An octahedral equal area partition of the sphere and near optimal configurations of points

We construct a new area preserving map from the unit sphere to the regular octahedron, both centered at the origin. Its inverse map allows the construction of uniform and refinable grids on a sphere, starting from any triangular uniform and refinable grid on the faces of the octahedron. We prove that our new grids are diameter bounded and then, for the resulting configurations of points we calculate some Riesz ss-energies and we compare them with the optimal ones. For some configurations, we also calculate the point energies and we list the minimum and maximum values, concluding that these values are very close. Finally, we show how we can map a hemisphere of the Earth onto a square, using our new area preserving projection. The simplicity and the symmetry of our formulas lead to fast computations.