Comment by thaumasiotes

If you're willing to use continuous dimensions, the surface is the same size as the volume; I don't see why the pigeonhole principle would present any problems.

Interestingly enough, I believe there is a result that space-filling curves cannot be one-to-one, but the implication there is just that, by virtue of the continuity of the one-dimensional curve, it contains more points than the two-dimensional space that it fills.