09-07-2026, 10:45 PM
Summary
Alexandrov’s theorem on polyhedra is a fundamental theorem in convex and discrete geometry. It states, roughly, that the intrinsic geometry of the surface of a convex polyhedron completely determines the polyhedron itself. Distances are measured along the surface rather than through three-dimensional space. Such a surface is locally Euclidean almost everywhere, except at its vertices, where there is positive angular defect.
If the face angles meeting at a vertex sum to $\theta$, then the angular defect is
$\delta = 2\pi - \theta$.
For a convex polyhedron,
$\delta > 0$,
and the total angular defect over all vertices satisfies
$\sum_i \delta_i = 4\pi$,
which is the polyhedral analogue of the Gauss–Bonnet theorem.
Alexandrov proved the converse: if a geodesic metric space is topologically a sphere, is locally Euclidean except at finitely many cone points, and all those cone points have positive angular defect, then this metric can be realized as the surface metric of a convex polyhedron in $\mathbb{R}^3$. Moreover, the polyhedron is unique up to rigid motions.
Thus, if two convex polyhedra have exactly the same intrinsic surface distances, then they must be congruent. Alexandrov’s theorem therefore combines both existence and rigidity: an appropriate abstract two-dimensional metric determines a unique convex three-dimensional polyhedron.
There are some qualifications. The resulting object may sometimes be degenerate, for example a doubly covered planar convex polygon. Also, uniqueness holds specifically among convex polyhedra; non-convex polyhedra can sometimes have the same intrinsic surface metric as a convex polyhedron.
Alexandrov’s original proof was nonconstructive and did not give an explicit algorithm for recovering the coordinates of the vertices. Later, Bobenko and Izmestiev developed an algorithmic method for approximately reconstructing the corresponding convex polyhedron.
The theorem generalizes classical rigidity results such as Cauchy’s rigidity theorem and has close analogues in the theory of smooth convex surfaces with positive Gaussian curvature.
Key takeaways
ARTICLE
Alexandrov’s theorem on polyhedra is a fundamental theorem in convex and discrete geometry. It states, roughly, that the intrinsic geometry of the surface of a convex polyhedron completely determines the polyhedron itself. Distances are measured along the surface rather than through three-dimensional space. Such a surface is locally Euclidean almost everywhere, except at its vertices, where there is positive angular defect.
If the face angles meeting at a vertex sum to $\theta$, then the angular defect is
$\delta = 2\pi - \theta$.
For a convex polyhedron,
$\delta > 0$,
and the total angular defect over all vertices satisfies
$\sum_i \delta_i = 4\pi$,
which is the polyhedral analogue of the Gauss–Bonnet theorem.
Alexandrov proved the converse: if a geodesic metric space is topologically a sphere, is locally Euclidean except at finitely many cone points, and all those cone points have positive angular defect, then this metric can be realized as the surface metric of a convex polyhedron in $\mathbb{R}^3$. Moreover, the polyhedron is unique up to rigid motions.
Thus, if two convex polyhedra have exactly the same intrinsic surface distances, then they must be congruent. Alexandrov’s theorem therefore combines both existence and rigidity: an appropriate abstract two-dimensional metric determines a unique convex three-dimensional polyhedron.
There are some qualifications. The resulting object may sometimes be degenerate, for example a doubly covered planar convex polygon. Also, uniqueness holds specifically among convex polyhedra; non-convex polyhedra can sometimes have the same intrinsic surface metric as a convex polyhedron.
Alexandrov’s original proof was nonconstructive and did not give an explicit algorithm for recovering the coordinates of the vertices. Later, Bobenko and Izmestiev developed an algorithmic method for approximately reconstructing the corresponding convex polyhedron.
The theorem generalizes classical rigidity results such as Cauchy’s rigidity theorem and has close analogues in the theory of smooth convex surfaces with positive Gaussian curvature.
Key takeaways
- The intrinsic surface geometry of a convex polyhedron determines it uniquely.
- A polyhedral metric is locally flat except at finitely many vertices.
- At each vertex the angular defect satisfies $\delta > 0$.
- The total angular defect is $\sum_i \delta_i = 4\pi$.
- Alexandrov’s theorem is both an existence theorem and a uniqueness theorem.
- It connects convex geometry, discrete geometry, metric geometry, and differential geometry.
ARTICLE
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