I find this (and the other comment referencing the volumes of hypercubes) to be interesting. Perhaps exactly because it is so unintuitive. Maybe I could reason

I find this (and the other comment referencing the volumes of hypercubes) to be interesting. Perhaps exactly because it is so unintuitive. Maybe I could reason it up myself, but what exactly do you see as the "volume" of the inner portion and the "volume" of the shell? Are we talking about the same units here? Are there generic formulae?