Alexandrov's uniqueness theorem

The Alexandrov uniqueness theorem is a rigidity theorem in mathematics, characterizing three-dimensional convex polyhedra in terms of the distances between points on their surfaces. It is named after Soviet mathematician Aleksandr Danilovich Aleksandrov, who published it in the 1940s.^[1][2][3]

Statement of the theorem

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; these paths are known as geodesics, and a space in which every distance is represented by a path is called a "geodesic space." The metric space formed in this way from a polyhedron is called its development.^[3]

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 be isometric to 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 neighborhood of a vertex is isometric to the neighborhood of the apex of a cone, 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 in the open interval from 0 to 2π. 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.^[3]

Alexandrov's theorem gives a converse to this description. It states that if a metric space (X,d) 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 (X,d) can be represented as the development of a convex polyhedron. 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.^[3]

Limitations

The polyhedron representing the given metric space may be degenerate: it may form a two-dimensional convex polygon rather than a fully three-dimensional polyhedron. In this case, its surface metric consists of two copies of the polygon (its two sides) glued together along corresponding edges.^[3]

The regular icosahedron has the same surface metric as a non-convex deltahedron in which one of its five-triangle pyramids is pushed in rather than protruding

Although Alexandrov's theorem states that there is a unique convex polyhedron whose surface has a given metric, it may also be possible for there to exist non-convex polyhedra with the same metric. An example is given by the regular icosahedron: if five of its triangles are removed, and are replaced by five congruent triangles forming an indentation into the polyhedra, the resulting surface metric stays unchanged.

The development of a polyhedron can be described concretely by a collection of two-dimensional polygons together with instructions for gluing them together along their edges to form a metric space, and the conditions of Alexandrov's theorem for such spaces are easily checked. However, 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. In 2008, Bobenko and Izmestiev such an algorithm.^[4] Their algorithm can approximate the coordinates arbitrarily accurately, in pseudo-polynomial time.^[5]

Related results

One of the first existence and uniqueness theorems for convex polyhedra is Cauchy's theorem, which states that a convex polyhedron is uniquely determined by the shape and connectivity of its faces. Alexandrov's theorem strengthens this, showing that even if the faces are allowed to bend or fold, their connectivity still determines the shape of the polyhedron. In turn, Alexandrov's proof of the existence part of his theorem uses a strengthening of Cauchy's theorem by Max Dehn to infinitesimal rigidity.^[3]

An analogous result to Alexandrov's holds for smooth convex surfaces: a two-dimensional smooth manifold whose total Gaussian curvature is 4π can be represented uniquely as the surface of a smooth convex body in three dimensions. This is a result of Stephan Cohn-Vossen from 1927. Aleksei Pogorelov generalized both these results, characterizing the developments of arbitrary convex bodies in three dimensions.^[3]

Another result of Pogorelov on the geodesic metric spaces derived from convex polyhedra is a version of the theorem of the three geodesics: every convex polyhedron has three simple closed quasigeodesics. 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.^[6]

References

1. Senechal gives a date of 1941, while O'Rourke lists 1948. See: Senechal, Marjorie (2013), Shaping Space: Exploring Polyhedra in Nature, Art, and the Geometrical Imagination, Springer, p. 62, ISBN 9780387927145. O’Rourke, Joseph (2011), How to Fold It: The Mathematics of Linkages, Origami and Polyhedra, Cambridge University Press, p. 134, ISBN 9781139498548.
2. Alexandrov, A. D. (2006), Convex Polyhedra, Springer Monographs in Mathematics, Springer, ISBN 9783540263401. Translated into English by N. S. Dairbekov, S. S. Kutateladze, and A.B. Sossinsky. The uniqueness part of the theorem is covered in Chapter 3, and the existence part is covered in Chapter 4.
3. Connelly, Robert (March 2006), "Convex Polyhedra by A. D. Alexandrov" (PDF), SIAM Review 48 (1): 157–160, doi:10.1137/SIREAD000048000001000149000001, JSTOR 204537.
4. Bobenko, Alexander I.; Izmestiev, Ivan (2008), "Alexandrov's theorem, weighted Delaunay triangulations, and mixed volumes", Université de Grenoble. Annales de l'Institut Fourier 58 (2): 447–505, MR 2410380
5. Kane, Daniel; Price, Gregory N.; Demaine, Erik D. (2009), "A pseudopolynomial algorithm for Alexandrov’s theorem", in Dehne, Frank; Gavrilova, Marina; Sack, Jörg-Rüdiger; Tóth, Csaba D., Algorithms and data structures. 11th International Symposium, WADS 2009, Banff, Canada, August 21-23, 2009, Proceedings, Lecture Notes in Computer Science 5664, Berlin: Springer, pp. 435–446, arXiv:0812.5030, doi:10.1007/978-3-642-03367-4_38, ISBN 978-3-642-03366-7, MR 2550627
6. Pogorelov, Aleksei V. (1949), "Quasi-geodesic lines on a convex surface", Matematicheskii Sbornik (in Russian), 25(62): 275–306, MR 0031767