Additional dimensions don't give you additional ways to be closer to something, they give you additional ways to be further away. If we're displaced from each

Additional dimensions don't give you additional ways to be closer to something, they give you additional ways to be further away. If we're displaced from each other by X, Y and Z units along axes in the usual 3 spatial dimensions, the closest we can be is if we're at a coincident position in each of the additional dimensions. Any displacement along those additional axes just moves us further away from each other.