  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.
