Alexandrov's uniqueness theorem
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The Alexandrov uniqueness theorem is a
rigidity theorem In mathematics, a rigid collection ''C'' of mathematical objects (for instance sets or functions) is one in which every ''c'' ∈ ''C'' is uniquely determined by less information about ''c'' than one would expect. The above statement do ...
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 space In mathematics, a metric space is a set together with a notion of '' distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general set ...
s 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 Aleksandr Danilovich Aleksandrov (russian: Алекса́ндр Дани́лович Алекса́ндров, alternative transliterations: ''Alexandr'' or ''Alexander'' (first name), and ''Alexandrov'' (last name)) (4 August 1912 – 27 July 19 ...
, who published it in the 1940s.


Statement of the theorem

The surface of any convex polyhedron in
Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean ...
forms a
metric space In mathematics, a metric space is a set together with a notion of '' distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general set ...
, in which the distance between two points is measured by the length of the
shortest path In graph theory, the shortest path problem is the problem of finding a path between two vertices (or nodes) in a graph such that the sum of the weights of its constituent edges is minimized. The problem of finding the shortest path between ...
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 Development or developing may refer to: Arts *Development hell, when a project is stuck in development *Filmmaking, development phase, including finance and budgeting *Development (music), the process thematic material is reshaped * Photograph ...
. The polyhedron can be thought of as being folded from a sheet of paper (a
net Net or net may refer to: Mathematics and physics * Net (mathematics), a filter-like topological generalization of a sequence * Net, a linear system of divisors of dimension 2 * Net (polyhedron), an arrangement of polygons that can be folded up ...
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 In topology and related areas of mathematics, a neighbourhood (or neighborhood) is one of the basic concepts in a topological space. It is closely related to the concepts of open set and interior. Intuitively speaking, a neighbourhood of a p ...
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 A cone is a three-dimensional geometric shape that tapers smoothly from a flat base (frequently, though not necessarily, circular) to a point called the apex or vertex. A cone is formed by a set of line segments, half-lines, or lines con ...
. 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 In geometry, the (angular) defect (or deficit or deficiency) means the failure of some angles to add up to the expected amount of 360° or 180°, when such angles in the Euclidean plane would. The opposite notion is the excess. Classically the defe ...
of the vertex; it is a positive number less than  2. The defect of a polyhedron vertex can be measured by subtracting the face angles at that vertex from 2. For instance, in a regular tetrahedron, each face angle is /3, and there are three of them at each vertex, so subtracting them from 2 leaves a defect of at each of the four vertices. Similarly, a cube has a defect of /2 at each of its eight vertices. Descartes' theorem on total angular defect (a form of the
Gauss–Bonnet theorem In the mathematical field of differential geometry, the Gauss–Bonnet theorem (or Gauss–Bonnet formula) is a fundamental formula which links the curvature of a surface to its underlying topology. In the simplest application, the case of a t ...
) states that the sum of the angular defects of all the vertices is always exactly 4. 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 4. 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 (necessarily summing to 4), 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 Congruence may refer to: Mathematics * Congruence (geometry), being the same size and shape * Congruence or congruence relation, in abstract algebra, an equivalence relation on an algebraic structure that is compatible with the structure * In mod ...
to each other as three-dimensional sets.


Limitations

The polyhedron representing the given metric space may be
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: it may form a doubly-covered two-dimensional convex polygon (a
dihedron A dihedron is a type of polyhedron, made of two polygon faces which share the same set of ''n'' edges. In three-dimensional Euclidean space, it is degenerate if its faces are flat, while in three-dimensional spherical space, a dihedron with flat ...
) 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. 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 In geometry, a regular icosahedron ( or ) is a convex polyhedron with 20 faces, 30 edges and 12 vertices. It is one of the five Platonic solids, and the one with the most faces. It has five equilateral triangular faces meeting at each vertex. It ...
: if five of its triangles are removed, and are replaced by five congruent triangles forming an indentation into the polyhedron, the resulting surface metric stays unchanged. The development of any 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 spaces described in this way are easily checked. However, the edges where two polygons are glued together could become flat and lie in the interior of faces of the resulting polyhedron, rather than becoming polyhedron edges. (For an example of this phenomenon, see the illustration of four hexagons glued to form an octahedron.) Therefore, even when the development is described in this way, it may not be clear what shape the resulting polyhedron has, what shapes its faces have, or even how many faces it has. Alexandrov's original proof does not lead to an
algorithm In mathematics and computer science, an algorithm () is a finite sequence of rigorous instructions, typically used to solve a class of specific problems or to perform a computation. Algorithms are used as specifications for performing ...
for constructing the polyhedron (for instance by giving coordinates for its vertices) realizing the given metric space. In 2008, Bobenko and Izmestiev provided such an algorithm. Their algorithm can approximate the coordinates arbitrarily accurately, in
pseudo-polynomial time In computational complexity theory, a numeric algorithm runs in pseudo-polynomial time if its running time is a polynomial in the ''numeric value'' of the input (the largest integer present in the input)—but not necessarily in the ''length'' of t ...
.


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, without stretching or shrinking, then 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 Max Wilhelm Dehn (November 13, 1878 – June 27, 1952) was a German mathematician most famous for his work in geometry, topology and geometric group theory. Born to a Jewish family in Germany, Dehn's early life and career took place in Germany. ...
to infinitesimal rigidity. An analogous result to Alexandrov's holds for smooth convex surfaces: a two-dimensional Riemannian manifold whose Gaussian curvature is everywhere positive and totals 4 can be represented uniquely as the surface of a smooth convex body in three dimensions. 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. Its existence was proven by Alexandrov, using an argument involving limits of polyhedral metrics.
Aleksei Pogorelov Aleksei Vasil'evich Pogorelov (russian: Алексе́й Васи́льевич Погоре́лов, ua, Олексі́й Васи́льович Погорє́лов; March 2, 1919 – December 17, 2002), was a Soviet and Ukrainian ...
generalized both these results, characterizing the developments of arbitrary convex bodies in three dimensions. Another result of Pogorelov on the geodesic metric spaces derived from convex polyhedra is a version of the
theorem of the three geodesics In differential geometry the theorem of the three geodesics, also known as Lyusternik–Schnirelmann theorem, states that every Riemannian manifold with the topology of a sphere has at least three simple closed geodesics (i.e. three embedded geode ...
: every convex polyhedron has at least 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 on both sides of them. 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.


References

{{Mathematics of paper folding Geodesic (mathematics) Mathematics of rigidity Theorems in convex geometry Theorems in discrete geometry Uniqueness theorems