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  "body": "\n## Statement\n\nLet $P$ and $P'$ be convex [[polyhedra|0325SGSJAR]] in $\\R^3$.\nSuppose there is a bijection between their faces that preserves the incidences of faces, edges and vertices.\nSuppose also that each face of $P$ is congruent to the corresponding face of $P'$.\nThen there is an [[isometry|02HPP5VCZH]] of $\\R^3$ that maps $P$ onto $P'$.\n\n## Remarks\n\nThe proof compares the dihedral angles at corresponding edges and marks each edge $+$, $-$ or $0$ according to the change.\nCauchy's arm lemma shows that around each vertex with a marked edge, the signs change at least four times.\nA 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.\nCauchy's original proof of the arm lemma had a gap, which was later repaired.\n\nConvexity is needed.\nTake a convex polyhedron with a vertex whose neighbours all lie in one plane.\nReflect the small pyramid at that vertex inward through that plane.\nThe new nonconvex polyhedron has the same combinatorics and congruent faces, but it is not congruent to the old one.\nFlexible polyhedra show that even a single nonconvex polyhedron can deform continuously with rigid faces.\n\n## Explanations\n\n### L1 human\n\nMake a convex solid from rigid cardboard faces joined by hinges at the edges.\nCauchy's theorem says the solid cannot wobble: the faces alone fix its whole shape.\nIn particular, a convex polyhedron is determined by its net of faces and the way they are glued.\n\n### L2 human\n\nThe theorem is a statement of global rigidity for convex polyhedral surfaces among all convex realizations.\nIt implies that a convex polyhedron cannot be deformed continuously with rigid faces.\nDehn later proved the stronger infinitesimal rigidity of convex polyhedra.\nFor nonconvex polyhedra rigidity can fail, as Connelly's flexible polyhedron shows, but the bellows conjecture says the volume is still preserved.\n",
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