I claim it's obviously recognizable as the entire surface of a 3-sphere projected onto 2 dimensions, and therefore is clearly an acceptable answer to the origin

I claim it's obviously recognizable as the entire surface of a 3-sphere projected onto 2 dimensions, and therefore is clearly an acceptable answer to the original question posed. I added some helpful and clarifying prose expounding upon which hemisphere is what, but could have left that as an exercise to the reader without loss of generality.