Math Monday: Try a Torus
by Glen Whitney
And if you do make a torus in this way, you really might want to paint it with the Ungar-Leech map shown above. Why? Because that map shows that unlike on a sphere, where any map can be colored with four colors, it takes at least seven colors to color certain maps on a torus. In particular, the Ungar-Leech map divides the torus into seven congruent regions, each of which touches all of the other six. So there’s no way to color it with fewer than seven colors.
This article first appeared on Make: Online, July 2, 2012.