  While "Dirac cone" dispersions can only be meaningfully defined in two
dimensional (2D) systems, the notion of a Dirac point can be extended to three
dimensional (3D) classical wave systems. We show that a simple cubic photonic
crystal composing of core-shell spheres exhibits a 3D Dirac point at the center
of the Brillouin zone at a finite frequency. Using effective medium theory, we
can map our structure to a zero refractive index material in which the
effective permittivity and permeability are simultaneously zero at the Dirac
point frequency. The Dirac point is six-fold degenerate and is formed by the
accidental degeneracy of electric dipole and magnetic dipole excitations, each
with three degrees of freedom. We found that 3D Dirac points at k=0 can also be
found in simple cubic acoustic wave crystals, but different from the case in
the photonic system, the 3D Dirac point in acoustic wave system is fourfold
degenerate, and is formed by the accidental degeneracy of dipole and monopole
excitations. Using effective medium theory, we can also describe this acoustic
wave system as a material which has both effective mass density and reciprocal
of bulk modulus equal to zero at the Dirac point frequency. For both the
photonic and phononic systems, a subset of the bands has linear dispersions
near the zone center, and they give rise to equi-frequency surfaces that are
spheres with radii proportional to (omega-omega_dirac).
