  Cellular checkerboard patterns are observed at many developmental stages of
embryos. We study an analytically tractable model for lateral inhibition and
show that a coupling coefficient with a negative value is sufficient to obtain
noisy or periodic checkerboard patterns. We solve the case of a linear chain of
cells explicitly and show that noisy anti-correlated patterns are available in
a post-critical regime $(\epsilon_c < \epsilon < 0)$. In the sub-critical
regime $(-\infty < \epsilon \leq \epsilon_c)$ a periodic and alternating steady
state is available, where pattern selection is determined by making an analogy
with the optical modes of phonons. For cells arranged in a hexagonal lattice,
the sub-critical pattern can be driven into three different states: two of
those states are periodic checkerboards and a third in which both periodic
states coexist.
