Distributed branch points and the shape of elastic surfaces with constant negative curvature

We develop a theory for distributed branch points and investigate their role in determining the shape and influencing the mechanics of thin hyperbolic objects. We show that branch points are the natural topological defects in hyperbolic sheets, they carry a topological index which gives them a degree of robustness, and they can influence the overall morphology of a hyperbolic surface without concentrating energy. We develop a discrete differential geometric (DDG) approach to study the deformations of hyperbolic objects with distributed branch points. We present evidence that the maximum curvature of surfaces with geodesic radius $R$ containing branch points grow sub-exponentially, $O(e^{c\sqrt{R}})$ in contrast to the exponential growth $O(e^{c' R})$ for surfaces without branch points. We argue that, to optimize norms of the curvature, i.e. the bending energy, distributed branch points are energetically preferred in sufficiently large pseudospherical surfaces. Further, they are distributed so that they lead to fractal-like recursive buckling patterns.