[xarlozhw] VR sphere double cover

You see a rotating sphere, perhaps manipulating it in virtual reality to control its rotation.  For simplicity, orthographic projection.  The sphere is paradoxical: it has much greater surface area than a regular sphere of that size.  Rotating it any direction 360 degrees does not put you back where you started; rather, somewhere less.

It seems impossible to nicely put the area of 2 spheres on 1, and we assume the result also holds for any larger integer (this is false, see update below).  Therefore, we consider imperfect solutions.

An infinite manifold of spherical curvature seems possible.

Double cover (or more) with discontinuities.  What are some elegant places to put the discontinuities?

Update: Density of a star polyhedron.  Many integer covering densities possible, but filtering out star polygon faces and inside-out faces leaves only great dodecahedron (density 3), great icosahedron (density 7), and great ditrigonal icosidodecahedron (density 6): project their faces to a sphere.  Previously on manifolds.  Also likely relevant: Schwarz triangles.

If we limit rotation to one axis only, then one can pack multiple covers of a cylinder.  This is equivalent in 2D to multiple covers of a circle on a circle.  We need to prevent seeing the entire neighborhood around a pole.