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Four color theorem

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The article is currently listed as A class by WikiProject Mathematics, but I think it could use improvement in the "fine writing" category. Also, things like style and formatting. I'd like to know how it reads to someone unfamiliar with the topic, preferably not too much into mathematics. I think the topic itself is more approachable than most advanced math topics, so I would like a "layman" perspective.

Thanks, C S (talk) 06:51, 15 March 2009 (UTC)[reply]


I think it would add, if you could provide a visual illustration of the simpler theorem that a 2D map divided only by lines running all the way across it (or is some further caveat needed? which cross other lines only at isolated points and share no line segments), can be colored by just 2 colors. doncram (talk) 02:34, 22 March 2009 (UTC)[reply]
I think that's correct. I happen to have Illustrator open while doing some figures, so I'll throw that into the queue. Oh, and thanks for badgering someone else into reviewing :-) --C S (talk) 22:40, 26 March 2009 (UTC)[reply]

Comment: even before I start to read it, I have trouble with this article.

  • The first image is captioned: "Example of a four-coloured map". This is not a map in any normal sense of the word (though it looks good as an Impressionist painting). Surely, "map" should read "diagram"?
  • Likewise, the caption to the second image, the USA map, is confusing. We are told to "note" something which cannot be noted from the map, we are not told the nature of the problem which might have occurred, or why in this case it didn't. On the whole, narrative captions are not a good idea - save the explanations for the text.

Brianboulton (talk) 09:38, 22 March 2009 (UTC)[reply]

(Later) In response to a request from Doncram, I have tried to read this article from what you call the "layman perspective" Perhaps I am unduly dim; I got to the fourth paragraph of the lead, and couldn't understand any of it. I did try to read on in the hopes that some understanding might develop, but it didn't. This, and the points I raised earlier concerning the opening two images, make me doubt seriously if this article will have any meaning to those without a mathematical background. The irritating thing is that I can understand the concept, in general terms; it is the mathematical expression of it that beats me. Perhaps someone else will find it easier. I will leave the article here in the backlog in the hopes that it will be picked up. Brianboulton (talk) 00:29, 25 March 2009 (UTC)[reply]
Hi Brian. Thanks for your comments. This is the type of feedback I was hoping for. Good points about those captions. I'll change them later tonight. I think I have an idea of why things are confusing...so I'll make some further changes in hopes of increasing readability, and I'll let you know when I've done so, if you're interested. --C S (talk) 03:13, 26 March 2009 (UTC)[reply]
The captions are certainly better. Could I also suggest that, in the first line of the article, you link "plane" to Plane (geometry), and also that you say "a plane" rather than "the plane"? On another point, I think your attempt in paragaph 4 to explain the proof is misplaced at this stage. This level of detail belongs later in the article – which would make the lead a bit less of a problem to understand. Looking through, I see that the article is totally uncited, which will require attention if you are proposing to take it on to FAC. That's really all the help I can offer (I see that Finetooth is a braver man than I am). Good luck with it anyway. Brianboulton (talk) 17:02, 29 March 2009 (UTC)[reply]
The article is completely cited, so I don't know how you can read it and say it is "totally uncited". But I do appreciate your comments and suggestions. Thanks. --C S (talk) 21:39, 29 March 2009 (UTC)[reply]
The article has a list of reference works, but no in-line citations. For example, take the sentence: "The conjecture was first proposed in 1852 when Francis Guthrie, while trying to color the map of counties of England, noticed that only four different colors were needed." What book did this come from, what page reference? Same applies to the next sentence, and the one after that, etc. Brianboulton (talk) 23:53, 29 March 2009 (UTC)[reply]
There are lots of inline citations, and every reference in the reference section is cited at least once, which is why it is there. The sentence you mentioned is followed by a quote which has a citation. I think it's a little variable whether one puts the citation after the very first sentence or at the end of a small block of sentences, as in the example you listed. This example is at the very beginning of the history section, but I'm sure you will notice more citations as you read on further. --C S (talk) 00:16, 30 March 2009 (UTC)[reply]
I should note that not every sentence ought to be followed by an inline citation. I don't believe this is even recommended practice. In some cases, the sentence may not be contentious, or a wikilink in the sentence may provide the necessary sources in another in-depth article (so for example, in a sentence referring to what the Heawood conjecture is and when it was proved, since the link provides much more verification). In other cases (like the one above), there might be one source in the paragraph that suffices for the whole thing. Although not every sentence does not have an inline citation, every paragraph has at least one, excepting those in the summary of proof section (which lists one basic source at the beginning), and a few short paragraphs, which have links to more verification, as I mentioned. I guess you didn't notice there are also some inline citations in the lede for assertions that are not mere paraphrasings of material from later on in the article.
Anyway, I suppose some may want more referencing, but it certainly is not the case, as you seem to think, that inline citations are nonexistent. It seems like you've only read the lede, which is fine, since you gave plenty of feedback on that, which I appreciate. If you want to look over the rest of the article to point out how the references should be placed (versus the way they are now), feel free. --C S (talk) 00:38, 30 March 2009 (UTC)[reply]
Sorry to have been a long time over this. I missed your in-line citations because I have never encountered that format before. You have round about 20 of these in the article, but some of them are to whole books, e.g. Wilson 2000. Many of the works listed as references are not cited in the article. I don't believe that every sentence needs specific citation, but I do believe that your level of citation is inadequate. At the very least you should identify the pages of the Wilson book that you are referring to. Brianboulton (talk) 23:17, 6 April 2009 (UTC)[reply]
Small correction: Probably the problematic part of the article with respect to sourcing is the "false disproof" section. I've been wondering what to do about that, and I was hoping someone would comment on it. --C S (talk) 00:55, 30 March 2009 (UTC)[reply]

Finetooth comments: Here are a few comments from a non-mathematician.

Lead"

  • "any separation of the plane" - Would "a" plane be better than "the" plane?
  • "if one region is surrounded by 3 regions" - Numbers smaller than 10 are usually written as words per WP:MOSNUM.
  • "A number of false proofs and disproofs" - Would it be more clear to say "a number of false proofs and false disproofs"?
  • "In 1976 Kenneth Appel and Wolfgang Haken showed a particular set of 1,936 maps had two properties – at least one must be included in any map (the set is unavoidable) and each one cannot be part of the smallest counterexample (each is reducible)." - I have trouble following this. I think I understand it until I get to the last clause: "and each one cannot be part of the smallest counterexample (each is reducible)". This seems to mean that each of the two properties cannot be part of the smallest counterexample, but I don't know what is meant by "counterexample". A further question: in the parenthetical phrase "the set is unavoidable, does "set" refer to the set of 1,936 maps or to the set of two properties? You can see from my questions why I might be flummoxed by the next sentence: "This shows there is no smallest counterexample, and hence no counterexample at all." I find myself in the unfortunate position of not understanding the material well enough to suggest an alternative way of putting it.
  • Except for the sentences mentioned above, the rest of the lead seems clear.
  • As I glance down through the rest of the article, I realize that I'm not going to be able to say much that is useful. The article appears to be generally well-written and well-sourced, and well-illustrated. I doubt that you can make the material entirely clear to most readers. Making it clear to mathematicians would then be the goal, and for that you'll need to find some editors who are also mathematicians. Finetooth (talk) 16:54, 25 March 2009 (UTC)[reply]
  1. Not sure. For whatever reason, "the plane" seems more common than "a plane"; the idea seems to be that there is really only one object, the 2D plane. Is this confusing, or does it just sound weird?
  2. Ok. Actually I know this, but often I find seeing the number is easier. I guess that's just a personal thing.
  3. Yes, I think that would be clearer.
  4. Actually, I had a feeling that was problematic. Maybe you can help fix it so you can understand it. What is going on is this: there are infinitely many maps, so this poses a problem on how to show they are all 4-colorable. The Haken--Appel proof starts by assuming there is a counterexample, i.e. a map that must be colored with at least 5 colors. First they show there is a finite set of 2000 or so maps with the property of "reducibility". This means that if a map contains a portion that is reducible, then if the rest of the map (outside that portion) is colorable by four colors, then the whole map is. So a smallest counterexample can never contain a reducible portion. This is because the whole map of a counterexample (by assumption) cannot be colored with four colors. But being minimal means that removing any portion of it gives a four colorable map. so in particular, removing the reducible portion gives a four colorable map. This means the whole map is four colorable, which is a contradiction to being a counterexample.
    But they also showed this finite set of reducible maps is also 'unavoidable': any map (counterexample or not) must contain a portion that looks look one of the finite set. So if there was a counterexample, there would be a minimal counterexample. It must contain one of these maps because they are unavoidable, but they are also reducible, and we already said a minimal counterexample can't contain a reducible map. So the existence of this set of unavoidable, reducible configurations implies the four color theorem. The fact that this set is actually finite means you can check each one for the two properties of unavoidability and reducibility (each property can be checked by either a human or computer). This verification procedure is exceedingly lengthy and so required a supercomputer for part of it.
I sincerely hope you will take a look at my explanation and possibly adjust your opinion that this is hopeless for the average reader. I'm not willing to give up on accessibility yet.
An idea I was thinking about is including the proof of the five color theorem, which is very simple and illustrates these basic ideas of unavoidability and reducibility. --C S (talk) 03:13, 26 March 2009 (UTC)[reply]
  • It did feel like a bit like a wimp-out on my part. I'm giving it another try. Your re-write of the lead seems much better to me than the earlier version. It makes the logic clear; the link to an explanation of counterexample is good, and the historical overview makes sense. However, in the sentence that says "at least one of the maps must be included in any map and each cannot be part of a smallest-sized counterexample to the four color theorem", don't you mean properties rather than maps? Perhaps this would work: "any map must have at least one of the properties, and the smallest-sized counterexample to the four-color theorem can have neither property". It may be though, that I still do not understand. Finetooth (talk) 15:11, 26 March 2009 (UTC)[reply]
  • Ok, I think I'm starting to understand the language confusion here, which I think I've unfortunately contributed to. The set of 2000 or so maps has the property of being 'unavoidable' (the elements of the set can be thought of as "mini" maps, because they are rather simple compared to how complicated an arbitrary map may look like). This means that any map you give me, there will be a piece of it (some configuration of regions in it) which must look like one of the "mini" maps in this set. That's an unavoidable set.
  • Reducibility is a property of a map, which ensures it cannot be part of a bigger map that is a minimal counterexample to the theorem. So reducibility of each of the 2000 or so maps is checked individually. Avoidability has to be checked for the entire set at once, and requires a process called "discharging". This discharging procedure is thus trickier to check and was originally checked by hand by Haken and Appel, but I believe the newer proof mentioned in the article also managed to implement most of that on computer too. --C S (talk) 18:41, 26 March 2009 (UTC)[reply]
I rewrote the lede using your input. Now it explains each property one at a time, instead of making the reader swallow too much at once. --C S (talk) 18:59, 26 March 2009 (UTC)[reply]
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