h2g2 The Hitchhiker's Guide to the Galaxy: Earth Edition

Entry: The Four Colour Theorem - A55292673
Author: Icy North - U225620


This is an entry on a very well-known mathematical problem. Like Fermat's Last Theorem, the problem is simple to understand, but the proof is very difficult. In this case it was only solved in the 1970s by two Americans, who used a computer.

I have drawn three diagrams which explain parts of the entry, and I have linked to them on a file share - I believe the entry needs them. (If you think it needs more diagrams, then let me know)

I haintend this to be understandable to non-mathematicians, so please let me know if it needs simplifying anywhere.

Comments and suggestions welcome

Icy

Icy - I take my hat off to you!

as a person of very little brain when it comes to maths, I found this quite easy to follow and very interesting to read.

I had heard of this theorem and the history of how it was eventually proved. As an ex-teacher, it used to come up in school when you had those bright kids needing a little brain stretching exercise.


Very worthwhile entry - I'm sure a mathematician will stroll through and give you some more in depth feedback!

"Until someone proved otherwise, it was always possible that some super-Nevada was out there that needed a fifth, purple crayon. The thought of some poor cartographer being caught short was not one to be entertained."

There are too many "some"s in there for it to read as smoothly as the rest of your work.

I understand it


Although I only have one colour here

lil xx

Thanks all

I've fixed one of those "some"s, Trout.

Icy

Amusingly, I think that the proof that a ring-shaped doughnut (A1134091) needs 7 colours had been proved for some time before the four colour theorem, and without computer. But I'm not really sure anyway

I think that proves how most maths is done in the coffee lounge.

...and Italian restaurants

This is a good entry, but I think they're yet to prove that the crayons need to be red, yellow, green and blue

Actually, I can prove that using h5ringer's Bistromaths. The last time I took a toddler to Pizza Hut, they gave us a colourable children's menu with a small plastic wallet containing four crayons: red, yellow, green and blue.

The Entry is perfectly good. You may want to make clearer--but it's pretty clear--the position of this theorem in mathematical history (as the first proof that relies upon an argument regarding a computer program). I would give examples of how this has become more routine, but I can't. I'm sure there are noteworthy cases of mathematical exhaustion out there, just none as famous or controversial as the first.

I concur with toy box, also, that the result on the torus was done before the standard problem. I'm nearly certain that's correct, but it doesn't look like I bought a book on the subject--I guess I decided I wouldn't read it--so I'm not absolutely positive.

Hi Julzes. Thanks for your comments.

Yes, I don't say in as many words that it's the first use of computers in mathematical proof, but I strongly imply it, particularly where I mention the philosophical debate which ensued. If anyone believes it needs to be more specific, then please suggest appropriate wording.

If anyone can confirm that the 7-colour torus was proved earlier, then please post a reference and I'll add a footnote. This is something which can always be added by curators later, if necessary.

Icy

This is the reference I have now: http://en.wikipedia.org/Four_Color_Theorem.

http://en.wikipedia.org/wiki/Four_color_theorem. That's better.

I also posted a historical question to their math reference desk on what notable computer-aided proofs have been done since. If anybody gets back to me with a good answer, I'll let you know.

Does that article mention when the 7-colour torus was proved? (I'm looking at a mirror site as wikipedia is blocked in the office, but can't see it in so many words)

Having read it, I'm glad I stuck to making it simple. I can't imagine anyone non-mathematical hoping to understand that Wiki article, comprehensive and well-illustrated though it is.

The closest the article (or any of the other sources I have looked at) comes to giving a date on the torus case is that a generalization (the Heawood Conjecture) was proven in 1968. I suspect the specific case of the torus came long before that. One of the relevant articles at Wolfram's Mathworld lists the most difficult cases in resolution of the Heawood Conjecture, so it's apparent that the project was spread over many years.

If I were writing the article, the following three sentences would find their way into it:
"Since Appel and Haken's proof, the use of sophisticated computer programs in proving results mathematicians consider important has become reasonably common, both where the computer's work can and cannot be checked. One relatively recent major case is what is known as the Kepler Conjecture, that three-dimensional balls cannot be packed any more tightly than in a regular lattice arrangement. A panel of twelve mathematicians associated with the journal Annals of Mathematics took four years (ending in 2003) to conclude they were "99% certain" of the correctness of the proof in this particular case, hedging that the computer might have done something wrong.

What I just said with the following change: "become reasonably common"-->"recurred numerous times"