It turns out that, not only is there a map on the torus that requires seven colors, there is one with just seven countries.
First we'll look at its dual.
Here it is on the unfolded torus
when you go off the right edge, you come back on the left
edge; same for top and bottom).
This graph has 7 points, and every point connects to every
(The points are just the places where the lines intersect.)
We made all the lines straight just to make it look pretty.
Here is a movie of this graph on a torus. See if you
can follow the loops. The torus is slightly transparent,
to help you. Go back and notice the loops on the square --
each color is actually one long line! The slopes of
the lines are
That last one loops in an interesting way.
And finally, here is the map itself, on a rotating torus. The dual graph is drawn in lightly over the map, so you can see how they relate. (No transparency here.)
This is a map on the torus of seven countries,
where every country touches every other country.
This proves that you need (at least) 7 colors.
As we mentioned, it turns out 7 colors is always enough. The proof is given here, along with a brief history. The seven-color theorem was first proved in 1890. The Four-Color Map Theorem turned out to be much harder; proving it took until 1976, and required computers.
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