Departamento de Gravitación y Teoría de Campos

                          Seminarios 2010

  

                          Instituto de Ciencias Nucleares, UNAM
[ ICN-UNAM]

Jueves 7 de octubre, a las 17:30 hrs en el Auditorio Principal del ICN

Jemal Guven (ICN, UNAM)

Defectos en papel

When a surface bends, it tends also to stretch. The latter will depend on the material properties of the surface which would explain why physicists abandoned interest in the problem over a hundred years ago, leaving it in the safe hands of engineers. This was a mistake. For if the surface is sufficiently thin, stretching will generally be far more costly energetically than bending so that--to a good approximation--it will be unstretchable almost everywhere. In this limit the surface is described pretty accurately by a geometrical field theory: the energy is quadratic in curvature; the metric is fixed. Stretching gets consigned to a set of sharp peaks connected by ridges, the familiar pattern in a crumpled sheet of paper. In this talk, I will focus on the physics of the simplest defect of this kind on a flat sheet: a cone, characterized by a deficit angle. I will show how the defect organizes the folding of the sheet. This provides a natural segue to cones with a surplus angle. In contrast to a circular disc with a deficit angle, whose equilibrium state--in the absence of external forces--is an unremarkable circular ice-cream cone, a disc with a surplus angle will spontaneously fold into one of an infinite number of--non-axisymmetric--states labeled by an integer n with an n-fold rotational symmetry. I will show how to construct these states explicitly and discuss their stability. There will be a critical value of the surplus angle for each of these states beyond which the cone comes into self-contact. Before this occurs, all states are stable. Upon contact, the n-fold symmetry will be spontaneously broken. As the surplus angle is increased further, a strangely intricate energy landscape begins to emerge. If a disc surrounding the apex is removed, something more dramatic occurs: all states are rendered unstable with respect to isometric deformations into some-- non-conical--developable surface. I will argue that these instabilities provide an intuitive explanation for the complex patterns that emerge during the growth of various plant tissues. Lettuce leaves or the flowers of the Angel's trumpet come to mind. A metaphor for the early universe will be entertained.