Three-dimensional odd elastic metamaterials

Non-reciprocity has received growing attention in out-of-equilibrium systems. However, achieving non-reciprocal behavior in three-dimensional active solids remains a major challenge. Here we present a systematic framework for designing three-dimensional odd elastic metamaterials. Our approach constructs various crystal structures from non-reciprocal springs and couples multiple floppy modes to achieve desired non-reciprocal behaviors. By manipulating the tessellation and fraction of active elements, we predict odd elastic moduli and experimentally observe coupled affine deformations. Furthermore, we demonstrate non-reciprocal compression-twisting responses that go beyond classic elasticity, extending the concept to odd micropolar elasticity. This framework paves the way for the rational design of three-dimensional active metamaterials and provides essential tools for developing adaptive and multimodal soft robotic systems.