We introduce the concept of ‘geometrical spine’ for 3-manifolds with natural metrics, in particular, for lens manifolds. We show that any spine of $L_{p,q}$ that is close enough to its geometrical spine contains at least $E(p,q)-3$ vertices, which is exactly the conjectured value for the complexity $c(L_{p,q})$. As a byproduct, we find the minimal rotation distance (in the Sleator–Tarjan–Thurston sense) between a triangulation of a regular $p$-gon and its image under rotation.