I am also interested in points on the sphere. I was thinking about using a projection based on mapping a square to an octahedron (along the general pattern of

I am also interested in points on the sphere. I was thinking about using a projection based on mapping a square to an octahedron (along the general pattern of the “quincuncial projection”, but perhaps preserving areas). However, the topology on the square is not a torus formed by associating opposite sides (as in your pattern) but instead has the two halves of each side of the square folded onto each-other. I suspect the points in the sequence won’t be quite as well distributed near those seams.