Talk:Four color theorem

Simple Proof

Discussion closed because it is not about improvements to the article based on reliably-published sources; see WP:TALK
The following discussion has been closed. Please do not modify it.
On a square grid start with a single square. Add a layer of squares around it so that it becomes a 3x3 square. Each square you add will touch at most 3 other squares, so will only require at most 4 colors to map it. Add layer after layer to infinity, similarly the criterion for 4 colors is met. Is this what was said to be the proof?GuildCompounder (talk) 03:14, 15 December 2020 (UTC) You are only colouring a particular map. Most maps don't consist of layers like this. McKay (talk) 04:42, 15 December 2020 (UTC) @GuildCompounder: Consider this ball. When you come to adding the last segment, it will touch five segments previously added. Does your method guarantee those five use no more than three colors? --CiaPan (talk) 15:47, 2 March 2021 (UTC) The four color theorem only applies to planar maps/graphs. It's well-known (and discussed in the article) that other topologies can need more colors. I don't see the relevance to this section. --mfb (talk) 04:18, 3 March 2021 (UTC) Sphere is equivalent to plane, actually. But McKay's original reply is correct -- building from layers of squares doesn't deal with any of the interesting maps. (And of course the observation that it sat unsolved for a century, and then required computer assistance, should be a good indication that any "simple" proof attempt is quite likely to be wrong!) Joule36e5 (talk) 05:47, 3 March 2021 (UTC) A map of the ball's surface @Mfb: Yes, other topologies can need more colors. But this is the same topology. If you exclude any point of a sphere (which may be an interior point of any "country" region, hence meaningless in map coloring), then the rest of the sphere is homeomorphic with a plane (see the Stereographic projection for an example of a continuous bijection between a punctured sphere and a plane), so any result of coloring a map on a sphere applies verbatim to a plane and vice versa. --CiaPan (talk) 10:46, 3 March 2021 (UTC) @Mfb: I have added a planar map corresponding to the ball's surface structure. Hopefully it makes it clear how the 'planar' theorem applies to the ball. --CiaPan (talk) 18:54, 3 March 2021 (UTC) I still don't see any relevance to the original question in this section. --mfb (talk) 04:51, 4 March 2021 (UTC) So look closer. Can you apply the proposed algorithm to the ball? Can you color it the way described? If not, then the reasoning does not guarantee this particular map can be properly filled with four colors, hence it's not a proof of the theorem (as the theorem applies to all planar maps, to this particular one among them). --CiaPan (talk) 07:06, 4 March 2021 (UTC) Of course you cannot, but why use a ball? The first reply already pointed out that the algorithm only works for specific maps - without needing to introduce a ball. --mfb (talk) 10:48, 5 March 2021 (UTC) Nope, the first reply pointed out that User:GuildCompounder applied the algorithm to a specific map, but User:McKay did not prove "that the algorithm only works for specific maps", i.e. the algorithm can't be applied to other kinds of maps. And it actually can, for example it works perfectly well also for hexagonal tiling, and for rhombitrihexagonal tiling, too. So I gave the ball as an example of another simple map, so OP can explain how their method applies to it, or try to strengthen their intended 'proof' by expanding the presented method so that it handles the ball, too. --CiaPan (talk) 18:37, 6 March 2021 (UTC) It recently occurred to me that layers can be added inwards instead of outwards. Start for example with a 9x9 layer, add a 7x7...3x3. All the squares in the layers touch at most 3 other squares making them 4 colorable. The exception is the 1x1 centre square which touches 4 squares. However, if opposite sides of the centre square touch each other, that would separate the other opposite sides of the centre square which could then be the same colour. That is why what works out for the 2 dimensional map does not work for the 3 dimensional map (which has no limit to the number of colours required).GuildCompounder (talk) 17:59, 13 March 2021 (UTC) Doesn't work, either. Assume two-'layers' honey-comb pattern. The central piece touches six outer pieces. No part of your proposal guarantees those six use three colors only. And if they use more, you can't color the central one. --CiaPan (talk) 21:13, 13 March 2021 (UTC) It seems to me that if the squares are relatively infinitesimals, they can cover patterns like the honey comb.GuildCompounder (talk) 20:43, 15 March 2021 (UTC) While interesting, this discussion does not really conform to the purpose of a talk page, which is confined to discussions of how to improve the article. Paul August ☎ 12:25, 14 March 2021 (UTC) Sorry but maybe there is some relevance? One more little thing...GuildCompounder (talk) 20:43, 15 March 2021 (UTC) THe "opposites cutoff" theorem I mentioned above is valid for convex objects like spheres and cylinders, but is blown for toroids. We can visualize 6 colours for the toroid by applying a diagonal slash through the failed theorem rectangle. Then delete the original 5 colour region allowing a 2 colour loop to touch in 2 places when it only needs 1 connection. This allows a gap at the far end of the loop. Now the 3 five colour regions all touch each other, requiring 7 colours. GuildCompounder (talk) 20:43, 15 March 2021 (UTC)

Web Maps

Perhaps add an example, like "Bing Maps political layer" one day when there is finally something out there users can zoom into to their hometown to see. Jidanni (talk) 18:29, 14 December 2022 (UTC)

flaw in colors of the map at the top of the four-color theorem article

In the picture at the top of the article, take the orang colored squarish item. to the upper left there is a blue section that is triangle ish shaped that should be yellow as it is now the blue is adjacent at a point to two other blue sections that are spikey. A point counts as they are touching. pls fix. 73.180.167.41 (talk) 12:26, 24 February 2023 (UTC)

Adjacency at a single point does not count as adjacency for the purposes of this theorem. As the article already clearly states: "regions are adjacent if they share a boundary segment; two regions that share only isolated boundary points are not considered adjacent." —David Eppstein (talk) 16:01, 24 February 2023 (UTC)

Use "colour" not "color" when quoting British mathematicians

There are several quotations here from British mathematicians which are not spelled as they would have originally been made, and which I think should be emended for veracity. 92.27.162.236 (talk) 09:41, 16 May 2023 (UTC)

Context: the above comment is an opening to the discuss step of WP:BRD, following my revert Special:Diff/1154889964 and my explanation at the IP editor's talk page User talk:92.27.162.236#A change of 'color' to 'colour' in Four color theorem. --CiaPan (talk) 12:14, 16 May 2023 (UTC)

There is no doubt that British mathematicians used the word "colour" in everything they wrote on the subject, not "color". Just because an American work is used as a source should not imply that the word was spelled "color" by any of the original British authors.

There must be original British sources which should be quoted first hand. "Color" is factually incorrect, and should be changed. 92.27.162.236 (talk) 03:09, 24 May 2023 (UTC)

Indeed, by looking at the facsimile of De Morgan's letter you will be able to read the words "coloured" and "colours" for yourselves. 92.27.162.236 (talk) 03:16, 24 May 2023 (UTC)

We should certainly use the original spelling, regardless of ENGVAR, as we do also for certain quotations using archaic spellings rather than attempting to modernize them. (Not that I think UK spelling is in any way archaic, but I think the precedent is similar.) —David Eppstein (talk) 06:51, 24 May 2023 (UTC)