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. 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.

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Gareth Branwyn

Gareth Branwyn

Gareth Branwyn is a freelancer writer and the former Editorial Director of Maker Media. He is the author or editor of over a dozen books on technology, DIY, and geek culture. He is currently a contributor for Boing Boing and WINK Books. And he has a new best-of writing collection and “lazy man’s memoir,” called Borg Like Me.


  • Henry

    Sliceform is a really easy and “123d make” makes it easy all you need is a printer because a torus is one of the sample models you can play with and make.
    is you want a more permanent torus you can make a sliceform torus with cardboard and cover it with clay.

  • http://david.rysdam.org/blog David Rysdam

    Why did he make a plaster mold sitting freeform on a board? Why not put it in a box? Seems a LOT easier.

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    Reblogged this on ilripassinodimatematica and commented:
    belle cose da fare…

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