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Problem of Apollonius

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This peer review discussion has been closed.
I've listed this article for peer review because I believe it to be ready for FAC. It covers an old but important problem in geometry, first solved by Apollonius of Perga and a test-bed for many mathematical methods, particularly in 19th century with the resurgence of geometry. I've been working on it for roughly five months, and others have contributed significantly as well. Several daughter articles have been expanded, numerous figures and tables have been created, and the article has been referenced thoroughly.

I would especially appreciate any advice or insight on the accessibility of the article, and how to make it more suitable for FAC.

Thank you for your help in improving the article, Willow (talk) 18:52, 24 June 2008 (UTC)[reply]


Comments by Jakob.scholbach (talk) 17:26, 26 June 2008 (UTC)[reply]

  • I think this is a nice article. It's very detailed. I have to confess that I didn't attempt to dive in the actual calculations and so on. So, here are some superficial thoughts:
Thank you, Jakob! I really appreciate you taking the time to do the review. Unfortunately, I've run out of time; I have to leave to help with my sister's wedding, and I can't follow up immediately on all of your suggestions. I'm leaving a few explanatory notes here, and hopefully someone else will be able to address your concerns. If not, I can always fix it once I return! Thanks again to everyone helping this little article, Willow (talk) 16:13, 27 June 2008 (UTC)[reply]
  • My main problem: (but I don't know what to do about that): the article tends to be closer to a elementary geometry textbook section than to an encyclopedic article. (The applications section is a welcome exception to this feeling).
That may result from the type of material being covered. I did try to cover aspects of the problem that aren't merely solution methods, e.g., discussing applications and placing it in a historical context, but the solution methods do form most of the story. Willow (talk) 16:13, 27 June 2008 (UTC)[reply]
I suppose the relevant question for us to ask is, "What should this article be about?" It needs to tell the reader what the Problem of Apollonius is, why it's interesting, how to solve the problem, and what the history of the solutions is. "What", "why", and "history" are relatively short. "How" is quite long.
Right now, "how" is broken up in two different ways: Either by solver (van Roomen, Viete, Gergonne) or by method (algebraic, inversion, Lie sphere). I think we need to pick one method of organization and stick with it. In my opinion that ought to be organization by method, because for a reader who wants to know how to solve the problem, the method used is more relevant that who discovered the method. The history can be put in its own section. I also have the feeling that it might be a good idea to move some of this information to subarticles. For instance, Solutions to the problem of Apollonius using circle inversion, Solutions to the problem of Apollonius using algebra, and Solutions to the problem of Apollonius using elementary geometry. Even if we decide against that move, it would be good if the article were organized so that it were possible. Ozob (talk) 17:53, 27 June 2008 (UTC)[reply]
  • the lead is pretty long, and seems to cover the article content in a slightly unbalanced manner: persons (and their names) take lots of space, but the actual solution methods only get one sentence "by a variety of geometric and algebraic methods, including transformations such as circle inversion", which seems too little from a glance at the TOC
Yes, I was concerned that the lead should convey the point of the problem clearly to non-mathematicians. But you're right, it does seem to give short shrift to the methods. Let me think about this, or others can take the plunge and do a re-write. Willow (talk) 16:13, 27 June 2008 (UTC)[reply]
  • "each of which include or exclude the given circles in diferent ways (Figure 1)" - this is not clear to me. (also a typo)
Typo fixed. When you say that it's not clear to you, do you mean that the sentence does not make sense, or that the figure does not seem to have this property? Ozob (talk) 23:48, 26 June 2008 (UTC)[reply]
I wanted to say that I don't understand the meaning of the sentence. Jakob.scholbach (talk) 09:47, 27 June 2008 (UTC)[reply]
It means "encircle/surround" versus exclude; the solution circle "includes" a given circle by surrounding it completely. It excludes it if the intersection of the two circles is the empty set. Willow (talk) 16:13, 27 June 2008 (UTC)[reply]
I changed it to "Each solution circle either contains or does not contain each given circle, and the eight solutions to Apollonius' problem correspond to the possible combinations of containment and non-containment (Figure 1)." Ozob (talk) 16:16, 27 June 2008 (UTC)[reply]
  • "This theorem was rediscovered three centuries later" - were they lost in between?
No, Steiner, Beecroft, Soddy, etc. just didn't do a very thorough literature search, apparently. They derived the same theorem independently of Descartes. Willow (talk) 16:13, 27 June 2008 (UTC)[reply]
  • "they [the circles] may even be points (circles of zero radius) or lines (circles of infinite radius)" - this may be distracting at this very early place. perhaps better in the paragraph "Apollonius' problem can be generalized in several ways ..."
That's a very good idea. I'll try to work it in somehow. Willow (talk) 16:13, 27 June 2008 (UTC)[reply]
  • "Candidate transformations must one Apollonius problem into another" lacks a verb.
I verbed it. Ozob (talk) 23:48, 26 June 2008 (UTC)[reply]
  • A general comment about the images: some are pretty complicated to digest on first sight (esp. the very first one) and hence have very lengthy captions. This suggests (to me) "separating" the images such that their meaning is conveyed in smaller steps. Also, it would be a plus if the three starting-circles would always have the same color in all images. Or use a dotted pen or something. Figure 2, e.g., is incredibly hard to figure out without reading the caption.
I just realize that many images do use the same colors. So, forget this comment. But still, none of the images jumps into my mind effortlessly. Jakob.scholbach (talk) 09:47, 27 June 2008 (UTC)[reply]
Sadly, as Euclid pointed out, there are no royal roads to Geometry. Readers will be required to think here, and to read captions. If anyone could improve the image to make it more immediately intelligibl, I'd be grateful, but I did the best I could with the skills I have, and it seems adequate. Willow (talk) 16:13, 27 June 2008 (UTC)[reply]
  • Figure 1 would be clearer (I guess) if the circles would not be shaded, but just outlined. It looks very nice, though.
It's great, no? That's the work of Melchoir. :) Willow (talk) 16:13, 27 June 2008 (UTC)[reply]
  • I don't understand why the section "Pairs of solutions by inversion" is placed under statement and motivation. This seems to be part of the solution process, right?
The interest of this section is that it shows why the number of solutions is usually even. There may be a better location for it, though. Ozob (talk) 23:48, 26 June 2008 (UTC)[reply]
It's helpful in some of the arguments below to know that solutions come in pairs. That's why it's introduced so early. It also warms the reader up for the inversive techniques later, so that they don't get hit with it all at once. I'd be open to moving it, but I did put some time and care into placing it where it is now. Willow (talk) 16:13, 27 June 2008 (UTC)[reply]
  • The special cases table 1 could perhaps be moved to the subpage? Also, do you count the CCC problem as the special case or as the general case? From a measure-theoretic point of view, CCC is the general case, but actually all cases show up under "special cases".
I've tried to finesse this point, but I've taken the position most easily understood by non-mathematicians, that the C in CCC consist of circles that have finite, nonzero radius. Hence C≠P and C≠L. To help in understanding some of the other work, however, I want to open readers' minds to the idea that lines are circles of infinite radius, and points circles of zero radius. Willow (talk) 16:13, 27 June 2008 (UTC)[reply]
  • Figure 7 would benefit from showing the (constant) centers of the points. Perhaps move the captions somewhere else then? Its caption "The tangency of a set of circles is preserved if their radii are changed by the same amount" is not that clear. If you also draw two radii (whose sum is constant over time), the whole caption could be simplified and would be more intuitive.
That's a good idea, thank you! I'll work on that. Willow (talk) 16:22, 27 June 2008 (UTC)[reply]
  • The caption inside Fig 9 (esp. the indices) are very small.
People do keep removing my pixel counts but here goes again. I've enforced the scale to be 350px, which should be legible to most readers. Willow (talk) 16:22, 27 June 2008 (UTC)[reply]
  • "being self-similar and having a dimension d that is roughly 1.3" - is it known exactly? also, a word explaining fractal dimension would be good, I guess.
It doesn't seem to be known exactly. Ozob (talk) 23:48, 26 June 2008 (UTC)[reply]
I believe that it's known only from computations, not analytically. Conceivably, it might depend on the starting configuration of circles, too, although I guess not. Willow (talk) 16:22, 27 June 2008 (UTC)[reply]

Comments by Geometry guy

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Fabulous article! I read it thoroughly again yesterday morning bright and early (UTC) and I understood it! Given my username, this may seem unsurprising, but in fact I have great difficultly getting my mind around these (sometimes literally) baroque solution methods, and when I first interacted with the article, I found it hard going, the Gergonne solution especially. Much credit to the editors who have added so much detail and clarity since then.

Anyway, this is peer review, so constructive criticism, not praise, is what editors are after, right? And since there is no obligation to follow my suggestions, and no icon to be awarded for success, I will be as critical and opinionated as I can be :-)

  • I agree with Jakob that there is a mismatch between the emphasis of the lead and the emphasis of the article, and that both need fixing to bring the article into balance: the lead has too little on the solution methods, while the rest of the article has too little else. On the other hand, the length of the lead is fine for an article of this size; it seems longer than it is because of the long wide lead image. At this stage of development, I believe it is best to get the body of the article right first; with luck, the lead will then write (right?) itself :-).
  • I agree with Jakob that more work is needed to make the article truly encyclopedic in style. In addition to the general issue of balance raised above (see the next point), there are also specific lapses, such as:
    Many geometrical constructions such as bisecting an angle or a line segment can be done "by eye", or improved iteratively from an approximate solution. Apollonius' problem is more difficult in this respect; it is difficult to see "by eye" where to place the center of the solution circle or how large to make its radius.
    According to whom?
  • I agree with Ozob that one way to improve the balance of the article would be to provide a separate history section, and organise the solution methods by approach. I realise that this would destroy a nice feature of the current article: at present the technical methods material is broken up by asides on historical context, personalities, applications. However, ultimately, the reader who wants to study the solution methods is going to have to concentrate, while other readers are more likely to prefer a separate discussion of history, people involved and relations to other mathematical ideas. I think this would result in a more encyclopedic approach and make the lead easier to write.
  • More specific comments (please don't interleave replies here - I'll just get confused :)...
    1. First sentence: it reads two ways. At first I wanted to add a comma after "the problem of Apollonius", but then I saw the other reading as a long noun phrase. However, if I can be dogmatic for a moment (with a touch of Orwellian irony): long noun phrases bad.
    2. Statement: I reordered the first paragraph; is that clearer?
    3. Limiting cases: the definition of "limiting cases" seems a bit, erm, limiting. From the point of view of Moebius geometry (which is the natural symmetry of the problem), the degeneration of a circle to a line isn't really a limiting case; on the other hand the case that two or more of the initial circles (etc.) become tangent clearly is a limiting case. Obviously we have to rely upon what the sources say here, but there surely must be sources which make this point. In terms of the algebra, the limiting cases are when some of the quadratic equations have repeated roots.
      As an aside, with an eye on the "comprehensiveness" criterion, surely someone must have studied the Apollonius problem over the complex numbers, where there should almost always be 8 solutions counted with multiplicity. The missing solutions in the real picture are actually complex conjugate. (Again, sources needed for that... although it is mentioned briefly later on.)
    4. Mutually tangent case: this is the first appearance of the signed radius issue (the radius and curvature can be negative). Also, is it relevant to say that Philip Beecroft was an amateur mathematician? The distinction is perhaps not so clear at this time in history.
    5. Solution methods: this first paragraph makes the case for a separate history section, in my view.
    6. Intersecting hyperbolas: although this has been mentioned in the lead, the reader may be surprised to discover that "modern" here means 1596 (date from O'Connor and Robertson)! At some point, the reader needs to be reminded that "modern" means "post classical antiquity" in this context.
    7. Viète's reconstruction: the number of lemmas involved is not relevant in my view; the use of lemmas is a matter of convenience only.
    8. Algebraic solutions: I've attempted to clarify the signs issue. Also the fact that no configurations have 7 solutions needs a cite (I think you have one somewhere else, but don't remember).
    9. Lie sphere geometry: the signed radius issue crops up again here, where it would be much nicer to allow the r's to have a sign to make it more obvious that the "unusual product" is a symmetric bilinear form.
    10. Gergonne's solution: now that I finally understand it, I agree it is pretty elegant! However, a single 1929 textbook is probably not sufficient support for the assertion that it "is widely considered to be the most elegant".

That's all folks! Thanks for your attention. Geometry guy 10:18, 28 June 2008 (UTC)[reply]

Comments from Ealdgyth (talk · contribs)

  • You said you wanted to know what to work on before taking to FAC, so I looked at the sourcing and referencing with that in mind. I reviewed the article's sources as I would at FAC.
Hope this helps. Please note that I don't watchlist Peer Reviews I've done. If you have a question about something, you'll have to drop a note on my talk page to get my attention. (My watchlist is already WAY too long, adding peer reviews would make things much worse.) 15:25, 28 June 2008 (UTC)
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