  Discretizations of the Laplacian operator on non-hypercubical lattices are
discussed in a systematic approach. It is shown that order $a^2$ errors always
exist for discretizations involving only nearest neighbors. Among all lattices
with the same density of lattice sites, the hypercubical lattices always have
errors smaller than other lattices with the same number of spacetime
dimensions. On the other hand, the four dimensional checkerboard lattice (also
known as the Celmaster lattice) is the only lattice which is isotropic at order
$a^2$. Explicit forms of the discretized Laplacian operators on root lattices
are presented, and different ways of eliminating order $a^2$ errors are
discussed.
