a list compiled by Alex Kasman (College of Charleston)

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 Problem in Geometry (1954) T.P. Caravan
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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:

 (quoted from Problem in Geometry) What we do, then, looking at it from our point of view, is warp the entire continuum down to a size we can travel across at a reasonable speed. The trouble is this: that warped-down space is full of kinks and twists put there by the presence of matter in it. ... No two parts of it are the same. Do you follow me? ... Now on order to plot a course through this sub-space, we have to work out its geometry, and that'w what the computor's used for. We have a bank of receptors which constantly measure the gravitational forces acting on your hull, and the computor, taking the apparent change in these forces as we move, uses this change to figure out the radius of curvature and all the other qualities of the geometry of our particular warp. From that information we plot our 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:

 (quoted from Problem in Geometry) "Are you a mathematician, Miss Dutton? Our computor has been destroyed and we have to find a way to navigate through subspace. The books say it can't be done." She settled herself primly in the chair. "The books are always right, young man. You can count on that. I've been teaching elementary algebra at the Decote School for Backward and Maladjusted Children now for almost forty years, and I've never found the text to be wrong yet." The captain moaned softly. "You teach algebra to idiots," he said. "And the purser's hobby is calculus." "Only differential,", said the purser. "Mostly I fool around with statistics." "Not idiots," said Miss Dutton, "morons." "And you took college math, but you flunked out." Sparks looked up, blushing. "I didn't really flunk," he said. "I quit to ship out on a freighter. I could of passed if I'd tried." "Um.". The captain turned to the chief mate. "And you and I took our math at the academy twenty or thirty years ago." "Yeah," said the mate, "and I don't remember mine. And even if I did, and even if I knew all there was to know about math, and even if I had all the books and tables I needed, even then I couldn't figure out a sub-space geometry."

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:

 (quoted from Problem in Geometry) The universe is infinite but bounded, isn't it? Like those reels of Eddie's? There's only a limit amount of film there, but he can keep playing it forever: one end's spliced to the other. OK. We've warped the universe down to size: let's keep right on going on the course we entered sub-space in. Let's go right on spang around the universe."

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.)

Works Similar to Problem in Geometry
According to my `secret formula', the following works of mathematical fiction are similar to this one:
1. The Island of Five Colors by Martin Gardner
2. The Last Magician by Bruce Elliott
3. Tiger by the Tail by A.G. Nourse
4. The Ifth of Oofth by Walter Trevis
5. Turnabout by Gordon R. Dickson
6. Star, Bright by Mark Clifton
7. Vanishing Point by C.C. Beck
8. Left or Right by Martin Gardner
9. The Moebius Room by Robert Donald Locke
10. Project Flatty by Irving Cox Jr.
Ratings for Problem in Geometry:
RatingsHave you seen/read this work of mathematical fiction? Then click here to enter your own votes on its mathematical content and literary quality or send me comments to post on this Webpage.
Mathematical Content:
3/5 (1 votes)
 . .
Literary Quality:
2/5 (1 votes)
 . .

Categories:
 Genre Science Fiction, Motif Math as Beautiful/Exciting/Useful, Topic Geometry/Topology/Trigonometry, Medium Short Stories,

Exciting News: The 1,600th entry was recently added to this database of mathematical fiction! Also, for those of you interested in non-fictional math books let me (shamelessly) plug the recent release of the second edition of my soliton theory textbook.

(Maintained by Alex Kasman, College of Charleston)