A construction system for mechanical metamaterials based on discrete assembly of a finite set of modular, mass-produced parts. A modular construction scheme enables a range of mechanical metamaterial properties to be achieved, including rigid, compliant, auxetic and chiral, all of which are assembled with a consistent process across part types, thereby expanding the functionality and accessibility of this approach. The incremental nature of discrete assembly enables mechanical metamaterials to be produced efficiently and at low cost, beyond the scale of the 3D printer. Additionally, a lattice structure constructed of two or more rigid, compliant, auxetic and chiral part types enable the creation of heterogenous macroscopic metamaterial structures.