  We examine a fundamental material property called Poisson's ratio, which
establishes the relationship for the relative deformation of a physical system
in orthogonal directions. Architects and engineers have designed advanced
systems using repeating patterns that can potentially exhibit auxetic behavior,
which is the property of having a negative Poisson's ratio. Because
two-dimensional cross sections of each of these patterns has an associated
wallpaper group, we can look for useful correlations between this geometric
information and the two-dimensional response as defined by Poisson's ratio. By
analyzing the data, we find two properties of the wallpaper group that
correlate with more effective Poisson's ratio required for applications. This
paper also contains an introduction to wallpaper groups and orbifold notation
and an appendix contains some literature references recreated to use our
preferred notation.
