Take two cubes, such that all linear sizes of the second cube are two times longer than those of the first one. Then, if the first cube has volume X, second cube has volume X * 2^3. Cubes are three-dimensional.

Take two squares, such that all linear sizes of the second square are two times longer than those of the first one. Then, if the first cube has area X, second cube has area X * 2^2. Squares are two-dimensional.

Take two Serpinsky's Triangles, such that all linear sizes of the second "triangle" are two times longer than those of the first one. Serpinsky's triangle doesn't have length or area or volume, but it has... well... size. Then, if the first "triangle" has the size of X, second "triangle" has size X * 2^log3base2 (which equals to X*3). Serpinsky's Triangles are log3base2-dimensional. At least in terms of a "fractal dimension".

Take two perfectly average cities, such that all linear sizes (such as average radius of a city or average length of a path from one random point in the city to another) from the second cube are roughly two times longer than those of the first one. Then, if the first cube has population X, second cube has population X * 2^y.

What is y ?

It's tempting to say thay y = 2, because cities appear to be pretty two-dimensional.
However, bigger cities tend to have higher buildings in the center, that are also packed together more tightly due to the defecit of the space, Meanwhile, smaller cities don't have the same deficit of space and may sprawl much further because they don't consider space defecit a problem at all, and have much lower profile because there are much less people able to pay for life in a skyscrapper.

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.

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?"

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.

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 ?"

You probably are right.

Oooh. Good one.

Haha what