Suppose you go around taking extremely close-up black-and-white pictures of randomly chosen natural and unnatural objects (rocks, trees, streams, buildings, etc.). What do they look like?

Well, each one looks like a patch of varying shades of gray, of course. But do some patches arise more than others? If each of your close-ups is, say, three pixels by three pixels, Which would you expect to see more of:

This? Or this?

Well, each 3 x 3 square of pixels is essentially a list of nine numbers (one for the darkness of each pixel, so that a pure black pixel is a “1”, a pure white pixel is a “0”, a nearly-black pixel is, say, a “.9”, etc.). A list of nine numbers specifies a location in nine-dimensional euclidean space (just as a list of three numbers specifies a location in the three-dimensional space in which we appear to live). So the question becomes: Where in nine-dimensional space do close-up patches of photos tend to live?

A natural expectation is that they’re scattered randomly — and at first blush that appears to be accurate. But it turns out that if you take a closer look, using a mathematical “lens” that lets you see more clearly into nine dimensions, they’re not scattered randomly at all. Instead, they’re clustered around (of all things!) a Klein bottle, which is a two-dimensional surface that can’t be squeezed into three dimensions, but fits perfectly well in nine (or for that matter in four).

(Actually it’s a somewhat thickened Klein bottle, and hence four-dimensional instead of two-dimensional, much as a solid hula hoop is a somewhat thickened circle and hence three-dimensional instead of one-dimensional.)

In other words, if you randomly photograph 10,000 objects, randomly choose 3-by-3 pixel patches, and plot the corresponding points in nine-dimensional space, what you’ll “see” (insofar as you can see in nine dimensions) is a somewhat blurry Klein bottle.

Who’da thunk?

Here is some of the original research and here is an expository (but pretty high-tech) paper on this and related matters.

Edited to add: The original computational work is here. A survey article by Gunnar Carlsson, one of the original researchers, is here. The research group’s webpage is here.