  We revisit the classic problem of the effective diffusion constant of a
Brownian particle in a square lattice of reflecting impenetrable hard disks.
This diffusion constant is also related to the effective conductivity of
non-conducting and infinitely conductive disks in the same geometry. We show
how a recently derived Green's function for the periodic lattice can be
exploited to derive a series expansion of the diffusion constant in terms of
the disk's volume fraction $\varphi$. Secondly we propose a variant of the
Fick-Jacobs approximation to study the large volume fraction limit. This
combination of analytical results is shown to describe the behavior of the
diffusion constant for all volume fractions.
