We study geometric frustration in two-dimensional lattice-based mechanical metamaterials comprised

of anisotropic triangular building blocks T, where each one possesses a nontrivial floppy mode of deformation. When

each triangle is oriented randomly neighboring triangles typically cannot deform self-consistently. On the one hand, we

analyze the conditions under which a non-periodic packing of these blocks form compatible, frustration-free largescale

structures, i.e., structures that exhibit a global floppy mode that is compatible with the local deformations of each

T. By mapping to an antiferromagnetic Ising model, we find an extensive number of possibilities to construct a

compatible structure: Ω₀ ~ exp(T). On the other hand, we study incompatible metamaterials in detail and we reveal two

distinct types of source of frustration (defects) which either highly localize the frustrated region to a small and finite

domain (local defects) or cause delocalized and long-ranged multi-stable conflicts (topological defects) whose multistability

scales as Ω ~ exp(T^1/2).