page_title: Alexandrov's uniqueness theorem
section_title: Statement of the theorem
hierarchical_section_title: Alexandrov's uniqueness theorem / Statement of the theorem
context_page_description: The Alexandrov uniqueness theorem is a rigidity theorem in mathematics, describing three-dimensional convex polyhedra in terms of the distances between points on their surfaces. It implies that convex polyhedra with distinct shapes from each other also have distinct metric spaces of surface distances, and it characterizes the metric spaces that come from the surface distances on polyhedra. It is named after Soviet mathematician Aleksandr Danilovich Aleksandrov, who published it in the 1940s.
context_section_description: The surface of any convex polyhedron in Euclidean space forms a metric space, in which the distance between two points is measured by the length of the shortest path from one point to the other along the surface. Within a single shortest path, distances between pairs of points equal the distances between corresponding points of a line segment of the same length; a path with this property is known as a geodesic.
  This property of polyhedral surfaces, that every pair of points is connected by a geodesic, is not true of many other metric spaces, and when it is true the space is called a geodesic space. The geodesic space formed from the surface of a polyhedron is called its development.
  The polyhedron can be thought of as being folded from a sheet of paper (a net for the polyhedron) and it inherits the same geometry as the paper: for every point p within a face of the polyhedron, a sufficiently small open neighborhood of p will have the same distances as a subset of the Euclidean plane. The same thing is true even for points on the edges of the polyhedron: they can be modeled locally as a Euclidean plane folded along a line and embedded into three-dimensional space, but the fold does not change the structure of shortest paths along the surface. However, the vertices of the polyhedron have a different distance structure: the local geometry of a polyhedron vertex is the same as the local geometry at the apex of a cone. Any cone can be formed from a flat sheet of paper with a wedge removed from it by gluing together the cut edges where the wedge was removed. The angle of the wedge that was removed is called the angular defect of the vertex; it is a positive number less than  2π. The defect of a polyhedron vertex can be measured by subtracting the face angles at that vertex from 2π. 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.
  Similarly, a cube has a defect of π/2 at each of its eight vertices. Descartes' theorem on total angular defect (a form of the Gauss–Bonnet theorem) states that the sum of the angular defects of all the vertices is always exactly 4π. 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.
  Alexandrov's theorem gives a converse to this description. It states that if a metric space is geodesic, homeomorphic to a sphere, and locally Euclidean except for a finite number of cone points of positive angular defect summing to 4π, then there exists a convex polyhedron whose development is the given space. Moreover, this polyhedron is uniquely defined from the metric: any two convex polyhedra with the same surface metric must be congruent to each other as three-dimensional sets.
caption: 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.