This is an attempt to generalize the concept of the geometrically regular shapes of the Platonic polyhedra to objects with curved edges and faces that are all identical, and where faces are not restricted to be topological disks. Some of the objects studied, are “globes” of genus zero, bi-pyramids, and Platonic edge-frames with twisted branches, as well as mathematical knots embedded as edges on the surface of handle-bodies of higher genus.