverse design of tunable
wave-trapping devices for classical waves.
