In the 1980s BMW began flexing their design muscles, forming an internal division called BMW Technik to develop new vehicle concepts and technologies. This division was to be…

…a kind of think tank in a cutting-edge company branch that would be completely isolated from all other development departments. The idea was to give highly skilled BMW engineers, technicians and designers free rein to work on turning their best creative concepts into reality themselves.

The Technik division's first project was the groundbreaking Z1 roadster. 

By Karrmann - Own work, CC BY-SA 3.0

The "Z" stood for Zukunft, "future" in German, and the car lived up to the name: It had a multi-link rear suspension (a big deal in 1985), xenon headlights (also a big deal in 1985), removable plastic body panels so that the driver could change the color of the entire car by swapping them out...

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...and most famously, these crazy doors:

The doors could even be open and closed on the fly, if the driver wanted a more open-air feel:

Here's a look at the mechanicals beneath the door skins:

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To help you understand what you're seeing, here's the description of the mechanism from BMW:

The mechanism for the doors and side windows was driven by two electric motors and a toothed belt. The motors incorporated a freewheel function to allow the doors and windows to be operated manually if necessary. To avoid burdening the actual door with additional weight, all of the electrical and mechanical components were integrated in the car's body. A double mechanical lock acting on the doors in both opened and closed position prevented rattling noise and excessive door movement while driving. The side windows could, of course, be lowered separately from the doors: after the doors had been opened and closed again, the windows automatically returned to their previous position.

And yes, the two mechanics in the latter two videos finally figured out how to solve the closing problem:

The Z1 was difficult to produce, taking nearly four years to bring to market in 1989. BMW claimed that they had 35,000 orders, but the car "[had] to be built largely by hand on account of its unusual design characteristics and the materials used," so they were never able to build more than 20 Z1s per day; during the entire production run, which lasted from '89 to '91, only 8,000 were produced. If you ever see one of these on the road, count yourself extremely lucky!

You can read more about the Z1 here.