Density of stars on celestial sphere The pattern of stars on the surface of the celestial sphere seems perceptually to have an universally looking structure. If you pick a region of the sky at random, for instance by using the (excellent) set of tools from the stellarium suite you get for example this:

There are bigger structures, like the milky way or more clustered areas, but on a smaller scale (like that you will see in binoculars with a field of view of the size of the moon), you will very often get this similar pattern. This looks to me very much like a self-similar fractal and I am curious to know if anybody knows more about it.
Assuming a given density of stars in this patch of the universe, and  omitting inhomogeneities at first,  can we derive a mathematical expression for describing the density of stars in a given patch of the sky ?
As a bonus, is there a map estimating this density for different areas of the celestial sphere? Clearly, the milky way is more crowded and would perhaps have more structure...
If stars are uniformly distributed with density $\rho$ per cubic parsec, out to a distance $d$ parsecs, then that volume contains $4\pi \rho d^3/3$ stars spread over the whole sky.
The angular size of the whole sky is 41,252.96125 square degrees, so you would divide by that number to estimate the stellar density in stars per square degree.
This assumes that you can see all the stars. In practice you cannot, because of dust extinction and because stars become fainter the further away they are. You can account for these in any calculation but the details depend on the luminosity function of the stars, the distribution of dust and the instrumental properties of the telescope/detector used to take the picture.