This lattice demonstrates how a solid might expand in the direction perpendicular to the one in which it is being pulled apart, rather than contract like a typical solid would.
The Poisson ratio characterizes how much a solid expands in one direction as it is compressed in another. Therefore, a material with a negative Poisson ratio will expand in one dimension as it expands in another. You can imagine pulling apart the left- and right-most arms to expand the interior of the lattice shown. Once the lattice cells reach a rectangular configuration, they can invert their Poisson ratio and adopt the hexagonal configuration shown below. If this lattice were compressed, it would expand in the perpendicular direction like a typical solid.
Modeled, rendered, and 3D printed using Autodesk's Fusion 360 software.