Symmetry-induced constraints on the isometries of thin periodic surfaces

Thin surfaces are ubiquitous in both natural and engineered systems, and understanding how they deform under external loads is important in the design of mechanical metamaterials. Because bending in thin structures costs far less energy than stretching, isometric deformations—those that preserve local distances—play a key role in determining their mechanical response. For generic surfaces, bending and stretching modes are intricately coupled and follow a symplectic relationship. However, many sheets possess spatial symmetries, such as mirror and rotational symmetry. Indeed, origami sheets such as the Miura ori depend on such symmetries in order to achieve folding modes. As we show, the presence of these same symmetries in a sheet leads to elegant geometric relationships between the different deformation modes of the sheet. For example, while a flat sheet has three linear bending modes the Miura ori origami pattern exhibits a purely planar deformation along with two bending modes. Moreover, the in-plane and associated out-of-plane modes have equal and opposite in-plane and out-of-plane Poisson's ratios, as is also widely observed in other origami metamaterials. In this work, we investigate how symmetries govern the existence, coupling, and decoupling of stretching and bending modes in thin surfaces, resolving how symmetry and inherent geometric mechanics of patterned sheets give rise to the opposite Poisson's ratios in many systems, and to other geometric phenomena. These findings establish a geometric framework for designing programmable, symmetry-protected thin surfaces with tunable flexibility and rigidity.