One possible way to fill R3 is to divide up R3 into onion-layers, and use some sort of "generalized spiral points" to fill each onion-layer -- kind of like winding a string from N pole to S pole to N pole to S pole.
In each layer, the spiral starts at the North pole and spirals out something like the "sunflower spiral", eventually crosses the equator. The Southern equator is (more or less) identical to the Northern equator. After the path crosses the equator, it spirals in smaller and tighter curves until it reaches the South pole. The whole spiral is vaguely similar to Sphere Spirals by M.C. Escher. This spiral is one solution to the "place n equally-spaced points on a sphere" problem -- it may not be "the best way to pixelize a sphere", it may not be the best solution to the "Thomson problem", but there is no one "best way to pixelize a sphere". Then you move out an onion-layer and start spiralling back from South to north.
Going at it from another direction: perhaps somehow number the spheres used to form Waterman polyhedra? That gives a much more regular and close-packed set of points (sphere centers) -- but it's not obvious to me if there is any nice formula to convert from sphere-number to xyz coordinates.