Talking about optimal representations can only be done if one fully defines the criterion for optimality; and, whatever it is, you will probably get an optimal representation that suffers in some other aspect. I think it's more interesting to talk about useful representations, instead.

There is a useful representation that is used all the time in Computer Graphics: cube mapping. If you have 32 bits to represent a diretion in 3D, you can think of the sphere as having an inscribed cube. Divide each face in 26754x26754 little squares and identify a direction with the little square it falls in. 26754*26754*6 = 4294659096, which means you are using more than 99.99% of the possible bit combinations. Conversion back and forth is pretty straight forward and the variation in resolution at different parts of the sphere is tolerable.

Talking about optimal representations can only be done if one fully defines the criterion for optimality, and whatever it is, you will probably get an optimal that suffers in some other aspect. I think it's more interesting to talk about useful representations, instead.

There is a useful representation that is used all the time in Computer Graphics: cube mapping. If you have 32 bits to represent a diretion in 3D, you can think of the sphere as having an inscribed cube. Divide each face in 26754x26754 little squares and identify a direction with the little square it falls in. 26754*26754*6 = 4294659096, which means you are using more than 99.99% of the possible bit combinations. Conversion back and forth is pretty straight forward and the variation in resolution at different parts of the sphere is tolerable.