It is shown that gravitational lensing by point masses arranged in an
infinitely extended regular lattice can be studied analytically using the
Weierstrass functions. In particular, we draw the critical curves and the
caustic networks for the lenses arranged in regular-polygonal -- square,
equilateral triangle, regular hexagon -- grids. From this, the mean number of
positive parity images as a function of the average optical depth is derived
and compared to the case of the infinitely extended field of randomly
distributed lenses. We find that the high degree of the symmetry in the lattice
arrangement leads to a significant bias towards canceling of the shear caused
by the neighboring lenses on a given lens position and lensing behaviour that
is qualitatively distinct from the random star field. We also discuss some
possible connections to more realistic lensing scenarios.Comment: to appear in Monthly Notices of RAS, including 17 figs, 1 appendix.
High-res figs and F95 code used available upon reques