By Glen Whitney for the Museum of Mathematics

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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. Suppose you didn’t happen to have access to a FabLab or a Modela MDX milling machine, so that you couldn’t follow these instructions to produce a wooden torus like this one:

torus

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.