a list compiled by Alex Kasman (College of Charleston)
|As the title suggests, this story by Charles Carroll Muñoz (writing under his usual pseudonym) uses a contrived science fiction scenario to set up an interesting problem in differential geometry whose solution involves topology.
A space ship is traveling quickly over interstellar distances using "overdrive" with the help of a computor (sic) to determine the curvature of the space and plot the course:
However, their computor has been completely destroyed in an explosion. Even though the receptors are still working, without the computor they cannot determine when they have reached their destination. If they just randomly turn off the overdrive, they are almost certain to simply be stranded in interstellar space.
At first, they try to determine whether anyone on the ship knows enough math to do the computor's work:
It seems as if they are doomed to either stay in overdrive forever or to die in some desolate location, but one crew member who was left out of the discussion above ends up inspiring a solution to the problem.
Continue reading below if you want to read my summary of the end of the story here on this webpage. (Stop reading if you don't care or would prefer to find a copy of the original story to read for full enjoyment. ; )
Spoiler Alert: I'm about to tell you how the story ends.
The ship's radio operator, who by coincidence was one of Miss Dutton's math students, was otherwise occupied during the earlier discussions by his hobby of watching a film that he has spliced into an endless loop. This inspires the purser to think of a clever (but not entirely satisfactory, IMHO) solution to their dilemma.
He points out that even without the computor, they can look at the readings from the receptors and compare them with the readings taken when they first left. Then, they can drop out of overdrive when the readings match and find themselves not at their intended destination but at least safe at home.
Others object to this suggestion because it would require them to turn the ship around and, without the computor to determine the sub-space geometry, there would be no way for them to ensure that they will return along the same path on which they came.
This is where the topology comes in. The purser says:
In other words, the idea is that since the universe (in this story) is known to have the topology of a 3-torus, if they keep going straight ahead they will eventually return to their starting point. Of course, we don't actually know the topology of the universe now in the 21st century as I write this, but supposing it will be known at some future date is reasonable for a science fiction story like this. Still, there is another problem I'd like to mention: Even though there are straight paths on a torus which "close" by returning to the starting point, most straight trajectories do not close. If you take a small straight line segment on a torus and extend it infinitely, you often get a quasi-periodic trajectory which wraps around over and over, never returning to the starting point. (Okay. You do get arbitrarily close to the starting point, which might be sufficient for the purposes of the story I suppose.) A quick internet search led me to these lecture notes which explain the idea of quasi-periodic trajectories in a torus in greater detail.
Thanks to Vijay Fafat for telling me about and sharing a copy of this story, which was published in Science Stories #4 (April 1954). (Note that some websites list the title of this story as "Problem ???????????? in ? Geometry" but I'm assuming that the question marks on the first page were intended as an illustration not actually part of the title itself.)
|(Note: This is just one work of mathematical fiction from the list. To see the entire list or to see more works of mathematical fiction, return to the Homepage.)|
(Maintained by Alex Kasman, College of Charleston)