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Unknown Things (1989)
Reginald Bretnor

Contributed by Vijay Fafat

A very short, well-written horror story about a collector, Andreas Hoogstraten. a wealthy man with an obsession with unusual objects. The narrator, one Mr. Dennison, a dealer in the antiques and the rare, happens to strike acquaintance with Andreas through a fellow dealer, and over time, starts selling Andreas objects whose functions or provenance cannot be ascertained or understood easily. In the process, Dennison falls madly in love with a fiendishly beautiful woman at Andreas’s house, whom he assumes to be Andreas’s wife. Dennison continues to try and find objects for Andreas simply to have short meetings of conversations with Mrs. Andreas.

Andreas is touched with a megalomania of a very different kind. He believes that all the mysterious objects coming into his possession through various dealers are not coincidental acquisitions but works of puzzles designed to test Andreas, presumably by some higher beings. As he articulates:

(quoted from Unknown Things)

“ “Dennison,” he said, “where do these things come from? Was it simply as a challenge to me, to my intellect? To see if I, Andreas Hoogstraten, have a breaking point?”

Over the years, he has managed to solve every one of his possessions except two or three. One of them is a 3-dimensional representation of a Klein bottle.

"Do you know what it is?"

Up close, it looked like grayish glass which surgeons look into our bodies, but with a higher lustre, and it was much, much heavier. Like any vase, it tapered to a neck, but there the resemblance ceased, for the neck doubled back on itself to penetrate the body halfway down and emerge again in a mouth melding with the other side.”

“It is a Klein bottle, Dennison. Are you familiar with the Moebius Strip?"

"You mean a strip of paper you give a sort of twist to and then join its ends so that it, in effect, has only one side?”

“Exactly. Well, a Klein Bottle is exactly like that, just in three dimensions (Reviewer: he must have meant either four dimensions or a three-dimensional surface)”

The Klein bottle model is filled with beautifully ground crystal artifacts, controls and structures which Andreas cannot understand or name. That bottle, Mrs. Andreas herself, and perhaps another object are things which have defeated Andreas’s understanding. That bothers him as well as excites him, giving him purpose in life.

What he does once he has solved a puzzle and how that relates to Mrs. Andreas is something we have to leave for the final, minor mystery for the reader rather than spoiling it here.

The story, admittedly, is barely mathematical fiction but I include it because the Mobius strip and the Klein bottle are two most commonly used topological objects used in fiction to capture the wonder and sometimes counter-intuitive aspects of mathematical results which challenge our common perceptions and expectations of the world around us. The author’s use of the Klein bottle as one of the mysterious objects to have pushed up against satisfying resolution for Andreas is apt and symbolic (he could also have used a coruscating 3-D model of a Mandelbrot set, I suppose). In the context of the horror story, I found its deployment sufficiently intriguing as a mathematical thought in short fiction.

This story was published in the February 1989 issue of Twilight Zone Magazine.

I agree with Vijay Fafat that the author was incorrect to say that a Klein bottle is just like a Möbius strip " just in three dimensions". Perhaps this is a good opportunity for an explanation of how the square, cylinder, Möbius strip and Klein bottle are related.

Picture a square. In fact, let's be more specific and imagine it as being the screen on which you are playing a video game. You have a little dot which you can move around. You can't get that dot to every point on the screen by moving it just horizontally. So, it is not one-dimensional. But you could get it to any point on the screen by moving it horizontally sometimes and vertically other times. It is because two directions of motion are necessary and sufficient that we say the screen is two-dimensional.

In some video games, you just bounce back into the screen if you try to go off one of the edges. If that's the case, then the playing field of the game really is just a square.

But, in other video games, you come back onto the screen in a different location. That changes the topology of the playing field.

For instance, suppose that if you go off the right edge of the screen, you appear at the left edge at the same height (but you still can't go off the top or bottom). This gives the playing field the topology of a cylinder. In fact, if you think of the dot moving on a square piece of paper, roll the paper into a tube and tape the left and right edges together, it would achieve that same effect. (This is topology. Ignore the fact that the paper is no longer flat and think only about where a dot moving on this cylinder would naturally appear if it was going off the right edge of the square which you had now taped to the left edge.)

The Möbius strip topology can also be obtained by thinking of a video game in which you appear at the left edge after going off the right edge, but now you don't show up at the same height. In this alternative version, if you go off the right edge of the screen x inches from the top, then you appear on the left edge x inches from the BOTTOM. In particular, going off just below the top right corner will result in the dot appearing just above the bottom left corner. Again, this can be achieved with a piece of paper, but this time you have to glue the left and right edges so that the top corner of one attaches to the bottom corner of the other. As I promised, this produces the familiar Möbius strip.

Finally, let's consider a game in which going off the right edge x inches from the top sends the dot to the left edge x inches from the bottom (as in the previous paragraph) but also going off the top x inches from the left causes the dot to reappear on the bottom x inches from the right. This is the topology of a Klein bottle. It would be easy to program a computer game to do that, but it is no longer easy to make a version of it out of paper, at least not in the universe as we know it. Technically, we say there is no way to embed the Klein bottle in three-dimensional space (without intersecting itself).

So, to summarize, the cylinder, Möbius strip, and Klein bottle are all 2-dimensional topological spaces which can be made from a square by adding rules about where a dot would reappear if it were to go off the edge. The cylinder and Möbius strip can both be made out of paper, but a real model of the Klein bottle cannot be produced unless you have access to higher-dimensional spaces. I do not see anything about the Klein bottle which would justify calling it a "three-dimensional analogue" of a Möbius strip as Bretnor seems to do. (BTW The Möbius strip and Klein bottle have in common that they are non-orientable, while the cylinder and square are. This is important not only if you want to avoid getting lost while walking around on one, but also if you are trying to compute a flux integral across one!)

(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 Unknown Things
According to my `secret formula', the following works of mathematical fiction are similar to this one:
  1. The Object by Alex Kasman
  2. Schwarzschild Radius by Connie Willis
  3. Solid Geometry by Ian McEwan
  4. The Book of Irrational Numbers by Michael Marshall Smith
  5. Geometria by Guillermo del Toro (Writer and Director)
  6. Danny’s Inferno by Albert Cowdrey
  7. Immortal Bird by H. Russell Wakefield
  8. The Judge's House by Bram Stoker
  9. Brain Dead by Charles Beaumont (writer) / Adam Simon (director)
  10. Old Fillikin by Joan Aiken
Ratings for Unknown Things:
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:
1/5 (1 votes)
Literary Quality:
/5 ( votes)

MotifMobius Strip/Nonorientability,
MediumShort Stories,

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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)