In condensed-matter physics, a Nambu-Goldstone mode is a particular low energy deformation of an ordered phase arising from a symmetry of the ground state. Examples are acoustic phonons and spin waves. In this paper, I use the fact that an isometry is a symmetry of the stretching energy of a shell to derive a set of low energy Goldstone-like deformations. This new approach to understanding the small deformations of shells is then used to revisit the problem of a pinched cylinder and used to study a pinched cone.

C.D. Santangelo, “Nambu-Goldstone modes and diffuse deformations in elastic shells,” to appear in Soft Matter (2013). [ARXIV]

We develop equations that describe the three-dimensional shape of a sheet upon which a series of concentric, curved folds have been inscribed. When there is no stretching outside the fold, the three-dimensional shape of a single fold prescribes the shape of the entire origami structure. To better explore these structures and their mechanical properties, we develop a description of the smooth surface intersecting all the mountain folds. We find that this surface is always saddle-shaped. A series of open, circular folds (as seen on the right figure) with constant fold angle generate a classical minimal surface known as the helicoid.

M.A. Dias and C.D. Santangelo, “The shape and mechanics of curved fold origami structures,” EPL 100, 54005 (2012). [JOURNAL] [ARXIV].