Traditional 3D printers, as well as most CNC routers, laser cutters, and most machine tools where the programmer doesn’t want to go crazy trying to get his or her head around the underlying code, work in the XY coordinate system.* This “theta printer,” as seen on Hackaday.io, eschews that paradigm and instead uses a rotational, or theta, axis and an R axis. This “R” tells the printer how far away from the center the extruder head is. That would be interesting enough, but it also uses not one, but four extruder heads.
This setup has several advantages, including that one can print with several types of material without negatively affecting performance. As each extrusion head is separate, the mass of each is smaller than if several were combined on a single XY gantry setup. Precision can also be improved, especially towards the center of the rotary table.
On the other hand, one would have to assume that more extruders and motors on a machine would equal greater expense, and possibly more maintenance issues. Either way, it’s a really innovative design, and you can see it in action below. I only see two extruders in action here, but this shows another advantage, in that it seems to be quite modular. Also, at least two identical parts can be printed at the same time with different materials.
*Yes, lathes work differently. Don’t derail my thought process!