By Glen Whitney for the Museum of Mathematics

Just suppose you wanted to make your own model of the Ungar-Leech map on the surface of a torus, like this one created by Norton Starr in 1972:

You’d probably want to start by making a torus. And just suppose you didn’t happen to have access to a Fab Lab or a Modela MDX milling machine, so that you couldn’t follow these instructions to produce a wooden torus like this one:

What could you do? Well, you could try making a plaster mold of a torus, as in the following detailed video showing the entire process from start to finish:

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.

More:
See all of our Math Monday columns