I am trying to study (finite) spherical buildings from a very combinatorial point of view : Every rank 3 spherical building is a finite simplicial complex of dimension 3, so one can define its density as the ratio #triangles/#vertices, expressed as a function of the number $n$ of vertices. Maybe it is the wrong name for a very classical notion but I did not find it anywhere. One of the first distinctions in buildings is the concept of thick buildings. In a way, this density measures the *thickness* of a given building, but I never saw this notion.

I am wondering what is the density of the spherical buildings, and which one are the densest ?

Starting from the classification of spherical buildings, it is enough to compute the density of the buildings of type $A_3$ and $C_3$. I know how to do it for $A_3$, since the building in this case is the flag complex of a projective space on $\mathbb{F}_q^4$, and it seems to give $\frac{[4]_q!}{\binom{4}{2}_q}=O(q^2)=O(\sqrt{n})$ where $n$ is the number of vertices, with the usual q-analog notations.

But $C_3$ looks scary to me, as there are multiple possible geometries, and I don't know of a systematic way to extract the (simple) combinatorial information out of the groups. Is this something well known, or is there a simple way to do this ?