# Talk:Alexandrov's theorem on polyhedra

Isn't there a convexity hypothesis missing in the statement of the theorem of Cohn-Vossen? Furthermore, why link to "smooth manifold" while the statement is presumably about Riemannian ones? I'd fix this but I am having trouble tracking the paper of Cohn-Vossen in question.Ventricule (talk) 11:27, 19 November 2021 (UTC)

Yes, the sentences "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." are quite confusing. Smooth manifolds do not have Gaussian curvature, and if it is to say "Riemannian manifold" then the total Gaussian curvature is already determined by Gauss-Bonnet theorem. The given ref seems to be unrelated. Perhaps these sentences are meant to refer to Hermann Weyl's problem on isometric embedding, of Riemannian metrics of positive Gaussian curvature on the 2-sphere, into R3? I believe the uniqueness in Weyl problem was indeed resolved by Cohn-Vossen (and/or Hans Lewy, depending on regularity assumptions), and that Louis Nirenberg and Pogorelov independently resolved the existence. Pogorelov later extended to more general three-dimensional spaces. Gumshoe2 (talk) 20:50, 19 February 2022 (UTC)
The 4π condition also makes it reminiscent of Minkowski problem, although this would be less directly related to this wiki page. Gumshoe2 (talk) 21:01, 19 February 2022 (UTC)
The given ref (Connelly's review of Alexandrov's book) is not unrelated. Did you even try to look it up, or are you just misjudging it by its title? Specifically, it states: "After Alexandrov’s first proof of this result appeared in the 1940’s [A**], A. V. Pogorelov [P**] generalized it considerably to the case where the surface was any sort of metric space homeomorphic to a two-dimensional sphere but still intrinsically convex, while at the same time extending the definition of intrinsic convexity to allow for the boundary of any convex set in three-space".  Also, it is incorrect that "the total Gaussian curvature is already determined by Gauss-Bonnet theorem": it is determined by Gauss-Bonnet + genus.  —David Eppstein (talk) 00:21, 20 February 2022 (UTC)
Hey, I really don't appreciate the aspersions here or in your edit summaries. As the saying goes, please assume good intentions. I did read the ref, and even re-read it a couple times to make sure I wasn't misunderstanding and wouldn't be posting time-wasting material on the talk page. Anyway, from the Guan-Li ref you added, I assume now that the material was intended to be about the Weyl problem. The quote you have just given here does not in any way suggest the smooth geometry which is the Weyl problem, nor does it suggest a generalization thereof by Pogorelov (his work on this problem being about the smoothness of certain convex bodies). So the Connelly ref is still not applicable for this paragraph. It is about a different result of Pogorelov's which is also of interest.
Anyway, the paragraph as written is now clearer but incorrect (referring to this version [1]). The manifold is specifically required to be the 2-sphere, and then (according to Gauss-Bonnet) the 4π condition is redundant. The Gaussian curvature is assumed to be positive, not non-negative. (As far as I know there is to this day no complete resolution of the Weyl problem in nonnegative curvature, the partial (non-smooth surfaces) results by Guan-Li notwithstanding.) Existence was not proven by Alexandrov; he proved existence of a limiting convex surface. Pogorelov proved existence by proving smoothness of Alexandrov's surface. Also, Louis Nirenberg proved the same thing (without Alexandrov's methods), so he should probably be mentioned as well.
Happy to clarify anything further, don't want to cause issues so I'll leave the page edits to someone else. Gumshoe2 (talk) 02:44, 20 February 2022 (UTC)
How does "The manifold is specifically required to be the 2-sphere, and then (according to Gauss-Bonnet) the 4π condition is redundant" differ in any substantial way from "The manifold is required to have total curvature 4π, and then (according to Gauss-Bonnet) the 2-sphere condition is redundant"? That is, why do you think it is necessary to remove the total curvature condition (and I assume say instead that it must be a 2-sphere, since we don't currently say that)? What improvement do you think would be made by that change? Why do you think one of these two formulations is correct and the other incorrect? —David Eppstein (talk) 02:59, 20 February 2022 (UTC)
Well, the 4π condition is at least just a little strange to single out or point out. It is much more usual to just say you have a metric on S2, with no further comment. But even formally, the present version does not clarify the compactness of the manifold, which actually does make a technical difference. As currently stated, it could be a metric on the plane. Gumshoe2 (talk) 03:04, 20 February 2022 (UTC)

I have some professional expertise in the PDE and differential-geometric analogue of these results, but for the polyhedral and discrete story I am only novice. So the following comments are from that perspective, having spent the last several hours looking through references and trying to understand the material. Following the present version of page: [2]

I previously [3] removed a wikilink to the differential-geometric notion of development, which seems to be unrelated to the present topic. David Eppstein reverted without explanation, but I still think it is the wrong wikilink.
(As a separate issue) The use of the word "development" on this wikipage seems to be slightly incorrect or non-standard.
On this wikipage, the development of a convex polyhedron is defined as its underlying metric space
In Alexandrov's book, a development is a polygonal set in the plane with edge gluings (pp.49-50), the quotient space (modulo gluings) of which has a natural metric space structure (p.51). A convex polyhedron has many developments associated with it. (pp.52-53) Each development, as a metric space, is isometric to the convex polyhedron.
The Connelly book review, which is the only ref given on the page for this, does not give precise definitions. The closest it comes is to say that a development of a convex polyhedron is a polygonal set in the plane obtained by cutting, which is certainly compatible with Alexandrov's precise definition. It also says that this is "essentially just the intrinsic metric surface" [emphasis mine] of the polyhedron, which seems to be the inspiration for the definition on this wikipage.
This wikipage suggests that Alexandrov's above definition of a development is actually called by Net (polyhedron).
I think this should be clarified; I do not know of anywhere where the presently given definition is made.
Actually, the statement of Alexandrov's existence theorem is less technical and easier to understand when made following his definition as given above: if a development (as polygonal set in plane) has sum of angles < 2π at each vertex, and if the Euler formula V-E+F=2 holds, then it arises from cutting of closed convex polyhedron (p.99). I think this is quite more accessible than the present wiki condition of having an abstract metric space which is geodesic, homeomorphic to a sphere, and locally Euclidean except for a finite number of cone points of positive angular defect summing to 4π. This is already partly present on the wikipage in section "Limitations"; I would strongly suggest making it fully explicit and moving to the main "statement of theorem" section - especially since the setup for it already has a rather good informal description there. This would also match the given Connelly ref (fourth paragraph), although the V-E+F condition is there formulated in terms of total angle 4π. (Actually, I don't know of where the present wiki condition is given as a formulation of Alexandrov's theorem. It seems to have unnecessary complications. The 4π condition is redundant by Descartes theorem, and I suspect that any polyhedral metric on the sphere is geodesic, but I am not 100% sure of the latter.)
The title of wikipage is Alexandrov's uniqueness theorem but his theorem is both existence and uniqueness, and it seems most text in Statement of the theorem section is relevant only to existence. Perhaps the title should be changed to better reflect the contents. Also, it seems that Pogorelov and Volkov have a more general and equally elementarily stated uniqueness theorem: two closed convex sets are congruent if and only if their metric space structures are isometric (footnote 45 on p.138). I do not know if the proof uses Alexandrov's more restricted theorem, but since proofs are not discussed on this wikipage anyway, perhaps it is better to make the Pogorelov-Volkov theorem a centerpiece result.
Page numbers are from Alexandrov's book. The above talk page comments about the Weyl problem paragraph are separate from those here. Gumshoe2 (talk) 17:11, 20 February 2022 (UTC)

If you have a better word for "metric space of surface distances" than development, I'd like to hear it, but Connelly's "The developement is essentially just the intrinsic metric surface of P" seems unambiguous to me. I think your suggestion that we can only define such spaces in a non-intrinsic way, by gluing together polygonal subsets of the Euclidean plane, is as un-mathematical as would be a suggestion that, say, we can only define vector spaces after we specify a basis and use coordinates in those bases. It is a way that we can define them, sure, but one that introduces all sorts of unnecessary and complicating information. For one thing, it requires a proof that all locally-Euclidean-with-cone-points metrics have nets, not in evidence (although I imagine that the star unfolding or source unfolding generalize to this setting). As for what to title the article: plenty of sources in the published literature call it "Alexandrov's uniqueness theorem". The Wikipedia standard is to follow the literature, not to make up new names for ourselves when we quibble with the accuracy of the usual names. —David Eppstein (talk) 20:29, 20 February 2022 (UTC)
Going through my listed points: #1 is unaddressed; #2 seems to be barely disputed (an openly informal "essentially just" in a book review hardly seems like the greatest source to point to on this); #3 is not my suggestion, it is what is given in Alexandrov's book (is he, of all people, un-mathematical?) and implicitly in the given Connelly ref; my point in #3 about how the given formulation of the theorem is not obviously the same as that in the immediate literature is unaddressed — as you have just now said yourself, the connection is "not in evidence" given that I am referring to how Alexandrov formulates things in the book where it is proved; for #4 I hardly suggested to invent a new name. When I look in the literature for "Alexandrov's uniqueness theorem" I see reference to his uniqueness theorem, not to his existence & uniqueness theorem. Maybe you can point me to where the opposite is the case; it is not in the Connelly ref, or in any of the others given from what I can see. Gumshoe2 (talk) 20:53, 20 February 2022 (UTC)
To be clear, I am saying that the presently given version of Alexandrov's existence theorem may well be true, but I do not know if it is and cannot verify it by looking at the given literature. It does not seem to be obviously the same as what is proved in Alexandrov's book (you seem to agree yourself, "not in evidence"), nor is it stated in Connelly's review. Gumshoe2 (talk) 21:06, 20 February 2022 (UTC)
We are not going to get very far if you insist on sentence-by-sentence responses to your diatribes, to which you respond with even-longer diatribes, and then infer from my refusal to engage that I must agree with everything I did not explicitly refute. That is not a reasonable way of engaging in discussion. If I did engage in that form of discussion, it would lead to a diverging sequences of ever-lengthening responses. I am tempted to leave it at that, and stop engaging with you altogether, as not worth the waste of my time when you are obviously more interested in arguing than in improving our articles. But on the off-chance that you can behave better, here's one last attempt at a partial response:
The coverage of this topic in Geometric Folding Algorithms suggests that the intrinsic formulation is how this was covered by Pogorelov. GFA starts with that, but then goes into a formulation where the surface is glued together from multiple polygons (this is NOT the same as the formulation you are trying to push where only a single polygon is glued to itself; for instance, it is much easier to find multiple-polygon gluings for the faces of convex polyhedra, while it is still a major open question whether convex polyhedra can be formed by gluing a single polygon so that the glued edges all lie along edges of the polyhdron). Geometric Folding Algorithms also calls the result "Alexandrov's theorem" without the uniqueness part, but I think there are lots of Alexandrov theorems (and even multiple Alexandrov uniqueness theorems, complicating searches). This theorem is called the "Alexandrov uniqueness theorem" for instance by Igor Pak in the 2008 _Monthly_. Pak gives yet another formulation: that the surface metric of a convex polyhedron cannot be shared by another non-congruent convex polyhedron. This formulation completely avoids the existence part of the problem, and also avoids the need to unfold anything; it just uses polyhedral metrics as they are. Some other sources call it "Alexandrov’s gluing theorem" but that presupposes that you are starting with unglued polygons and gluing them, which is not an essential part of the theorem. —David Eppstein (talk) 21:28, 20 February 2022 (UTC)
Thank you for these references. It seems like Geometric Folding Algorithms would be a very valuable ref to add to the page. Some comments:
I see that GFA states their theorem 23.3.1 with the same polygonal set approach as in Alexandrov's book, which I think supports what I said earlier about it being a good approach to use (as well as not being "un-mathematical"). It looks like I caused confusion with my overly vague word choice "polygonal set", by which I meant multiple polygons (as in Alexandrov's book definition of "development"). Apologies!
GFA quotation of Alexandrov and Pogorelov's statements of the Alexandrov theorem on p.345 is very satisfactory. So I suggest the ref for the present wikipage theorem statement be altered to GFA or Pogorelov's book, since I still have not been able to find this formulation in Alexandrov's book or Connelly's review. (Note that GFA's citation of Alexandrov's book, as opposed to his original research article, is for the polygonal set formulation, not for the formulation now on the wikipage.)
Pak's article uses phrase "Alexandrov uniqueness theorem" and proceeds to quote the uniqueness half of the existence & uniqueness theorem. I do not understand how this shows that Alexandrov's existence theorem is ever referred to as "Alexandrov uniqueness theorem" or part thereof.
Also, I have been very surprised and a little disturbed by the aggressiveness of your responses here and in your first edit summary on the wikipage. I'll leave this page and talk page alone for a while. I hope you will revisit my points #1 and 2 above at some point in the future; mathematicians like Connelly usually don't say "A is essentially just B" when they mean for B to be a definition of A. The Weyl problem paragraph is also still problematic for the reasons given in the previous talk page section. Gumshoe2 (talk) 01:25, 21 February 2022 (UTC)
A couple more references: Guo and Tian, Proc AMS 1992 formulate this theorem as "any spherical cone metric of positive curvature at the vertices is isometric to the boundary of a spherical convex polytope that may be degenerated to be the doubling of a spherical polyhedron. Furthermore, the corresponding spherical convex polytope is unique up to isometry." Fillastre and Izmestiev formulate it a little more generally as "Let g be a metric of constant curvature K with conical singularities of positive singular curvature on the 2–sphere S. Then (S,g) can be realized as a convex polyhedral surface in the 3–dimensional Riemannian space-form of curvature K. The realization is unique up to an ambient isometry." Both credit to Alexandrov, and Fillastre and Izmestiev call it "famous", but neither give it a specific name. Note the complete non-mention of gluings. —David Eppstein (talk) 22:55, 21 February 2022 (UTC)

@David Eppstein As an alternative to folding 4 hexagons into an octahedron as shown in the picture in this article, the same hexagons can also be folded into a non-convex kiscube (Kleetope formed by erecting a square pyramid on each face of a cube) with equilateral triangle faces. You can get back and forth between the two by creasing each octahedron face along each median, and then trying to flatten the original octahedron edges while making "ridge" folds along the crease line from the midpoint to vertices of each octahedron face, and "valley" folds along the crease lines from each face midpoint to side-midpoints. This might be a fun extra counterexample to use, though it might be easier to make a paper model of than draw with a computer. –jacobolus (t) 09:29, 18 January 2024 (UTC)

Is there a source for this? —David Eppstein (talk) 16:55, 18 January 2024 (UTC)
You can find a picture in my paper here,
Rus, Jacob (2017), "Flowsnake Earth", in Swart, David; Séquin, Carlo; Fenyvesi, Kristóf (eds.), Bridges Waterloo Conference Proceedings, Bridges 2017, Waterloo, Ont., Canada, 27–31 July 2017, Phoenix: Tessellations Publishing, pp. 237–244
"With four hexagons, we can connect a hexagon to each of the other three along two adjacent edges. [...] If we crease and fold these flat hexagons into a polyhedron, we can make either a (non-convex) tetrakis hexahedron (a shape formed by gluing a square pyramid to each face of a cube, see Figure 6), or an octahedron, depending on where we place the creases. Unfortunately neither of these shapes is easy to visualize from line drawings; I urge readers to make paper models."
though I wouldn't claim this to be an original idea. –jacobolus (t) 17:46, 18 January 2024 (UTC)
Ok, added. —David Eppstein (talk) 19:31, 18 January 2024 (UTC)
Tangentially related, @David Eppstein you might enjoy Purser, R. James; Rančić, Miodrag (2011), "A standardized procedure for the derivation of smooth and partially overset grids on the sphere, associated with polyhedra that admit regular griddings of their surfaces. Part I: Mathematical principles of classification and construction", Office Note 467, NOAA. This is one of the more complete sources I know about surface spaces of polyhedra that can be folded from squares or equilateral triangles (therefore promoting the drawing of regular grids); see the last few pages full of colorful figures. These NOAA scientists are interested this topic because grids of such polyhedra can be conformally mapped to the sphere, which can then be used for solving various differential equations in weather prediction etc. –jacobolus (t) 09:48, 19 January 2024 (UTC)

The section in which the theorem is stated is long and meandering, even if it provides important background information.
It would be good if someone knowledgeable about this subject can include in the article a terse statement of the theorem. (With all relevant definitions somewhere nearby, but without forcing the reader to plow through a ton of text before the theorem is ever stated.)
— Preceding unsigned comment added by 2601:200:c082:2ea0:e89a:bdcb:d077:e35d (talk • contribs) 04:31, 30 January 2024 (UTC)

Spherical 2-manifolds that are locally Euclidean except for finitely many points of positive angular deficit, summing to 4π, can be represented geometrically either as the surface of a convex polyhedron in Euclidean space or as a double-covered Euclidean convex polygon, and this representation is unique up to Euclidean congruences.
But I think it would make the article much less readable to non-experts to state it in that way before the meandering definition of terms. —David Eppstein (talk) 07:47, 30 January 2024 (UTC)
One thing that might help could be more pictures trying to show e.g. why a vertex is considered cone-like, or how at a vertex there is some behavior which differs compared to the Euclidean plane; for example we might try plotting a polar coordinate system originating from a point near a vertex and showing how two geodesic lines emanating from the same point can cross on the other side of the vertex. Figuring out what to draw and how to make it clear isn't entirely trivial though. –jacobolus (t) 09:34, 30 January 2024 (UTC)
I added another illustration of an inflated double-covered square, showing its cone-like corners. —David Eppstein (talk) 20:52, 30 January 2024 (UTC)
That picture is nice. I wonder if there's a freely available higher resolution image of the same idea anywhere. –jacobolus (t) 22:31, 30 January 2024 (UTC)
If one could be found it could also be a replacement at paper bag problem, where I took it from. —David Eppstein (talk) 08:11, 31 January 2024 (UTC)
@Dedhert.Jr sorry I slightly misinterpreted what your edit did. I saw the multi-image and I got confused with the side-by-side folded octahedron picture (which is just a single image). I made that one a bit narrower; my experience is that once floating images get wider than about upright=1.5, on narrower viewports they start getting more likely to cause layout wonkiness. Sorry if my inaccurate revert edit summary was confusing. I don't think the square pillow shape image needs to be moved to the left. It's okay if a stack of a few images spill out of their immediate section as long as they don't start colliding with images in the following section. There are generally fewer layout issues with images going down the page on the right than there are with floating images sandwiching text between them. The sandwich style leads to problems when the viewport gets narrower.
Conceivably we could try to make some more dedicated illustrations for this article and put some of them as block-level elements in the text instead of as floating images, but given the current layout I think floating to the right looks okay when I try resizing the window to arbitrary widths. I added a defensive {{clear}} template before the references just in case; it rarely if ever has an effect but it also doesn't hurt anything. –jacobolus (t) 07:23, 1 February 2024 (UTC)
@Jacobolus I don't mind that actually. It is merely a problem of image positions, and I prefer to find the comfort of their appearances beyond the writings—that was the reason I tried to avoid the possibility of pushing the images down each other. I do think that every user has a different feature on their screen, and I think it depends on the zoom screen as well. Hopefully, there is an alternative option to solve this. Dedhert.Jr (talk) 13:20, 1 February 2024 (UTC)
I'm happy to leave the final word about this to David Eppstein; my contributions to this article are pretty trivial. –jacobolus (t) 15:49, 1 February 2024 (UTC)

In an unrelated discussion elsewhere with User:GreenLipstickLesbian, User: Gumshoe2 has brought up the question of whether this article is appropriately titled, noting that only half of the theorem is uniqueness and the other half is existence, and suggesting instead the titles "Alexandrov's existence and uniqueness theorem" (more accurate but a little cumbersome and not supported by sources that use that name) or "Alexandrov's theorem (convex surfaces)" (maybe too general as Alexandrov is only about polyhedra and maybe problematic as a disambiguated title that is not ambiguous). 
Maybe this is a better place to hold that discussion. Can we find a name that is simultaneously accurate, supported by use in published sources, and not overly cumbersome (pick any two)?
To get things going, I agree with the accuracy complaint but I don't think that is necessarily a big problem as many things have inaccurate names that are nevertheless their names. However, I don't have a strong opinion that the name should stay as it is, and it's reasonable to hope that it's possible to find a better name. Anyone have more suggestions for that better name? One idea from arXiv:2212.10479 is "Alexandrov's embedding theorem". Another is "Alexandrov's theorem on polyhedra".
Another possibility would be to choose both a different name and a different focus of the article, something like existence and uniqueness of convex surfaces, and to expand some of the "related results" section into that refocused article. Although that might make sense I'm not convinced I'm the right person to do that expansion; I'm much more familiar with the polyhedral material than its generalization, am unsure of the correct formalization of non-embedded convex surfaces that are not smooth but whose non-smoothness is not confined to isolated points (like say the edge of a cone), and cannot read sources like the Russian-language paper of Pogorelov cited but not detailed here. —David Eppstein (talk) 00:13, 9 May 2025 (UTC)

The current name seems okay, as does "Alexandrov's theorem on polyhedra". If someone wants to reframe the article they should make a more complete proposal. As an aside, machine translation is passably good for translating math papers from Russian to English. You can copy-paste from the PDF at https://www.mathnet.ru/eng/sm6003 into Google Translate or the like. –jacobolus (t) 01:17, 9 May 2025 (UTC)
Thanks for opening discussion, I appreciate it.
One observation, which I feel a bit foolish for not noticing until now: "Alexandrov's uniqueness theorem" as a title is itself ambiguous, since he has a completely different famous uniqueness theorem, saying that the round spheres are the unique compact surfaces in Euclidean space of constant mean curvature. Some examples of explicitly calling this "Alexandrov's uniqueness theorem": [4] [5] [6] [7] [8]
In terms of scope, I think it's very suitable to have a page solely centered on Alexandrov theorem, with no need for major expansion or change. If you don't object, I can expand the second paragraph of the 'related results' section slightly, I think to about twice its current length — but I think it would be hard combine well with the other material if it's pushed past that.
For title, I like any of the following, in increasing order of how much I like them:
Alexandrov's embedding theorem
Alexandrov's theorem (convex surfaces)
Alexandrov's theorem on polyhedra
Alexandrov's theorem (convex polyhedra)
I agree with the criticism of my previous suggestion "Alexandrov's existence and uniqueness theorem." I like "Alexandrov's theorem (convex polyhedra)" the most since I think it is both non-cumbersome and (the most) precise, and clearly compatible with virtually the whole literature. As per your comment "disambiguated title that is not ambiguous", bear in mind the page Alexandrov theorem (which could be disambiguated to "Alexandrov theorem (mathematical analysis)" or "Alexandrov theorem (convex analysis)").  Gumshoe2 (talk) 03:43, 9 May 2025 (UTC)
Please do go ahead and expand that related result paragraph. The existence of another theorem that can reasonably be called a uniqueness theorem is a good reason for finding a better title for this article (just like the fact that "Alexandrov's theorem" without qualification often refers to something else is a good reason for not calling it that).
Incidentally, I removed the development wikilink that you wanted removed some time ago. Looking at it again, I agree that it is not really the same concept. I don't think we have an article on the concept called development here, so maybe it's better left unlinked. —David Eppstein (talk) 05:24, 9 May 2025 (UTC)
Thank you! Unfortunately I have had no time to edit recently, but once I do I'll follow through with the edit of second paragraph of Related Results. Gumshoe2 (talk) 19:41, 4 June 2025 (UTC)