What do the regions described here look like? I heard in the news a while back that the four-color theorem had been proved. I understand that the proof applies to the four-color theorem about coloring the nodes of a graph. I think it's also true that a "normal-looking" map where all boundaries can be represented by some finite number of joined line segments of finite length and where there are no discontinuous areas that have to be the same color (so no discontinuous countries and also no discontinuous bodies of water with a common color) will always have an equivalent graph where the nodes represent the countries and the edges represent the boundaries.
This question actually isn't about graph colorings, but about some alternative type of coloring that I don't understand how to formalize.
I just read a confusing comment on cs.stackexchange.com about a purported counterexample to the four-color theorem if we omit "certain well-behavedness assumptions". The commenter says

It was in a Scientific American circa 2003 or so. The contrary case involved a border line defined by y = sin (1/x) as well as two more border lines going through the origin and the fifth being a box around the whole thing.

– Joshua
I don't understand the "map" this describes and how it works as a counterexample even to a version of the four color theorem that is not formalized in terms of graph theory.
Here are the steps I went through when trying to analyze the example:

*

*The function $y = \sin(1/x)$ has a gap at the origin, since $1/0$ is undefined.


*When I googled the function, I found various sources explaining that there is also no limit of $\sin(1/x)$ as $x$ approaches $0$.


*This makes me confused about how any boundary between a finite number of countries could be defined using $\sin(1/x)$ and border lines passing through the origin. It seems like the lines would not intersect with $\sin(1/x)$ at the origin (although they would intersect infinitely many times at other points if they have any defined slope). So I don't understand how this map could have closed boundaries for all of the regions near the origin
Can someone explain how this creates an example that could be meaningfully described as a map of countries that requires more than four colors? I found a brief mention of sin 1/x in Formal Proof—The Four Color Theorem, Georges Gonthier, Notices of the AMS Volume 55, Number 11, but I'm not sure if it is at all relevant:

Definition 3. Two regions of a map are adjacent if
their respective closures have a common point that
is not a corner of the map.
Definition 4. A point is a corner of a map if and
only if it belongs to the closures of at least three
regions.
The definition of “corner” allows for contiguous points, to allow for boundaries with accumulation points, such as the curve sin 1/x.

(page 1388)
 A: Thanks to saulspatz's comment, I found out about the article "Four Colors Do Not Suffice," by Hud Hudson, from The American Mathematical Monthly May 2003 issue. This may be the article that the cs.stackexchange.com commenter was thinking of.
The jstor version of Hudson's article had some grayscale figures that helped me out a bit in picturing the "map" presented there. Something I hadn't realized from the commenter's description is that the map is partially open, missing a line segment region in the middle that passes through the origin. That line segment is what Hudson treats as the "common boundary" of the six regions.



The archives for the Wikipedia page on the Four Color theorem seem to have some relevant discussion.
An IP editor writes that

What Gonthier proves is that "The regions of any simple planer map can be coloured with only four colours, in such a way that any two adjacent regions have different colours", where "a planer map" is "a set of pairwise disjoint subsets of the plane called regions and "a simple map" is "a map whose regions are connected open sets", and where "two regions of a map are adjacent" if "their respective closures have a common point that is not a corner of the map" and "a point is a corner of a map" if and only if "it belongs to the closure of at least three regions". Importantly the "corners of a map" need not be isolated points and can even form line segments. In Hudson's example, all the regions would technically be considered non-adjacent because all the border points are shared with at least three regions, and therefore could all be coloured with one colour. Though Hudson's example is a good illustration that maybe the technical statement of the four colour theorem doesn't live up to our informal expectations.

The same point was made by Hagen von Eitzen in a comment here regarding the "Lakes of Wada" construction.
Actually, Hudson's map does seem to have borders not shared by all regions, but they allow for coloring with fewer than 5 colors: 3 colors work, one for LG and LB, one for DG and DB, and one for R and B, given that the "border" consisting of the vertical line-segment in the middle counts as a corner by Gonthier's definition.
