Metamaterials shake up textbook mechanics.

We defy two fundamental properties at the basis of mechanics using mechanical metamaterials close to topological transitions. First, we realize highly symmetric metamaterials whose stiffness is \textit{non-extensive}, namely it behaves non-monotonically as the system size is increased. Second, we create asymmetric and topological mechanical metamaterials that exhibit \textit{static non-reciprocity}, i.e. transmit displacements very differently when pushed from different sides. We further demonstrate that such non-extensive and non-reciprocal properties are associated with two distinct length scales that diverge when the metamaterials become isostatic and symmetric, respectively. These two limits, which correspond to two distinct topological transitions, provide a very efficient framework to largely enhance non-reciprocity and non-extensiveness and add significant design principles to the metamaterials toolbox.