Cities are useful because of connectivity. We maximize population density to maximize the benefits of this connectivity. (Of course, it also drives city dwellers mad, but that’s another story.)
The use of the third dimension is for now limited by elevator technology: As skyscrapers get taller you lose more and more space for more and more elevator shafts.
An obvious solution is having threedimensional elevator networks – cabins that move in all directions.
The German conglomerate ThyssenKrupp has a thing called the Multi for this:
And amazingly, there’s a fascinating elevator in Genoa that does a little bit of that already.
On 2017 Jun 22 ThyssenKrupp wants to put a full size MULTI elevator in operation in their testing tower in Rottweil. German source, photos.