# Talk:Double bubble theorem

I'm very glad that there's a photo that accompanies the article.  Most folks appreciate abstract ideas better when they can visualize them.  Good illustrations are hard to come by.
However, this photo does not show a minimal surface.  The smaller bubble should push/bend into the larger bubble, but the illustration shows a flat surface between them.  I mention it because this is a common misconception, and the illustration perpetuates that mistaken mental image.
Does anyone know what software was used to create the illustration in File:Double bubble.png?  Thanks, 24.240.67.157 (talk) 01:33, 14 June 2015 (UTC)

I don't know, but maybe a better question is: what software can we use to get a good image that doesn't have this problem? I agree that it is a problem. —David Eppstein (talk) 02:33, 14 June 2015 (UTC)
Update: I replaced the bad illustration by a prettier photo. The two bubbles in the photo are quite unbalanced in size, making it more obvious that the surface between them is not flat. —David Eppstein (talk) 04:53, 14 June 2015 (UTC)

This review is transcluded from Talk:Double bubble theorem/GA1. The edit link for this section can be used to add comments to the review.
Reviewer: Kusma (talk · contribs) 15:11, 18 June 2022 (UTC)
Will take this one, shouldn't take too long unless my other reviews on hold all come back at once :) —Kusma (talk) 15:11, 18 June 2022 (UTC)


### Progress box and general comments
Let's get started. A very nice topic (probably more accessible than most of geometric measure theory, and there is a nice story to tell about the proofs). I have seen Frank Morgan give a talk about it ca. 20 years ago, but I don't remember much of what he said, just that he enjoyed advertising his undergraduates :) 

Images are free and appropriately licensed. Might be nicer to have one where the volumes are closer to equal, and/or a diagram, but the images work.
I like the lead image, because (1) it is pretty, (2) it shows a real double bubble without wind distortion making it non-spherical, and (3) you can clearly see in it that the central membrane is curved. I don't think we have a lot of other good double bubble photos on commons. But a 2d diagram should be possible. Equal areas is very easy to draw but non-equal but less unbalanced might be more informative. —David Eppstein (talk) 18:52, 19 June 2022 (UTC)
The new one is excellent and shows very well the interplay between the volumes, the radii and 2pi/3. —Kusma (talk) 11:00, 20 June 2022 (UTC)
All sources are high quality reliable sources. Formatting is fine except that the Frank Morgan book (which is probably the single best resource for the interested non-expert) should have an ISBN or OCLC or other data that helps to find it.
Updated to 5th ed and added Google Books, ISBN ids, and page numbers. —David Eppstein (talk) 18:52, 19 June 2022 (UTC)
Some page numbers could help, but I'll comment on that in detail in the prose review.
Standard copyvio tests are clear.
Stable and neutral.
More later! —Kusma (talk) 09:51, 19 June 2022 (UTC)


### Content and prose review
General mathematical comment: What you explain here is that if there is a minimal solution to the double bubble problem, it is the standard double bubble. What is missing completely is that there is such a thing as a minimal solution. (This is the content of Chapter 13 in Morgan's book). OK, your statement includes that, but none of the things you say about the proof contains existence of a solution to the problem. You should at least mention that it goes through a rather generalized definition of surfaces that then gets reduced again to ordinary surfaces by Taylor's theorem (13.9 in Morgan).
Existence of a solution is trivial: just surround the two volumes by disjoint spheres or cubes or whatever. I guess you must mean the fact that there is an area-minimizing solution, rather than a sequence of solutions that converges to but never reaches the minimum. I added a paragraph to the start of the "Statement" section outlining this issue and its solution in very general terms, with an expanded reference to Morgan's book. —David Eppstein (talk) 19:11, 20 June 2022 (UTC)
It is existence of admissible double bubbles that is trivial, but as I said, existence of a minimal one is far from trivial. (One simple related counterexample I know is Weierstraß' example that is mentioned in Dirichlet's principle). Famously, Steiner believed he had proved the optimality of the circle in the Isoperimetric inequality by Steiner symmetrisation, but Perron's paradox shows that his reasoning is not valid unless there is an existence proof. —Kusma (talk) 19:45, 20 June 2022 (UTC)
Since you seem to have missed this, and are responding as if I were arguing against your point, let me repeat: I added a paragraph to the start of the "Statement" section outlining this issue and its solution. —David Eppstein (talk) 20:39, 20 June 2022 (UTC)
It is better now.
Lead: Try to include an extra sentence about the proof and maybe about generalizations to make it a better summary of the article?
Added a paragraph-length sentence about the ingredients of the proof. —David Eppstein (talk) 19:29, 20 June 2022 (UTC)
Very nice.
Statement: This section kind of starts with observation (Plateau's laws) stated about soap bubbles and then cited to Taylor's paper about minimal surfaces (with a highly technical definition in typical terms of geometric measure theory). We then have Young-Laplace, which is physics motivation for the equation for the radii. Then we get a definition of the standard double bubble, and then the statement of the theorem seems almost an afterthought. Could you untangle this a bit? I'm concerned that it looks like using physics to prove mathematics. Also, it might be easier on the reader to cite Plateau's laws also to a more accessibly written reference.
Mentioned Taylor's role in-text, moved the Taylor footnote there, and sourced the general statement of Plateau's laws to Morgan instead. Added a little text to the paragraph on double bubbles distinguishing physical principles (pressure difference) from mathematical principles (the equation obeyed by the three radii). —David Eppstein (talk) 19:29, 20 June 2022 (UTC)
Fine, although there is an issue with "division by zero. Solving "1/r=0" isn't really a division by zero, it is a proper (or degenerate, depending on your point of view) limit case. But maybe I am splitting hairs here?
The way I had in mind for solving for the middle radius was to rearrange the given radius formula into 
  
    
      
        
          r
          
            2
          
        
        =
        1
        
          
            /
          
        
        
          
            (
          
        
        
          
            1
            
              r
              
                1
              
            
          
        
        −
        
          
            1
            
              r
              
                2
              
            
          
        
        
          
            )
          
        
        .
      
    
    {\displaystyle r_{2}=1{\Big /}{\Bigl (}{\frac {1}{r_{1}}}-{\frac {1}{r_{2}}}{\Bigr )}.}
  
 This really does give a division by zero. But I think putting this additional formula into the article explicitly would unbalance the coverage, giving unnecessary prominence to a minor technicality. —David Eppstein (talk) 17:55, 21 June 2022 (UTC)
In my head, "division by zero" is solving 
  
    
      
        0
        x
        =
        1
      
    
    {\displaystyle 0x=1}
  
, but you are right that 
  
    
      
        0
        =
        
          x
          
            −
            1
          
        
      
    
    {\displaystyle 0=x^{-1}}
  
 is also accurately described as division by zero, so I withdraw that comment. —Kusma (talk) 18:22, 21 June 2022 (UTC)
History: if you mention 2D double bubbles, you could also mention 2D isoperimetry here.
Should have mentioned 3D isoperimetry much earlier. Added to lead and statement. —David Eppstein (talk) 19:35, 20 June 2022 (UTC)
Good.
Generalization to higher dimensions: you could mention the group of undergraduates that includes Reichardt that did the 4D case a few years before he did the general case.
Done. —David Eppstein (talk) 19:49, 20 June 2022 (UTC)
Gaussian double bubbles: you could consider naming the authors and say that they proved something.
Are the names important for the article text? —David Eppstein (talk) 19:49, 20 June 2022 (UTC)
No, but possibly that the study led to a theorem.
Proof: Here we are missing the existence bit.
The standard double bubble obviously exists. The proof proves that nothing is better. Therefore there is an optimal solution, the standard double bubble. —David Eppstein (talk) 19:49, 20 June 2022 (UTC)
This is the fallacy exposed in Perron's paradox. You know, the proof that 1 is the largest integer? Obviously, 1 is an integer (you said "obviously the double bubble exists"). If n is the largest integer, then if n>1 we have n2 > n ("nothing is better"), so we must have n=1 . —Kusma (talk) 20:21, 20 June 2022 (UTC)
Ok, fine, I also added another sentence at the end of the proof section clarifying how the proof of existence comes into it. —David Eppstein (talk) 20:39, 20 June 2022 (UTC)
Ok. There's just one thing now: I'm not convinced "solution" is the right word for a non-minimal competitor double bubble.
Changed to "candidate surface", as part of some other copyedits. —David Eppstein (talk) 17:55, 21 June 2022 (UTC)
Works for me. —Kusma (talk) 18:23, 21 June 2022 (UTC)
Brian White: According to the source, it is an "idea by White, written up by Foisy and Hutchings".
Yes? So it is a lemma of White. We cannot copy the exact wording of our sources; that would be plagiarism. —David Eppstein (talk) 19:51, 20 June 2022 (UTC)
I'm concerned people might look for it in Brian White's works, but I won't argue.
Rest of the proof section is excellent.
(except for the "solution" issue above). —Kusma (talk) 11:11, 21 June 2022 (UTC)
Related problems: Is Sullivan talking about equal volumes or more generally?
More generally. Clarified. —David Eppstein (talk) 21:14, 20 June 2022 (UTC)
Good.
Kelvin conjecture: Would it make sense to try an "as of" date for the fact that we don't know whether Weaire-Phelan is optimal?
I'd prefer not. That tends to cause more difficulties than it solves: we would need to continually find more recent sources to update the date for which it remains unknown, and then well-meaning gnomes would keep updating the "as of" date to the present without finding those sources. If it ever gets solved, we would need to do a thorough search through the encyclopedia to find mentions in need of updates, regardless of whether they were dated or not. So it doesn't help the readers (because the gnome updates will mean that it contains no more actual information than an undated statement), doesn't help editors change the text if it ever needs changing, and causes ongoing maintenance and verifiability issues. —David Eppstein (talk) 21:14, 20 June 2022 (UTC)
Curve shortening flow: As the volumes are not constant in this flow, I am not convinced that this is related enough.
Anyway, looking at the source I think you should clarify what they say: the actual evolution of course goes to the smaller volume disappearing, but if you go and blow up the solution to constant volume anyway, it will look like the vesica piscis. This is explained better over at Curve-shortening_flow#Gage–Hamilton–Grayson_theorem.
Added rescaling to preserve area. —David Eppstein (talk) 21:18, 20 June 2022 (UTC)
You could make it clearer that the triple bubble problem has been done in 2D but it is open in 3D and higher, and Morgan even calls it "inaccessible".
Quote added. —David Eppstein (talk) 21:23, 20 June 2022 (UTC)
External links are OK, even if MathWorld is a bit out of date.
I think that's all. Other than perhaps talking about existence, mostly very small issues. Thank you for the new image! —Kusma (talk) 11:00, 20 June 2022 (UTC)

Much better! I think there's just the division by zero (maybe) and the "solution" vs "competitor" or "trial solution" issues. —Kusma (talk) 11:11, 21 June 2022 (UTC)
Happy now, promoting. —Kusma (talk) 18:23, 21 June 2022 (UTC)