Neat application of geometry, though.

That’s using the lathe as a mill and thus wouldn’t be guaranteed concentric, as mentioned in the article.

Powered lead screws are pretty common even on smaller cheaper machines, and it’s fairly obvious from the video how he’s built this attachment so it would not be rocket surgery to add it to almost any lathe.

Using a car CV joint / driveshaft is pretty clever, high quality for low cost.

Btw, the paths cut are exactly like the paths traced out by a Spirograph. The only difference is the frame of reference. In a Spirograph, the “workpiece” (paper) is still while the “cutter” (pen) both rotates and moves around it; in the lathe it’s the opposite-ish. The workpiece is rotating, while the cutter’s center is held steady (it still rotates, though).

That video does a good job of showing the paths cut out and demonstrating that there are multiple ways to achieve the same polygon, with some resulting in flatter sides than others. In fact, you can choose to have sides that are concave or convex, basically letting you make torx bits and other interesting shapes. He wrote a program and made a webpage to let you experiment and see all the possible outcomes: https://ridethegeartrain.com (scroll down to get to polygon cutting)