In 1950, Armin Deutsch wrote a story about the Boston subway called ‘A Subway Named Möbius’, for which spoilers will follow almost immediately.
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In this story, a tunnel is added to the MBTA subway, which curently looks something like this:
Sounds like a story which would at best only appeal to train spotters, right? Not a bit of it. The tunnel in question involves Boylston and changes the topology of the network in such a way that one of the trains vanishes along with everyone on it and only reappears several weeks later. When it does reappear, another train disappears. The story was adapted into the 1996 Argentine film ‘Moebius’, set on the Buenos Aires metro, Subte, whose current layout is:
There is a small amount of technical mathematical terminology in the original story, which is here. I’m not sure if this is more than the use of jargon to impress the reader, but it is genuine, valid, mathematical vocabulary relating to graph theory. Graphs in this sense are not the likes of line graphs, histograms or pie charts, but more like the maps shown above, in which pairs of items are related to each other in some way. In general, such a set of items with their relationships could be represented by dots (nodes) and lines (edges) on a piece of paper, or rather, some of them could be. I’m not sure all of them could, for instance the surface of a torus might not work. In the story, the addition of the Boylston shuttle has caused the graph to have “infinite connectivity”, and it’s this which I find most dubious.
Connectivity has a couple of meanings in mathematics. In graph theory, connectivity is the minimum number of elements (nodes or edges) which need to be removed to separate the network into at least two isolated networks. If this is the meaning used here, it sounds very doubtful that this could really happen because there is not an infinite number of either nodes or edges, and there couldn’t be in a mass-transit system. There are at least two other meanings, both topological. One is that it’s possible to move between any two points in the space, which is of course true in many mass-transit systems, but presumably not all because I would expect there to be ones which either have yet to be linked up or can’t practically be connected. The physical correspondent of either graph will be a different shape and the constraints on movement include the fact that the trains move along rails in tunnels of impenetrable brick, concrete or stone. There are other ways of considering this network – for example, flooding would effectively block off the lower layers. It also works as a template. Model railways could be constructed from these maps. They are also, famously, topological rather than geographic. This famous map:
. . . is geographically like this:
However, for the purposes of the story and the film, the topology is the relevant aspect, and it’s also the relevant aspect for passengers. Presumably for the people running the system, geography is more important for various reasons, such as power consumption, timetabling and location of depots.
The Glasgow Underground is something like the second or third oldest of its kind and despite protests has never deviated from its original plan:
I presume that most people know what a Möbius strip, also known as an Afghan Band, is, but in case you don’t, take a strip of paper, put an odd number (such as one!) of twists in it and glue the ends together. Pretty simple stuff of course, but unlike a similar strip without twists or an even number thereof, such a band has only one edge and one side. This is Wikipedia’s illustration:
It might seem contrary to attempt to claim that this shape has only one side and one edge because when it comes down to it it’s just a strip of material with a twist in it, but in fact it is exactly that. If you trace the edge with your finger, you will have to go round twice to return to the original spot and if you colour it in on one side, you’ll have to do the same. As an actual physical object which takes its thickness and the interior of the sheet of paper it’s made of, this isn’t what it is, but even then it’s a three dimensional shape with only two faces, two fewer than a tetrahedron, which we usually assume to have the minimum possible number of four, but as far as I can tell, either one or two edges.
One of the peculiar features of Möbius strips is that if you cut them in half down the middle, you get a longer, narrower strip, but if you cut them a third of the way from the edge instead you get two linked strips. Hence there are three “lines of interest”, as it were. A line near the edge describes a single edge, which however is at both the top and the bottom of the strip. A line near the centre appears to loop round twice. Lines around a third of the way in start near the top, plunge down near the bottom and return to near the top. All of the lines make two complete circuits of the strip before meeting themselves. The central line doesn’t change level, assuming that the topology of the strip corresponds to minimised distances between the lines.
Now imagine each one of those lines is an underground rail tunnel, like the Glasgow underground or the London tube circle line. A cross-section of the strip at any point would appear to show five tunnels, which would be vertically arranged at one point and horizontally arranged 180° away from that point. However, there are in fact only three tunnels, even though every transverse section seems to pass through five. A relatively simple case of this system involves them vertically oriented on one side, gradually rotating to an horizontal arrangement on the other, but this is only one version of the “geographical” arrangement. They could all be rotated through right angles so the tunnels pass near the surface along half of their route and then all plunge deep underground along the other half, they could be inside a narrow tower, or each of them could be imperfect circles and meander around. However, whatever else happens, to conform to the topology of a Möbius strip, they must at some point “twist” around to the opposite side an odd number of times, the simplest case being once.
In order to make this simple, I want to imagine this system to consist of apparently five tunnels arranged vertically on one side and horizontally a semicircle away along their route. I also want there to be a system of lift shafts linking them all together and to the surface, perhaps ten of them, plus an extra set of five vertical shafts where they reach their horizontal orientation. Also at that point, the so-called “lift shaft” is horizontal, so it’s more a walkway or travelator, or perhaps a supplementary train tunnel since it is running horizontally. If we were talking about Glasgow here, which is the closest to this arrangement but still not very close, if the clockwise and anticlockwise routes were in tunnels at different levels, only one more tunnel would be needed to complete the arrangment and then each of the fifteen stations could have five different levels joined by stairs, elevators or lift shafts. If it were the Tube, the Circle Line is the most obvious, or it would be if it was actually even topologically a circle, which it isn’t:
Ignoring the western branch though, which I’m sure has an official name but I have no idea what it is, the “Circle” Line has two and a third dozen stations, by contrast with the mere fifteen of Glasgow’s. However, both systems score over Subte and the MBTA in actually having circular routes, topologically speaking.
The question arises of what the point of any of this is. Why would anyone bother to design an underground with three different levels of completely superimposed tracks? However, I kind of want to look past this. Thinking of the shops and other concessions associated with the pedestrian subways in London near the Tube stations, maybe this is an entire underground city, or rather village, with a couple of hundred subterranean chambers, some residential, some business and some administrative, and the like. But for whatever reason, this subterranean railroad exists. In my mind, anyway.
Although it would be possible for the tracks to maintain the trains at the same angle, i.e. upright, it would also be possible to make them monorails, both suspended and on a rail, with overlapping rails and beams, with the carriages rolling around in circular frames to keep them in the same position and avoid moving the passengers around in ways they might not like. Alternatively, they could all just be strapped in very securely, or perhaps in padded pods, with four doors, two in the walls, one in the ceiling and one in the floor. Magnetic levitation is another option. For safety reasons, only the doors facing the station platforms should open.
Although this does all seem rather pointless, there would be advantages to such a system. For instance, it would save wear and tear on tracks or other equipment if twice the length of tunnels were required to return to the original point. It could also allow for train arrivals and departures to be staggered so that five times as many trains were operating on three times as many tracks, although this would work against the wear and tear advantage. Trains on the same track would have more mean spacing between them, reducing the probability of collisions. However, there are also more significant electrical advantages. A resistor in such a shape will do so without causing magnetic interference. Nikola Tesla patented a similar electromagnetic device in 1894 for the wireless transmission of electricity. It’s also possible that this will enable high-temperature semiconductors, meaning that if the tracks are indeed linear induction motors, such a shape would be ideal for them. See also this video:
There is a connection between superconductivity and Möbius strips in any case because of the nature of quantum spin as mentioned previously. The way spin behaves for fermions, the particles of which matter as opposed to forces are made, can be envisaged using a strip of this kind, because as previously mentioned, the magnetic field of a fermion has to be turned through 720° to return to its original polarity, which is the same as an arrow pointing towards the edge of the strip. At low temperatures, fermions such as electrons can pair up, but the members of these pairs can be thousands of atoms apart, and because each electron is a fermion, the two together act as a boson, with integral spin. Also, because of the distance, many such bosons can occupy the same space. Because they are bosons rather than fermions in such a condition, they can have the same energy states, and this makes superconductivity possible, although not all superconductors work like this.
It also occurs to me, and this is probably nothing, that plasma of protium (ordinary hydrogen) could be suspended to form such a strip electromagnetically and this might “do something”, but this is all very vague. It’s probably nothing, but this plasma would consist entirely of fermions.
A suggestion made by a mathematician on a rather related subject was the minimum number of edges a shape could have. I only remember this very vaguely, but just as a Möbius strip has just one edge despite appearing to have to, this topologist stated that there was no known reason why a shape shouldn’t have zero edges. This is true in any case of certain shapes such as spheres and tori, but the idea was that there could be no sides or edges, and this could be constructed from the theoretical infinitely elastic “plasticine” which topological shapes are made of. It’s easy to imagine, say, taking a triangular shape with three strips, twisting each a different odd number of times (1, 3, and 5 for example), gluing them together and ending up with something weird. But intuition tells us it’s impossible, as do the laws of physics, because if you imagine folding something in the right way causing it to disappear in, to quote Douglas Adams, “a puff of unsmoke”, that clearly wouldn’t happen because matter has to go somewhere. Another way of thinking of it is to imagine it entering hyperspace, and therefore only seeming to vanish, but prima facie that doesn’t seem any more feasible. However, if such a shape started off with more than three dimensions, or it was a distortion of space itself rather than being made of matter as such, as might be achieved by negative mass or actual mass, maybe something else could arise. It’s possible that the Universe itself has a “twist” in it somewhere which makes it effectively into a multidimensional analogue of a Möbius strip, and if that is the case, a trip round the Universe through such a spatial anomaly would bring objects back as mirror images of themselves.
There are macroscopic consequences of fermion spin converting to boson spin, such as the aforementioned superconductivity but also superfluidity. What isn’t clear to me is whether a direct macroscopic manifestation of something like a Möbius strip could happen. Maybe it could. Quantum computers exist, for example. So do superfluids, superconductors and Bose-Einstein condensates. A magnification of non-integral spin would involve something like a gyroscope which needed to be inverted twice before it was spinning the other way, or indeed a tunnel which would cause a magnetically levitated train to turn upside down only if it went through it twice. It’s also rather imponderable what state such a train would be in if it had only gone through it once. Would it then be potentially inverted so that it would only need to go through it once to turn upside down? Would it mean that a passenger who changed trains or got off after one circuit would then carry the potential to turning upside down themselves if they travelled through the tunnel again? It’s very difficult to contemplate even as a thought experiment.
Armin Deutsch was primarily an astronomer. His story could itself be seen as a loop, as could the film ‘Moebius’, as a train is discovered to have vanished just after the original one disappears. The film, of course, could literally have been made as a strip with a twist in it if it were on one reel, and become a never-ending story. Alternatively, there is a way to make a story a Möbius strip by having the characters swap identities as the plot proceeds. I feel that the possibilities of the Möbius strip with respect to story writing have yet to be explored. That would be a real twist in the tale.
A Box Of Chocolates 