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 cutoutfoldup.com. 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

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