Number of faces and edges on a manifold surface mesh

Question

Why the number of edges is in the same order of magnitude as the number of faces?

Answer 1

Because $v+f-e$ (with $v=$ vertices, $e=$ edges, $f=$ faces) is an invariant of the manifold, we have $f\sim e-v$. If the average vertex has $\rho\ge 3$ edges, then $2e=\rho v$, hence $e-v=\frac{\rho-2}\rho e$ and the factor $\frac{\rho-2}\rho=1-\frac2\rho$ is between $\frac13$ and $1$.