Particles that are embedded in fluid membranes or plates can induce bending if they impose a nonzero angle of contact. This bending mediates complicated effective particle-particle interactions. In the absence of tension, these interactions are nonpairwise additive and can result in clusters of particles with specific configurations that give rise to zero total membrane bending energy. Here, we consider an infinite periodic lattice of such membrane inclusions. Upon summing the nonpairwise interactions within a regular lattice, we find an unexpected infinite number of periodic lattices that preserve zero membrane bending energy. Elliptically shaped membrane inclusions further increase the phase space of this degeneracy.