  We present a study of the shear modulus of the crystalline color
superconducting phase of quark matter, showing that this phase of dense, but
not asymptotically dense, quark matter responds to shear stress as a very rigid
solid. This phase is characterized by a gap parameter $\Delta$ that is
periodically modulated in space and therefore spontaneously breaks
translational invariance. We derive the effective action for the phonon fields
that describe space- and time-dependent fluctuations of the crystal structure
formed by $\Delta$, and obtain the shear modulus from the coefficients of the
spatial derivative terms. Within a Ginzburg-Landau approximation, we find shear
moduli which are 20 to 1000 times larger than those of neutron star crusts.
This phase of matter is thus more rigid than any known material in the
universe, but at the same time the crystalline color superconducting phase is
also superfluid. These properties raise the possibility that the presence of
this phase within neutron stars may have distinct implications for their
phenomenology. For example, (some) pulsar glitches may originate in crystalline
superconducting neutron star cores.
