# Four color map theorem...disproven?

I believe that the picture in the OP is intended to represent a knotted curve in which the red and blue meet under the black and similarly for the yellow and green. It is simply not a 2-dimensional surface and the theorem makes no claims. As stated above there is no limit in 3 and higher dimensions.

The computations have now been carried out on so many different computers with so many different programming languages using so many different operating systems that it is, in my opinion, in less doubt than most proofs done by hand. The original proof was based on the study of some 1800 “minimal unavoidable configurations” and the Haken and Appell children were being paid to find errors in the data entry to describe those configurations, a dollar an error, until that became too expensive. This did not create great confidence in the result at the time, but no serious doubt remains.

But what about numbers of dimensions between 2 and 3?

Don’t be bringing fractals into this now. :o

In the usual formulation of the theorem, the map is assumed to be composed of pairwise disjoint connected open sets (in the plane). It doesn’t matter how fractal the boundaries might be, as long as you take care to keep track which regions are adjacent (ones whose closures have a point in common that is not a corner point). Or am I missing something?

The counterexample in the article cited by Wikipedia has these fractal objects “touching”:

``````

aaaaaaaa    aaaaaaaa a    bbbb  bbbbbbbbbbbbbbbb
aaaaaaaa    aaaaaaaa a    b bb  bbbbbbbbbbbbbbbb
aaaaaaaa    aaaa  aa a    b bb  bbbbbbbbbbbbbbbb
aaaaaaaa    aaaa  aa a    b bb  bbbbbbbbbbbbbbbb
aaaaaaaaaaaaaaaa  aa a    b bb  bbbb    bbbbbbbb
aaaaaaaaaaaaaaaa  aa a    b bb  bbbb    bbbbbbbb
aaaaaaaaaaaaaaaa  aa a    b bbbbbbbb    bbbbbbbb
aaaaaaaaaaaaaaaa  aaaa    b bbbbbbbb    bbbbbbbb

``````

The shape continues to get skinnier and tighter each repetition by a factor of 2 each time. Neither shape would hit the line exactly between them, but they’re arbitrarily close to each other. Therefore, they need different colors.

Furthermore, if you make four countries around the edge of the map, one in each corner of the map, and let them annex all the land on their side of the weird areas, kinda like this:

``````
111111222222
1a1aaab2bbb2
1a1a3ab2bbb2
3aaa3ab2b4b4
3aaa3abbb4b4
333333444444
``````

…then all of the regions are arbitrarily close to the others. Therefore, this map requires* 6 colors. (As you might expect from a peer-reviewed article, the math is more rigorous than I remember how to duplicate 25+ years removed from my topology class.)

Actually no. As Ceci’'s column says, the proof was done with the aid of a computer. I heard a talk by Appel and Haken when I was a grad student at Illinois righr after they announced the proof.
The computer was used to enumerate the possibilities, IIRC. The particular computer was a CDC Cyber7600 which ran PLATO, and was used at night when the only load was when a bunch of us played a multiplayer Star Trek game - this in 1976.

A point in the closure of more than two regions is, by definition, a “corner point” in the usual formulation of the theorem. You can jam together however many regions you want at their corners. In that example, the set of “corner points” is not discrete; in particular, it contains a whole segment.

That was then. This is now.