Alexandrov's uniqueness theorem

For instance, in a regular tetrahedron, each face angle is π/3, and there are three of them at each vertex, so subtracting them from 2π leaves a defect of π at each of the four vertices.

In summary, the development of a convex polyhedron is geodesic, homeomorphic (topologically equivalent) to a sphere, and locally Euclidean except for a finite number of cone points whose angular defect sums to 4π.

Six triangles meet at each additional vertex introduced by this refolding, so they have zero angular defect and remain locally Euclidean.

Alexandrov's original proof does not lead to an algorithm for constructing the polyhedron (for instance by giving coordinates for its vertices) realizing the given metric space.

Alexandrov's theorem strengthens this, showing that even if the faces are allowed to bend or fold, without stretching or shrinking, then their connectivity still determines the shape of the polyhedron.

The uniqueness of this representation is a result of Stephan Cohn-Vossen from 1927, with some regularity conditions on the surface that were removed in later research.

[10] Aleksei Pogorelov generalized both these results, characterizing the developments of arbitrary convex bodies in three dimensions.

These are curves that are locally straight lines except when they pass through a vertex, where they are required to have angles of less than π on both sides of them.

[11] The developments of ideal hyperbolic polyhedra can be characterized in a similar way to Euclidean convex polyhedra: every two-dimensional manifold with uniform hyperbolic geometry and finite area, combinatorially equivalent to a finitely-punctured sphere, can be realized as the surface of an ideal polyhedron.

Four regular hexagons can be folded and glued to form the surface of a regular octahedron. [ 4 ] In this example the edges of the hexagons do not fall along edges of the octahedron. The same gluing pattern can also produce a non-convex polyhedron with 24 triangular faces. [ 5 ]
Two square sheets joined along their edges form a degenerate flat polyhedron, with four points of angular deficiency π at its four corners. It can be inflated without stretching into this non-convex shape , making the conical nature of the corners more apparent.