Unknown Things

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:

“ “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, 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.