Isn't this a bit pretentious? Rather than going on about "dimensions in a city" you are literally asking, "Hey, what's the equation for the population density of a city?"

Yes.

However, I've statred to think about it in a context of dimensions.

A person said to me: "This city has 15 times more people living in it, so all distances in it will be, like, 15 times longer !"

And I replied: "No, that's not right ! Cities are two-dimensional, so all distances will only be about 4 times longer !"

To which that person said something like "WTF you are talking about ? Cut your bulls..t !"

Which I did.

But then later, I start to think about cities, and realize: "Wait, that's probably not quite right either... Are cities REALLY really two-dimensional with respect to population ?"

There's also a lot of variation between cities of the same size. Some cities are spread out over a large amount of space, but have a small population and vice versa. The best you can hope for is to find a formula to produce Y.

You probably are right.

I mean, the question seems to be more "what's the growth rate of city population as a factor of its length in a single direction?" Which isn't quite the same as asking how many dimensions a city has, it's less philosophical and more a matter of civic engineering. As TPman says, nothing's consistent: some cities build up, others build out. I live in Los Angeles, which has a land area of 468 square miles and a population of about 4 million. New York City has only about 300 square miles of land area, but a population of more like 8.6 million. So, clearly, you can't derive the population of a city from its land area alone

I think the best way to model a city would be as a dome, where the radius of the circular cross-section is such that the area of that cross section has the appropriate area, and a height above this plane at its zenith one standard deviation above the mean height of buildings (possibly residences) in the city. Find the volume of this dome and you might be able to back of the envelope estimate city population as being similar to the population of cities with a similar dome area -- but while the plane for Los Angeles would be broader than that of New York, its dome height would be lower, and ultimately it would have less volume.

Oooh. Good one.