mechanistic analysis of lambertian and isotropic radiators

Hello Physics Forums
I need a little help wrapping my head around the concept of lambertian emitters as compared to isotropic emitters.

As I understand it a lambertian emitter emits less and less photons as the angle of emission gets further and further away from the surface normal. An isotropic emitter however emits the same number of photons in all directions.

The idea of a lambertian emitter has been a little confusing to me mathematically, but I've found it is a very intuitive concept when thought of in practical terms. A piece of paper will look just as white when viewed from any angle, because it is a lambertian surface. However, when viewed from an oblique angle the apparent size of the piece of paper will be smaller. The same brightness and smaller area compound to a smaller number of photons being emitted into oblique angles.

All examples of isotropic radiators I have found so far are spheres. A sphere appears to be the same size when viewed from any angle. So, even if a sphere is a lambertian emitter it appears to be isotropic.

So I'm wondering, if a surface was composed of lambertian spheres (for instance sand) would it exhibit a lambertian or an isotropic emission profile? Intuitively it would of course still be a lambertian emitter, because its apparent surface area would decrease at oblique angles. But why does the emittance of all these isotropic radiators add up to a non-isotropic pattern?

Thanks for your input, and happy 4th of July!