Universal response of two-dimensional flexible structures

Nature and Humanity have both discovered that simple structural motifs can generate strong yet flexible structures that offer a wide range of mechanical phenomenology. At the same time, subtle interactions can have disastrous consequences, from collapsed bridges to misfolded proteins. I will discuss recent progress towards developing unifying principles governing the complex mechanical response of flexible structures. In contrast with a conventional solid with a unique shape, such a structure possesses a rich family of ground states defined by the mathematical compatibility conditions that permit elements in different states to be joined together like jigsaw pieces. This approach captures the real-world response of various systems, including origami sheets and planar mechanical metamaterials, with differing symmetry, coordination (bond) number, symmetry and dimensionality. These give rise to a rich set of phenomena, including mechanical dualities, holographic principles, exceptional points and topologically protected states.