  We demonstrate that photonic and phononic crystals consisting of closely
spaced inclusions constitute a versatile class of subwavelength metamaterials.
Intuitively, the voids and narrow gaps that characterise the crystal form an
interconnected network of Helmholtz-like resonators. We use this intuition to
argue that these continuous photonic (phononic) crystals are in fact
asymptotically equivalent, at low frequencies, to discrete capacitor-inductor
(mass-spring) networks whose lumped parameters we derive explicitly. The
crystals are tantamount to metamaterials as their entire acoustic branch, or
branches when the discrete analogue is polyatomic, is squeezed into a
subwavelength regime where the ratio of wavelength to period scales like the
ratio of period to gap width raised to the power 1/4; at yet larger wavelengths
we accordingly find a comparably large effective refractive index. The fully
analytical dispersion relations predicted by the discrete models yield
dispersion curves that agree with those from finite-element simulations of the
continuous crystals. The insight gained from the network approach is used to
show that, surprisingly, the continuum created by a closely packed hexagonal
lattice of cylinders is represented by a discrete honeycomb lattice. The
analogy is utilised to show that the hexagonal continuum lattice has a
Dirac-point degeneracy that is lifted in a controlled manner by specifying the
area of a symmetry-breaking defect.
