By folding and gluing the edges of a collection of one or more regular hexagons, there are several ways to build closed polyhedra. The surface of the sphere can be mapped conformally to the faces of these polyhedra, and therefore onto the hexagons. The Gosper island is a two-dimensional fractal shape generated from a hexagon which seamlessly tiles the plane in a triangular lattice, and can be recursively subdivided into 7 congruent Gosper island shapes similar to the original. The flowsnake is a space-filling fractal curve which fills the Gosper island. When we combine these ideas we can construct several map projections from the sphere onto a collection of Gosper island shapes, each yielding as an inverse a “sphere-filling” fractal curve, and a recursive subdivision of the sphere into smaller Gosper island shapes in a succession of triangular lattices.