Cauchy's theorem (geometry)

That is, any polyhedral net formed by unfolding the faces of the polyhedron onto a flat surface, together with gluing instructions describing which faces should be connected to each other, uniquely determines the shape of the original polyhedron.

One can "push in" a vertex to create a nonconvex polyhedron that is still combinatorially equivalent to the regular icosahedron; that is, one can take five faces of the icosahedron meeting at a vertex, which form the sides of a pentagonal pyramid, and reflect the pyramid with respect to its base.

The result originated in Euclid's Elements, where solids are called equal if the same holds for their faces.

This version of the result was proved by Cauchy in 1813 based on earlier work by Lagrange.

An error in Cauchy's proof of the main lemma was corrected by Ernst Steinitz, Isaac Jacob Schoenberg, and Aleksandr Danilovich Aleksandrov.