Sometimes it's fun to try to take the most ludicrously-abstract and pointless geometrical results and to try to turn them into something useful. It's a fun game, and the more abstract the thing is, the higher the chance that nobody's actually brainstormed it properly before you. The "square-cutting" exercise ended up as a possible idea for new storage media for hydrogen-powered cars, so after uploading the "Cutting up Doughnuts" post, I was scratching my head to try to think of some real-world application for the "Villarceau Circle" result, that might turn the pastry-cutting exercise into something with actual physics applications.
The best I could come up with was a variable-geometry magnetic containment device.
If we take our two interlocking Villarceau circles, and delete one of them, we're left with a simple ring that wraps once around the torus limb and its central void. This counts as a special-case toroidal winding. We can interleave a series of these single angled rings around the torus, intersecting, without any of them clashing or colliding. If current is circulated around each ring (perhaps by "breaking" the rings and wiring them in series), you have yourself a rather unusual toroidal coil.
What's unusual about it that it has variable geometry. Each circular ring-segment can be a rigid wound coil, and by tilting the angle of these coils we can create a larger torus with arbitrary proportions (major axis radius fixed, minor axis radius variable). Okay, so there's a limit to how fat or thin we'd be able to go due to the finite thickness of the rings that we're using to construct it, but essentially, we have something that looks like a toroidal accelerator and containment device, that can actually change shape while it's running. Provided that the "open" configurations of the resulting toroidal coil aren't too open, this might let you prototype a device without having to calculate the ideal proportions beforehand - you'd be able to adjust the torus shape while the device was actually operating.
Now, suppose for the sake of argument that you wanted a containment device that allowed you to open it out, fire high-energy particles into it in low-energy mode, then close the coils, squash the plasma density to encourage some sort of reaction, and then open the coils again to allow the reaction products to spill out into the surrounding coolant. You could have a system that "breathes", and holds different shapes for different parts of its cycle.
Okay, I'm trying not to be too glib here – because nuclear physics is NOT my specialist field – but this thing would look awfully like a cross between the "cage fusor devices" and the "tokamak" configurations that people use for nuclear fusion. When it's closed you have something that looks like a tokamak, and when it's open you have something that looks (superficially) more like a fusor cage. One of the annoyances of the tokamak designs is that once you've built them, they're usually locked into particular configuration – with a Villarceau coil, the variable geometry means that you should be able to get some pretty significant changes in internal volume and field strength without having to vary the current flow to the coils. And if the internal pressure gets too great, the thing's going to have a tendency to self-adjust by opening out like a flower-bud, reducing internal pressure and temperature, and releasing excess plasma into the surrounding coolant in a semi-controlled way (rather than being all bottled up until things go more badly wrong).
Anyhow ... bottom line is, that even if this configuration is no damned use at all for conventional nuclear fusion, it'd still look damned cool as a piece of hardware.
Designers and art directors for science fiction movies take note. Remember how cool people though the Big Scary Spinny Machine was in Contact (1997)? Well, this configuration would be a really nice thing to use next time you have to design a cool fictional device for a spaceship reactor or engine pod. Shiny silver interlocking steel circles that tilt and swivel, with a whizzy blue plasma glow inside. Mmmm.
I want to see this cool thing in a movie NOW ! :) Who's going to be first? PS: I did spent the last couple of weeks seriously consider building one of these as a toy, sticking it in a small vacuum chamber and whacking a high-tension voltage into it, as a version of those plasma balls that you find in gadget shops. I figured that with that, plus a set of circular coil units, and I might have a cool little device that could spin plasma (or bits of shiny silver paper) in an amusing way. I got as far as looking up coil formers. But sanity prevailed. Plus, I think my current landlord might take a dim view of his tenants trying to build small prototype nuclear fusion reactors on the premises.