If you've never done this before, you should definitely try it out: take a trip of paper about an inch wide and a foot long. Hold the two ends and make one half-twist. Join the two ends (with a staple, tape, or glue). Now, starting anywhere on the strip, draw a line in the middle and follow the strip. After a while you should be right back to where you started. This in and of itself isn't very surprising---any circular strip will make this possible. What is cool about this is that both "sides" of the paper will have the line drawn across it. In other words, there is only side to this strip of paper!
Argentinian film maker Gustavo Mosquera and his students present a grandiose vision in his film Moebius involving the Moebius strip described above, infinity, perpetual motion machines, metaphoric political statements, and general philosophy about life. Unfortunately a bit more research should've been done, in my view, to fully realise this vision. That said, Moebius does touch upon some interesting existential concepts that are probably better off explored by yourself within your own mind rather than being spoon-fed by a film.
Inspired by the short story A Subway Called Moebius by A. J. Deutsch, Moebius is the story of Daniel Pratt (Goillermo Angelelli), a mathematician who specialises in topological theory. Pratt is called upon to investigate missing trains in the Buenos Aires subway system. He soon discovers that the engineers kept adding tracks to the system without realising that they had constructed a set of tracks that formed a Moebius strip. As a result, a train carrying 30 passengers that embarks on a journey on this strip never returns, though it is heard every so often. As Pratt tries to find a solution to this problem, he suddenly finds himself confronted in a universe right out of an M.C. Escher painting.
The story, the direction, the cinematography, and the philosophy has a strong Borgesian feel to it (a rather blatant reference is made in the film). The stark and bleak visuals and the minimal dialogue contribute greatly to this aesthetic. The movie is bold, unconventional, subversive, and definitely in a class of its own. The one complaint I have about the plot is the presence of a young girl who makes Pratt realise what's happening with the missing trains but, for the most part, is completely incidental to the story.
As I say above, the topics presented in this film are thought provoking: the concept of a train moving along an infinite path is interesting, but not very exciting because nothing more than a simple Moebius strip is alluded to in the film (i.e., the same effect could have been achieved with a circular track). Carrying on with the experiment I describe above, cut the piece of paper along the line you just drew---you should have one long strip. If one half-twist represents one dimension, then you have a strip with four dimensions (our world). Cut this long strip in the middle and keep repeating this process. As you keep doing this, you will have more and more strips formed each of which is interlocked with every other strip. That the strips are separate is an illusion: they can all be reduced back to the original strip! In this way, the Moebius strip explores the nature of infinity on many different levels. I think the film could've incorporated more information about the nature of infinity, including the different "levels" or "sizes" of infinity.
The train that can be heard but not seen is definitely an allegory for the disappearance of the 30,000 or so "political dissidents" in pre-Democratic Argentina in the 70s, something that has long occupied the minds of the Argentinian public. The movie engages the viewer into thinking about concepts that are not easily explainable to the common man (there's definitely a Jorge Luis Borges influence here). "No one ever listens", cites Pratt's mentor, with regards to trying to explain the disappearance of the trains because of a Moebius strip (this could also serve as a political statement). This sort of philosophical commentary is peppered throughout the movie, sometimes subtly, and sometimes very explicitly, like at the end of the film.