This blew my mind wide open today. This guy, he goes by the name of Aaron M. Hoover. He makes things. He made this model of a Möbius Gear. The black outer ring was laid down layer by layer in thin plastic slices with a computer controlled machine. The white Möbius ring in the center and the blue gears were both constructed from silicone rubber in wax molds also built with computer controlled machines. 3D Printing is very cool. Check it out.
Aaron's design demonstrates multi-part fabrication principles and the ways to go about fabricating a prototype widget quickly and with as little waste as possible. This model was first built as a three dimensional model and then fabricated using separate materials so that the whole thing could fit together properly. The black ring was fabricated using an additive process. Layers of plastic were deposited down on top of one another to build the model slice by slice. The other common machine shop technique to fabricate parts is to start with a chunk of material, plastic in this case, and use a computer controlled router to cut away at the chunk until the piece is formed.
In any kind of computer controlled process like this the neat little Calculus bit is translating all of the 3D model data into what are known as tool paths. Every single movement of that computer controlled nib is translated from the 3D model into the language of the 3D printer, which talks in equations defining curves along which the machine will deposit material to build the part. Is that confusing enough?
This is a video of an unrelated 3D fabricated part to show the process in action.