  Periodicity is usually assumed to be the necessary and sufficient condition
for the formation of band gaps, i.e., energy bands with a suppressed density of
states. Here, we check this premise by analyzing the band gap properties of
three structures that differ in the degree of periodicity and ordering. We
consider a photonic crystal, disordered lattice, and ordered but nonperiodic
quasicrystalline structure. A real-space metric allows us to compare the degree
of periodicity of these different structures. Using this metric, we reveal that
the disordered lattice and the ordered quasicrystal can be attributed to the
same group of material structures. We examine the density of their photonic
states both theoretically and experimentally. The analysis reveals that despite
their dramatically different degrees of periodicity, the photonic crystal and
the quasicrystalline structure demonstrate an almost similar suppression of the
density of states. Our results give new insight into the physical mechanisms
resulting in the formation of band gaps.
