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Mathematics Goes to the Movies

by Burkard Polster and Marty Ross


Moebius (1996)

Daniel Pratt is a mathematician sent by the builder Decker to investigate the disappearance of a train in subway network.


DECKER: You like problems. You specialized in mathematics, right?
DANIEL: Topology. The fascination with the analysis of surfaces. Formula, calculation and you put it into a book. I find it as odd as the Director General’s questions.

Decker gives Daniel a puzzle.
DANIEL: What is that?
DECKER: I don’t know. They say it strengthens perception. Work it out.

Daniel introduces himself to Blasi, the Director General of the subway system.
DANIEL: I am a topologist, a mathematician.
BLASI: What is that?
DANIEL: A branch of mathematics that studies surfaces. And turns them into formulas.

BLASI: Lights detect nonexistent trains. That’s to say, they fail. There are unplanned changes of rails. And a whole train is missing. There are so many tunnels that we can’t find it.

Daniel tries to find his old professor of topology Mistein at the university.
PROFESSOR (not Mistein): Don’t forget we are talking about pure theory which hasn’t been applied so far.
DANIEL (asking a student): Is this Topology III?
Student: Yes.
PROFESSOR: The conclusion is that both spaces, though not formally equal, are the same as those usually applied. If an astromomer presented a theory which was acceptable. To the common man he’d probably be wrong.
But if someone said time freezes in certain areas we should at least listen. He might be right.
Science and philosophy are not so farfetched yet. Thanks. The class is over.
DANIEL: May I ask you a question?
PROFESSOR: Are you a student.
DANIEL: No, I’m a graduate.
PROFESSOR: What can I do for you.
DANIEL: I am looking for a PROFESSOR who taught topology for many years.
PROFESSOR: Don’t waste your time. When did you last see him?
DANIEL: Some five years ago.
PROFESSOR: He left the university soon after that.
PROFESSOR: He’d lost interest in teaching. He had something else in mind.
DANIEL: Do you know where to find him?
PROFESSOR: I must have his old phone number and address. We used to call him. He didn’t answer. We never knew why. I wonder what ‘s become of him. Here, we liked him

Professor Mistein’s apartment.

Escher print in the background.


Closeup of puzzle.

Daniel talking to some city officials.
DANIEL: We have a train with people in it. It’s missing but the system is closed. Trains cannot leave it. Train 86 is in the system. The whole system’s been checked and it’s not here.
OFFICIAL: You think we are hiding it?
OFFICIAL: Shut up and let him make his point.
DANIEL: We know it was going to Plaza mayo on the morning of the 4th. But it never got there.
OFFICIAL: You don’t say! And when it got to Maldonado strem it turned into a boat and it’s sailing to Africa.
Engineer: Excuse me. How can it have been running since March 4th if nobody’s seen it? It’s almost five days.
DANIEL: That’s just it. The train cannot be seen. That train, number 86 with no technical problems and the system of 300 trains working at full somehow disappeared into thin air. It hit a node.
OFFICIAL: A node? What is that? The railway is clean. There’s nothing but trains on it.
DANIEL: You still don’t understand. A node is not an obstruction. In topology, a node is a peculiarity. A higher degree pole.
OFFICIAL: I don’t understand a word of what you are saying.
OFFICIAL: Let him finish.
DANIEL: I think it’s the new line.
OFFICIAL: What has that got to do with it?
DANIEL: The whole system is a very complex network. That was true even before its expansion. But the Peripheral has made it pretty peculiar. I’m not sure but I think it has integrated the whole system at such a high level that I don’t even know how to calculate it. I suppose it’s become infinite. If I’m right, gentlemen we can conclude that the system’s working like a Mobius band.
Demonstrates with a strip of paper

Please, if you look at this band you may see it has two sides. But if you put its ends together like this, we get a very peculiar strip. It’s a one-sided strip along which we could run for ever. Endlessly. If you translate this to a subway network, it should have an infinite number of singularities. Can you imagine the features of a system like that? I cannot either. In fact, the whole system is beyond my grasp now. I can do nothing but guess.
OFFICIAL: This is unbelievable.
OFFICIAL: We have 150 miles of tunnels. How can a finite surface become infinite?
DANIEL: Good question. However, it has happened, it has.
OFFICIAL: What about its passengers?
DANIEL: I have no idea.
OFFICIAL: Wait a second. If the Peripheral Line is causing this, we could close it and then resume normal operation without any danger.
DANIEL: If you close it the train will never reappear. But anyway, it might reappear running in the wrong direction. In that case there might be an accident.
OFFICIAL: Mr. BLASI, how many people are there on that train?
BLASI: Some 30 or 40 people.
OFFICIAL: Listen, Pratt! Do you want us to believe that the train has plunged into another dimension and is not in the system any longer?
DANIEL: That is one way of saying it.
OFFICIAL: And you mean this peculiar behaviour has been caused by certain mathematical properties that have to do with the Peripheral?
DANIEL: That’s it.
OFFICIAL: Isn’t there anything we can do to bring the train back?
DANIEL: Not that I know.
Nobody believes any of this.

Engineer: Pratt, a question. Could we keep the Peripheral open with no other trains running so that if the missing train appears there cannot be a collision?
DANIEL: No. With the new line, the whole system has infinite linkage. 1:00:19
Studies the map of the Peripheral at the roller coaster. Finds some more notes of Mistein.
Estimate maximum number of bends. 1:02:27
DANIEL is figuring out when the ghost train will arrive at the platform he is waiting at.

1:08 eclipse
After catching the ghost train finds the professor.
MISTEIN: It wasn’t easy to get here, was it?
MISTEIN: Relax. I know you have a lot of questions to ask me.
DANIEL: We’re travelling at an impossible speed.
MISTEIN: A simple change of railway. The train crosses a junction after a bend. The right combination at the right time. And the properties of a Moebius band apply.
You’ve invented a perfect machine.
MISTEIN: Don’t be blasphemous, son. Man has invented a lot of machines. But forgets that he himself is a much more complex one.
DANIEL: There’ll be no limits now
MISTEIN: There never were any. Man neither knows his limits nor his possibilities. He doesn’t even know that he doesn’t know himself. We are so busy looking for external values we don’t see what really matters.
DANIEL: This realy matters! Everything would change if your just said it.
MISTEIN: You did. You explained it perfectly well. I passed by Parque station and I saw you trying to explain Moebius theory. Did anyone believe you?
DANIEL: No. But they would believe you.
MISTEIN: No. I would have said exactly the same as you. The same truth. But we’re living in a world that won’t listen, dear Pratt.
DANIEL: What will you do then?
MISTEIN: Nothing.
DANIEL: Nothing?
MISTEIN: Don’t worry. The time will come.
DANIEL (looking at passengers): Don’t they understand what is happening.
MISTEIN: They? They’ll never wake up before realizing they are asleep. What are you afraid of?
DANIEL: Vertigo. That’s normal. Nobody can confront infinity without feeling vertigo. Nobody can experience it without feeling deeply puzzled.
MISTEIN: We are travelling at the speed of thought. How could one enjoy this life lacking in attraction, naiveté and spontaneity? How not choose these shadows when the sea of deafness out there carries us to helpless disgrace.
DANIEL: This can’t get lost.
MISTEIN: Neither men nor time disappear without leaving traces. They stick to our souls.