In materials science, the notion of auxetic behavior relies on elasticity theory and is an expression 
of negative Poisson's ratios. Simply phrased, auxetic behavior means becoming laterally wider when stretched and 
thinner when compressed. We show that, for periodic framework structures, a purely geometric theory of auxetic 
one-parameter deformations can be based on the evolution of the periodicity lattice. A deformation path will be auxetic 
when the Gram matrix for a basis of periods gives a curve with all tangents in the positive semidefinite cone,  
analogous to a causal line in special relativity. We review, from this new perspective, several examples of
auxetic behavior in phase transitions of crystalline materials. Precise relations between expansive and auxetic
trajectories are explained in dimension two.