When a material is elongated or compressed along one axis, the Poisson's ratio, <math>\nu</math>, of a material is the ratio of the Strain in the directions perpendicular to the axis, divided by the Strain along that axis. Essentially, this dimensionless number describes the tendency of a contraction or expansion in one dimension to cause contraction or expansion in other dimensions. Most materials have a positive Poisson's ratio, meaning other dimensions will contract in response to elongation along one dimension, and vice versa. Figure 1 demonstrates this concept.
Most materials have a Poisson's ratio near 0.3, with more rubbery materials approaching 0.5.  Auxetic materials, or materials with a negative Poisson's ratio were first reported in 1987 by Lakes. Example materials of this type include some foams, honeycomb structures produced by Prall and Lakes, and patterned arrays of circular and elliptical holes in an elastomer (Bertoldi). In these last two cases, local buckling of the structure (i.e., repeatable collapse) leading to volume reduction is directly observed to cause a negative Poisson's ratio. Figure 1 shows a schematic of local buckling in Bertoldi's material. The material compresses uniformly until a critical applied strain is reached. At this strain, an elastic instability develops, and the structure experiences uniform, repeatable local buckling of the walls. Figures 3 and 4 show Prall and Lakes' honeycomb structures. Figure 3 shows a "regular" honeycomb shape, with a positive Poisson's ratio, and a "reentrant" honeycomb shape, with a negative Poisson's ratio. Figure 4 shows the same groups chiral honey comb structures in a) a schematic and b) a photograph of the fabricated structure. 
For an isotropic, unconstrained three dimensional elastic material, Poisson's ratio must range from -1 to 1/2.
 '"Negative Poisson’s Ratio Behavior Induced by an Elastic Instability". Katia Bertoldi, Pedro M. Reis, Stephen Willshaw, and Tom Mullin. Adv. Mater. 2009, 21, 1–6.
 "Properties of a chiral honeycomb with a Poisson's ratio -1" D. Prall, R. S. Lakes. Int. J. of Mechanical Sciences, 39, 305-314, (1996)