  The interplay of hopping parameters that can give rise to flat bands in
consequence of quantum interference in electronic, photonic, and other
interesting materials has become an extensively studied topic. Most of the
recognized structures having flat bands are the lattices that can be understood
by the mathematical theory of line graphs, such as the Lieb, kagome, and
checkerboard lattices. Here, we demonstrate that the structures that can
realize the same kind of flat bands given by those well-known lattices hosting
exotic quantum phases are more flexible. The flat bands belonging to the
recognized structures can be ideally embedded into the new structures that
cannot be considered as the original ones in terms of a unitary transformation.
The uncovered mechanism enriches the understanding of physics behind the
localized quantum states and broadens the choice of materials that can be used
for designing electronic and photonic devices from the zero band dispersion.
