Cauchy's rigidity theorem

Remarks

The proof compares the dihedral angles at corresponding edges and marks each edge $+$, $-$ or $0$ according to the change. Cauchy's arm lemma shows that around each vertex with a marked edge, the signs change at least four times. A count of sign changes with Euler's formula $V - E + F = 2$ then shows that no edge can be marked, so all dihedral angles agree. Cauchy's original proof of the arm lemma had a gap, which was later repaired.

Convexity is needed. Take a convex polyhedron with a vertex whose neighbours all lie in one plane. Reflect the small pyramid at that vertex inward through that plane. The new nonconvex polyhedron has the same combinatorics and congruent faces, but it is not congruent to the old one. Flexible polyhedra show that even a single nonconvex polyhedron can deform continuously with rigid faces.

Going deeper

For a student

The theorem is a statement of global rigidity for convex polyhedral surfaces among all convex realizations. It implies that a convex polyhedron cannot be deformed continuously with rigid faces. Dehn later proved the stronger infinitesimal rigidity of convex polyhedra. For nonconvex polyhedra rigidity can fail, as Connelly's flexible polyhedron shows, but the bellows conjecture says the volume is still preserved.