We discuss, in precise geometric language, the notions of expansive and auxetic deformations for

periodic bar-and-joint frameworks. Expansive implies auxetic. In dimension two, expansive periodic structures

can be understood in terms of pseudo-triangulations. From this characterization we derive an effective procedure to generate infinitely many planar periodic expansive designs. We also present a few principles to generate infinite auxetic families in higher dimensions. We illustrate these techniques with an abundance of new auxetic blueprints in dimension three, which is most important for metamaterial design.

This is joint work with Ciprian Borcea.