Paper torus with Villarceau circles

I made another thing!

This is a torus, made from 24 crescent-shaped pieces of paper with slots cut into them so they interlock with each other. I followed these instructions on There is also a template with some ideas for nice variations here.

The idea of this model is to highlight Villarceau circles. Everyone knows how to find circular cross-sections of a torus, right? You can cut it horizontally (like cutting a bagel in half) in which case you get two concentric circles:

Or you can cut it vertically (like cutting a donut in half, if you wanted to share it with someone else), in which case you get two separate circles:

But it turns out there is yet another way to cut a torus to get circular cross-sections, by cutting it with a diagonal plane:

Note that the plane is tangent to the torus at the two places where the circles intersect. These circles are called Villarceau circles, named after French mathematician Yvon Villarceau, who published a paper about them in 1848. The paper model highlights these circles: each circle is composed of edges from two differently-colored crescents; the points of the crescents come together at the tangent points where two Villarceau circles intersect.

If you want to try making this model yourself, be warned: it is quite difficult! The five-star difficulty rating is no joke. The time from when I first printed out the templates for cutting until the time I finally finished it was several months. Partly that was because I only worked off and on and often went weeks without touching it. But partly it was also because I gave up in frustration a few times. The first time I attempted to assemble all the pieces it ended up completely falling apart. But the second time went much better (using what I had learned from my first failed attempt).

About Brent

Associate Professor of Computer Science at Hendrix College. Functional programmer, mathematician, teacher, pianist, follower of Jesus.
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