I’ve spent a lot of time talking about short stories about invisibility, but my searches have occasionally reminded me of some of my other favorite, non-invisibility-related, science fiction stories. Today I thought I’d take a short look at “A Subway Named Mobius,” written by Armin Joseph Deutsch and which appeared in the December 1950 issue of Astounding Science Fiction.

This story is one of the true classics of science fiction, all the more remarkable because I believe it is the only story that Deutsch ever published! It is a semi-humorous story based on one of the more challenging subjects in mathematics, topology: the study of those properties of sets or objects that are invariant under continuous deformations of the object. A classic example of this line of thinking is the topological similarity of a coffee cup and a donut; both of them have a single hole, and if we imagine them made out of clay, one could be molded into the other, as long as we obey the rule that we can stretch and compress and warp the clay, but not tear it.

Another classic structure from topology is the Möbius strip, a one-sided surface, that gives the story its name. It is created by taking a ribbon of paper, giving one end a half-twist, and then taping the ends together. It is distinct from a cylinder, which could be created by taping the ends of the paper together without the half-twist.

So let’s take a look at “A Subway Named Mobius!” You can read the entire story from its original source at this link.

Armin Joseph Deutsch‘s main career was astronomy. He received his PhD from the University of Chicago (my undergraduate alma mater) in 1946, and moved to Harvard University in 1947, where he stayed until 1951.

Presumably it was his time exposed to Boston’s intricate subway system that inspired his clever short story, which begins as a new line, the Boylston Shuttle, is opened on March 3rd, adding numerous new connections to travel through the city. Then, on March 4th, an entire subway car, Number 86, disappears without a trace during its route. It takes a day before anyone realizes that the train and all of its occupants seem to have gone missing, some 350 people, including the motorman and conductor. The disappearances by themselves are noticed by the press, and it doesn’t take long before they are connected to the missing train.

The newspaper reports draw the attention of Roger Tupelo, a Harvard mathematician. He contacts Kelvin Whyte, the General Manager of the subway system, and presents him with an incredible theory of what has happened to the train: the introduction of the new Boylston Shuttle line has pushed the overall mathematical connectivity of the subway system to infinity, and the train has disappeared into a non-spatial part of the system! This theory only becomes more plausible as time passes, but Tupelo can provide no solution to the problem. He notes that he is not a specialist in topology, and though there is a specialist in topology at Harvard, Merritt Turnbull, he was riding on the Number 86 train on March 4th…

I first read this story some decades ago, and really enjoyed it, in part because it finds a fun way to present ideas from topology to a science fiction audience!

I am definitely not an expert in topology, but it appears that Deutsch is drawing a lot of his ideas from the subject known as graph theory. in which the relationships between a set of vertices (or nodes) and the lines (or edges) that connect them are studied. A couple of random graphs I drew to illustrate the point are shown below:

Graph theory is a very powerful mathematical tool that can unify seemingly disparate fields of mathematics. For example, studies of geometric objects like cubes and dodecahedrons can be simplified by literally treating the vertices of the objects as nodes and the edges of the objects as lines in a graph. Or, in our case, we can imagine the nodes as stations in a subway system and the lines are the rails that connect them.

I am not sure that Deutsch was drawing on any official theorem of graph theory to write his story, or if he just made something up (he was an astronomer, not a mathematician, after all), but the idea seems to be something as follows. If you look at the two graphs I drew above, you can think about all the different ways that pairs of nodes are connected by lines without retracing paths. With only three nodes, there are only two connections between one pair: for example, if you want to look at the connections between 1 and 3, you can either follow the straight line between them or go from 1 to 2 to 3. But when the graph includes more nodes and lines, the number of connections increases dramatically! There are many, many more ways to go from 1 to 2 in the second graph I drew: 1 to 2, 1 to 3 to 2, 1 to 6 to 3 to 2, 1 to 6 to 3 to 4 to 2, and 1 to 3 to 4 to 2! In his story, Deutsch speculates that the number of connections grows so rapidly with the number of nodes and lines that it is possible to cross a threshold into an infinite number of connections, at which point weird stuff happens!

As for the “non-spatial” place where the subway goes when this critical threshold is passed? Deutsch hints at, though doesn’t explicitly name, the phenomenon of a Riemann surface. In the field of complex analysis, there are some functions that have multiple values, and where ordinary functions can be described by a single flat surface, multivalued functions must be described by multiple surfaces, which are connected together in a nontrivial way; this combined multiple-layer structure is what is known as a “Riemann surface.” For example, you may be familiar with the fact that the square root function is often said to have two roots, a positive and a negative one. For example, the square root of 4 can be +2 or -2, because the square of both +2 and -2 is 4. When working with complex numbers, these two roots become two surfaces, that are connected together by ramp-like structures. (The following figure is from my math methods textbook.)

I should probably talk about these Riemann surfaces in more detail sometime; one helpful way to think about it is that each “singularity” in a multivalued function creates a ramp like a parking deck, where going around the singularity takes you from one level to another. This seems to be what Deutsch is implying happens to the subway in his story: when the connectivity of the subway system reached a critical level, the number of singularities in the system exploded, and the missing subway car took a wrong turn that took it to another Riemann sheet!

This is, at least, how I envision what’s happening in Deutsch’s tale. It is a fun story to ponder, and I come back to it often just for the sheer cleverness of it. Well worth a read! I wish Deutsch had written more stories like it.