A New Metric on Spherical f-Tilings

The continuous deformation of any spherical isometric folding into the standard spherical folding f s , defined by f s (x, y, z) = (x, y, |z|), has been an open problem since 1989. Some relations between the deformation of spherical isometric foldings and the deformation of spherical f-tilings are analyzed, as they are closely related. The natural way to deform f-tilings is based on the Hausdorff metric on compact sets. However, this metric does not induce a continuous deformation on its associated isometric foldings. A new metric on spherical f-tilings will be introduced and a new contribution to the deformation of isometric foldings (via deformation of f-tilings) will be given.