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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.
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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
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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
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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).
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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
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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
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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
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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
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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
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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.
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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
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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.
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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?
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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
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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.
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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. —
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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. —
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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.
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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.
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The standard double bubble obviously exists. The proof proves that nothing is better. Therefore there is an optimal solution, the standard double bubble. —
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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. —
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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".
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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.
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of the "Statement" section outlining this issue and its solution in very general terms, with an expanded reference to Morgan's book. —
1162:. You know, the proof that 1 is the largest integer? Obviously, 1 is an integer (you said "obviously the double bubble exists"). If
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Lead: Try to include an extra sentence about the proof and maybe about generalizations to make it a better summary of the article?
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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?
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Much better! I think there's just the division by zero (maybe) and the "solution" vs "competitor" or "trial solution" issues. —
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Ok, fine, I also added another sentence at the end of the proof section clarifying how the proof of existence comes into it. —
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Ok. There's just one thing now: I'm not convinced "solution" is the right word for a non-minimal competitor double bubble.
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by
Steiner symmetrisation, but Perron's paradox shows that his reasoning is not valid unless there is an existence proof. —
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I think that's all. Other than perhaps talking about existence, mostly very small issues. Thank you for the new image! —
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Curve shortening flow: As the volumes are not constant in this flow, I am not convinced that this is related enough.
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Yes? So it is a lemma of White. We cannot copy the exact wording of our sources; that would be plagiarism. —
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be possible. Equal areas is very easy to draw but non-equal but less unbalanced might be more informative. —
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The way I had in mind for solving for the middle radius was to rearrange the given radius formula into
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The new one is excellent and shows very well the interplay between the volumes, the radii and 2pi/3. —
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Will take this one, shouldn't take too long unless my other reviews on hold all come back at once :) —
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Gaussian double bubbles: you could consider naming the authors and say that they proved something.
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Some page numbers could help, but I'll comment on that in detail in the prose review.
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Should have mentioned 3D isoperimetry much earlier. Added to lead and statement. —
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is also accurately described as division by zero, so I withdraw that comment. —
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Added a paragraph-length sentence about the ingredients of the proof. —
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Updated to 5th ed and added Google Books, ISBN ids, and page numbers. —
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Does anyone know what software was used to create the illustration in
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Changed to "candidate surface", as part of some other copyedits. —
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Note: this represents where the article stands relative to the
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External links are OK, even if MathWorld is a bit out of date.
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302:? Thanks,
241:Mathematics
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96:check views
1443:Categories
1171:1 we have
847:Very nice.
514:ref layout
370:Authorship
356:GA toolbox
37:under the
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137:Knowledge
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1099:Done. —
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54:Review
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1312:Good.
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1089:Good.
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